title: On the power of Chatterjee rank correlation
authors: Hongjian Shi Mathias Drton Fang Han
text: 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  On the power of Chatterjee's rank correlation
      
  Hongjian ShiDepartment of Statistics, University of Washington, Seattle, WA 98195, USA; e-mail: hongshi@uw.edu,  
  Mathias DrtonDepartment of Mathematics, Technical University
    of Munich, 85748 Garching b. München, Germany; e-mail: mathias.drton@tum.de,  and 
  Fang HanDepartment of Statistics, University of Washington, Seattle, WA 98195, USA; e-mail: fanghan@uw.edu.
  
      
  =====================================================================================================================================================================================================================================================================================================================================================================================
  
  
  
  
  
  
    <cit.> introduced a simple new rank correlation
    coefficient that has attracted much recent attention.  The
    coefficient has the unusual appeal that it not only estimates a
    population quantity first proposed by <cit.> 
    that is zero if and only if the underlying pair
    of random variables is independent, but also is
    asymptotically normal under independence.  This paper compares
    Chatterjee's new correlation coefficient to three established rank correlations
    that also facilitate consistent tests of independence, namely,
    Hoeffding's D, Blum–Kiefer–Rosenblatt's R, and
    Bergsma–Dassios–Yanagimoto's τ^*.  We contrast their
    computational efficiency in light of recent advances, and
    investigate their power against local rotation and mixture alternatives.  
    Our main results show that Chatterjee's coefficient is unfortunately rate
    sub-optimal compared to D, R, and τ^*.  The situation is
    more subtle for a related earlier estimator of
    <cit.>.   
    These results favor D, R, and τ^* over
    Chatterjee's new correlation coefficient for the purpose of testing
    independence.
  
  
  Keywords: 
  Dependence measure; Independence test; Le Cam's third
  lemma; Rank correlation; Rate-optimality.
  
  
  
  
  § INTRODUCTION
  
  
  
  
  
  Let X^(1),X^(2) be two real-valued random variables 
  defined on 
  a common probability space.
  
  We will be 
  concerned with testing the null hypothesis
  
      H_0: X^(1) and X^(2) are independent,
  
  based on a sample from the joint distribution
  
  of (X^(1),X^(2)).
  This classical problem has seen
  
  
  
  revived interest in recent years as independence tests constitute a
  key component in modern statistical methodology such as, e.g., methods
  for causal discovery <cit.>.
  
  
  
  
  
  
  
  The problem of testing independence has been examined from a
    number of different perspectives; see, for example, the work of
  <cit.>, <cit.>, and
  <cit.>, and the references therein.  In this paper, our 
   focus will be on testing H_0 via rank correlations that measure ordinal
  association. Rank correlations are
  particularly attractive for continuous distributions for which 
  they are distribution-free under H_0.  Early
  proposals of rank correlations include the widely-used ρ of
  <cit.> and τ of <cit.>, but also the footrule 
  of <cit.>, the γ
  of <cit.>, and the β of <cit.>.
  
  
  
  Unfortunately,
  all five of these rank correlations fail to give a
  consistent test of independence.  Indeed, each correlation coefficient consistently estimates a population correlation measure
  that takes the same value under H_0 and certain fixed alternatives to H_0.  This fact leads to 
  trivial power at such alternatives.
  
  
  
  
  
  
  In order to arrive at a consistent test of independence,
  <cit.> proposed a correlation measure that, for
  absolutely continuous bivariate distributions, vanishes if and only if
  H_0 holds.
  
  <cit.> considered a modification that
  is
  consistent against all dependent bivariate alternatives 
  <cit.>.   <cit.> proposed a
  new test of independence and showed its consistency for bivariate
  distributions that are discrete, absolutely continuous, or a mixture
  of both types.  As pointed out by <cit.>, mere
  continuity of the marginal distribution functions is sufficient for
  consistency of their test.  This follows from a relation discovered by
  <cit.> who implicitly considers the correlation of <cit.> when proving a
  conjecture of <cit.>.
  
  All three aforementioned correlation measures admit natural efficient
  estimators in the form of U-statistics that depend only on ranks.
  However, in each case, the U-statistic is degenerate and has a
  non-normal asymptotic distribution under H_0.  In light of this
  fact, it is interesting that <cit.> were able to construct
  a consistent correlation measure ξ 
  which is also able to detect perfect functional dependence
  (see also ) 
  
  and in a recent paper that received much attention,
  <cit.> gives a very simple rank correlation, with no tuning parameter involved, that 
  surprisingly estimates ξ and has an asymptotically normal
  null distribution.
  
  This paper  compares Chatterjee's and also
  Dette–Siburg–Stoimenov's rank correlation coefficients to the three
  obvious competitors given by the D of <cit.>, the R of
  <cit.>, and the τ^* of <cit.>.  Our
  comparison considers  three  criteria:
  
  
    * Statistical consistency of the independence test.  A correlation measure μ
    assigns to each joint distribution of (^(1),^(2)) a real
    number μ(^(1),^(2)).  Such a correlation measure is
    consistent in a family of distributions  if for all pairs
    (^(1),^(2)) with joint distribution in , it holds that
    μ(^(1),^(2))=0 if and only if ^(1) is
    independent of ^(2). 
  Correlation measures that are consistent within a large nonparametric
  family are able to detect non-linear, non-monotone relationship, and facilitate consistent tests of independence.
  If a correlation measure μ is consistent, then the consistency of
  tests of independence based on an estimator μ_n of μ is
  guaranteed by the consistency of that estimator.
  
    * Computational efficiency.  Computing ranks requires
    O(nlog n) time.  With a view towards large-scale applications, we
    prioritize rank correlation coefficients that are computable
    without much additional effort, that is, also in
    O(nlog n) time.  This is easily seen to be the case
    for Chatterjee's coefficient but, as we shall survey in
    Section <ref>, recent advances clarify that D, R, and
    τ^* can be computed similarly efficiently.
  
    * Statistical efficiency of the independence test.
    Our final criterion is optimal efficiency in the statistical sense
    <cit.>.  To assess this, we use different
    local alternatives inspired from work of <cit.> and of
    <cit.>; the latter type of alternatives was
    further developed in <cit.>.  We then call an
    independence test rate-optimal (or rate sub-optimal) against a family
    of local alternatives if within this family the test achieves the
    detection boundary up to constants  (or not).
  
  
  
  The main contribution of this paper pertains to statistical
  efficiency.   Chatterjee's derivation of asymptotic normality for his
  rank correlation coefficient relies on a reformulation of his statistic and then invoking a type of permutation central limit theorem that was established in <cit.>. We found that a direct use of this technique to analyse the local power is hard. In recent
  related work we were able to overcome a similar issue in a related
  multivariate setting <cit.>
  by developing a suitable Hájek representation theory
  <cit.>. Applying this philosophy here, we  build a particular form of the projected statistic that was introduced in <cit.> to provide an alternative proof of Theorem 2.1 in
  <cit.> that gives an asymptotic representation. 
  Integrating the representation into Le Cam's third lemma and employing
  further a version of the conditional multiplier central limit theorem 
  (cf. Chapter 2.9 in ), we are then
  able to show that the test based on Chatterjee's rank correlation
  coefficient is in fact rate sub-optimal against the two
  considered local alternative families; recall point (iii) above.  Our theoretical analysis thus echos Chatterjee's empirical observation,
  that is, his test of independence can suffer from low power; 
  see Remark <ref> below.
  
  
  
  
  In contrast, the tests based on the more
  established coefficients D, R, and τ^* are all rate-optimal
  for all considered local alternative families.  We therefore consider the latter
  more suitable for testing independence
  than Chatterjee's test. On the other hand, the test based on Dette–Siburg–Stoimenov's coefficient is empirically observed to have non-trivial power against certain alternatives in finite-sample simulations. A theoretical study of this phenomenon, however, has to be left to the future due to involved technical difficulties.
  
  
  
  
  
  
  
  
  
  
  
  
  
  The proofs of our claims, including details on examples, are given in 
  the supplementary material. 
  
  
  
  
  
  
  
  As we were completing the manuscript, we became aware of independent
  work by <cit.>, who accomplished a similar local
  power analysis for Chatterjee's 
  correlation coefficient and presented a result that is similar to our
  Theorem <ref>, Claim (<ref>). 
  The local alternatives considered in their
  paper are, however, different from ours. 
  
  
  
  
  
  
  In addition, the two papers differ in
  their focus.  The work of  concentrates on correlation
  measures that are 1 if and only if one variable is a
  shape-restricted function of the other variable, while our interest
  is in comparing consistent tests of independence. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  § RANK CORRELATIONS AND INDEPENDENCE TESTS
  
  
  
  
   §.§ Considered rank correlations and their computation
  
  
  When considering correlations, we will use the term correlation
    measure to refer to population quantities, which we write using
  Greek or Latin letters.  The term
  correlation coefficient is reserved for sample
  quantities, which are written with an added subscript n.
  
  
  
  
  
  The symbol F denotes a joint bivariate distribution function for
  the considered pair of random variables  (X^(1),X^(2)), and
  F_1 and F_2 are the respective marginal distribution functions.
  Throughout,
  
  (X^(1)_1,X^(2)_1),…,(X^(1)_n,X^(2)_n) is a sample
  comprised of n independent copies of (X^(1),X^(2)).
  
  We now introduce in precise terms the five types of rank correlations
  we consider in this paper.  We begin by specifying the correlation
  measure and coefficients from <cit.> and
  <cit.>.  To this end, let
  (X^(1)_[1],X^(2)_[1]),…,(X^(1)_[n],X^(2)_[n]) be
  a rearrangement of the sample such that
  X^(1)_[1]≤⋯≤ X^(1)_[n], with ties, if existing,
  broken at random.  Define
  
      r_[i]≡∑_j=1^n(X^(2)_[j]≤
        X^(2)_[i])
  
   
  with I(·) representing the indicator function, and
  ℓ_[i]≡∑_j=1^n(X^(2)_[j]≥ X^(2)_[i]).
  We emphasize that if F_2 is continuous, then there are almost surely
  no ties among X^(2)_1,…,X^(2)_n, in which case r_[i]
  is simply the rank of X^(2)_[i] among
  X^(2)_[1],…,X^(2)_[n].
  
  
  
  
  The correlation coefficient of <cit.> is
  
      ξ_n≡1-n∑_i=1^n-1| r_[i+1]-r_[i]|/2∑_i=1^nℓ_[i](n-ℓ_[i]).
  
  If there are no ties among X^(2)_1,…,X^(2)_n, it holds
  that
  
      ξ_n =1-3∑_i=1^n-1| r_[i+1]-r_[i]|/n^2-1.
  
  <cit.> proved that ξ_n estimates the correlation measure
  
      ξ≡∫[{(X^(2)≥ x)| X^(1)}] F̣_2(x)/∫{(X^(2)≥ x)}F̣_2(x).
  
  This measure was in fact first proposed in <cit.>; cf. r(X,Y) in their
  Theorem 2.  We thus term ξ the
  Dette–Siburg–Stoimenov's rank correlation measure.
  
  
  We note that ξ was also considered by
  <cit.>; see the Cramér–von Mises index S^v_2,CVM
  before their Properties
  3.2.  For estimation of ξ, <cit.> proposed the
  following coefficient; denoted r_n in their Equation (15).
  
  
  
  
    
    Let K be a symmetric and twice continuously differentiable kernel
    with compact support, and let
    K(x)≡∫_-∞^xK(t)ṭ.  Let h_1, h_2>
    0 be
    bandwidths that are chosen such that they tend to zero with
    
      nh_1^3→∞,    
          nh_1^4→ 0,    
          nh_2^4→ 0,    
          nh_1h_2→∞
  
    as n→∞.  Define
  
      ζ_n(u^(1),u^(2))≡1/nh_1∑_i=1^nK(u^(1)-i/n/h_1)K(u^(2)-r_[i]/n/h_2)
  
  with r_[i] as in (<ref>).  Then the Dette–Siburg–Stoimenov's
   correlation coefficient is
  
      ξ_n^*≡ 6∫_0^1∫_0^1{ζ_n(u^(1),u^(2))}^2ụ^(1)ụ^(2) -2.
  
  
  
  
  
  
  
  
  
  Next we introduce two classical rank correlations of
  <cit.> and <cit.>, both of which assess dependence
  in a very intuitive way by integrating squared deviations between the
  joint distribution function and the product of the marginal
  distribution functions. 
  
  
  
  Hoeffding's correlation measure is defined as
  
      D≡∫{F(x^(1),x^(2))-F_1(x^(1))F_2(x^(2))}^2
        F̣(x^(1),x^(2)).
  
  It is unbiasedly estimated by the correlation coefficient 
  
      D_n≡    1/n(n-1)⋯(n-4)∑_i_1… i_51/4
         [{(X^(1)_i_1≤ X^(1)_i_5)-(X^(1)_i_2≤ X^(1)_i_5)}{(X^(1)_i_3≤ X^(1)_i_5)-(X^(1)_i_4≤ X^(1)_i_5)}]
         [{(X^(2)_i_1≤ X^(2)_i_5)-(X^(2)_i_2≤ X^(2)_i_5)}{(X^(2)_i_3≤ X^(2)_i_5)-(X^(2)_i_4≤ X^(2)_i_5)}],
  
  which is a rank-based U-statistic of order 5. 
  
  
  
  
  
  Blum–Kiefer–Rosenblatt's correlation measure is defined as
  
      R≡∫{F(x^(1),x^(2))-F_1(x^(1))F_2(x^(2))}^2 
      F̣_1(x^(1))F̣_2(x^(2)).
  
  It is unbiasedly estimated by the Blum–Kiefer–Rosenblatt's correlation coefficient 
  
      R_n≡    1/n(n-1)⋯(n-5)∑_i_1… i_61/4
         [{(X^(1)_i_1≤ X^(1)_i_5)-(X^(1)_i_2≤ X^(1)_i_5)}{(X^(1)_i_3≤ X^(1)_i_5)-(X^(1)_i_4≤ X^(1)_i_5)}]
         [{(X^(2)_i_1≤ X^(2)_i_6)-(X^(2)_i_2≤ X^(2)_i_6)}{(X^(2)_i_3≤ X^(2)_i_6)-(X^(2)_i_4≤ X^(2)_i_6)}],
  
  which is a rank-based U-statistic of order 6. 
  
  
  More recently, <cit.> introduced the following rank
  correlation, which is connected to work by
  <cit.>.  We refer the reader to
  <cit.> for a motivation via con-/disconcordance of 4-point patterns and connections to Kendall's tau.
  
  
    
    Write (x_1,x_2<x_3,x_4)≡(max{x_1,x_2}<min{x_3,x_4}).  The
    Bergsma–Dassios–Yanagimoto's correlation measure is
    
      τ^*≡     4(X^(1)_1,X^(1)_3< X^(1)_2,X^(1)_4 , 
                 X^(2)_1,X^(2)_3< X^(2)_2,X^(2)_4)
         +4(X^(1)_1,X^(1)_3< X^(1)_2,X^(1)_4 , 
                 X^(2)_2,X^(2)_4< X^(2)_1,X^(2)_3)
         -8(X^(1)_1,X^(1)_3< X^(1)_2,X^(1)_4 , 
                 X^(2)_1,X^(2)_4< X^(2)_2,X^(2)_3).
  
  It is unbiasedly estimated by a U-statistic of order 4, namely, the
  Bergsma–Dassios–Yanagimoto's correlation coefficient
  
      τ^*_n≡    1/n(n-1)(n-2)(n-3)∑_i_1… i_4
         {(X^(1)_i_1,X^(1)_i_3< X^(1)_i_2,X^(1)_i_4)
               +(X^(1)_i_2,X^(1)_i_4< X^(1)_i_1,X^(1)_i_3)
                 -(X^(1)_i_1,X^(1)_i_4< X^(1)_i_2,X^(1)_i_3)
               -(X^(1)_i_2,X^(1)_i_3< X^(1)_i_1,X^(1)_i_4)}
         {(X^(2)_i_1,X^(2)_i_3< X^(2)_i_2,X^(2)_i_4)
               +(X^(2)_i_2,X^(2)_i_4< X^(2)_i_1,X^(2)_i_3)
                 -(X^(2)_i_1,X^(2)_i_4< X^(2)_i_2,X^(2)_i_3)
               -(X^(2)_i_2,X^(2)_i_3< X^(2)_i_1,X^(2)_i_4)}.
  
  
  
  
  
   
  As conveyed by Equation (6.1) in <cit.>, as long as n≥ 6 and there are
  no ties in the data, it holds that 12D_n+24R_n=τ^*_n.
  Consequently, 12D+24R=τ^* given continuity but not necessarily
  absolute continuity of F; compare page 62 of <cit.>.
  
  
  
  
  
  
  
  At first sight the computation of the different correlation
  coefficients appears to be of very different complexity.  However,
  this is not the case due to recent developments, which yield nearly
  linear computation time for all
  coefficients except ξ^*_n.
  
  
    
    If data have no ties, then ξ_n, D_n, R_n, and τ^*_n can
    all be computed in O(nlog n) time.
  
  
    It is evident from its simple form that  ξ_n can be computed in
    O(nlog n) time <cit.>.  The result
    about D_n is due to <cit.>; see also
    <cit.>.  The claim about τ^*_n is based on
    recent new methods due to <cit.> 
    and <cit.>; for
    an implementation see <cit.>.
    The claim about R_n then follows from the relation given in
    Remark <ref>.
  
  
  
    The definition of ξ^*_n involves an integral over the unit
    square [0,1]^2.  How quickly the integral can be computed depends
    on smoothness properties of the considered kernel and the bandwidth
    choice.  <cit.> suggests a time
    complexity of O(n^5/3).  Indeed, for a symmetric and four times
    continuously differentiable kernel K that has compact support,
    there is a choice of bandwidths h_1,h_2 that satisfies the
    requirements of Definition <ref> and for which ξ_n^*
    can be approximated with an absolute error of order o(n^-1/2) in
    O(n^5/3) time.
  
    To accomplish this we may choose h_1=h_2=n^-1/4-ϵ for
    small ϵ>0 and apply Simpson's rule to the two-dimensional
    integral in the definition of ξ^*_n.  By assumptions on K, the
    function ζ_n^2 has continuous and compactly supported fourth
    partial derivatives that are bounded by a constant multiple of
    h_1^-5.  The error of Simpson's rule applied with a grid of
    M^2 points in [0,1]^2 is then O(h_1^-5/M^4).  With
    M^2=O(h_1^-5/2n^1/4+ϵ/2)=O(n^7/8+3ϵ), this
    error becomes O(n^-1/2-ϵ)=o(n^-1/2).  Due to the
    compact support of K, one evaluation of ζ_n requires O(nh_1)
    operations.  The overall computational time is thus
    O(nh_1M^2)=O(n^13/8 + 2ϵ), which is O(n^5/3) as long
    as ϵ≤1/48.
  
  
  
   
    When the data can be considered as generated from a continuous
    distribution but featuring a small number of ties due to rounding,
    then ad-hoc breaking of ties poses little problem.  In contrast, if
    ties arise due to discontinuity of the data-generating distribution,
    then the situation is more subtle.  In this case, Chatterjee's
    ξ_n is to be computed in the form from (<ref>),
    but the computational time clearly remains O(nlog n). In
    contrast, ξ_n^* is no longer a suitable estimator of
    ξ. Hoeffding's formulas for D_n continue to apply with ties,
    keeping the computation at O(nlog n) but, as we shall emphasize in Section <ref>, 
    the estimated D may lose some of its appeal.  Bergsma–Dassios–Yanagimoto's τ^*_n is
    suitable also for discrete data, but the available implementations
    that explicitly account for data with ties <cit.> are
    based on the O(n^2log n) algorithm of <cit.>
    or the slighly more memory intensive but faster O(n^2) algorithm
    of <cit.>.  Computation of R_n 
    with ties is also O(n^2) <cit.>.  
  
  
  
  
  
  
   §.§ Consistency
  
  
  In the rest of this section as well as in Section <ref>, we
  will always assume that the joint distribution function F is
  continuous, though not necessarily jointly absolutely continuous with regard to the Lebesgue measure.
  
  Accordingly, both 
  X^(1)_1,…,X^(1)_n and X^(2)_1,…,X^(2)_n are free of ties
  with probability one. 
  To clearly state the following results, we introduce three families of
  bivariate distributions specified via their joint distribution
  function F:
  
      ^   ≡{F: F is continuous as a bivariate function},
      
      
      
         ≡{F: F is absolutely continuous with regard to the Lebesgue measure},
      
      
      
      ^ DSS   ≡{F∈^: F has a copula
                     C(u^(1),u^(2)) that  is three and two times continuously 
                 differentiable
      with respect to the arguments u^(1) and u^(2), respectively}.
  
  Recall that the copula of F satisfies F(x^(1),x^(2))=C{F_1(x^(1)),F_2(x^(2))}.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  We first discuss the large-sample consistency of the correlation coefficients as
  estimators of the corresponding correlation measures.  Convergence in
  probability is denoted ⟶_p.
  
  
  
  
    For any F∈^ and  n→∞, we have
    
      ξ_n⟶_p ξ,
             D_n⟶_p D,
             R_n⟶_p R, 
             and   τ^*_n⟶_p τ^*.
  
  If in addition F∈^ DSS and K,h_1,h_2 satisfy all assumptions stated in Definition <ref>, then also ξ^*_n⟶_p ξ.
  
  
    The claim about ξ_n is Theorem 1.1 in <cit.>,
    and the one about ξ_n^* is proved in the supplement Section <ref> based on a revised version of Theorem 3 in <cit.>.  The
    remaining claims are immediate from U-statistics theory (e.g.,
    Proposition 1 in , Theorem 5.4.A in
    ). 
  
  
  Next, we turn to the correlation measures themselves.  
  
  
  
  It is clear that ξ, D, and R are always nonnegative, and that
  the same is true for τ^* when applied to F∈^; this
  follows from Remark <ref>.  The consistency
  properties for continuous observations can be summarized as follows.
  
  
    Each one of the correlation measures ξ, R, and τ^* is consistent for the entire class
    ^, that is, if
    F∈^, then ξ=0 (or R=0 or τ^*=0) if
    and only if the pair (X^(1),X^(2)) is independent.
    Hoeffding's D is consistent for ^ but not ^.
  
  
     The consistency of ξ
    is Theorem 2 of <cit.>, and Theorem 1.1 of
    <cit.>.  The consistency of R is shown in detail
    in Theorem 2 of <cit.>; see also p. 490 in
    <cit.>.  The consistency of τ^* was established for
    ^ in
    Theorem 1 in <cit.>, and that for ^ can be shown
    via Remark <ref>; compare  Theorem 6.1 of
    <cit.>.  Finally, the claim about D follows from
    Theorem 3.1 of <cit.> and its generalization in
    Proposition 3 of <cit.>. 
  
  
  
  
  
  
  
   §.§ Independence tests
  
  
  For large samples, computationally efficient independence tests may be
  implemented using the
  asymptotic null distributions of the correlation coefficients, which are summarized below.
  
  
  
  
  
  
  
  
  Suppose F∈^ has X^(1) and X^(2) independent. As n→∞, it holds that
  
  
    * for Chatterjee's correlation coefficient ξ_n, n^1/2ξ_n→ N(0,2/5) in distribution (Theorem 2.1 in ); 
  
  
  
  
    * for Dette–Siburg–Stoimenov's correlation coefficient ξ_n^*, 
  n^1/2ξ_n^*→ 0 in probability
  assuming that F∈^ DSS
  and K,h_1,h_2 satisfy all assumptions stated in
  Definition <ref> (revised version of Theorem 3 in
    ; see Section <ref> of
    the supplementary material); 
  
    * for μ∈{D, R, τ^*}, 
  
      nμ_n→∑_v_1,v_2=1^∞λ^μ_v_1,v_2(ξ_v_1,v_2^2-1)   in distribution,
  
  where 
  
      λ^μ_v_1,v_2=
      1/(π^4 v_1^2 v_2^2)     when μ=D,R,
      
      36/(π^4 v_1^2 v_2^2)     when μ=τ^*,
  
  for v_1,v_2=1,2,…,  
  
  
  
  
  
  
  
  
  
  and {ξ_v_1,v_2} as independent standard normal random variables 
  (Proposition 7 in , Proposition 3.1 in ). 
  
  
  
  For a given significance level α∈(0,1), let z_1-α/2 be
  the (1-α/2)-quantile of the standard normal distribution.  Then the
  asymptotic test based on Chatterjee's ξ_n is
  
      T^ξ_n_α≡{n^1/2|ξ_n| >(2/5)^1/2· 
                              z_1-α/2}.
  
  The tests based on μ_n with μ∈{D, R, τ^*} take the form
  
      T^μ_n_α≡(n μ_n > q^μ_1-α),   
      q^μ_1-α≡inf[x:{∑_v_1,v_2=1^∞λ^μ_v_1,v_2(ξ_v_1,v_2^2-1)≤ x}≥ 1-α],
  
  where λ^μ_v_1,v_2 and ξ_v_1,v_2,
  v_1,v_2=1,…,n,… were presented in Proposition
  <ref>.  We note that <cit.> gives a routine to
  compute the needed quantiles.
  It is unclear how to implement the test based on
    Dette–Siburg–Stoimenov's ξ_n^* without the need for simulation
    or permutation as a non-degenerate limiting null distribution is
    currently unknown. 
  
  Given the distribution-freeness of ranks for the class ^,
  Proposition <ref> yields uniform asymptotic validity
  of the tests just defined.  Moreover,
  Propositions <ref>–<ref> yield
  consistency at fixed alternatives.  We summarize these facts below.
  
  
  
  
  
  
  
  
  
  
    The tests based on the correlation coefficients μ_n ∈{ξ_n, D_n, R_n, τ^*_n} are 
    uniformly valid in the sense that
    
      lim_n→∞sup_F∈^(T^μ_n_α=1| H_0)=α.
  
  Moreover, these tests are consistent, i.e., for fixed F∈^ such that X^(1) and X^(2) are
  dependent and μ_n∈{ξ_n,R_n,τ^*_n}, it holds that
   
      lim_n→∞(T^μ_n_α=1| H_1)=1.
   
  The conclusion (<ref>) holds for μ_n=D_n if assuming further that F∈. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  § LOCAL POWER ANALYSIS
  
  
  This section investigates the local power of the four rank
  correlation-based tests of H_0 introduced in Section
  <ref>.  To this end, we consider two classical  and well-used
  families of alternatives to the null hypothesis of independence: rotation alternatives (Konijn alternatives;
  ) and mixture alternatives (Farlie-type
    alternatives; ; see also ). 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  (A) Rotation alternatives. 
  
  
  
  
  Let Y^(1) and Y^(2) be two real-valued independent random
  variables that have mean zero and are absolutely continuous with Lebesgue-densities f_1 and f_2, respectively.  
  For  Δ∈(-1,1), consider 
  
      = (X^(1)
       X^(2))
      ≡(1    Δ
      Δ    1) 
                    (Y^(1)
       Y^(2))
      = _Δ(Y^(1)
       Y^(2))
      = _Δ.
  
   
   
  For all Δ∈(-1,1), the matrix _Δ is clearly full
  rank and invertible.
  
  For any Δ∈(-1,1), let f_(;Δ) denote the
  density of =_Δ.  We then make the following assumptions on Y^(1),Y^(2).
  
  
  It holds that 
  
  
  
  
    *  the distributions of  have a common support for all Δ∈(-1,1), so that without loss of generality ≡{: f_(;Δ) > 0} is independent of Δ; 
  
    *  the density f_k is absolutely continuous with non-constant
    logarithmic derivative
    ρ_k≡ f'_k/f_k,
    
  
  
    k=1,2; 
  
    *  the Fisher information of  relative to Δ at the point 0, denoted _(0), is strictly positive, and {(Y^(k))^2} <∞, [{ρ_k(Y^(k))}^2] <∞ for k=1,2.
  
  
  
  
  Assumption <ref><ref>,<ref> implies {ρ_k(Y^(k))}=0 and _(0)<∞. 
  
  
  
  Suppose f_k(z) is absolutely continuous and positive for all real numbers z, k=1,2. If 
  
      (Y^(k)) =0,   {(Y^(k))^2} <∞,    [{ρ_k(Y^(k))}^2] <∞,   for k=1,2,
  
  
  
  then Assumption <ref> holds.  As a special case,
  Assumption <ref> holds if ^(1) and ^(2) are
  centred and follow normal distributions or t-distributions with not necessarily integer-valued degrees of
  freedom greater than two.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  (B) Mixture alternatives. 
  
  
  
  
  
  Consider the following mixture alternatives that were used in
  <cit.>.  Let F_1 and F_2 be fixed univariate
  distribution functions that are absolutely continuous with
  Lebesgue-density functions f_1 and f_2, respectively.  Let
  F_0(x^(1),x^(2))=F_1(x^(1))F_2(x^(2))
  be the product distribution function yielding independence, and let
  G F_0 be a fixed bivariate distribution function that is
  absolutely continuous and such that (X^(1),X^(2)) are dependent
  under G. 
  Let the density functions of F_0 and G, denoted by f_0 and g,
  respectively, be continuous and have compact supports.  Then define the
  following alternative model for the distribution of =(X^(1),X^(2)):
  
      F_≡(1-Δ)F_0+Δ G,
  
  with 0≤Δ≤ 1. 
  
  We make the following additional assumptions on F_0 and G.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  It holds that 
  
  
    *  the distribution G is absolutely continuous with respect to
    F_0 and s(x)≡ g(x)/f_0(x)-1 is continuous; 
  
    *  the conditional expectation {s(Y)|Y^(1)}=0 almost surely for Y=(Y^(1),Y^(2))∼ F_0;
    
  
  
  
  
    *  the function s is not additively separable, i.e., there
    do not exist univariate functions h_1 and h_2 such that
    s(x)=h_1(x^(1))+h_2(x^(2));
    
  
  
  
    *  the Fisher information _(0)>0. 
  
  
  
  
  In this model, g(x)/f_0(x) is continuous and has compact support, which guarantees that _(0)<∞. 
  
  
  
  
  
  
  (Farlie alternatives)
  Let G in (<ref>) be given as
  
      G(x^(1),x^(2))≡ F_1(x^(1))F_2(x^(2))[1+{1-F_1(x^(1))}{1-F_2(x^(2))}].
  
  Then Assumption <ref> is satisfied <cit.>. Notice also that {s(Y)|Y^(2)}=0 almost surely for Y=(Y^(1),Y^(2))∼ F_0. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
    Let the density f_2 
    be symmetric around 0, and consider two univariate functions
    h_1 and h_2 that are both non-constant and bounded by
    1 in magnitude, with
    h_2 additionally being an odd function.  Let
    f_1 be a density such that ∫ f_1(x^(1))h_1(x^(1)) x̣^(1) 0.  Then the bivariate density
    g can be chosen such that s(x)=h_1(x^(1))h_2(x^(2)) and
    
    
    
    
    
    then Assumption <ref> holds. 
    
    
    
    
    For example, we can take 
    f_1(t)=f_2(t)=1/2×(-1≤ t≤ 1), 
    h_1(t)=|1-2Ψ(t)|, and h_2(t)=1-2Ψ(t), 
    where Ψ denotes the distribution function 
    of the uniform distribution on [-1,1]. 
    In this case, {s(Y)|Y^(2)} is not almost surely
    zero for Y=(Y^(1),Y^(2))∼ F_0. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  For a local power analysis in any one of the two considered
  alternative families, we examine the asymptotic power along a respective sequence of
  alternatives obtained as
  
      H_1,n(Δ_0):Δ=Δ_n,   where Δ_n≡ n^-1/2Δ_0
  
  with some constant Δ_0> 0.
  
  
  
  
  
  
  
  
  
  
  We obtain the following results on 
  the discussed tests.
  
  
  
    Suppose the considered sequences of local alternatives are formed
    such that Assumption <ref> or <ref>
    holds when considering a family of type (A) or (B), respectively.  Then concerning any sequence  of alternatives given in (<ref>), 
  
  
  
    * for any one of the two types of alternatives (A) or (B), and
    any fixed constant Δ_0>0,
  
      lim_n→∞{T^ξ_n_α=1| H_1,n(Δ_0)}=α;
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
    * for any local alternative family and any number β>0, 
  there exists some sufficiently large constant 
  C_β>0 only depending on β such that, as long as Δ_0>C_β,
  
      lim_n→∞{T^μ_n_α=1| H_1,n(Δ_0)}≥1-β,
  
  where μ_n∈{D_n, R_n, τ^*_n}. 
  
  
  
  
  
  
  
  In contrast to Theorem <ref>, Proposition <ref>
    below shows that the power of any size-α test can be
    arbitrarily close to α when Δ_0 is sufficiently small
    in the local alternative model H_1,n(Δ_0). This result
    combined with (<ref>) and (<ref>) manifests that 
  the size-α tests based on one of D_n,R_n,τ^*_n 
  are rate-optimal against the considered local alternatives, 
  while the size-α test based on Chatterjee's correlation coefficient, with only trivial power against the local alternative model H_1,n(Δ_0) for any fixed Δ_0, 
  is rate sub-optimal. 
  
  
  
  Concerning any one of the two local alternative families and any
  sequence  of alternatives given in (<ref>), as
  long as the corresponding 
  Assumption <ref> or <ref> holds, we have that  for any number β>0 satisfying α+β<1 there exists a constant c_β>0 only depending on β such that
  
      inf_T_α∈_α{T_α=0| H_1,n(c_β)}≥ 1-α-β
  
  for all sufficiently large n. Here the infimum is
  taken over all size-α
  tests. 
  
  
  
  
    
      Assumptions <ref> and <ref> are
  technical conditions imposed to ensure that (i) the two considered
  sequences of alternatives are all locally asymptotically normal
  <cit.>, i.e., the log likelihood ratio processes
  admit a quadratic expansion; (ii) the conditional expectation of the
  score function given the first margin is almost surely zero. Here the
  second requirement was invoked to allow for a use of the conditional
  multiplier central limit theorem (cf. Chapter 2.9 in )
  that appears to be the key in
  analysing the power of Chatterjee's correlation coefficient.  In
  addition to their generality, we would like to emphasize that these
  technical assumptions are indeed satisfied by important models such as
  Gaussian rotation and Farlie alternatives, which are commonly used to
  investigate local power of independence tests.
      
      
  
  
  
  
  
  
  
  
  
  
  
  
  We note that the linear, step function, W-shaped,
  sinusoid, and circular alternatives considered in
  <cit.> can all be viewed as generalized rotation
  alternatives. The proof techniques used in this paper are hence
  directly applicable to these five alternatives by means of a re-parametrization. 
  To illustrate this point, consider, for example, the following alternative motivated by <cit.>:
  
  
  
  
  
  
      X^(1) = Y^(1)   and   
      X^(2) = Δ g(Y^(1))+ Y^(2),
  
  where Y^(1) and Y^(2) are independent and absolutely
  continuous with respective densities f_1,f_2. Notice that model (<ref>) and the one used in <cit.> are equivalent for rank-based tests as ranks are scale invariant. Assume then that
  
  
    * the distributions of =(X^(1),X^(2)) have a common support for all Δ∈(-1,1);
  
    * the density f_2 is absolutely continuous with non-constant logarithmic derivative ρ_2≡ f'_2/f_2 with 0<[{ρ_2(Y^(2))}^2] <∞; 
  
    * the function g is non-constant and measurable such that 0<[{g(Y^(1))}^2] <∞. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  Claims (<ref>) and (<ref>) will then hold for the
  alternatives (<ref>) in observation of arguments similar to
  those made in the proof of Theorem <ref> for the rotation alternatives (A). 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
    <cit.> performed a local power
    analysis for Chatterjee's ξ_n 
    under a set of assumptions that differs from ours.  
    The goal of our local power analysis was to exhibit explicitly the, 
    at times surprising, differences in power of the independence tests
    given by the four rank correlation coefficients from
    Definitions <ref>, <ref>–<ref>.  To this end, we focused on
    rotation and mixture 
    alternatives from the literature.  However, from the proof techniques in
    Section <ref> of the supplementary material, it is evident that
    Claims (<ref>) and (<ref>) hold for further types
    of local alternative families.  For the former claim, which concerns
    lack of power of
    Chatterjee's ξ_n, this point has been pursued in Section 4.4 of
    
    
    <cit.>.
    
    
    
    
    
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  § RANK CORRELATIONS FOR DISCONTINUOUS DISTRIBUTIONS
  
  
  In this section, we drop the continuity assumption of F made in Sections <ref>–<ref>, and allow for ties to exist with a nonzero probability. 
  Among the five correlation coefficients, ξ_n^* is no longer an
  appropriate estimator when F is not continuous. 
  We will only discuss the properties of the other four estimators ξ_n, D_n, R_n, and τ_n^*.  
  
  
  Recall that the computation issue has been address in Remark <ref>. Our first result in this section focuses on approximation consistency of the correlation coefficients ξ_n, D_n, R_n and τ^*_n to their population quantities. 
  To this end, we define 
  the families of distribution more general than the ones considered so
  far as follows:  
  
      ≡{F: F is a bivariate distribution function},
      ^*   ≡{F: F_k is not degenerate, i.e., F_k(x) I(x ≥ x_0) for any real number x_0 for k=1,2},
      ^τ^*   ≡{F: F is discrete, continuous, or a mixture of
                 discrete and jointly absolutely continuous distribution functions}.
  
  
  
  For the estimators ξ_n, D_n, R_n, and τ_n^*, the following result on consistency can be given.
  
  
  
  
  
  As n→∞, we have
  
  
    * for F∈^*, ξ_n converges in probability to ξ (Theorem 1.1 in ); 
  
    * for F∈, μ_n converges in probability to μ for μ∈{D,R,τ^*} (Proposition 1 in , Theorem 5.4.A in ). 
  
  
  
  
  
  The following proposition is a generalization of Proposition <ref>.
  
   
  
  
  
  
  
  
  The following are true:
  
  
    * for F∈^*, ξ≥ 0 with equality if and only if the pair is independent (Theorem 1.1 in );
  
    * for F∈, D≥ 0; for F∈, D=0 if and only if the pair 
  is independent (Theorem 3.1 in , Proposition 3 in );  
  
    * for F∈, R≥ 0 with equality if and only if the pair is independent (page 490 of );
  
    * for F∈^τ^*, τ^*≥0 where equality holds
  if and only if the variables are independent (Theorem 1 in , Theorem 6.1 in ).
  
  
  
  
  The asymptotic distribution theory from
  Section <ref> can also be extended.  
  
  As the continuity requirement is dropped, the central limit theorems for
  Chatterjee's ξ_n still holds.  However, the asymptotic variance
  now has a more complicated form and is not necessarily constant across
  the null hypothesis of independence (Theorem 2.2 in
  ).  A similar phenomenon arises for the
  limiting null distributions of D_n, R_n and τ_n^* when one or
  two marginals are not continuous; see Theorem 4.5 and Corollary 4.1 in
  <cit.> for further discussion.  As a result, permutation
  analysis, which is unfortunately computationally much more intensive, is typically invoked to implement a test outside the realm of
  continuous distributions.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  § SIMULATION RESULTS
  
  
  
  In order to further examine the power of the tests, we simulate data as  a sample comprised of n independent copies of
  (X^(1),X^(2)), for which we consider a suite of different
  specifications based on mixture, rotation, and generalized rotation alternatives.
  
  
    For the distribution of (X^(1),X^(2)) we choose the six
    alternatives.  In their specification, Y^(1) and Y^(2) are
    always independent random variables and
    Δ≡ n^-1/2Δ_0.
  
  
    *   The pair (X^(1),X^(2)) is given by the
    rotation alternative (<ref>), where Y^(1),Y^(2)
    are both standard Gaussian and Δ_0=2.  This is an instance of
    our Example <ref>.
  
    *   The pair (X^(1),X^(2)) is given by the
    mixture alternative (<ref>), where
  
      F_0(x^(1),x^(2))   ≡Ψ(x^(1))Ψ(x^(2)),
      
      G(x^(1),x^(2))   ≡Ψ(x^(1))Ψ(x^(2))[1+{1-Ψ(x^(1))}{1-Ψ(x^(2))}],
  
  Ψ(·) denotes the distribution function of the uniform
  distribution on [-1,1], and Δ_0=10.  This is in accordance
  with our Example <ref>.
  
    *   The pair (X^(1),X^(2)) is given by the
    mixture alternative (<ref>), where the density functions of F and G, denoted by f_0 and g, are given by
  
      f_0(x^(1),x^(2))   ≡ψ(x^(1))ψ(x^(2)),
      
      g(x^(1),x^(2))   ≡ψ(x^(1))ψ(x^(2))[1+|1-2Ψ(x^(1))|{1-2Ψ(x^(2))}],
  
  ψ(t)≡1/2×(-1≤ t≤ 1), and Δ_0=20.  This
  is an instance of our Example <ref>.
  
    *  The pair (X^(1),X^(2)) is given by the
    generalized rotation alternative (<ref>), where Y^(1) is uniformly distributed on [-1,1], Y^(2) is standard Gaussian, g takes values -3, 2, -4, and -3 in the intervals [-1, -0.5), [-0.5, 0), [0, 0.5), and [0.5, 1], respectively, and Δ_0=3.
  
    *  The pair (X^(1),X^(2)) is given by (<ref>), where Y^(1) is uniformly distributed on [-1,1], Y^(2) is standard Gaussian, g(t)≡ |t+0.5|(t<0)+|t-0.5|(t≥0), and Δ_0=60.
  
    *  The pair (X^(1),X^(2)) is given by (<ref>), where Y^(1) is uniformly distributed on [-1,1], Y^(2) is standard Gaussian, g(t)≡cos(2π t), and Δ_0=12. 
  
  
  
  As indicated, the first three simulation settings are taken from
  Examples <ref>–<ref>.  The latter three are motivated by 
  step function, W-shaped, and 
  sinusoid settings in which Chatterjee's 
  correlation coefficient performs well; see 
  <cit.>.
  
  Our focus is on comparing the empirical performance of the five tests
  T^ξ_n_α, T^ξ_n^*_α,
  T^D_n_α, T^R_n_α,
  T^τ^*_n_α.  The first four tests are conducted using the
  asymptotics from Proposition <ref>.  The last test is
  implemented with bandwidths chosen as h_1=h_2=n^-3/10 following the suggestion in Section 6.1 of <cit.> and using a finite-sample critical value, which we approximate
  via 1000 Monte Carlo simulations. The nominal significance level
  is set to 0.05, and the sample size is chosen as
  n∈{500,1000,5000,10000}.  For each of the six settings and four sample
  sizes, we conduct 1000 simulations.
  
  Before turning to statistical properties, we contrast the computation 
  times for calculating the five considered rank correlation coefficients first.  
  Table <ref> shows times in the considered rotation setting (a); 
  the results for other settings are essentially the same. 
  The calculations of ξ_n and ξ_n^* are by our own implementation, 
  and those of D_n, R_n, τ_n^* are made using the functions 
  , , 
  and  from R package  
  <cit.>, respectively.  All experiments are conducted on a laptop with a 2.6
  GHz Intel Core i5 processor and a 8 GB memory.
  One observes the clear computational advantages
  of ξ_n, D_n, R_n, and τ_n^* over <cit.>'s
  estimator ξ_n^*.  
  
  
  
  
  The difference in computation time between Chatterjee's coefficient ξ_n and
  Hoeffding's D_n is insignificant.   Both ξ_n and D_n are slightly
  faster to compute than Blum–Kiefer–Rosenblatt's R_n and
  Bergsma–Dassios–Yanagimoto's τ^*_n; computation times differ by
  a factor less than 2.5.
  
  
  Table <ref> shows the empirical powers 
  of the five tests. 
  
  The results confirm our earlier theoretical
  claims on the powers of the  different tests in the different models, 
  that Hoeffding's D, Blum–Kiefer–Rosenblatt's R, and Bergsma–Dassios–Yanagimoto's τ^* outperform Chatterjee's correlation coefficient in all the settings considered. 
  
  
  Interestingly, the simulation results suggest that the test based on ξ_n^* may have
  non-trivial power against certain alternatives; 
  see results for Example <ref><ref>,<ref> 
  in Table <ref>.
  
  
  
  
  
  
  
   
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  § DISCUSSION
   
  
  
  In this paper we considered independence tests based on the five rank
  correlations from Definitions <ref>–<ref>.  As we
  surveyed in Section <ref>, recent advances lead to little
  difference in the efficiency of known algorithms to compute these
  correlation coefficients.  For continuous distributions, i.e., data
  without ties, all correlations except for Dette–Siburg–Stoimenov's ξ_n^* can be
  computed in nearly linear time.  Moreover, all but Hoeffding's D
  give consistent tests of independence for arbitrary continuous
  distributions; consistency of D can be established for all
  absolutely continuous distributions.
  
  Our main new contribution is a local power analysis for continuous
  distributions that revealed interesting differences in the power of
  the tests.  This analysis features subtle differences but the
  take-away message is that ξ_n is suboptimal for
  testing independence, whereas the more classical D_n, R_n, and
  τ^*_n are rate optimal in the considered setup.  This said, ξ_n
  and ξ_n^* have very appealing properties that do not pertain to
  independence but rather detection of perfect functional dependence.  We refer
  the reader to <cit.> and <cit.> as well as <cit.>.
  
  We summarize the properties discussed in our paper in
  Table <ref>.  When referring to independence tests in this
  table we assume continuous observations, i.e., F∈^c.
  Moreover, when discussing ξ_n^*, we assume additionally that the
  kernel K and bandwidths h_1,h_2 satisfy all assumptions stated in
  Definition <ref>.  The table features two rows for
  computation, where the first pertains to continuous observations free
  of ties and the second pertains to arbitrary observations. The third 
  row of the table concerns consistency of correlation measures; refer to 
  (<ref>) and (<ref>) for the definitions of 
  table entries. The fourth row concerns consistency of independence 
  tests assuming F∈^c. Finally, we summarize the rate-optimality 
  and rate sub-optimality of five independence tests under two 
  local alternatives (A) and (B) considered in Section <ref>.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  § PROOFS
  
  
  
  
  Throughout the proofs below, all the claims regarding conditional
  expectations, conditional variances, and conditional covariances are in the almost sure sense. 
  
  
  
  
  
  
  
  
   §.§ Proof of Proposition <ref> (ξ^*_n)
  
  
  
  
  Equation (21) in <cit.> states that
  
      B̂_2n-B̃_2n-C_1n-C_2n=o_p(n^-1/2),
  
  but tracking a glitch in signs the equation should in fact be 
      B̂_2n-B̃_2n+C_1n+C_2n=o_p(n^-1/2).
    
  Accordingly, a revised version of Equations (24)–(26) in <cit.> shows that, 
  
      n^1/2(ξ_n^*-ξ)=12/n^1/2∑_i=1^n(Z_i- Z_i)+o_p(1)
  
  where Z_i≡ Z_i,1-Z_i,2-Z_i,3 with 
  
      Z_i,1   ≡∫_0^1{F_Y^(2)(Y^(2)_i)≤ u^(2)}τ(F_Y^(1)(Y^(1)_i), u^(2))ụ^(2),
      
      Z_i,2   ≡∫_0^1∫_0^1{F_Y^(1)(Y^(1)_i)≤ u^(1)}τ(u^(1), u^(2))∂/∂ u^(1)τ(u^(1), u^(2))ụ^(1)ụ^(2),
      
      Z_i,3   ≡∫_0^1∫_0^1{F_Y^(2)(Y^(2)_i)≤ u^(2)}τ(u^(1), u^(2))∂/∂ u^(2)τ(u^(1), u^(2))ụ^(1)ụ^(2),
  
  τ(u^(1), u^(2))=∂C(u^(1), u^(2))/∂ u^(1), 
  and C(u^(1), u^(2)) is the copula of (Y^(1), Y^(2)). 
  Since the first term on the right hand side of (<ref>) has finite variance (see computation on pages 34–35 of <cit.>), we deduce that
  
      ξ^*_n⟶_p ξ.
  
  This completes the proof.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   §.§ Proof of Proposition <ref>(ii)
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  Applying (<ref>), 
  it holds under the null that
  
      C(u^(1), u^(2))=u^(1)u^(2),    τ(u^(1), u^(2))=u^(2).
  
   Accordingly,
  
      Z_i,1=Z_i,3=∫_0^1{F_Y^(2)(Y^(2)_i)≤ u^(2)}u^(2)ụ^(2)=1/2[1-{F_Y^(2)(Y^(2)_i)}^2]
         and   Z_i,2=0,
  
  which yields
  
      n^1/2ξ_n^*→ 0    in probability.
  
  This completes the proof.
  
  
  
  
  
  
  
  
  
  
  
  
  
   §.§ Proof of Remark <ref>
  
  
  
  Recall that f_(;Δ) denotes the density of  with Δ. Denote
  
      L(;Δ)≡f_(;Δ)/f_(;0)   and   
      L'(;Δ)≡∂/∂ΔL(;Δ).
  
  These definitions make sense by
  Assumption <ref><ref>,<ref>, and we may write  _(0)=[{L'(;0)}^2]. Notice that  is distributed as  with Δ=0. 
  Since =_Δ^-1 is an invertible linear transformation, the density of  can be expressed as  
  
      f_(;Δ)=|(_Δ)|^-1f_(_Δ^-1),
  
  where 
  f_()=f_(y^(1),y^(2))=f_1(y^(1))f_2(y^(2)). 
  Direct computation yields
  
      L(;Δ)=|(_Δ)|^-1f_(_Δ^-1)/f_(),
      and      L'(;0)=-x^(1){ρ_2(x^(2))}
                 -x^(2){ρ_1(x^(1))}.
  
  Thus {(Y^(k))^2} <∞ and [{ρ_k(Y^(k))}^2] <∞ for k=1,2 will imply _(0)=[{L'(;0)}^2]<∞ under the Konijn alternatives. Also, [{ρ_k(Y^(k))}^2] <∞ implies that {ρ_k(Y^(k))}=0 by Lemma A.1 (Part A) in <cit.>. 
  
  
  
  
  
  
  
  
  
  
  
  
   §.§ Proof of Example <ref>
  
  
  
  
  
  
  Assumption <ref><ref> is satisfied since f_k(z)>0, k=1,2 for all real z. 
  Assumption <ref><ref> holds in view of
  (<ref>); notice that L'(;0) can never always be 0. 
  For Assumption <ref><ref>, if ρ_k(z) is constant, then f_k(z) is either constant or proportional to e^Cz with some constant C for all real z, which is impossible. Then Assumption <ref> is satisfied. 
  
  Regarding the special case, 
  without loss of generality, we can assume Y_1 and Y_2 to be standard normal or standard t-distributed. For the standard normal, we have ρ_k(z)=-t and thus (<ref>) is satisfied. For the standard t-distribution with ν_k degrees of freedom, we have 
  ρ_k(z)=-z(1+1/ν_k)/(1+z^2/ν_k). It is easy to check (<ref>) is satisfied when ν_k>2.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   §.§ Proof of Remark <ref>
  
  
  
  
  Let f_(;Δ) denote the density of  with Δ. Denote
  
      L(;Δ)≡f_(;Δ)/f_(;0)   and   
      L'(;Δ)≡∂/∂ΔL(;Δ),
  
  then we can write _(0)=[{L'(;0)}^2], where  is distributed as  with Δ=0. 
  Direct computation yields 
  
      L(;Δ)=(1-Δ)f_0()+Δ g()/f_0(),   
      L'(;0)=g()-f_0()/f_0(),
  
  and thus
  
      _(0)   =[{L'(;0)}^2]=[{g(Y)/f_0(Y)-1}^2]
         =[{s(Y)}^2]=χ^2(G,F_0)≡∫ (G̣/F̣_0-1)^2 F̣_0.
  
  Since s(x)=g(x)/f_0(x)-1 is continuous and both g and f_0 have compact support, s(x) is bounded. Hence _(0)<∞. 
  
  
  
  
  
  
  
  
  
  
  
  
   §.§ Proof of Example <ref>
  
  
  
  
  To verify Assumption <ref> for the Farlie alternatives, 
  we first prove that G is a bonafide joint distribution function.  The corresponding density g is given by
  
      g(x^(1),x^(2))=f_1(x^(1))f_2(x^(2))[1+{1-2F_1(x^(1))}{1-2F_2(x^(2))}],
  
  which is a bonafide joint density function <cit.>. 
  Then we have 
  
      s(x)=g(x)/f_0(x)-1={1-2F_1(x^(1))}{1-2F_2(x^(2))}
  
  and find that
  
      [s(Y)|Y^(1)]={1-2F_1(Y^(1))}×{1-2F_2(Y^(2))}=0
      and      [s(Y)|Y^(2)]={1-2F_1(Y^(1))}×{1-2F_2(Y^(2))}=0.
  
  The proof is completed. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   §.§ Proof of Example <ref>
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  We first verify that g is a bonafide joint density function. First since both h_1 and h_2 are bounded by 1, 
  
      | g(x)/f_0(x)-1|=| h_1(x^(1))h_2(x^(1))|≤1,
  
  and thus g(x)≥0. Then we write
  
      g(x^(1),x^(2))=
      f_1(x^(1))f_2(x^(2)) +
      f_1(x^(1))h_1(x^(1))f_2(x^(2))h_2(x^(2))
  
  and 
  
      ∫_-∞^∞∫_-∞^∞ g(x^(1),x^(2))x̣^(1)x̣^(2)
      =    ∫_-∞^∞f_1(x^(1))x̣^(1)×∫_-∞^∞f_2(x^(2))x̣^(2)
         +
       ∫_-∞^∞f_1(x^(1))h_1(x^(1))x̣^(1)×∫_-∞^∞f_2(x^(2))h_2(x^(2))x̣^(2)=1,
  
  where
  
      ∫_-∞^∞f_1(x^(1))h_1(x^(1))x̣^(1)<∞   and   ∫_-∞^∞f_2(x^(2))h_2(x^(2))x̣^(2)=0
  
  since h_1(x^(1)),h_2(x^(2)) are bounded by 1 and f_2(x^(2))h_2(x^(2))=-f_2(-x^(2))h_2(-x^(2)). We also have
  
      [s(Y)|Y^(1)]=h_1(Y^(1))×[h_2(Y^(2))]=h_1(Y^(1))∫_-∞^∞f_2(x^(2))h_2(x^(2))x̣^(2)=0,
      and      [s(Y)|Y^(2)]=[h_1(Y^(1))]× h_2(Y^(2))   with [h_1(Y^(1))]=∫_-∞^∞f_1(x^(1))h_1(x^(1))x̣^(1) 0.
  
  The proof is completed. 
  
  
  
  
  
  
  
  
  
  
  
  
  
   §.§ Proof of Theorem <ref>(i)
  
  
  
  
  (A) This proof uses all of Assumption <ref>.
  
  
  Let _i=(Y^(1)_i,Y^(2)_i), i=1,…,n be independent copies of . 
  
  Recall that f_(;Δ) is the density of  with Δ. 
  Denote
  
      L(;Δ)≡f_(;Δ)/f_(;0),   
      L'(;Δ)≡∂/∂ΔL(;Δ),
  
  and define Λ_n=∑_i=1^nlog L(_i;Δ_n) and T_n≡Δ_n∑_i=1^nL'(_i;0). 
  These definitions make sense by Assumption <ref><ref>,<ref>. 
  
  To employ a corollary to Le Cam's third lemma, 
  we wish to derive the joint limiting null distribution of (-n^1/2ξ_n/3,Λ_n).
  Under the null hypothesis, it holds that Y^(2)_[1],…,Y^(2)_[n] are still independent and identically distributed, where [i] is  such that Y^(1)_[1]<⋯<Y^(1)_[n].
  
  In view of <cit.>, we have that under the null, 
  
      (-n^1/2ξ_n/3)
      
      -n^-1/2∑_i=1^n-1Ξ_[i]→ 0    in probability,
  
  where 
  
      Ξ_[i]≡| F_Y^(2)(Y^(2)_[i+1])-F_Y^(2)(Y^(2)_[i])|   +F_Y^(2)(Y^(2)_[i+1]){1-F_Y^(2)(Y^(2)_[i+1])}
         +F_Y^(2)(Y^(2)_[i]){1-F_Y^(2)(Y^(2)_[i])}-2/3,
  
  and F_Y^(2) is the cumulative distribution function for Y^(2). One readily verifies |Ξ_[i]|≤ 1. 
  
  Using (<ref>), the limiting null distribution of (-n^1/2ξ_n/3,Λ_n) will be the same as that of 
  (n^-1/2∑_i=1^n-1Ξ_[i],Λ_n). 
  To find the limiting null distribution of (n^-1/2∑_i=1^n-1Ξ_[i],Λ_n), using the idea from <cit.>, we first find the limiting null distribution of 
  
      (n^-1/2∑_i=1^n-1Ξ_[i],T_n)
         =(n^-1/2∑_i=1^n-1Ξ_[i],n^-1/2Δ_0∑_i=1^nL'(_i;0))
         =(n^-1/2∑_i=1^n-1Ξ_[i],n^-1/2Δ_0∑_i=1^nL'(_[i];0)),
  
  where _[i]≡ (^(1)_[i],^(2)_[i]). 
  
  
  To employ the Cramér–Wold device, we aim to show that under the null, for any real numbers a and b, 
  
      an^-1/2∑_i=1^n-1Ξ_[i]+ bn^-1/2Δ_0∑_i=1^nL'(_[i];0)
      → N(0,2a^2/45+b^2Δ_0^2_(0))    in distribution.
  
  The idea of the proof is to first show a conditional central limit result
  
      an^-1/2∑_i=1^n-1Ξ_[i]+ bn^-1/2Δ_0∑_i=1^nL'(_[i];0)| Y^(1)_1,…,Y^(1)_n→ N(0,2a^2/45+b^2Δ_0^2_(0))
         in distribution, for almost every sequence Y^(1)_1,…,Y^(1)_n,…,
  
  and secondly deduce the desired unconditional central limit result. 
  
  To prove (<ref>), we follow the idea put forward in the proof of Lemma 2.9.5 in <cit.>. 
  According to the central limit theorem for 1-dependent random
  variables (see, e.g., the Corollary in ), 
  the statement (<ref>) is true if the following conditions hold: for almost every sequence Y^(1)_1,…,Y^(1)_n,…, 
  
      ^(2)( W_[i])= 0, 
      
         1/n^(2){(∑_i=1^n
        W_[i])^2}→ 2a^2/45+b^2Δ_0^2_(0), 
      
         ∑_i=1^n^(2)( W_[i] ^2)/^(2){(∑_i=1^n
        W_[i])^2}   is bounded,
      
      and      1/n∑_i=1^n^(2){ W_[i] ^2
      ×( n^-1/2| W_[i]| > ϵ)}→ 0
         for every ϵ>0,
  
  where ^(2) denotes the expectation conditionally on Y^(1)_1,…,Y^(1)_n, and
  
      W_[i]≡ aΞ_[i]
       +bΔ_0L'(Y_[i];0)
         for i=1,…,n-1,
         and   
      W_[n]≡ 
        bΔ_0L'(Y_[n];0).
  
  
  
  We verify conditions (<ref>)–(<ref>) as follows, starting from (<ref>).
  Under the null hypothesis, conditionally on Y^(1)_1,…,Y^(1)_n, we have that Y^(2)_[1],…,Y^(2)_[n] are still independent and identically distributed as Y^(2), which implies that ^(2)(Ξ_[i])=0. 
  We also deduce, by (<ref>) and Assumption <ref><ref>, that 
  
      {L'(Y;0)| Y^(1)}=0,
  
  and thus ^(2){L'(Y_[i];0)}=0. 
  Then (<ref>) follows by noticing that
  
      ^(2)(Ξ_[i])=0
         and   ^(2){L'(Y_[i];0)}=0.
   
  
  For (<ref>) and (<ref>), we first claim that 
  
      ^(2){n^-1/2∑_i=1^n-1Ξ_[i],n^-1/2Δ_0∑_i=1^nL'(_[i];0))}=0,
   
  where ^(2) denotes the covariance conditionally on Y^(1)_1,…,Y^(1)_n. 
  Recall that, under the null hypothesis, Y^(2)_[1],…,Y^(2)_[n] are still independent and identically distributed as Y^(2), conditionally on Y^(1)_1,…,Y^(1)_n.
  We obtain
  
      ^(2){|F_Y^(2)(Y^(2)_[i+1])-F_Y^(2)(Y^(2)_[i])|, L'(Y_[i+1];0)}
      
      =    ^(2)[1/2{F_Y^(2)(Y^(2)_[i+1])}^2
      +1/2{1-F_Y^(2)(Y^(2)_[i+1])}^2, L'(Y_[i+1];0)]
  
  by taking expectation with respect to Y^(2)_[i],
  
      ^(2){|F_Y^(2)(Y^(2)_[i+1])-F_Y^(2)(Y^(2)_[i])|, L'(Y_[i];0)}
      
      =    ^(2)[1/2{F_Y^(2)(Y^(2)_[i])}^2
      +1/2{1-F_Y^(2)(Y^(2)_[i])}^2, L'(Y_[i];0)]
  
  by taking expectation with respect to Y^(2)_[i+1], and
  
      ^(2){|F_Y^(2)(Y^(2)_[i+1])-F_Y^(2)(Y^(2)_[i])|, L'(Y_[j];0)}=0   for all j i, i+1,
  
  since Y^(2)_[i],Y^(2)_[i+1] are independent of Y^(2)_[j] with j i, i+1, conditionally on Y^(1)_1,…,Y^(1)_n. 
  Taking into account (<ref>)–(<ref>), it follows that
  
      ^(2){n^-1/2∑_i=1^n-1Ξ_[i],n^-1/2Δ_0∑_i=1^nL'(Y_[i])}
      
      =    n^-1Δ_0(∑_i=2^n^(2)[1/2{F_Y^(2)(Y^(2)_[i])}^2
      +1/2{1-F_Y^(2)(Y^(2)_[i])}^2, L'(Y_[i];0)]
               +∑_i=1^n-1^(2)[1/2{F_Y^(2)(Y^(2)_[i])}^2
      +1/2{1-F_Y^(2)(Y^(2)_[i])}^2, L'(Y_[i];0)]
               +∑_i=2^n^(2)[F_Y^(2)(Y^(2)_[i]){1-F_Y^(2)(Y^(2)_[i])}, L'(Y_[i];0)]
      
               +∑_i=1^n-1^(2)[F_Y^(2)(Y^(2)_[i]){1-F_Y^(2)(Y^(2)_[i])}, L'(Y_[i];0)])
      
      =    n^-1[∑_i=2^n^(2){1/2, L'(Y_[i];0)}+∑_i=1^n-1^(2){1/2, L'(Y_[i];0)}]
      
      =    n^-1[-^(2){1/2, L'(Y_[1];0)}-^(2){1/2, L'(Y_[n];0)}] = 0,
  
  where 
  
  
  
  
  
  
  
  
  we notice that
  
      {n^-1| L'(Y_[j];0)|}≤{n^-1∑_i=1^n| L'(Y_i;0)|}
      =  | L'(Y;0)|
      <  ∞,
   
  for any given j.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  Then using (<ref>)–(<ref>) we can prove (<ref>) as follows:
  
      1/n^(2){(∑_i=1^nW_[i]   )^2}
      =1/n^(2)[{a∑_i=1^n-1Ξ_[i]+ bΔ_0∑_i=1^nL'(_[i];0)}^2]
      
      =    1/n^(2)[(a∑_i=1^n-1Ξ_[i])^2+ {bΔ_0∑_i=1^nL'(_[i];0)}^2]
      
      =    1/n^(2)[(a∑_i=1^n-1Ξ_[i])^2+ {bΔ_0∑_i=1^nL'(_i;0)}^2]
      
      =    2a^2(n-1)/45n+ 1/n∑_i=1^n^(2)[{bΔ_0L'(_i;0)}^2]
      →2a^2/45+b^2Δ_0^2_(0),
  
  where the last step holds for almost all sequences Y^(1)_1,…,Y^(1)_n,… by the law of large numbers. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  To verify (<ref>), recalling (<ref>) and using (<ref>),(<ref>), we obtain
  
      ^(2){Ξ_[i]×Δ_0L'(Y_[i];0)}
      =^(2){Ξ_[i],Δ_0L'(Y_[i];0)}=0,
  
  and moreover,
  
      1/n∑_i=1^n^(2)( W_[i] ^2)
         =1/n(∑_i=1^n-1^(2)[{aΞ_[i]
       +bΔ_0L'(Y_[i];0)}^2]
      +^(2)[{bΔ_0L'(Y_[n];0)}^2])
         =1/n(∑_i=1^n-1^(2){(aΞ_[i])^2}
      +∑_i=1^n^(2)[{bΔ_0L'(Y_[i];0)}^2])
         =1/n(2a^2(n-1)/45
      +∑_i=1^n^(2)[{bΔ_0L'(Y_i;0)}^2]).
  
  Hence we have, recalling (<ref>),
  
      ∑_i=1^n^(2)( W_[i] ^2)/^(2){(∑_i=1^n
        W_[i])^2}=1.
  
  
  For proving (<ref>), we recall that as given in (<ref>)
  
      L'(Y_[i];0)=
      -Y_[i]^(1){ρ_2(Y_[i]^(2))}
      -Y_[i]^(2){ρ_1(Y_[i]^(1))},
  
  where ρ_k(z)≡ f'_k(z)/f_k(z). 
  The existence of finite second moments assumed in Assumption <ref><ref>, {(Y^(1))^2} <∞ and [{ρ_1(Y^(1))}^2] <∞, implies that
  
      max_1≤ i≤ nn^-1/2| Y^(1)_i|→ 0
         and   max_1≤ i≤ nn^-1/2|ρ_1(Y^(1)_i) |→ 0
  
  for almost all sequences Y^(1)_1,…,Y^(1)_n,… <cit.>. Since |Ξ_[i]|≤ 1, we have
  
      ( n^-1/2| W_[i]| > ϵ)
      ≤( | a| n^-1/2 > ϵ/3 )
         +  {| b|×(max_1≤ i≤ nn^-1/2| Y^(1)_i|)
                       ×|ρ_2(Y_[i]^(2))| > ϵ/3 }
         +  {| b|×(max_1≤ i≤ nn^-1/2|ρ_1(Y^(1)_i) |)
                       ×| Y_[i]^(2)| > ϵ/3 }.
  
  Then for every ϵ>0, 
  
      1/n∑_i=1^n^(2){ W_[i] ^2
      ×( n^-1/2| W_[i]| > ϵ)}
      ≤    1/n∑_i=1^n^(2)( 3[a^2 +
      {Y_[i]^(1)ρ_2(Y_[i]^(2))}^2 +
      {Y_[i]^(2)ρ_1(Y_[i]^(1))}^2]
              ×[
         ( | a| n^-1/2 > ϵ/3 )+  {| b|×(max_1≤ i≤ nn^-1/2| Y^(1)_i|)
                       ×|ρ_2(Y_[i]^(2))| > ϵ/3 }
                                 +  {| b|×(max_1≤ i≤ nn^-1/2|ρ_1(Y^(1)_i) |)
                       ×| Y_[i]^(2)| > ϵ/3 }]).
  
  Here in (<ref>) we have by (<ref>) and dominated convergence theorem that
  
      1/n∑_i=1^n^(2)[{| b|×(max_1≤ i≤ nn^-1/2| Y^(1)_i|)
                       ×|ρ_2(Y_[i]^(2))| > ϵ/3 }]
      
      =    ^(2)[{| b|×(max_1≤ i≤ nn^-1/2| Y^(1)_i|)
                       ×|ρ_2(Y_1^(2))| > ϵ/3 }]→ 0,
  
  where
  
      {| b|×(max_1≤ i≤ nn^-1/2| Y^(1)_i|)
                       ×|ρ_2(Y_1^(2))| > ϵ/3 }]→ 0    in probability,
  
  for almost all sequences Y^(1)_1,…,Y^(1)_n,…. We also have
  
      1/n∑_i=1^n^(2)[{Y_[i]^(1)ρ_2(Y_[i]^(2))}^2 
      ×{| b|×(max_1≤ i≤ nn^-1/2| Y^(1)_i|)
                       ×|ρ_2(Y_[i]^(2))| > ϵ/3 }]
      
      =    1/n∑_i=1^n(Y_[i]^(1))^2^(2)[{ρ_2(Y_[i]^(2))}^2 
      ×{| b|×(max_1≤ i≤ nn^-1/2| Y^(1)_i|)
                       ×|ρ_2(Y_[i]^(2))| > ϵ/3 }]
      
      =    (1/n∑_i=1^n(Y_[i]^(1))^2)(^(2)[{ρ_2(Y_1^(2))}^2 
      ×{| b|×(max_1≤ i≤ nn^-1/2| Y^(1)_i|)
                       ×|ρ_2(Y_1^(2))| > ϵ/3 }])
      
      =    (1/n∑_i=1^n(Y_i^(1))^2)(^(2)[{ρ_2(Y_1^(2))}^2 
      ×{| b|×(max_1≤ i≤ nn^-1/2| Y^(1)_i|)
                       ×|ρ_2(Y_1^(2))| > ϵ/3 }])
      →    0,
  
  where for almost all sequences Y^(1)_1,…,Y^(1)_n,…,
  
      1/n∑_i=1^n(Y_i^(1))^2→{(Y^(1))^2}
  
  by the law of large numbers, and
  
      ^(2)[{ρ_2(Y_1^(2))}^2 
      ×{ |b|×(max_1≤ i≤ nn^-1/2| Y^(1)_i|)
                       ×|ρ_2(Y_1^(2))| > ϵ/3 }]→ 0
  
  by (<ref>) and the dominated convergence theorem.  
  We can deduce similar convergences for all the other summands in (<ref>). 
  Hence for almost all sequences Y^(1)_1,…,Y^(1)_n,…, all conditions (<ref>)–(<ref>) are satisfied. This completes the proof of (<ref>). Moreover, the desired result (<ref>) follows. 
  
  
  
  Finally, the Cramér–Wold device yields that under the null,
  
      (n^-1/2∑_i=1^n-1Ξ_[i],T_n)
      →
      N_2((0
      0),
      (2/45     0
      
      0    Δ_0^2_(0)))    in distribution.
  
  Furthermore, using ideas from <cit.>
  (see also ), we have under the null, 
  
      Λ_n-T_n+Δ_0^2_(0)/2→ 0    in probability,
  
  and thus under the null, 
  
      (n^-1/2∑_i=1^n-1Ξ_[i],Λ_n)
      →
      N_2((0
      -Δ_0^2_(0)/2),
      (2/45     0
      
      0    Δ_0^2_(0)))    in distribution,
  
  and (-n^1/2ξ_n/3,Λ_n) has the same limiting null distribution by (<ref>).  
  Finally, we employ a corollary to Le Cam's third lemma () to obtain that, under the considered local alternative H_1,n(Δ_0) with any fixed Δ_0>0, -n^1/2ξ_n/3→ N(0,2/45) in distribution, and thus
  
      n^1/2ξ_n
      →
      N(0,2/5)    in distribution.
  
  This completes the proof for family (A). 
  
  
  
  
  
  
  
  (B) This proof proceeds with only Assumption <ref><ref>,<ref>,<ref>.
  Let _i=(Y^(1)_i,Y^(2)_i), i=1,…,n be independent copies of  (distributed as  with Δ=0).
  
  
  Denote
  
      L(;Δ)≡f_(;Δ)/f_(;0),   
      L'(;Δ)≡∂/∂ΔL(;Δ),
  
  and define Λ_n=∑_i=1^nlog L(_i;Δ_n) and T_n≡Δ_n∑_i=1^nL'(_i;0). Direct computation yields 
  
      L(;Δ)≡(1-Δ)f_0()+Δ g()/f_0(),   
      L'(;0)=g()-f_0()/f_0(),
  
  and thus
  
      _(0)   =[{L'(;0)}^2]=[{g(Y)/f_0(Y)-1}^2]
         =[{s(Y)}^2]=χ^2(G,F_0)≡∫ (G̣/F̣_0-1)^2 F̣_0.
  
  
  
  
  
  Similar to the proof for family (A), we proceed to determine the limiting
  null distribution of (-n^1/2ξ_n/3,Λ_n).  To this end, 
  in view of the proof of Theorem 2 in <cit.>, 
  we first find the limiting null distribution of 
  (n^-1/2∑_i=1^n-1Ξ_[i],T_n). 
  The idea of deriving it is still to first show (<ref>), 
  then (<ref>), and thus (<ref>). 
  
  
  Next we verify conditions (<ref>)–(<ref>)
  for family (B). Notice that when we verify conditions
  (<ref>)–(<ref>) for family (A) (from
  (<ref>) to (<ref>)), we only use that 
  
  
    * 
  under the null hypothesis, Y^(2)_[1],…,Y^(2)_[n] are still independent and identically distributed as Y^(2), conditionally on Y^(1)_1,…,Y^(1)_n, 
  
    * {L'(Y;0)| Y^(1)}=0, and
  
    * 0<_(0)<∞. 
  
  The first property always holds under the null hypothesis. 
  The latter two are assumed or implied in Assumption <ref><ref> and Assumption <ref><ref>,<ref>, respectively. 
  Hence we can verify conditions (<ref>)–(<ref>) for family (B) using the same arguments. 
  The only difference lies in proving (<ref>). Since s(x)=g(x)/f_0(x)-1 is continuous and has compact support, it is bounded by some constant, say C_s>0. We have by definition of W_[i] in (<ref>), 
  
      | W_[i]|≤| a| + | b|Δ_0 C_s,
  
  and thus
  
      ( n^-1/2| W_[i]| > ϵ)=0
         for all n>(|a|+|b|Δ_0 C_s/ϵ)^2.
  
  Then (<ref>) follows by the dominated convergence theorem. 
  
  
  We have proven (<ref>) for family (B). Furthermore, 
  in the proof of Theorem 2 in <cit.>, they showed that under the null,
  
      Λ_n-T_n+Δ_0^2_(0)/2→ 0    in probability.
  
  Thus under the null, we have (<ref>) as well. The rest of the proof is 
  to employ a corollary to Le Cam's third lemma () 
  to obtain (<ref>). 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   §.§ Proof of Theorem <ref>(ii)
  
  
  
  (A) This proof uses all of Assumption <ref>.
  
  
  Let _i=(Y^(1)_i,Y^(2)_i) and _i=(X^(1)_i,X^(2)_i), i=1,…,n be independent copies of  and , respectively.
  Here  depends on n with Δ=Δ_n=n^-1/2Δ_0. 
  Let F^(0) and F^(a) be the (joint) distribution functions of (_1,…,_n) and (_1,…,_n), respectively. 
  Denote
  
      L(;Δ)≡f_(;Δ)/f_(;0),   
      L'(;Δ)≡∂/∂ΔL(;Δ),
  
  and define Λ_n=∑_i=1^nlog L(_i;Δ_n) and T_n≡Δ_n∑_i=1^nL'(_i;0). 
  These definitions make sense by Assumption <ref><ref>,<ref>. 
  
  In this proof we will consider the Hoeffding decomposition of μ_n under the null:
  
      μ_n
      =∑_ℓ=1^m^μnℓ^-1∑_1≤ i_1<⋯< i_ℓ≤ nm^μℓ h^μ_ℓ{(Y^(1)_i_1,Y^(2)_i_1),…,
        (Y^(1)_i_ℓ,Y^(2)_i_ℓ)
        }_ H^μ_n,ℓ,
  
  where
  
      h^μ_ℓ(_1,…,_ℓ) 
      ≡ h^μ_ℓ(_1,…,_ℓ)- h^μ-∑_k=1^ℓ-1∑_1≤ i_1<⋯<i_k≤ℓh^μ_k(_i_1,…,_i_k),
         h^μ_ℓ(_1…,_ℓ)≡ h^μ(_1…,_ℓ,_ℓ+1,…,_m^μ),
          h^μ≡ h^μ(_1,…,_m^μ),
  
  and _1,…,_m^μ are m^μ independent copies of .  
  Here h^μ is the “symmetrized” kernel 
  and m^μ is the order of the kernel function h^μ
  for μ∈{D,R,τ^*} related to (<ref>), (<ref>), or (<ref>):
  
      h^D(_1,…,_5)
      ≡    1/5!∑_1≤ i_1⋯ i_5≤ 51/4
            [{(y^(1)_i_1≤ y^(1)_i_5)-(y^(1)_i_2≤ y^(1)_i_5)}{(y^(1)_i_3≤ y^(1)_i_5)-(y^(1)_i_4≤ y^(1)_i_5)}]
      
            [{(y^(2)_i_1≤ y^(2)_i_5)-(y^(2)_i_2≤ y^(2)_i_5)}{(y^(2)_i_3≤ y^(2)_i_5)-(y^(2)_i_4≤ y^(2)_i_5)}],
      
      h^R(_1,…,_6)
      ≡    1/6!∑_1≤ i_1⋯ i_6≤ 61/4
            [{(y^(1)_i_1≤ y^(1)_i_5)-(y^(1)_i_2≤ y^(1)_i_5)}{(y^(1)_i_3≤ y^(1)_i_5)-(y^(1)_i_4≤ y^(1)_i_5)}]
            [{(y^(2)_i_1≤ y^(2)_i_6)-(y^(2)_i_2≤ y^(2)_i_6)}{(y^(2)_i_3≤ y^(2)_i_6)-(y^(2)_i_4≤ y^(2)_i_6)}],
      
      h^τ^*(_1,…,_4)
      ≡    1/4!∑_1≤ i_1⋯ i_4≤ 4{(y^(1)_i_1,y^(1)_i_3< y^(1)_i_2,y^(1)_i_4)
               +(y^(1)_i_2,y^(1)_i_4< y^(1)_i_1,y^(1)_i_3)
         125mu
               -(y^(1)_i_1,y^(1)_i_4< y^(1)_i_2,y^(1)_i_3)
               -(y^(1)_i_2,y^(1)_i_3< y^(1)_i_1,y^(1)_i_4)}
         120mu
      {(y^(2)_i_1,y^(2)_i_3< y^(2)_i_2,y^(2)_i_4)
               +(y^(2)_i_2,y^(2)_i_4< y^(2)_i_1,y^(2)_i_3)
         125mu
               -(y^(2)_i_1,y^(2)_i_4< y^(2)_i_2,y^(2)_i_3)
               -(y^(2)_i_2,y^(2)_i_3< y^(2)_i_1,y^(2)_i_4)},
  
  and m^D=5, m^R=6, m^τ^*=4. 
  We will omit the superscript μ in m^μ, h^μ,
  h^μ_ℓ, h^μ_ℓ, and H^μ_n,ℓ
  hereafter if no confusion is possible. 
  
  
  The proof is split into three steps. 
  First, we prove that F^(a) is contiguous to F^(0) in order to employ Le Cam's third lemma <cit.>.
  Next, we find the limiting null distribution of (nμ_n, Λ_n). 
  Lastly, we employ Le Cam's third lemma to deduce the alternative distribution of (nμ_n, Λ_n).
  
  
  Step I. In view of <cit.>, Assumption <ref> is sufficient for the contiguity: we have that F^(a) is contiguous to F^(0). 
  
  
  Step II. Next we need to derive the limiting distribution of (nμ_n, Λ_n) under null hypothesis. 
  To this end, we first derive the limiting null distribution of (nH_n,2, Λ_n), where H_n,2 is defined in (<ref>). 
  We write by the Fredholm theory of integral equations <cit.> that 
  
      H_n,2=
      1/n(n-1)∑_i j∑_v=1^∞λ_vψ_v(Y^(1)_i,Y^(2)_i)
        ψ_v(Y^(1)_j,Y^(2)_j),
   
  where {λ_v,v=1,2,…} is an arrangement of {λ_v_1,v_2, v_1,v_2=1,2,…}, and ψ_v is the normalized eigenfunction associated with λ_v. 
  For each positive integer K, define the “truncated” U-statistic as
  
      H_n,2,K≡1/n(n-1)∑_i j∑_v=1^Kλ_vψ_v(Y^(1)_i,Y^(2)_i)
        ψ_v(Y^(1)_j,Y^(2)_j).
  
  
  Notice that nH_n,2 and nH_n,2,K can be written as
  
      n H_n,2   =n/n-1(∑_v=1^∞λ_v{n^-1/2∑_i=1^n ψ_v(Y^(1)_i,Y^(2)_i)}^2
      -∑_v=1^∞λ_v[n^-1∑_i=1^n {ψ_v(Y^(1)_i,Y^(2)_i)}^2]),
      
      n H_n,2,K   =n/n-1(∑_v=1^Kλ_v{n^-1/2∑_i=1^n ψ_v(Y^(1)_i,Y^(2)_i)}^2
      -∑_v=1^Kλ_v[n^-1∑_i=1^n {ψ_v(Y^(1)_i,Y^(2)_i)}^2]).
  
  For a simpler presentation, let S_n,v denote n^-1/2∑_i=1^n ψ_v(Y^(1)_i,Y^(2)_i) 
  
  hereafter.
  
  To derive the limiting null distribution of (nH_n,2, Λ_n), 
  we first derive the limiting null distribution of (nH_n,2,K, T_n) for each integer K.
  Observe that
  
      (S_n,v)    = 0,   (S_n,v)  = 1,   (S_n,v, T_n)→ d_vΔ_0,
        (T_n)        = 0,   (T_n)      = _(0),
  
  where d_v≡{ψ_v(),L'(;0)}
  and 0<_X(0)<∞ by Assumption <ref>. 
  There exists at least one v≥ 1 such that d_v≠ 0.  
  Indeed, applying Theorem 4.4 and Lemma 4.2 in <cit.> yields 
  
      {ψ_v(),v=1,2,…}
      ={ψ_v_1^(1)(x^(1))
             ψ_v_2^(2)(x^(2)),v_1,v_2=1,2,…},
  
  where
  
      ψ_v_1^(1)(x^(1))ψ_v_2^(2)(x^(2))
      ≡2cos{π v_1 F_Y^(1)(x^(1))}cos{π v_2 F_Y^(2)(x^(2))}
  
  is associated with eigenvalue λ^μ_v_1,v_2 defined in Proposition <ref>.
  Since 
  
      Y^(k)={ρ_Y^(k)(Y^(k))}=0,
   
  {ψ_v(),v=1,2,…} forms a complete orthogonal basis for the family of functions of the form (<ref>): d_v=0 for all v thus entails 
  
      _X(0)=[{L^'(;0)}^2]=[{∑_v=1^∞d_vψ_v(^(1),^(2))}^2]=∑_v=1^∞d_v^2=0,
  
  which contradicts  Assumption <ref>(iii). 
  Therefore,  d_v^*0 for some v^*. 
  Applying the multivariate central limit theorem <cit.>, 
  we deduce that under the null,
  
      (S_n,1, …,S_n,K, T_n)
      →
      (ξ_1, …,ξ_K, V_K)   in distribution,
  
  where
  
      (ξ_1, …,ξ_K, V_K)
      ∼ N_K+1((0_K
       0),
      (_K    Δ_0
      Δ_0^   Δ_0^2)) .
  
  Here 0_K denotes a zero vector of dimension K, _K denotes an identity matrix of dimension K,  is short for _(0), and =(d_1,…,d_K).
  Thus V_K can be expressed as 
  
      (Δ_0^2)^1/2{∑_v=1^Kc_vξ_v + c_0,Kξ_0},
  
  where c_v≡^-1/2d_v, c_0,K≡ (1-∑_v=1^Kc_v^2)^1/2, 
  and ξ_0 is standard Gaussian and independent of ξ_1,…,ξ_K.
  Then by the continuous mapping theorem <cit.> 
  and Slutsky's theorem <cit.>, we have under the null,
  
      (nH_n,2,K, T_n)
      →(∑_v=1^Kλ_v(ξ_v^2-1), (Δ_0^2)^1/2(∑_v=1^Kc_vξ_v + c_0,Kξ_0))    in distribution.
  
  Moreover, we claim that under the null, 
  
      (nH_n,2, T_n)
      →(∑_v=1^∞λ_v(ξ_v^2-1), (Δ_0^2)^1/2(∑_v=1^∞c_vξ_v + c_0,∞ξ_0))    in distribution,
  
  with c_0,∞≡ (1-∑_v=1^∞c_v^2)^1/2 via the following argument. 
  Denote
  
      M_K   ≡∑_v=1^Kλ_v(ξ_v^2-1),      
      V_K   ≡(Δ_0^2)^1/2(∑_v=1^Kc_v ξ_v + c_0,Kξ_0),
      
      M   ≡∑_v=1^∞λ_v(ξ_v^2-1),    and   
      V   ≡(Δ_0^2)^1/2(∑_v=1^∞c_v ξ_v + c_0,∞ξ_0).
  
  To prove (<ref>), it suffices to prove that for any real numbers a and b, 
  
      |{exp( anH_n,2+ bT_n)} -
      {exp( aM+ bV)}|→ 0   as n→∞,
  
  where  denotes the imaginary unit. 
  We have
  
      |{exp( anH_n,2+ bT_n)} -
      {exp( aM+ bV)}|
      ≤    |{exp( anH_n,2+ bT_n)} -
      {exp( anH_n,2,K+ bT_n)}|
         +|{exp( anH_n,2,K+ bT_n)} -
      {exp( aM_K+ bV_K)}|
         +|{exp( aM_K+ bV_K)} -
      {exp( aM+ bV)}|≡ I+II+III,   say,
  
  where in view of page 82 of <cit.> and Equation (4.3.10) in <cit.>, 
  
      I 
         ≤|exp{ an(H_n,2-H_n,2,K)}-1|≤{| an(H_n,2-H_n,2,K)|^2}^1/2
      =(2na^2/n-1∑_v=K+1^∞λ_v^2)^1/2
  
  and
  
      III 
         ≤|exp{ a(M_K-M)+ b(V_K-V)}-1|≤{| a(M_K-M)+b(V_K-V)|^2}^1/2
         ≤{2(2a^2∑_v=K+1^∞λ_v^2 + 
      2b^2Δ_0^2∑_v=K+1^∞c_v^2)}^1/2.
  
  Since by Remark 3.1 in <cit.>, 
  
      ∑_v=1^∞λ_v^2=
      1/8100    when μ=D,R,
      1/225    when μ=τ^*,
         and   ∑_v=1^∞c_v^2=^-1∑_v=1^∞d_v^2= 1,
  
  we conclude that, for any ϵ>0, there exists K_0 such that I<ϵ/3 and III<ϵ/3 for all n and all K≥ K_0. For this K_0, we have II<ϵ/3 for all sufficiently large n by (<ref>). These together prove (<ref>).
  We also have, using the idea from <cit.>
  (see also ), that under the null
  
      Λ_n-T_n+Δ_0^2/2→ 0    in probability.
  
  Combining (<ref>) and (<ref>) yields that under the null, 
  
      (nH_n,2, Λ_n)
      →(∑_v=1^∞λ_v(ξ_v^2-1), (Δ_0^2)^1/2(∑_v=1^∞c_vξ_v + c_0,∞ξ_0)-Δ_0^2/2)   in distribution.
  
  Using the fact H_n,1=0 and Equation (1.6.7) in <cit.> yields that (nμ_n, Λ_n) has the same limiting distribution as (<ref>) under the null.
  
  Step III. 
  
  
  Finally employing Le Cam's third lemma <cit.>
  we obtain that under the local alternative
  
      {nμ_n≤ q_1-α| H_1,n(Δ_0)}
      →    [{∑_v=1^∞λ_v(ξ_v^2-1)≤ q_1-α}×exp{(Δ_0^2)^1/2(∑_v=1^∞c_v ξ_v + c_0,∞ξ_0)-Δ_0^2/2}]
      ≤    [{|ξ_v^*|≤(q_1-α+∑_v=1^∞λ_v/λ_v^*)^1/2}×exp{(Δ_0^2)^1/2(∑_v=1^∞c_v ξ_v + c_0,∞ξ_0)-Δ_0^2/2}]
      
      =    [{|ξ_v^*|≤(q_1-α+∑_v=1^∞λ_v/λ_v^*)^1/2}×exp{(Δ_0^2)^1/2(c_v^*ξ_v^* + (1-c_v^*^2)^1/2ξ_0)-Δ_0^2/2}]
      
      =    Φ{(q_1-α+∑_v=1^∞λ_v/λ_v^*)^1/2 -c_v^*(Δ_0^2)^1/2} 
      - Φ{-(q_1-α+∑_v=1^∞λ_v/λ_v^*)^1/2 -c_v^*(Δ_0^2)^1/2}
      ≤     2(q_1-α+∑_v=1^∞λ_v/λ_v^*)^1/2φ{| c_v^*|(Δ_0^2)^1/2-(q_1-α+∑_v=1^∞λ_v/λ_v^*)^1/2},
  
  for some v^* such that  c_v^*=^-1/2d_v^*0 and
  
      Δ_0 ≥| c_v^*|^-1^-1/2(q_1-α+∑_v=1^∞λ_v/λ_v^*)^1/2,
  
  where Φ and φ are the distribution function and density function of the standard normal distribution, respectively. 
  
  Note that the right-hand side of (<ref>) is monotonically decreasing as Δ_0 increases given (<ref>). 
  There exists a positive constant C_β such that (<ref>) is smaller than β/2 as long as Δ_0≥ C_β, regardless of whether c_v^* is positive or negative. This concludes the proof. 
  
  
  
  
  
  
  
  (B) This proof uses Assumption <ref><ref>,<ref>,<ref>.
  Let _i=(Y^(1)_i,Y^(2)_i), i=1,…,n be independent copies of  (distributed as  with Δ=0).
  
  
  Denote
  
      L(;Δ)≡f_(;Δ)/f_(;0),   
      L'(;Δ)≡∂/∂ΔL(;Δ),
  
  and define Λ_n=∑_i=1^nlog L(_i;Δ_n) and T_n≡Δ_n∑_i=1^nL'(_i;0). Direct computation yields 
  
      L(;Δ)≡(1-Δ)f_0()+Δ g()/f_0(),   
      L'(;0)=g()-f_0()/f_0(),
  
  and thus
  
      _(0)   =[{L'(;0)}^2]=[{g(Y)/f_0(Y)-1}^2]
         =[{s(Y)}^2]=χ^2(G,F_0)≡∫ (G̣/F̣_0-1)^2 F̣_0.
  
  
  
  
  
  This is similar to the proof for family (A). The only difference lies in
  proving the existence of at least one v≥ 1 such that d_v≠ 0, 
  where d_v≡[ψ_v(),L'(;0)]. 
  Now L'(x;0)=s(x) is not of the form (<ref>), and 
  {ψ_v(),v=1,2,…} does not necessarily form a complete orthogonal basis for the family of functions of s(x). 
  However, recall that
  
      {ψ_v(),v=1,2,…}
      ={ψ_v_1^(1)(x^(1))
             ψ_v_2^(2)(x^(2)),v_1,v_2=1,2,…},
  
  where
  
      ψ_v_1^(1)(x^(1))ψ_v_2^(2)(x^(2))
      ≡2cos{π v_1 F_Y^(1)(x^(1))}cos{π v_2 F_Y^(2)(x^(2))}.
  
  Since
  
      {ψ_v_1^(1)(x^(1))
            ψ_v_2^(2)(x^(2)),v_1,v_2=0,1,2,…}
  
  forms a complete orthogonal basis of the set of square integrable functions, d_v=0 for all v≥1 thus entails s(x)=h_1(x^(1))+h_2(x^(2)) for some functions h_1,h_2, where h_k(x^(k)) depends only on x^(k) for k=1,2. This contradicts Assumption <ref><ref>. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   §.§ Proof of Proposition <ref>
  
  
  
  (A) This proof uses all of Assumption <ref>.
  Let _i=(Y^(1)_i,Y^(2)_i) and _i=(X^(1)_i,X^(2)_i), i=1,…,n be independent copies of  and  with Δ=Δ_n=n^-1/2Δ_0, respectively. 
  Let F^(0) and F^(a) be the (joint) distribution functions of (_1,…,_n) and (_1,…,_n), respectively, 
  and let F^(0)_i and F^(a)_i be the distribution functions of _i and _i, respectively. 
  
  The total variation distance between two distribution functions G and F on the same real probability space is defined as 
  
      (G,F)≡sup_A|_G(A)-_F(A)|,
  
  where A is taken over the Borel field and _G,_F are respective probability measures induced by G and F. Furthermore, if G is absolutely continuous with respect to F, the Hellinger distance between G and F is defined as
  
      (G,F)≡[∫ 2{1-(G̣/F̣)^1/2}F̣]^1/2.
  
  By Assumption <ref><ref>, (F^(a),F^(0)) is well-defined. 
  It suffices to prove that for any small 0<β<1-α, there exists Δ_0=c_β such that, for all sufficiently large n, (F^(a),F^(0))<β, which is implied by 
  (F^(a),F^(0))<β using the relation <cit.>
  
      (F^(a),F^(0))≤(F^(a),F^(0)).
  
  We also know that <cit.>
  
      1-1/2^2(F^(a),F^(0))=∏_i=1^n{1-1/2^2(F^(a)_i,F^(0)_i)}.
  
  We then aim to evaluate ^2(F^(a),F^(0)) in terms of _(0) and Δ_0. By definition,
  
      1/2^2(F^(a)_i,F^(0)_i)
      =[1-{L(_i;Δ_n)}^1/2].
  
  Given Assumption <ref>, we deduce in view of <cit.> that
  
      n[1-{L(_i;Δ_n)}^1/2]
      =(∑_i=1^n[1-{L(_i;Δ_n)}^1/2])
      →Δ_0^2_(0)/8.
  
  Therefore, 
  
      1-1/2^2(F^(a),F^(0)) →exp{-Δ_0^2_(0)/8}.
  
  The desired result follows by taking c_β>0 such that
  
      exp{-c_β^2_(0)/8}=1-β^2/8.
  
  
  
  
  
  
  
  
  
  
  
  
  
  (B) This proof requires Assumption <ref><ref>,<ref>.
  This is similar to the proof for family (A), but here we will use the relation <cit.>
  
      (F^(a),F^(0))≤{χ^2(F^(a),F^(0))}^1/2,
  
  where the chi-square distance between two distribution functions G and F on the same real probability space such that G is absolutely continuous with respect to F is defined as 
  
      χ^2(G,F)≡∫(G̣/F̣-1)^2 F̣.
   
  Here χ^2(F^(a),F^(0)) is well-defined by Assumption <ref><ref>. 
  We also know that <cit.>
  
      1+χ^2(F^(a),F^(0))
      =∏_i=1^n{1+χ^2(F^(a)_i,F^(0)_i)}.
  
  Next we aim to evaluate χ^2(F^(a),F^(0)) in terms of
  _(0)=χ^2(G,F_0) and Δ_0. Here
  0<_(0)<∞ by
  Assumption <ref><ref>,<ref>. We have by
  definition that
  
      χ^2(F^(a)_i,F^(0)_i)=χ^2((1-Δ_n)F_0+Δ_n G, F_0)=Δ_n^2χ^2(G,F_0)=n^-1Δ_0^2χ^2(G,F_0).
  
  Therefore, it holds that
  
      1+χ^2(F^(a),F^(0)) →exp{Δ_0^2χ^2(G,F_0)}.
  
  The desired result follows by taking c_β>0 such that
  
      exp{c_β^2χ^2(G,F_0)}=1+β^2/4.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  This completes the proof.
  
  
  
  
  
  
  
  
  
  § ACKNOWLEDGEMENT
  
  
  We thank Holger Dette for pointing out related literature, and Sky Cao, Peter Bickel, and Bodhisattva Sen for many helpful comments. We thank the editor, the associate editor, and three anonymous referees for their very detailed and constructive comments and suggestions, which greatly helped improve the paper quality.
  
  This project has received funding support from the United States NSF Grant SES-2019363 
  and European Research Council (ERC) 
  under the European Union's Horizon 2020 research 
  and innovation programme (grant agreement No 883818). 
  
  
  
  
  
  
  
  
  
  
  
  apalike
  
  
  
  
  
  
  