Monotonicity-Constrained Nonparametric Estimation and Inference for First-Price Auctions
Jun Ma Vadim Marmer Artyom Shneyerov Pai Xu





Monotonicity-Constrained Nonparametric Estimation and Inference
for First-Price AuctionsThis version: December 30, 2023
    Jun MaSchool of Economics, Renmin University of ChinaVadim MarmerVancouver School of Economics, University of British ColumbiaArtyom ShneyerovDepartment of Economics, Concordia UniversityPai XuSchool of Economics and Finance, University of Hong Kong
    December 30, 2023
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We propose a new nonparametric estimator for first-price auctions
with independent private values that imposes the monotonicity constraint
on the estimated inverse bidding strategy. We show that our estimator
has a smaller asymptotic variance than that of Guerre, Perrigne and
Vuong’s (2000) estimator. In addition to establishing
pointwise asymptotic normality of our estimator, we provide a bootstrap-based
approach to constructing uniform confidence bands for the density
function of latent valuations. 




§ INTRODUCTION


Shape restrictions on infinite-dimensional parameters have received
much attention in econometric research. The roles of shape restrictions
in the literature include facilitating identification, providing testable
implications and improving estimation and inference. See <cit.>
for a recent review. 

This paper focuses on the first-price sealed-bid auction model with
symmetric bidders, which is the same as that studied in <cit.>.
GPV's estimation strategy uses a nonparametrically estimated inverse
bidding function to generate pseudo valuations. However, the true
bidding strategy must be strictly increasing and the plug-in nonparametric
estimator of GPV ignores this shape restriction. Imposing such a constraint
at the estimation stage nonparametrically is interesting. For this
purpose, we may use the methods in the monotone nonparametric regression
literature.[See <cit.> for a comprehensive survey of the
literature on monotone nonparametric regression.] Some of these methods can be easily adapted to producing constrained
estimators for the inverse bidding strategy in the auction setting.
E.g., <cit.> take the constrained re-weighting
approach pioneered by <cit.>, which was originally
used to build a monotone estimator for the conditional expectation
function. <cit.> impose monotonicity by using
the greatest convex minorant of the integrated quantile function of
values.

In this paper, we pursue a different approach. We investigate the
asymptotic properties of a new monotonicity-constrained estimator
based on the smooth rearrangement approach of <cit.>
and propose a uniform confidence band around this monotonicity-constrained
nonparametric estimator. We show that this rearrangement-based monotonicity-constrained
estimator is asymptotically normal with an asymptotic variance smaller
than the unconstrained estimator as in GPV. Since the asymptotic variance
plays an important role in determining the width of a uniform confidence
band in large samples, the fact that the rearrangement-based estimator
has a smaller asymptotic variance will result in sharper inference.
Our estimator also has substantial computational advantage over <cit.>.[See <cit.> for simulation studies that compare
the rearrangement and reweighting approaches to monotone regression.
<cit.> notice that the rearrangement approach
has computational advantage since the reweighting approach requires
solving constrained optimization.] 

As a by-product, our method also produces a simple estimator for the
true bidding function. Note that GPV's procedure is based on the inverse-bidding
strategy, which has a simple form. On the other hand, the bidding
function has an integral expression that depends on the unknown distribution
of latent valuations and constructing its direct plug-in type estimator
would be cumbersome. Our simple estimator of the bidding function
can be of interest on its own, as it can be used in practical applications
for computing counterfactual bids. 

We compare the finite sample performances of the confidence bands
based on the rearrangement-based and the unconstrained estimators
in Monte Carlo experiments. We find that the confidence band based
on the rearrangement-based estimator tends to be narrower without
sacrificing the coverage accuracy.

The literature on structural econometrics of auctions is vast. See
<cit.> for a recent review; see also the reviews
of the literature in <cit.> and <cit.>.
In their seminal paper, GPV demonstrate nonparametric identification
of the first-price auction model with independent private values,
and propose two-step nonparametric estimation of the density of latent
valuations. This paper improves on the GPV estimator by incorporating
the monotonicity constraint in the nonparametric estimation procedure. 

In a recent paper, <cit.> describe
the asymptotic distribution of the GPV estimator and propose a valid
bootstrap procedure based on the GPV estimator. They also propose
a procedure for constructing uniform confidence bands for the density
of latent valuations. This paper builds on their results.

The GPV estimator has been widely used in the empirical literature.
For examples of applications, see the literature review in <cit.>
and <cit.>. The GPV approach has been also
utilized in models with risk aversion (,
), unobserved heterogeneity (),
bidder asymmetry and affiliated values (), common
values (, ),
and entry (, , ).

The recent related literature includes <cit.>, who
propose a test for the monotonicity of the bidding function, <cit.>,
who propose a procedure for comparing valuation distributions, <cit.>,
<cit.>, <cit.>, who
propose quantile based methods in the context of auctions. Our paper
is also related to the econometrics literature on two-step nonparametric
estimation. See, e.g., <cit.>. 

The rest of the paper is organized as follows. Section <ref>
introduces the empirical auction model studied in this paper and its
estimation technique, including the GPV estimator and a new monotonicity-constrained
estimator. In Section <ref>, we show that
the new estimator is asymptotically normal with a smaller asymptotic
variance, compared to the unconstrained estimator. Section <ref>
provides an estimator for the asymptotic variance and a uniform confidence
band around the new monotonicity-constrained estimator. We extend
the proposed estimation and inference method to an auction model with
observed auction heterogeneity in Section <ref>.
Section <ref> reports Monte Carlo simulation
results. Proofs are collected in the appendix. 

 “a b” is understood as “a is
defined by b”. “a b” is understood as “b
is defined by a”. 1(·) denotes the
indicator function, and we also denote 1_A1(·∈ A).
Let ℓ^∞(A) be the class of bounded functions
defined on A. For any f∈ℓ^∞(A), let ‖ f‖ _Ax∈ Asup|f(x)|
be the sup-norm.



§ THE AUCTION MODEL AND ESTIMATION


In this section, we consider an auction model for homogeneous goods
and with a fixed number of bidders. A model with observed covariates
capturing auction-specific heterogeneity will be considered in Section
<ref>. The econometrician observes bids
from L auctions, with a fixed number of bidders in each auction:

    { B_il:i=1,…,N, l=1,…,L} .

Bidders' valuations

    { V_il:i=1,…,N,l=1,…,L}

are not unobservable to the econometrician. We assume the distribution
of the valuations satisfy the following assumption.
[Data Generating Process]



Assumption <ref>(a) assumes that the bidders are symmetric
and the auctions are identical. Assumption <ref> is similar
to Assumptions A1 and A2 of GPV and Assumption 1 of MMS. The object
of interest is the PDF of the valuations at interior points of [v,v].
Suppose that v_l>v, v_u<v and I[v_l,v_u]
is an inner closed sub-interval of [v,v]. 

We assume that the observed bids are generated from the valuations
and by the Bayesian Nash equilibrium (BNE) bidding strategy:

    B_il=s(V_il) V_il-1/F(V_il)^N-1∫_v^V_ilF(u)^N-1du.

The BNE requires that the bidding strategy s is strictly increasing.
Moreover, GPV show that s is at least three times continuously
differentiable on (v,v). The
inverse of the BNE bidding strategy can be written as

    ξ(b) s^-1(b)=b+1/N-1G(b)/g(b),

where G and g are CDF and PDF of the bids, respectively. Let
b s(v) and b s(v)
denote the boundaries of support for the observed i.i.d. bids. GPV
show that g is three-times continuously differentiable and also
bounded away from zero on its support [b,b]:

    C_gb∈[b,b]infg(b)>0.


Let G be the empirical CDF of the bids:

    G(b)1/N· L∑_i,l1(B_il≤ b)

 and g be the kernel density estimator of g:

    g(b)1/N· L∑_i,l1/h_gK_g(B_il-b/h_g)

with some bandwidth h_g>0 and kernel K_g. Therefore, the
plug-in nonparametric estimator of the inverse bidding strategy ξ
is

    ξ(b) b+1/N-1G(b)/g(b).

 For each B_il, we construct a pseudo valuation by V_ilξ(B_il).
There is a boundary bias issue when ordinary kernel density estimator
as in (<ref>) is used. Thus, the pseudo valuations
corresponding to bids in boundary regions are contaminated. GPV propose
to trim off bids that lie in [b,b+h_g)∪(b-h_g,b],
where

    b   max{ B_il:i=1,...,N,l=1,...,L}
    b   min{ B_il:i=1,...,N,l=1,...,L} .


We modify the kernel density estimator g(b)
in the boundary region b∈[b,b+h_g)∪(b-h_g,b]
to avoid boundary bias and trimming.[See <cit.> for more discussion on why we
should avoid trimming when estimating the auction model using GPV
approach.] Another concern is that, in order to remove more bias, we need to
make use of the fact that the bid density g is smoother than the
valuation density f and use a higher-order kernel when estimating
the inverse bidding strategy.[See Remark 2.3 of MMS.] A
suitable choice is the local quadratic minimum contrast estimator
(MCE). See <cit.>.[See <cit.> and <cit.> for more recent
applications of MCE in econometrics.] Similar to the local polynomial regression, the local quadratic MCE
automatically adapts to the boundary so that the rate of the bias
is the same in the boundary region and the interior.[Note that the MCE requires knowledge of the locations of the endpoints
b and b. Since the estimators b
and b are super-consistent: b=b+O_p(log(L)/L)
and b=b+O_p(log(L)/L),
we can replace the unknown endpoints in the MCE with these estimators
without affecting the validity of the asymptotic results.] The local quadratic MCE coincides with the kernel density estimator
(<ref>) with a fourth-order kernel, and therefore
we achieve desired bias removal in the interior region [b+h_g,b-h_g].
The GPV estimator is 

    f_GPV(v)1/N· L∑_i,l1/h_fK_f(V_il-v/h_f),

for some bandwidth h_f>0 and kernel K_f. We note again that
trimming can be avoided. 

Another important observation is that the plug-in estimator (<ref>)
may not be monotone in finite samples, although its population counterpart
ξ is strictly increasing under the assumption that the empirical
auction model is correctly-specified. We apply smooth rearrangement
to build a new monotonicity-constrained estimator of ξ on the
plug-in estimator ξ. Define 

    s(t)∫_b^b∫_-∞^t1/h_rK_r(ξ(b)-u/h_r)dudb+b, t∈ℝ,

for some bandwidth h_r>0 and (second-order) kernel function K_r.
Denote K_r(u)∫_-∞^uK_r(t)dt.
Alternatively we can write

    s(t)=∫_b^bK_r(t-ξ(b)/h_r)db+b, t∈ℝ.

(<ref>) contains less challenge in computation
as the expression of K_r is always available for
standard kernel functions. 

It is clear that s is increasing on ℝ and

    s(t)=b   if 
    b   if

It is easy to see that s can be viewed as an estimator
of the bidding function s (see Lemma <ref>). Note
that this estimator is of interest on its own, as the bidding function
s is an important structural object. Moreover, constructing an
estimator of s directly from its expression in (<ref>)
can be cumbersome.

Let s^-1 denote the pseudo inverse of s:

    s^-1(b)inf{ u∈ℝ:s(u)≥ b} .

s^-1 is a rearrangement-based estimator of ξ with
the monotonicity constraint imposed. A new modified GPV estimation
procedure now can be proposed: First, we construct ξ,
the plug-in nonparametric estimator of the inverse bidding strategy
ξ. To avoid trimming, we use the local quadratic MCE instead
of the ordinary kernel density estimator. Then, we construct the monotonicity-imposed
estimator of the inverse bidding strategy: s^-1 and
generate monotonicity-constrained pseudo valuations V_il^†s^-1(B_il)
for i=1,...,N, l=1,...,L. A new estimator, the rearrangement-based
GPV (RGPV), is

    f_RGPV(v)1/N· L∑_i,l1/h_fK_f(V_il^†-v/h_f).


The RGPV estimation procedure is computationally more involved than
the standard GPV procedure. When implementing it in practice, the
integral in (<ref>) can be approximated by an
upper Riemann sum. Let M∈ℕ be a very large number. Let
d(b-b)/M.
We can approximate ∫_b^bK_r((t-ξ(b))/h_r)db
by

    ∑_i=1^MK_r(t-ξ(b+i· d)/h_r)d.

However, unlike the estimator in <cit.>, it is
not required to solve a constrained optimization problem. Thus the
RGPV estimation procedure is computationally less demanding than <cit.>'s
estimation procedure, which is based on constrained reweighting. 

An alternative, and closely related, approach is to impose the monotonicity
restriction through the “non-smooth” rearrangement. Instead of
using (<ref>), we can define 

    s_0(t)∫_b^b1(ξ(b)≤ t)db+b, t∈ℝ,

and use s_0^-1 as a monotonicity-constrained estimator
of ξ to generated pseudo valuations. If the empirical auction
model is correctly specified so that ξ is strictly increasing,
s_0^-1 is always an improvement over the plug-in
estimator ξ, in the sense that s_0^-1
has a strictly smaller (finite-sample) integrated mean square error
whenever ξ is not monotonic. See .
However, for the structural auction model, the parameter of interest
is the density f. It is unclear whether the density estimator based
on pseudo valuations generated by s_0^-1 is an improvement
over the unconstrained GPV estimator theoretically. In this paper,
we focus on smooth rearrangement and in the next section, we show
that based on the RGPV estimator we could potentially achieve sharper
inference in large samples, which can be regarded as theoretical advantage
over the unconstrained GPV estimator.



§ ASYMPTOTIC PROPERTIES


The following assumptions are imposed on the kernel functions and
the bandwidths, respectively.
[Kernel]



[Bandwidth]



Assumption <ref> implies that the kernel
functions are of second order. When the kernel used in the local quadratic
MCE of the bid density is second-order, at the interior points, the
MCE is the same as the ordinary kernel density estimator (<ref>)
with K_g being fourth-order. Assumption <ref>
is similar to Assumption 3 in MMS. 

It is shown in MMS that under assumptions <ref>-<ref>,

    (Lh_f^2h_g)^1/2(f_GPV(v)-f(v))→_dN(0,V_GPV(v)),

where

    V_GPV(v)1/N(N-1)^2F(v)^2f(v)^2/g(s(v))^3∫{∫ K_f'(u)K_g(w-λ_f/λ_gs'(v)u)du} ^2dw.

MMS show that the asymptotic variance (<ref>)
can be consistently estimated by some estimator V_GPV(v)
and an asymptotically valid confidence interval for f(v)
can be constructed:

    [f_GPV(v)-z_1-α/2√(V_GPV(v)/Lh_f^2h_g),f_GPV(v)+z_1-α/2√(V_GPV(v)/Lh_f^2h_g)],

where z_1-α/2 denotes the 1-α/2 quantile of the
standard normal distribution. Note that V_GPV(v)
and its probabilistic limit V_GPV(v) play
important roles in determining the length of the confidence interval.
Furthermore, a bootstrap uniform confidence band is given by 

    CB_GPV(v)[f_GPV(v)-ζ_GPV,α√(V_GPV(v)/Lh_f^2h_g),f_GPV(v)+ζ_GPV,α√(V_GPV(v)/Lh_f^2h_g)], for ,

where ζ_GPV,α is some bootstrap critical value. It can
be shown that 

    P[f(v)∈ CB_GPV(v), for all ]→1-α, as .


Next, we show that a similar asymptotic normality result holds for
the RGPV estimator, but with a smaller asymptotic variance. The proof
uses the same arguments as in the proof of Theorem 2.1 in MMS. First,
we derive the following asymptotic representation:

    f_RGPV(v)-f(v)=1/(N-1)·1/(N· L)^2∑_i,l∑_j,kℳ(B_il,B_jk;v)+o_p((Lh^3)^-1/2),

where the remainder term is uniform in v∈ I . Moreover, 

    ℳ(b',b;v)-1/h_f^2K_f'(ξ(b')-v/h_f)ξ'(b')∫_b^b1/h_rK_r(ξ(b')-ξ(u)/h_r)G(u)/g(u)^2(1/h_gK_g(b-u/h_g)-g(u))du.

Note that this “kernel” is different from that of the unconstrained
GPV estimator. See Equation (2.6) of MMS.

Define 

    ℳ_1(b;v)∫ℳ(b,b';v)dG(b'),
       ℳ_2(b;v)∫ℳ(b',b;v)dG(b') and μ_ℳ(v)∫∫ℳ(b,b';v)dG(b)dG(b').

Note that μ_ℳ(v)=E[ℳ_1(B_11;v)]=E[ℳ_2(B_11;v)].
Applying Hoeffding decomposition to the leading term in (<ref>)
and techniques from the theories of empirical processes and U processes,
we can show that

    f_RGPV(v)-f(v)=1/N-1·1/N· L∑_i,l(ℳ_2(B_il;v)-μ_ℳ(v))+o_p((Lh^3)^-1/2),

where the remainder term is uniform in v∈ I. ℳ_2(B_il;v)-μ_ℳ(v),
i=1,...,N, l=1,...,L are independent, zero-mean but dependent
on the bandwidths. This term can be shown asymptotically normal. 




In the proof of Theorem <ref>, we show
that 

    E[Lh_f^2h_g/(N-1)^2(1/N· L∑_i,l(ℳ_2(B_il;v)-μ_ℳ(v)))^2]
    
    =    1/N(N-1)^2h_f^2h_g∫{∫ K_f'(ξ(b')-v/h_f)ξ'(b')∫_b^b1/h_rK_r(ξ(b')-ξ(u)/h_r)G(u)/g(u)^2K_g(b-u/h_g)dudG(b')} ^2dG(b)
        +O(h^3)
       V_ℳ(v)+O(h^3),

where the remainder term is uniform in v∈ I. The asymptotic variance
is the limit of the leading term of (<ref>)
as h↓0:

    h↓0lim1/h_f^2h_g∫{∫ K_f'(ξ(b')-v/h_f)ξ'(b')∫_b^b1/h_rK_r(ξ(b')-ξ(u)/h_r)G(u)/g(u)^2K_g(b-u/h_g)dudG(b')} ^2dG(b)
    
    =    F(v)^2f(v)^2/g(s(v))^3∫{∫∫ K_f'(u)K_r(u-z)K_g(w-λ_f/λ_gs'(v)z)dzdu} ^2dw.

A consistent estimator of the asymptotic variance V_RGPV(v)
can be derived based on the sample analogue of V_ℳ(v).


The proof of Theorem <ref> also incorporates
the bias term: 

    (Lh_f^2h_g)^1/2(f_RGPV(v)-f(v)-ι(v))→_dN(0,V_RGPV(v)),

where 

    ι(v)1/2f”(v)(∫ K_f(u)u^2du)h_f^2+1/2(s”'(v)f(v)+s”(v)f'(v))s'(v)-s”(v)^2f(v)/s'(v)^2(∫ K_r(u)u^2du)h_r^2.

A comparison of ι(v) with the bias given in Remark
2.3 of MMS shows that the smooth rearrangement incurs additional bias.
For inference, we take the “under-smoothing” approach to select
sufficiently small bandwidths so that these bias terms become negligible.

MMS show that the GPV estimator
has a smaller asymptotic variance than the quantile-based estimator
of <cit.>. The proof of (<ref>)
uses similar arguments. It is easy to show that the inequality (<ref>)
is strict for all v∈(v,v),
if the kernel function satisfies K'(u)<0 for all u∈(0,1)
and K'(u)>0 for all u∈(-1,0). 

Suppose that the valuations are drawn from the family of distributions

    F(v)=
    0,     v<0
    
    v^θ,     0≤ v≤1
    
    1,     v>1

supported on [0,1] with some parameter θ>0.
The Bayesian Nash equilibrium bidding strategy in this example is

    s(v)=(1-1/θ(N-1)+1)v,

which is linear in v. Thus, the ratio V_GPV(v)/V_RGPV(v)
is independent from v, by the definitions of V_RGPV(v)
and V_GPV(v). In this example, we choose
the triweight kernel

    K(u)=35/32(1-u^2)^31(|u|≤1).

We can analytically evaluate the multi-dimensional integrals in (<ref>)
and (<ref>) and calculate V_GPV(v)/V_RGPV(v)
in this example. We experiment with different combinations of (θ,N),
and find that the ratio V_GPV(v)/V_RGPV(v)
can be quite large in my cases. For instance, in the case of (θ,N)=(1,5),
V_GPV(v)/V_RGPV(v)
is approximately 1.587.



§ INFERENCE


If the asymptotic variance V_RGPV(v) in (<ref>)
can be consistently estimated by some estimator V_RGPV(v),
Theorem <ref> shows that we can construct an asymptotically
valid confidence interval for f(v):

    [f_RGPV(v)-z_1-α/2√(V_RGPV(v)/Lh_f^2h_g),f_RGPV(v)+z_1-α/2√(V_RGPV(v)/Lh_f^2h_g)].

As shown in (<ref>), our rearrangement-based
estimator has a smaller asymptotic variance than the GPV estimator.
Therefore, the confidence intervals based on our estimator in (<ref>)
should be shorter than those based on the GPV estimator in (<ref>)
in large samples. 

An estimator of V_RGPV(v) is derived based
on the sample analogue of V_ℳ(v),
(see (<ref>)):

    V_RGPV(v)   1/N(N-1)^2h_f^2h_g1/(N· L)(N· L-1)(N· L-2)∑_i,l∑_(j,k)≠(i,l)∑_(j',k')≠(i,l), (j',k')≠(j,k)η_il,jk(v)η_il,j'k'(v),
    η_il,jk(v)    K_f'(V_jk^†-v/h_f)1/s'(V_jk^†)∫_b^b1/h_rK_r(V_jk^†-ξ(u)/h_r)G(u)/g(u)^2K_g(B_il-u/h_g)du,

where we use the fact ξ'(b)=1/s'(ξ(b))
and

    s'(v)∫_b^b1/h_rK_r(ξ(b)-v/h_r)db.

The integral can be approximated by an upper Riemann sum in practice.
The following result provides uniform consistency of the variance
estimator and also an estimate of its uniform rate of convergence.
The proof uses the same arguments as in the proof of Theorem 3.1 of
MMS. 




An alternative approach to constructing pointwise confidence intervals
is based on bootstrapping. Let

    { B_il^*:i=1,…,N,l=1,…,L}

denote a set of independent random variables drawn from the original
sample (<ref>) with replacement. G^*
and g^* denote the bootstrap analogues of G
and g respectively. Let ξ^* be the bootstrap
analogue of ξ. ξ^* is defined by replacing
G and g with G^* and g^*.
We further define 

    s^*(t)∫_b^b∫_-∞^t1/h_rK_r(ξ^*(b)-u/h_r)dudb+b, t∈ℝ

and bootstrap analogues of V_il^†, denoted
by V_il^†*, i=1,…,N, l=1,…,L
by using the pseudo inverse of s^*. Lastly, we construct
a bootstrap analogue of f_RGPV:

    f_RGPV^*(v)1/N· L∑_i,l1/h_fK_f(V_il^†*-v/h_f).

The percentile bootstrap pointwise confidence interval for f(v)
is 

    [q_α/2^*(v),q_1-α/2^*(v)],

where q_τ^*(v) is the τ-th quantile of
the conditional distribution of f_RGPV^*(v)
given the original sample.

The pointwise inference results for f(v) described above can be
extended for the inference on the optimal reserve price, as the latter
is a function of the density at the reserve price. Such an extension
is discussed in Section 8 in MMS. Similarly, a function of the density
(1-F(v))/f(v) is of interest in applications as it represents the
markup of the bidder with value v. Again, since relatively to GPV's
our rearrangement estimator has a smaller asymptotic variance, basing
inference on optimal reserve price or the markup on our estimator
would result in more powerful tests and shorter confidence intervals.

Next, we show that a bootstrap-based uniform confidence band for { f(v):v∈ I}
centered at the rearrangement-based estimator can be constructed by
using intermediate Gaussian approximation pioneered by <cit.>.
Consider the following bootstrap process

    Z^*(v)f_RGPV^*(v)-f_RGPV(v)/(Lh_f^2h_g)^-1/2V_RGPV(v)^1/2, v∈ I.

Let P^*[·] denote the probability conditional
on the original sample and

    ζ_RGPV,αinf{ z∈ℝ:P^*[‖ Z^*‖ _I≤ z]≥1-α}

be the (1-α)-quantile of the conditional distribution
of ‖ Z^*‖ _I given the original sample.
A uniform confidence band around the rearrangement-based estimator
is


    CB_RGPV(v)[f_RGPV(v)-ζ_RGPV,α√(V_RGPV(v)/Lh_f^2h_g), f_RGPV(v)+ζ_RGPV,α√(V_RGPV(v)/Lh_f^2h_g)], for .

The following theorem establishes the asymptotic validity of CB_RGPV.
Its proof uses the same arguments as in the proof of Corollary 4.3
of MMS. 




    P[f(v)∈ CB_RGPV(v), for all ]→1-α, as .





§ AUCTION-SPECIFIC HETEROGENEITY


In previous sections, we focused on the case of identical auctions
with a fixed number of bidders. In this section, we consider auction
models with auction-specific heterogeneity. The econometrician observes
data from L auctions. Let X_l denote the d-dimensional
relevant characteristics for the object in the l-th auction. Let
N_l denote the number of bidders in the l-th auction. Let
B_il denote the bid submitted by the i-th bidder in the l-th
auction. The data observed by the econometrician is given by

    {(B_il,X_l,N_l):i=1,...,N_l, l=1,...,L} .

Unobserved bidders' valuations are denoted by

    { V_il:i=1,...,N_l, l=1,...,L} .


Assume that {(X_l,N_l):l=1,...,L}
are i.i.d. and for each l=1,...,L, given X_l=x
and N_l=n, the valuations { V_il:i=1,...,n}
are i.i.d. with conditional PDF f(·|x).
We follow the literature and assume that the valuations and the number
of bidders N_l are conditionally independent given the observed
characteristics X_l.[For a test of this assumption, see <cit.>.]
Also assume that the conditional probability mass function of N_l
given X_l has a known support {n,...,n}.

The observed bid B_il is assumed from the Bayesian Nash equilibrium
bidding for risk-neutral bidder i submitted in the l-th auction.
Let G(·|x,n) denote the conditional
CDF of B_il given X_l=x and N_l=n.
Let g(·|x,n) be the conditional PDF.
The inverse bidding strategy in this context becomes

    V_il=ξ(B_il,X_l,N_l) B_il+1/N_l-1G(B_il|X_l,N_l)/g(B_il|X_l,N_l).


For estimation and inference, we can generate pseudo valuations from
(<ref>) by replacing the true conditional
CDF and PDF by their kernel estimators. The GPV estimator can be defined
analogously in this general context. Asymptotically valid pointwise
confidence intervals and uniform confidence bands for f(·|x)
can be constructed. See Section 5 of MMS for more details. 

Following <cit.>, we use a semi-parametric
approach to homogenize the bids.[The homogenization approach is used extensively in the literature,
e.g. <cit.>, <cit.>, <cit.>,
and many others.] Let 

    {ϵ_il:i=1,...,N_l, l=1,...,L}

denote positive i.i.d. idiosyncratic values that are independent from
the auction-specific characteristics X_l, l=1,...,L.
Let F_ϵ be its CDF and [ϵ,ϵ]
be its support. Define

    B_il=s(ϵ_il,N_l)ϵ_il-1/F_ϵ(ϵ_il)^N_l-1∫_ϵ^ϵ_ilF_ϵ(u)^N_l-1du.

Let

    G(b|n)P[B_il≤ b|N_l=n]

be the conditional CDF and g(b|n) be the
corresponding conditional PDF. Note that we have 

    ϵ_il=s(ϵ_il,N_l)+1/N_l-1G(s(ϵ_il,N_l)|N_l)/g(s(ϵ_il,N_l)|N_l).


The approach of <cit.> assumes that, for
some parametric function  to be specified below, V_il=(X_l)ϵ_il
for i=1,...,N_l and l=1,...,L. It then can be shown that the
conditional CDF and PDF of (X_l)B_il
(denoted by G_ and g_,
respectively) satisfy

    G_(b|X_l,N_l)P[(X_l)B_il≤ b|X_l,N_l]=G(b/(X_l)|N_l)

and 

    g_(b|X_l,N_l)=g(b/(X_l)|N_l)1/(X_l).

It is clear from these results and (<ref>) that 

    (X_l)ϵ_il=(X_l)s(ϵ_il,N_l)+1/N_l-1G_((X_l)s(ϵ_il,N_l)|X_l,N_l)/g_((X_l)s(ϵ_il,N_l)|X_l,N_l),

which implies that B_il=(X_l)B_il,
for i=1,...,N_l and l=1,...,L.

Now, we write 

    log(B_il)=α(N_l)+log((X_l))+U_il,

where 

    α(N_l)E[log(s(ϵ_il,N_l))|N_l] and -α(N_l).

It is easy to check that E[U_il|X_l,N_l]=0.
Since N_l is discrete, we can write 

    α(N_l)=∑_n=n^nα_n1(N_l=n).

We assume that the function  is log-linear in parameters:
log((X_l))=X_l^Tβ
for some unknown β. Now (<ref>)
can be written as 

    log(B_il)=∑_n=n^nα_n1(N_l=n)+X_l^Tβ+U_il.


Regressing the log-bids on the covariates and the indicators for the
number of bidders yields an estimator β
of β. Then, the homogenized bids are given by

    B_il^0exp(log(B_il)+x_0^Tβ-X_l^Tβ), for  and .

These bids can be interpreted as the bid that would have been submitted
by the i-th bidder if the covariates were equal to x_0,
in the l-th auction. Suppose we are interested in inference on
f(·|x_0) for some fixed x_0.[For example, x_0 can be taken to be the sample mean
L^-1∑_l=1^LX_l. See <cit.>.] 

Let 

    G(b,n)1/L∑_l=1^L1(N_l=n)1/N_l∑_i=1^N_l1(B_il^0≤ b),
    g(b,n)1/L∑_l=1^L1(N_l=n)1/N_l∑_i=1^N_l1/h_gK_g(B_il^0-b/h_g) and
    ξ(b,n) b+1/n-1G(b,n)/g(b,n).

 s(·,n) can also be defined analogously
and s^-1(·,n) is its pseudo inverse.
Let V_il^0†s^-1(B_il^0,N_l)
be the monotonicity-constrained pseudo valuations.

A semi-parametric estimator of f(·|x_0)
is 

    f(v|x_0)1/L∑_l=1^L1/N_l∑_i=1^N_l1/h_fK_f(V_il^0†-v/h_f),

which is asymptotically normal. Its asymptotic variance can be consistently
estimated by

    V_RGPV(v)       ∑_n=n^n1/n(n-1)^2h_f^2h_g
          ×1/L(L-1)(L-2)∑_l=1^L∑_k≠ l∑_k'≠ k,k'≠ l1(N_l=n,N_k=n,N_k'=n)1/N_l∑_i=1^N_lη_il,k(v)η_il,k'(v),

where 

    η_il,k(v)1/N_k∑_j=1^N_kK_f'(V_jk^0†-v/h_f)1/s'(V_jk^0†,N_k)∫_b_N_k^0^b_N_k^01/h_rK_r(V_jk^0†-ξ(u,N_k)/h_r)G(u,N_k)/g(u,N_k)^2K_g(B_jk^0-u/h_g)du,
    s'(v,n)∫_b_n^0^b_n^01/h_rK_r(ξ(b,n)-v/h_r)db

and 

    b_n^0(i,l):N_l=nmaxB_il^0 and .


A pointwise confidence interval for f(v|x_0)
that is of the same form as (<ref>) can
be proved to be asymptotically valid. For bootstrap resampling, we
treat the homogenized bids (<ref>) as our observed
bids and apply the two-step resampling procedure provided by <cit.>.
In each bootstrap replication, we first randomly draw L observations
from { N_l:l=1,...,L} with replacement. Next, we
randomly draw bids with replacement from bids corresponding to the
selected number of bidders. If for the l-th observation in the
bootstrap sample, we have N_l^*=N_l', then let { B_il^0*:i=1,...,N_l^*}
be i.i.d. draws from all the bids in auctions with number of bidders
being N_l' with replacement. Now the bootstrap sample is (B_il^0*,N_l^*),
i=1,...,N_l^* and l=1,...,L. Then it is straightforward
to construct the bootstrap analogue of f(·|x_0)
and a bootstrap-based uniform confidence band for f(·|x_0)
can be constructed analogously. 



§ MONTE CARLO SIMULATIONS


In this section, we assess the finite-sample performances of the uniform
confidence bands based on both the unconstrained GPV estimator and
the rearrangement-based monotonicity-constrained estimator proposed
in this paper. Our simulation design follows <cit.>
and the DGP is described in Remark <ref>. We consider
θ=1 and draw 2100 independent valuations from F_θ.
In all these cases, the number of bidders N is constant. The number
of auctions is determined by N· L=2100. We choose K_r=K_f
to be the second-order triweight kernel. For estimation of the inverse
bidding strategy, we use the second-order triweight kernel in the
MCE. In this case, the kernel K_g in the expressions of the asymptotic
variances is the fourth-order triweight kernel. 

We use the same bandwidths as in GPV. We take h_g=3.72·σ_b·(N· L)^-1/5
when estimating the inverse bidding strategy. σ_b
is the estimated standard deviation of the observed bids. We use h_f=3.15·σ_v·(N· L)^-1/5
as the second-step bandwidth, where σ_v is the
estimated standard deviation of the (unconstrained) pseudo valuations.
The constants 3.72 and 3.15 are Silverman's rule-of-thumb constants
corresponding to fourth-order and second-order triweight kernels.
When imposing monotonicity, we take h_r=h_f. We consider different
numbers of bidders N∈{ 3,5,7}, and also the density
function over the ranges v∈[0.2,0.8] and v∈[0.3,0.7].
When computing the bootstrap-based critical values, we set the number
of bootstrap replications to 499 and use grid maximization over the
grid [v_l:0.001:v_u].[We also tried a finer grid [v_l:0.0001:v_u], which
produced similar results.]

The main result of this paper is that imposing monotonicity using
smooth rearrangement results in more efficient inference. Therefore,
in addition to assessing coverage accuracy, we also report the ratio
of the supremum widths of the confidence bands: 

    W_GPVv∈ Isup 2·ζ_GPV,α√(V_GPV(v)/Lh_f^2h_g) and W_RGPVv∈ Isup 2·ζ_RGPV,α√(V_RGPV(v)/Lh_f^2h_g).




Table <ref> reports the coverage probabilities as well
as the relative supremum width of the confidence bands based on the
GPV and our rearrangement estimators. The coverage probabilities of
both methods are similar and accurate. However, our approach can produce
considerably smaller confidence with the reduction in supremum width
ranging from 6.8% to 33.4%. More substantial reductions in supremum
width are obtained for smaller numbers of bidders. This is due to
the inverse relationship between the number of bidders and the asymptotic
variance of the estimators.



§ CONCLUSION


The GPV nonparametric identification and estimation approach is an
indispensable working tool in structural econometrics of auctions.
This paper contributes to the literature by showing how one can reduce
the asymptotic variance of GPV-type estimators by incorporating the
monotonicity constraint in estimation. While monotonicity-constrained
estimators have been previously considered in the auction literature,
to the best of our knowledge ours is the first to obtain a reduction
in asymptotic variance in this context. Our method is simple to implement,
and as a by-product, it also produces a simple estimator for the bidding
function. We also discuss construction of uniform confidence bands
for the density of valuations. In a simulation study, we show that
by applying our approach one can increase the precision of the confidence
bands without sacrificing their coverage in finite samples. 

chicago







PART: 
*Appendix

Let  denote an inequality up to a universal constant that
does not depend on the sample size L. For a sequence of classes
of functions ℱ_L (that may depend on the sample size)
defined on [b,b]^d, for some
d≥1, let N(ϵ,ℱ_L,‖·‖ _Q,2)
denote the ϵ-covering number, i.e., the smallest integer
m such that there are m balls of radius ϵ centered
at points in ℱ_L, with respect to the metric induced
by the norm ‖·‖ _Q,2, where ‖ f‖ _Q,2(∫|f|^2dQ)^1/2,
f∈ℱ_L. A function F_L:[b,b]^d→ℝ_+
is an envelope of ℱ_L if F_L≥f∈ℱ_Lsup|f|.
We say that ℱ_L is a (uniform) Vapnik-Chervonenkis-type
(VC-type) class with respect to the envelope F_L (see, e.g.,
) if there exist
some positive constant C_1 and C_2 that are independent
of L such that

    N(ϵ‖ F_L‖ _Q,2,ℱ_L,‖·‖ _Q,2)≤(C_1/ϵ)^C_2, for all ,

for all finitely discrete probability measure Q on [b,b]^d.
Note that the all function classes that appear later are dependent
on L. We suppress the dependence for notational simplicity.

“With probability approaching 1” is abbreviated as “w.p.a.1”.
For notational simplicity, in the proofs, i,lmax
is understood as (i,l)∈{ 1,...,N}×{ 1,...,L}max.
(2)∑ is understood as ∑_(j,k)≠(i,l)
and (3)∑ is understood as 

    ∑_i,l∑_(j,k)≠(i,l)∑_(j',k')≠(i,l), (j',k')≠(j,k),

i.e., summing over all distinct indices. (N· L)_2
is understood as (N· L)(N· L-1) and
(N· L)_3 is understood as (N· L)(N· L-1)(N· L-2). 

Fix 

    δ_0min{(v-v_u)/2,(v_l-v)/2} .





§ PROOFS OF THE MAIN RESULTS


It follows from
Lemma <ref> that

    (Lh_f^2h_g)^1/2(f_GPV(v)-f(v))=1/N^1/2(N-1)1/(N· L)^1/2∑_i,lh_fh_g^1/2(ℳ_2(B_il;v)-μ_ℳ(v))+o_p(1).


Denote 

    ℳ^(b',b;v)-1/h_f^2K_f'(ξ(b')-v/h_f)ξ'(b')∫_b^b1/h_rK_r(ξ(b')-ξ(u)/h_r)G(u)/g(u)^21/h_gK_g(b-u/h_g)du
    ℳ_2^(b;v)-∫1/h_f^2K_f'(ξ(b')-v/h_f)ξ'(b')∫_b^b1/h_rK_r(ξ(b')-ξ(u)/h_r)G(u)/g(u)^21/h_gK_g(b-u/h_g)dudG(b')
    and μ_ℳ^(v)∫∫ℳ^(b,b';v)dG(b)dG(b')=∫ℳ_2^(b;v)dG(b).

Then it is clear that 

    ℳ_2(B_il;v)-μ_ℳ(v)=ℳ_2^(B_il;v)-μ_ℳ^(v) for all  and

and 

    μ_ℳ^(v)=μ_ℳ(v)-∫∫_b^b1/h_f^2K_f'(ξ(b')-v/h_f)ξ'(b')1/h_fK_r(ξ(b')-ξ(u)/h_f)G(u)/g(u)dudG(b').


Note that we assumed h_r=h_f. By change of variables,

    ∫∫_b^bK_f'(ξ(b')-v/h_f)ξ'(b')1/h_fK_r(ξ(b')-ξ(u)/h_f)G(u)/g(u)dudG(b')
    
    =     h_f∫_v-v/h_f^v-v/h_f∫_v-v/h_f^v-v/h_fK_f'(w)K_r(w-z)G(s(h_fz+v))/g(s(h_fz+v))s'(h_fz+v)g(s(h_fw+v))dzdw
    
    =     h_f∫_v-v/h_f^v-v/h_f∫_v-v/h_f^v-v/h_fK_f'(w)K_r(w-z){G(s(v))s'(v)/g(s(v))+{ g(s(v̈))s'(v̈)^2+G(s(v̈))s”(v̈)} g(s(v̈))/g(s(v̈))^2h_fz.
       .-G(s(v̈))s'(v̈)^2g'(s(v̈))/g(s(v̈))^2h_fz}{ g(s(v))+g'(s(v̇))s'(v̇)h_fw}dzdw,

where v̇ and v̈ are mean values that are dependent
on w and z with |v̇-v|≤ h_f|w|
and |v̈-v|≤ h_f|z|.

Since when h_f is small enough,

    ∫_v-v/h_f^v-v/h_f∫_v-v/h_f^v-v/h_fK_f'(w)K_r(w-z)dzdw=0,

it follows that

    ∫∫_b^bK_f'(ξ(b')-v/h_f)ξ'(b')1/h_fK_r(ξ(b')-ξ(u)/h_f)G(u)/g(u)dudG(b')=O(h^2),

uniformly in v∈ I. v∈ Isup|μ_ℳ^(v)|=O(1)
follows from this result and Lemma <ref>.

Define

    U_il(v)1/N^1/2(N-1)1/(N· L)^1/2h_fh_g^1/2(ℳ_2^(B_il;v)-μ_ℳ^(v)),

and

    σ(v)(∑_i,lE[U_il(v)^2])^1/2=(1/N(N-1)^2h_f^2h_gE[(ℳ_2^(B_11;v)-μ_ℳ^(v))^2])^1/2.


By change of variables,

    E[h_f^2h_gℳ_2^(B_11;v)^2]     =    ∫1/h_g{∫_b^b∫_b^b1/h_fK'(ξ(b')-v/h_f)1/s'(ξ(b'))1/h_fK(ξ(b')-ξ(u)/h_f)G(u)/g(u)^2.
          .× K(b-u/h_g)g(b')dudb'} ^2dG(b)
        =    ∫_b-s(v)/h_g^b-s(v)/h_gρ_h(y)^2g(h_gy+s(v))dy.

where

    ρ_h(y)       ∫_v-v/h_f^v-v/h_f∫_v-v/h_f^v-v/h_fK_f'(w)K_r(w-z)G(s(h_fz+v))/g(s(h_fz+v))^2K_g(y-s(h_fz+v)-s(v)/h_g)
          × s'(h_fz+v)g(s(h_fw+v))dzdw.

We can easily verify that the dominated convergence theorem applies
to the integral on the right hand side of (<ref>)
(as h↓0) for each y∈ℝ:

    h↓0limρ_h(y)=G(s(v))s'(v)/g(s(v))∫∫ K_f'(w)K_r(w-z)K_g(y-s'(v)λ_f/λ_gz)dzdw.

Since K_f' and K_r are compactly supported on [-1,1],
by reverse triangle inequality, 

    |ρ_h(y)|       ∫_v-v/h_f^v-v/h_f∫_v-v/h_f^v-v/h_f|K'(w)||1(|z|≤1+|w|)|1(|y|≤1+|s(h_fz+v)-s(v)/h_g|)dzdw
          1(|y|≤1+2λ_f/λ_gC_s'),

where C_s'v∈[v,v]sups'(v). 

Now the dominated convergence theorem applies and 

    h↓0lim∫_b-s(v)/h_g^b-s(v)/h_gρ_h(y)^2g(h_gy+s(v))dy
    
    =    G(s(v))^2s'(v)^2/g(s(v))∫{∫∫ K'(w)K(w-z)K(y-s'(v)λ_f/λ_gz)dzdw} ^2dy
    
    =    F(v)^2f(v)^2/g(s(v))^3∫{∫∫ K'(w)K(w-z)K(y-s'(v)λ_f/λ_gz)dzdw} ^2dy

where the second equality follows from the relation f(v)=g(s(v))s'(v).
It follows from this result and v∈ Isup|μ_ℳ^(v)|=O(1)
that

    E[h_f^2h_g(ℳ_2^(B_11;v)-μ_ℳ^(v))^2]
    
    =    E[h_f^2h_gℳ_2^(B_11;v)^2]-h_f^2h_gμ_ℳ^(v)^2
    
    =    F(v)^2f(v)^2/g(s(v))^3∫{∫∫ K_f'(w)K_r(w-z)K_g(y-s'(v)λ_f/λ_gz)dzdw} ^2dy+o(1)

and thus 

    σ(v)=1/N^1/2(N-1){F(v)^2f(v)^2/g(s(v))^3∫{∫∫ K_f'(w)K_r(w-z)K_g(y-s'(v)λ_f/λ_gz)dzdw} ^2dy} ^1/2+o(1).

It is easy to adapt the proofs to show that the remainder terms are indeed uniform in v∈ I. See the proof of Theorem 2.1 of MMS.

By the c_r inequality (see, e.g., ),
we have

    ∑_i,lE[|U_il(v)/σ(v)|^3]     =    σ(v)^-3(N-1)^-3(N· L)^-1/2E[h_f^3h_g^3/2|(ℳ_2^(B_11;v)-μ_ℳ^(v))|^3]
          σ(v)^-3(N· L)^-1/2(h_f^3h_g^3/2E[|ℳ_2^(B_11;v)|^3]+h_f^3h_g^3/2|μ_ℳ^(v)|^3).

Then it is easy to verify that the Lyapunov's condition holds:

    L↑∞lim∑_i,lE[|U_il(v)/σ(v)|^3]=0.

By Lyapunov's central limit theorem,

    ∑_i,lU_il(v)/σ(v)→_dN(0,1), as .


For the second part, it suffices to show 

    ∫{∫∫ K_f'(u)K_r(u-z)K_g(w-λ_f/λ_gs'(v)z)dzdu} ^2dw≤∫{∫ K_f'(u)K_g(w-s'(v)λ_f/λ_gu)du} ^2dw.

For each w∈ℝ, since K_r and K_f' are assumed
to be bounded and compactly supported, the Fubini-Tonelli theorem
applies and therefore,

    ∫∫ K_f'(u)K_r(u-z)K_g(w-λ_f/λ_gs'(v)z)dzdu=∫(∫ K_f'(u)K_r(u-z)du)K_g(w-λ_f/λ_gs'(v)z)dz.

Since K_r and K_f' are supported on [-1,1],
by integration by parts,

    ∫_-1^1K_f'(u)K_r(u-z)du-∫_-1^1K_f(u)K_r'(u-z)du=0.


Now,

    {∫∫ K_f'(u)K_r(u-z)K_g(w-λ_f/λ_gs'(v)z)dzdu} ^2    =    {∫(∫ K_f(u)K_r'(u-z)du)K_g(w-λ_f/λ_gs'(v)z)dz} ^2
        =    {∫∫ K_f(u)K_r'(u-z)K_g(w-λ_f/λ_gs'(v)z)dzdu} ^2
       ≤   ∫ K_f(u){∫ K_r'(u-z)K_g(w-λ_f/λ_gs'(v)z)dz} ^2du,

for each w∈ℝ, where the first equality follows from
(<ref>), the second equality follows
from the Fubini-Tonelli theorem and the inequality follows from Jensen's
inequality since K_f is a probability density function. 

Since the inequality (<ref>) holds for
all w∈ℝ, by the Fubini-Tonelli theorem, 

    ∫{∫∫ K_f'(u)K_r(u-z)K_g(w-λ_f/λ_gs'(v)z)dzdu} ^2dw
    ≤   ∫ K_f(u)∫{∫ K_r'(u-z)K_g(w-λ_f/λ_gs'(v)z)dz} ^2dwdu.

Now for any fixed (u,w)∈ℝ^2, by change
of variables, 

    ∫ K_r'(u-z)K_g(w-λ_f/λ_gs'(v)z)dz=-∫ K_r'(y)K_g(w-λ_f/λ_gs'(v)(y+u))dy.

Then,

    ∫ K_f(u)∫{∫ K_r'(u-z)K_g(w-λ_f/λ_gs'(v)z)dz} ^2dwdu
    
    =    ∫ K_f(u)∫{∫ K_r'(y)K_g(w-λ_f/λ_gs'(v)(y+u))dy} ^2dwdu.

It follows from change of variables that 

    ∫{∫ K_r'(y)K_g(w-λ_f/λ_gs'(v)(y+u))dy} ^2dw=∫{∫ K_r'(y)K_g(w-λ_f/λ_gs'(v)y)dy} ^2dw,

for all u∈ℝ. The conclusion follows from this result,
(<ref>) and the assumption K_r=K_f.

The proof
is very similar to that of Theorem 3.1 of MMS. Let 


    V(v)   1/N(N-1)^2h_fh_g^21/(N· L)_3∑_(3)𝕋_jkη_il,jk(v)𝕋_j'k'η_il,j'k'(v),
    with η_il,jk(v)    K_f'(V_jk^†-v/h_f)1/s'(V_jk^†)∫_b^b1/h_rK_r(V_jk^†-ξ(u)/h_r)G(u)/g(u)^2K_g(B_il-u/h_g)du

and 

    V(v)   1/N(N-1)^2h_fh_g^21/(N· L)_3∑_(3)η_il,jk(v)η_il,j'k'(v),
    with η_il,jk(v)    K_f'(V_jk^†-v/h_f)1/s'(V_jk^†)∫_b^b1/h_rK_r(V_jk^†-ξ(u)/h_r)G(u)/g(u)^2K_g(B_il-u/h_g)du.


By the arguments used in the proof of Lemma <ref>, 

    V_RGPV(v)=1/N(N-1)^2h_fh_g^21/(N· L)_3∑_(3)𝕋_jkη_il,jk(v)𝕋_j'k'η_il,j'k'(v), for all , w.p.a.1.

By using Lemma <ref>, Lemma <ref>, (<ref>),
Taylor expansion, tedious algebra and empirical process techniques
invoked in the proof of Theorem 3.1 of MMS, one can show that 

    v∈ Isup|1/N(N-1)^2h_fh_g^21/(N· L)_3∑_(3)𝕋_jkη_il,jk(v)𝕋_j'k'η_il,j'k'(v)-V(v)|=O_p((log(L)/Lh^3)^1/2+h^2)

and

    v∈ Isup|V(v)-V(v)|=O_p((log(L)/Lh^3)^1/2+h^2).


Now note that {V(v):v∈ I}
is a U process with V_ℳ(v)=E[V(v)].
One can apply Hoeffding decomposition to V(v)
and apply techniques invoked in the proof of Theorem 3.1 of MMS from
empirical process and U process theory to derive the uniform rate
of convergence of V(v)-V_ℳ(v).
Then the conclusion follows.

First,
it can be verified by standard arguments that the function class {ℳ_2^(·;v):v∈ I}
is (uniformly) VC-type with respect to a constant envelope that is
a multiple of h_f^-2h_g^-1. Then it essentially follows
from Lemma <ref> and Theorem <ref>
that the process

    Z(v)f_RGPV(v)-f(v)/(Lh_f^2h_g)^-1/2V_RGPV(v)^1/2, v∈ I

can be approximated by 

    (v)1/(N· L)^1/2∑_i,lℳ_2^(B_il;v)-μ_ℳ^(v)/Var[ℳ_2^(B_11;v)]^1/2, v∈ I

uniformly in v∈ I, with an estimated rate of uniform approximation
error. See the proof of Lemma B.4 for details. 

By adapting the proofs of Lemmas B.5 - B.9 of MMS, we can show that
the bootstrap process { Z^*(v):v∈ I}
can be approximated by the bootstrap analogue of (v)
uniformly in v∈ I. The rest of the proof is identical to that
of Corollary 4.3 of MMS. We can show that the difference between the
distribution of ‖ Z‖ _I and that of ‖_G‖ _I,
where {_G(v):v∈ I} is an
intermediate Gaussian process that has the same covariance structure
as that of  , converges to zero uniformly. See Theorem
4.3 of MMS and its proof. Then it can be shown that the difference
between the distribution of ‖ Z^*‖ _I and
that of ‖_G‖ _I converges to zero
uniformly in the bootstrap world. See Theorem 4.4 of MMS and its proof.
The conclusion follows easily from these observations. See the proof
of Corollaries 4.2 and 4.3 of MMS.



§ LEMMAS


Suppose that Assumptions <ref> - <ref>
are satisfied. Then we have

    b∈[b,b]sup|ξ(b)-ξ(b)|=O_p((log(L)/Lh)^1/2+h^3)

and

    b∈[b,b]sup|ξ(b)-ξ(b)+1/N-1G(b)/g(b)^2(g(b)-g(b))-1/N-1G(b)-G(b)/g(b)|=O_p(log(L)/Lh+h^6).



Standard arguments
(see Lemma 1 of ) yield

    b∈[b,b]sup|G(b)-G(b)|=O_p((log(L)/L)^1/2).

The bias of the local quadratic MCE E[g(b)]-g(b)
is O(h^3) uniformly over the entire support b∈[b,b].
The stochastic part g(b)-E[g(b)]
can be shown to be O_p(log(L)^1/2(Lh)^-1/2)
uniformly over the entire support b∈[b,b],
by using standard arguments (see, e.g., ).
Therefore, 

    b∈[b,b]sup|g(b)-g(b)|=O_p((log(L)/Lh)^1/2+h^3).


Applying the identity a/b=a/c-a(b-c)/c^2+a(b-c)^2/bc^2,

    ξ(b)-ξ(b)     =    1/N-1{ -G(b)(g(b)-g(b))/g(b)^2+G(b)-G(b)/g(b)+G(b)/g(b)(g(b)-g(b))^2/g(b)^2.
          .-(G(b)-G(b))(g(b)-g(b))/g(b)^2} .

By using (<ref>), (<ref>)
and (<ref>), 

    b∈[b,b]sup|G(b)(g(b)-g(b))/g(b)^2|=O_p((log(L)/Lh)^1/2+h^2)

and

    b∈[b,b]sup|G(b)-G(b)/g(b)|=O_p((log(L)/L)^1/2).

Since b∈[b,b]sup|g(b)-g(b)|=o_p(1)
w.p.a.1, b∈[b,b]supg(b)^-1<(C_g/2)^-1,
w.p.a.1 and consequently, 

    b∈[b,b]sup|G(b)/g(b)(g(b)-g(b))^2/g(b)^2|       b∈[b,b]sup(g(b)-g(b))^2
        =     O_p(log(L)/Lh+h^6).

The conclusion follows from this result, (<ref>),
(<ref>) and (<ref>).

Suppose that Assumptions <ref> - <ref>
are satisfied. Then, (a). s is strictly increasing on
[ξ(b),ξ(b)],
w.p.a.1; (b). 

    s(ξ(b)+h_r)-b=O_p(h) and

(c). 

    z∈[ξ(b)+h_r,ξ(b)-h_r]sup|s(z)-s(z)|=O_p((log(L)/Lh)^1/2+h^2);

(d). 

    z∈[ξ(b)+h_r,ξ(b)-h_r]sup|s'(z)-s'(z)|=O_p((log(L)/Lh^3)^1/2+h^2);

(e). For any inner closed sub-interval [b_l,b_u]
of [b,b], 

    b∈[b_l,b_u]sup|s^-1(b)-ξ(b)|=O_p((log(L)/Lh)^1/2+h^2).



By the definition
of s, we have

    s'(t)=∫_b^b1/h_rK_r(ξ(b)-t/h_r)db.

For any t∈[ξ(b),ξ(b)],
s'(t)>0 if the measurable set { b∈[b,b]:|ξ(b)-t|≤ h_r}
has positive measure. Let 

    r_ξb∈[b,b]sup|ξ(b)-ξ(b)|.

Clearly,

    { b∈[b,b]:|ξ(b)-t|+r_ξ≤ h_r}⊆{ b∈[b,b]:|ξ(b)-t|≤ h_r} .

 By Lemma <ref>, r_ξ=o_p(h). Consequently,
r_ξ≤ h_r/2 w.p.a.1, and

    { b∈[b,b]:|ξ(b)-t|≤ h_r/2}⊆{ b∈[b,b]:|ξ(b)-t|≤ h_r} ,for all , w.p.a.1.

Since ξ is continuous and strictly increasing on [b,b],
{ b∈[b,b]:|ξ(b)-t|≤ h_r}
has positive measure.

For Part (b), by change of variables and Fubini-Tonelli theorem, we
have

    s(ξ(b)+h_r)-b=    ∫_b^b∫_-∞^∞1(w≤ξ(b)+h_r-ξ(b)/h_r)K_r(w)dwdb
    ≤   ∫_-∞^∞∫_b^b1(ξ(b)≤ r_ξ+ξ(b)+(1-w)h_r)dbK_r(w)dw
    
    =    ∫_-∞^∞max{ s(r_ξ+ξ(b)+(1-w)h_r)-b,0} K_r(w)dw.

By a mean value expansion, we have 

    ∫_-∞^∞max{ s(ξ(b)+(1-w)h_r+r_ξ)-b,0} K(w)dw
    ≤   ∫_-∞^∞|s(ξ(b)+(1-w)h_r+r_ξ)-b|K(w)dw
       ∫_-∞^∞|(1-w)h_r+r_ξ|K(w)dw
    
    =     O_p(h),

where the last equality holds since r_ξ=o_p(h).
Therefore, s(ξ(b)+h_r)-b=O_p(h).
The proof of b-s(ξ(b)-h_r)=O_p(h)
is similar. 

For Part (c), by the triangle inequality and a second-order Taylor
expansion, we have

    |s(z)-s(z)|    ≤   |∫_b^bK_r(z-ξ(b)/h_r)db-∫_b^bK_r(z-ξ(b)/h_r)db|+|∫_b^bK_r(z-ξ(b)/h_r)db+b-s(z)|.
       ≤   |∫_b^b1/h_rK_r(z-ξ(b)/h_r)(ξ(b)-ξ(b))db|+|1/2∫_b^b1/h_r^2K_r(z-ξ̇(b)/h_r)(ξ(b)-ξ(b))^2db|
           +|∫_b^bK_r(z-ξ(b)/h_r)db+b-s(z)|,

where ξ̇(b) is the mean value satisfying |ξ̇(b)-ξ(b)|≤|ξ(b)-ξ(b)|
for each b∈[b,b].
Then we have

    z∈[ξ(b)+h_r,ξ(b)-h_r]sup|∫_b^b1/h_rK_r(z-ξ(b)/h_r)(ξ(b)-ξ(b))db|    ≤   {z∈[ξ(b)+h_r,ξ(b)-h_r]sup∫_b^b1/h_rK_r(z-ξ(b)/h_r)db} r_ξ
       ≤   {z∈[ξ(b)+h_r,ξ(b)-h_r]sup∫_b^b1/h_r1(|z-ξ(b)|≤ h_r)db} r_ξ
        =     O_p((log(L)/Lh)^1/2+h^3),

where the equality follows from Lemma <ref> . 

For all z∈[ξ(b)+h_r,ξ(b)-h_r],

    |∫_b^b1/h_r^2K(z-ξ̇(b)/h_r)(ξ̂(b)-ξ(b))^2db|    ≤   {∫_b^b1/h_r^2K_r(z-ξ̇(b)/h_r)db} r_ξ^2
          {∫_b^b1/h_r^21(|z-ξ(b)|≤ h_r+r_ξ)db} r_ξ^2
       ≤   {∫_b^b1/h_r^21(|z-ξ(b)|≤2h_r)db} r_ξ^2,

where the last inequality holds w.p.a.1 since r_ξ=o_p(h).
Now by (<ref>),
(<ref>) and
(<ref>),

    z∈[ξ(b)+h_r,ξ(b)-h_r]sup|∫_b^bK_r(z-ξ(b)/h_r)db-∫_b^bK_r(z-ξ(b)/h_r)db|=O_p((log(L)/Lh)^1/2+h^3).


For all z∈[ξ(b)+h_r,ξ(b)-h_r],
since K_r(u)=∫_-∞^uK_r(t)dt
and K_r is supported on [-1,1],

    |∫_b^bK_r(z-ξ(b)/h_r)db+b-s(z)|     =    |∫_b^s(z-h_r)K_r(z-ξ(b)/h_r)db+∫_s(z-h_r)^s(z+h_r)K_r(z-ξ(b)/h_r)db+b-s(z)|
        =    |s(z-h_r)+∫_s(z-h_r)^s(z+h_r)K_r(z-ξ(b)/h_r)db-s(z)|.

By change of variables and integration by parts,

    ∫_s(z-h_r)^s(z+h_r)K_r(z-ξ(b)/h_r)db     =    ∫_-1^1K_r(-u)h_rs'(h_ru+z)du
        =     -s(z-h_r)+∫_-1^1s(z+h_ru)K_r(u)du.

Now it follows that 

    z∈[ξ(b)+h_r,ξ(b)-h_r]sup|∫_b^bK_r(z-ξ(b)/h_r)db+b-s(z)|     =    z∈[ξ(b)+h_r,ξ(b)-h_r]sup|∫_-1^1K_r(u)s(z+h_ru)du-s(z)|
        =     O_p(h^2),

where the second equality follows from Taylor expansion and the fact
∫ uK_r(u)du=0. The conclusion of Part
(c) follows from (<ref>),
(<ref>) and
(<ref>).

For Part (d), note 

    |s'(z)-s'(z)|≤|∫_b^b1/h_rK_r(ξ(b)-z/h_r)db-∫_b^b1/h_rK_r(ξ(b)-z/h_r)db|+|∫_b^b1/h_rK_r(ξ(b)-z/h_r)db-s'(z)|.

By arguments that are similar to those used to prove (<ref>),

    z∈[ξ(b)+h_r,ξ(b)-h_r]sup|∫_b^b1/h_rK_r(ξ(b)-z/h_r)db-∫_b^b1/h_rK_r(ξ(b)-z/h_r)db|=O_p((log(L)/Lh^3)^1/2+h^2).

By integration by parts and Taylor expansion, 

    z∈[ξ(b)+h_r,ξ(b)-h_r]sup|∫_b^b1/h_rK_r(ξ(b)-z/h_r)db-s'(z)|     =    z∈[ξ(b)+h_r,ξ(b)-h_r]sup|∫_-1^1K_r(u)s'(z+h_ru)du-s'(z)|
        =     O_p(h^2).

The conclusion of Part (d) therefore follows.

For Part (e), Part (b) implies that [b_l,b_u] is
contained in the interior of [ξ(b)+h_r,ξ(b)-h_r]
w.p.a.1. Then Part (a) implies that 

    s^-1(b)∈[ξ(b)+h_r,ξ(b)-h_r] and b=s(s^-1(b)), for all  w.p.a.1.

Therefore, for any b∈[b_l,b_u], 

    s(s^-1(b))-s(ξ(b))=s'(v̇)(s^-1(b)-ξ(b))

for some mean value v̇ with |v̇-ξ(b)|≤|s^-1(b)-ξ(b)|.
Then, since s' is bounded away from zero (see Lemma A1 of GPV),

    b∈[b_l,b_u]sup|s^-1(b)-ξ(b)|       b∈[b_l,b_u]sup|s(s^-1(b))-s(ξ(b))|
       ≤   z∈[ξ(b)+h_r,ξ(b)-h_r]sup|s(z)-s(z)|
        =     O_p((log(L)/Lh)^1/2+h^2),

where the second inequality follows from (<ref>) and
holds w.p.a.1. 



Suppose that Assumptions <ref> - <ref>
hold. Let 𝕋_il1(V_il∈[v-δ_0,v+δ_0]).
Then we have

    f_RGPV(v)-f(v)     =    1/N· L∑_i,l𝕋_il1/h_f^2K_f'(V_il-v/h_f)(V_il^†-V_il)+1/2f”(v)(∫ K_f(u)u^2du)h_f^2
           +O_p(log(L)/Lh^3+(log(L)/Lh)^1/2+h^2),

where the remainder term is uniform in v∈ I.


Write

    f_RGPV(v)=1/N· L∑_i,l{𝕋_il1/h_fK_f(V_il^†-v/h_f)+(1-𝕋_il)1/h_fK_f(V_il^†-v/h_f)} .

Now for any v∈[ξ(b)+h_f+h_r,ξ(b)-h_f-h_r],

    |1/N· L∑_i,l(1-𝕋_il)1/h_fK_f(V_il^†-v/h_f)|
    ≤   1/N· L∑_i,lh_f^-1(1-𝕋_il)1(|V_il^†-v|≤ h_f)
    ≤   1/N· L∑_i,lh_f^-11(B_il>s(v+δ_0))1(B_il∈[s(v-h_f),s(v+h_f)])
        +1/N· L∑_i,lh_f^-11(B_il<s(v-δ_0))1(B_il∈[s(v-h_f),s(v+h_f)])
    ≤   1/N· L∑_i,lh_f^-11(B_il>s(v+δ_0))1(B_il∈[s(v-h_f)-r_s,s(v+h_f)+r_s])
        +1/N· L∑_i,lh_f^-11(B_il<s(v-δ_0))1(B_il∈[s(v-h_f)-r_s,s(v+h_f)+r_s]),

where r_sz∈[ξ(b)+h_r,ξ(b)-h_r]sup|s(z)-s(z)|
and the second inequality holds w.p.a.1. Therefore,

    f_RGPV(v)=1/N· L∑_i,l𝕋_il1/h_fK_f(V_il^†-v/h_f), for all , w.p.a.1.


Let f denote the infeasible estimator that uses the
unobserved true valuations:

    f(v)=1/N· L∑_i,l1/h_fK_f(V_il-v/h_f).

It now follows that

    f_RGPV(v)-f(v)=1/N· L∑_i,l𝕋_il1/h_f(K_f(V_il^†-v/h_f)-K_f(V_il-v/h_f)), for all , w.p.a.1.

By a second-order Taylor expansion of the right-hand side of the above
equality,

    f_RGPV(v)-f(v)=1/N· L∑_i,l𝕋_il1/h_f^2K_f'(V_il-v/h_f)(V_il^†-V_il)+1/2·1/N· L∑_i,l𝕋_il1/h_f^3K_f”(V̇_il-v/h_f)(V_il^†-V_il)^2,

for some mean value V̇_il that lies on the line joining
V_il^† and V_il. 

Lemma <ref>(e) implies that 

    v∈ Isup i,lmax 𝕋_il|V_il^†-V_il|=O_p((log(L)/Lh)^1/2+h^2).

Since K_f” is compactly supported on [-1,1] and
bounded, by the triangle inequality, 

    v∈ Isup|1/N· L∑_i,l𝕋_il1/h_f^3K_f”(V̇_il-v/h_f)(V_il^†-V_il)^2|
       {v∈ Isup 1/N· L∑_i,l𝕋_ilh_f^-31(|V̇_il-v|≤ h_f)}{v∈ Isup i,lmax 𝕋_il(V_il^†-V_il)^2}
    ≤   {v∈ Isup 1/N· L∑_i,l𝕋_ilh_f^-31(|V_il-v|≤2h_f)}{v∈ Isup i,lmax 𝕋_il(V_il^†-V_il)^2} ,

where the last inequality holds w.p.a.1, since v∈ Isup i,lmax 𝕋_il|V̇_il-V_il|=o_p(h).
It follows that

    v∈ Isup|1/N· L∑_i,l𝕋_il1/h_f^3K_f”(V̇_il-v/h_f)(V_il^†-V_il)^2|=O_p(log(L)/Lh^3+h^2).


By standard arguments for kernel density estimation,

    f(v)-E[f(v)]=O_p((log(L)/Lh)^1/2) and E[f(v)]-f(v)=1/2f”(v)(∫ K_f(u)u^2du)h_f^2+o(h^2),

where the remainder terms are uniform in v∈ I. The conclusion
follows. 

Suppose that Assumptions <ref> - <ref>
hold. Let

    s(t)∫_b^b∫_-∞^t1/h_rK_r(ξ(b)-u/h_r)dudb+b, t∈ℝ.

Then,

    f_RGPV(v)-f(v)     =     -1/N· L∑_i,l1/h_f^2𝕋_ilK_f'(V_il-v/h_f)s(V_il)-s(V_il)/s'(V_il)+1/2f”(v)(∫ K_f(u)u^2du)h_f^2
           +1/2(s”'(v)f(v)+s”(v)f'(v))s'(v)-s”(v)f(v)s”(v)/s'(v)^2(∫ K_r(u)u^2du)h_r^2
           +O_p(log(L)/Lh^3+(log(L)/Lh)^1/2+h^2),

where the remainder term is uniform in v∈ I.


It follows from Lemma
<ref>(b) that [s(v_l-δ_0),s(v_u+δ_0)]
is an inner closed sub-interval of [s(ξ(b)+h_r),s(ξ(b)-h_r)]
w.p.a.1. By Lemma <ref>(a), s is strictly
increasing on [ξ(b)+h_r,ξ(b)-h_r]
w.p.a.1 and for all B_il satisfying B_il∈[s(v_l-δ_0),s(v_u+δ_0)],
we have the following expansion by <cit.>:

    V_il^†-V_il=-(s-s/s')∘ξ(B_il)+χ_1,il+χ_2,il

where

    χ_1,il-2(s-s/s'+λ_il(s'-s')·s'-s'/s'+λ_il(s'-s'))∘(s+λ_il(s-s))^-1(B_il)

and

    χ_2,il{s-s/s'+λ_il(s'-s')·(s-s)(s”+λ_il(s”-s”))/(s'+λ_il(s'-s'))^2}∘(s+λ_il(s-s))^-1(B_il)

for some λ_il∈(0,1) that depends on B_il.
By Lemma <ref>(b),

    (s+λ_il(s-s))^-1(B_il)∈[ξ(b)+h_r,ξ(b)-h_r], w.p.a.1,

for all B_il satisfying B_il∈[s(v_l-δ_0),s(v_u+δ_0)].
Next, write

    1/NL∑_i,l𝕋_il1/h_f^2K_f'(V_il-v/h_f)(V_il^†-V_il)
    
    =     -1/NL∑_i,l1/h_f^2𝕋_ilK_f'(V_il-v/h_f)s(V_il)-s(V_il)/s'(V_il)-1/NL∑_i,l1/h_f^2𝕋_ilK_f'(V_il-v/h_f)s(V_il)-B_il/s'(V_il)
        +1/NL∑_i,l1/h_f^2𝕋_ilK_f'(V_il-v/h_f)χ_1,il+1/NL∑_i,l1/h_f^2𝕋_ilK_f'(V_il-v/h_f)χ_2,il.


By Lemma <ref>(c) and Lemma <ref>(d), 

    i,lsup𝕋_il|χ_1,il|=O_p(log(L)/Lh^2+h^4).

For χ_il^2, since K_r' is supported on [-1,1],

    t∈[ξ(b)+h_r,ξ(b)-h_r]sup|s”(t)|       t∈[ξ(b)+h_r,ξ(b)-h_r]sup∫_b^b1/h_r^21(|t-ξ(b)|≤ h_r+r_ξ)db
        =     O_p(h^-1).

It follows from this result, Lemma <ref>(c) and Lemma
<ref>(d) that

    i,lsup𝕋_il|χ_2,il|=O_p(log(L)/Lh^2+h^3).

Now it is clear that

    v∈ Isup|1/N· L∑_i,l1/h_f^2𝕋_ilK_f'(V_il-v/h_f)χ_1,il|    ≤   (v∈ Isup1/N· L∑_i,l|1/h_f^2K_f'(V_il-v/h_f)|)(i,lsup𝕋_il|χ_1,il|)
        =     O_p(log(L)/Lh^3+h^3)

and similarly,

    v∈ Isup|1/NL∑_i,l1/h_f^2𝕋_ilK_f'(V_il-v/h_f)χ_2,il|=O_p(log(L)/Lh^3+h^2).


By the definition of s, when h_f is small enough,

    1/NL∑_i,l1/h_f^2𝕋_ilK_f'(V_il-v/h_f)s(V_il)-B_il/s'(V_il)
    
    =    1/NL∑_i,l1/h_f^2K_f'(V_il-v/h_f)1/s'(V_il)(∫_b^bK_r(V_il-ξ(b)/h_r)db+b-s(V_il)).

By change of variable and integration by parts, 

    ∫_b^bK_r(V_il-ξ(b)/h_r)db=    ∫_v^vK_r(V_il-u/h_r)s'(u)du
    
    =    b·K_r(V_il-v/h_r)-b·K_r(V_il-v/h_r)+∫_v^v1/h_rK_r(V_il-u/h_r)s(u)du
    
    =    ∫_v^v1/h_rK_r(V_il-u/h_r)s(u)du-b,

when h_r is small enough, for all V_il satisfying V_il∈[v_l-δ_0,v_u+δ_0]. 

Denote

    β_s(w)∫_v^v1/h_rK_r(w-u/h_r)s(u)du-s(w)

and write

    1/N· L∑_i,l1/h_f^2K_f'(V_il-v/h_f)1/s'(V_il)(∫_v^v1/h_rK_r(V_il-u/h_r)s(u)du-s(V_il))
    
    =    {1/N· L∑_i,l1/h_f^2K_f'(V_il-v/h_f)β_s(V_il)/s'(V_il)-E[1/h_f^2K_f'(V_11-v/h_f)β_s(V_11)/s'(V_11)]} +E[1/h_f^2K_f'(V_11-v/h_f)β_s(V_11)/s'(V_11)].


By standard argument for kernel density estimation (see, e.g., ),
since s is three-times continuously differentiable,

    β_s(w)=h_r^2/2s”(w)∫ u^2K_r(u)du+o(h^2)
    β_s'(w)=h_r^2/2s”'(w)∫ u^2K_r(u)du+o(h^2),

where the remainder terms are uniform in v∈ I.

By change of variables,

    E[1/h_f^2K_f'(V_11-v/h_f)β_s(V_11)/s'(V_11)]
    
    =    ∫_v-v/h_f^v-v/h_f1/h_fK_f'(z)β_s(h_fz+v)/s'(h_fz+v)f(h_fz+v)dz
    
    =    ∫_v-v/h_f^v-v/h_f1/h_fK_f'(z){β_s(v)f(v)/s'(v)+(β_s'(v̇)f(v̇)+β_s(v̇)f'(v̇))s'(v̇)-β_s(v̇)f(v̇)s”(v̇)/s'(v̇)^2h_fz}dz,

where v̇ is the mean value depending on z with |v̇-v|≤ h_f|z|.
It is clear that for small enough h_f, 

    β_s(v)f(v)/s'(v)(∫_v-v/h_f^v-v/h_fK_f'(z)dz)=0 for all .

Now it follows from (<ref>) and (<ref>)
that

    E[1/h_f^2K_f'(V_11-v/h_f)β_s(V_11)/s'(V_11)]=h_r^2/2(s”'(v)f(v)+s”(v)f'(v))s'(v)-s”(v)f(v)s”(v)/s'(v)^2∫ u^2K_r(u)du+o(h^2),

where the remainder term is uniform in v∈ I.

Denote 

    𝒮(z;v)1/h_f^2K_f'(z-v/h_f)β_s(z)/s'(z)

and thus

    1/N· L∑_i,l1/h_f^2K_f'(V_il-v/h_f)β_s(V_il)/s'(V_il)-E[1/h_f^2K_f'(V_11-v/h_f)β_s(V_11)/s'(V_11)]=1/N· L∑_i,l𝒮(V_il;v)-E[𝒮(V_11;v)].

By standard arguments (see the proof of Lemma B.3 of MMS), it can
be easily verified that {𝒮(·;v):v∈ I}
is (uniformly) VC-type with respect to a constant envelope F_𝒮
which satisfies F_𝒮 h_f^-2v∈[v_l-δ_0,v_u+δ_0]sup|β_s(v)|
when h_f is small enough. The applying a maximal inequality ()
yields 

    E[|1/N· L∑_i,l𝒮(V_il;v)-E[𝒮(V_11;v)]|]≤ L^-1/2|F_𝒮|=O(L^-1/2),

when h is small enough. The conclusion follows from this result,
(<ref>), (<ref>),
(<ref>) and (<ref>).


Suppose that Assumptions <ref> - <ref>
hold. We have

    f_RGPV(v)-f(v)     =    1/(N-1)1/(N· L)^2∑_i,l∑_j,kℳ(B_il,B_jk;v)+1/2f”(v)(∫ K_f(u)u^2du)h_f^2
           +1/2(s”'(v)f(v)+s”(v)f'(v))s'(v)-s”(v)^2f(v)/s'(v)^2(∫ K_r(u)u^2du)h_r^2
           +O_p(log(L)/Lh^3+(log(L)/Lh)^1/2+h^2),

where the remainder term is uniform in v∈ I.


First we show

    -1/N· L∑_i,l1/h_f^2𝕋_ilK_f'(V_il-v/h_f)s(V_il)-s(V_il)/s'(V_il)
    
    =     -1/N-11/N· L∑_i,l1/h_f^2𝕋_ilK_f'(V_il-v/h_f)1/s'(V_il)∫_b^b1/h_rK_r(ξ(b)-V_il/h_r)G(b)/g(b)^2(g(b)-g(b))db
        +O_p(log(L)/Lh^3+h^3+(log(L)/Lh)^1/2).


Since by the Borel-Cantelli lemma we have

    |b-b|=O_p(log(L)/L), |b-b|=O_p(log(L)/L),

therefore,

    1/N· L∑_i,l𝕋_il1/h_f^2K_f'(V_il-v/h_f)s(V_il)-s(V_il)/s'(V_il)
    
    =    1/N· L∑_i,l𝕋_il1/h_f^2K_f'(V_il-v/h_f)1/s'(V_il)∫_b^b(K_r(V_il-ξ(b)/h_r)-K_r(V_il-ξ(b)/h_r))db+O_p(log(L)/Lh),

where the remainder term is uniform in v∈ I. 

By a second-order Taylor expansion, we have

    1/N· L∑_i,l𝕋_il1/h_f^2K_f'(V_il-v/h_f)1/s'(V_il)∫_b^b(K_r(V_il-ξ(b)/h_r)-K_r(V_il-ξ(b)/h_r))db
    
    =    1/N-11/N· L∑_i,l𝕋_il1/h_f^2K_f'(V_il-v/h_f)1/s'(V_il)∫_b^b1/h_rK_r(V_il-ξ(b)/h_r)G(b)/g(b)^2(g(b)-g(b))db
        -1/N-11/N· L∑_i,l𝕋_il1/h_f^2K_f'(V_il-v/h_f)1/s'(V_il)∫_b^b1/h_rK_r(V_il-ξ(b)/h_r)G(b)-G(b)/g(b)db
        +I_1(v)+I_2(v)

where

    I_1(v)        -1/N· L∑_i,l𝕋_il1/h_f^2K_f'(V_il-v/h_f)1/s'(V_il)∫_b^b1/h_rK_r(V_il-ξ(b)/h_r)
          ×{(ξ(b)-ξ(b))+1/N-1G(b)/g(b)^2(g(b)-g(b))-1/N-1G(b)-G(b)/g(b)}db

and 

    I_2(v)1/N· L∑_i,l𝕋_il1/h_f^2K_f'(V_il-v/h_f)1/s'(V_il)∫_b^b1/h_r^2K_r'(V_il-ξ̇(b)/h_r)(ξ(b)-ξ(b))^2db

for some mean value ξ̇(b) with |ξ̇(b)-ξ(b)|≤|ξ(b)-ξ(b)|
for each b∈[b,b]. 

Then, by Lemma <ref>,

    v∈ Isup|I_1(v)|       {v∈ Isup1/N· L∑_i,l1/h_f^2|K_f'(V_il-v/h_f)|}{v∈[v_l-δ_0,v_u+δ_0]sup∫_b^b1/h_rK_r(v-ξ(b)/h_r)db}
          ×b∈[b,b]sup|(ξ(b)-ξ(b))+1/N-1G(b)/g(b)^2(g(b)-g(b))-1/N-1G(b)-G(b)/g(b)|
        =     O_p(log(L)/Lh^2+h^3)

and

    v∈ Isup|I_2(v)|       {v∈ Isup1/N· L∑_i,l1/h_f^2|K_f'(V_il-v/h_f)|}{v∈[v_l-δ_0,v_u+δ_0]sup∫_b^b1/h_r^2K(v-ξ̇(b)/h_r)db} r_ξ^2
        =     O_p(log(L)/Lh^3+h^3),

when the inequalities hold when h_r is small enough. 

Define 

    𝒢(b,b';v)1/h_f^2K_f'(ξ(b)-v/h_f)1/s'(ξ(b))∫_b^b1/h_rK_r(ξ(b)-ξ(z)/h_r)1(b'≤ z)-G(z)/g(z)dz.

By the definition of 𝒢, we have 

    1/N· L∑_i,l𝕋_il1/h_f^2K_f'(V_il-v/h_f)1/s'(V_il)∫_b^b1/h_rK_r(V_il-ξ(b)/h_r)G(b)-G(b)/g(b)db
    
    =    1/(N· L)^2∑_(2)𝒢(B_il,B_jk;v)+1/(N· L)^2∑_i,l𝒢(B_il,B_il;v), for all ,

when h_f is small enough. The kernel 𝒢 satisfies

    𝒢_1(b;v)∫𝒢(b,b';v)dG(b')=0 and μ_𝒢(v)∫∫𝒢(b,b';v)dG(b')dG(b)=0, for all .

Also define 

    𝒢_2(b;v)∫𝒢(b',b;v)dG(b').

Hoeffding decomposition gives 

    1/(N· L)_2∑_(2)𝒢(B_il,B_jk;v)=1/N· L∑_i,l𝒢_2(B_il;v)+1/(N· L)_2∑_(2){𝒢(B_il,B_jk;v)-𝒢_2(B_il;v)} .

By standard arguments (see the proof of Lemma B.2 of MMS for details),
it can be easily verified that {𝒢(·,·;v):v∈ I}
is (uniformly) VC-type with respect to a constant envelope F_𝒢
which satisfies F_𝒢 h_f^-2. This implies

    v∈ Isup|1/(N· L)^2∑_i,l𝒢(B_il,B_il;v)|=O_p((Lh^2)^-1).

Application of a maximal inequality ()
and Markov's inequality gives 

    v∈ Isup|1/(N· L)_2∑_(2){𝒢(B_il,B_jk;v)-𝒢_2(B_il;v)}|=O_p((Lh^2)^-1).


Next, we show that 

    v∈ Isup E[𝒢_2(B_11;v)^2] h^-1,

when h is small enough. Denote 

    τ(z)∫_b^b1/h_f^2K_f'(ξ(b)-v/h_f)ξ'(b)1/h_rK_r(ξ(b)-ξ(z)/h_r)g(b)db.

Then by change of variables and the Fubini-Tonelli theorem,

    ∫𝒢_2(b;v)^2dG(b)
    ≤   ∫_b^b{∫_b^bτ(z)1(b≤ z)/g(z)dz} ^2g(b)db
    
    =    ∫_b^b∫_b^bτ(z)/g(z)τ(z')/g(z')G(min{ z,z'})dzdz'
    
    =     h_r^2∫_v-v/h_r^v-v/h_r∫_v-v/h_r^v-v/h_rτ(s(h_rw+v))/g(s(h_rw+v))τ(s(h_rw'+v))/g(s(h_rw'+v))G(min{ s(h_rw+v),s(h_rw'+v)})s'(h_rw+v)s'(h_rw'+v)dwdw'
    
    =    2/h_f^2∫_v-v/h_r^v-v/h_rs'(h_rw'+v)/g(s(h_rw'+v))∫_v-v/h_r^v-v/h_rK_f'(u')K_r(λ_f/λ_ru'-w')g(s(h_fu'+v))du'
       ×∫_v-v/h_r^w's'(h_rw+v)G(s(h_rw+v))/g(s(h_rw+v))∫_v-v/h_r^v-v/h_rK_f'(u)K_r(λ_f/λ_ru-w)g(s(h_fu+v))dudwdw',

where the last equality follows from symmetry. It follows from integration
by parts that 

    ∫∫_-∞^w'{∫ K_f'(u')K_r(λ_f/λ_ru'-w')du'}{∫ K_f'(u)K_r(λ_f/λ_ru-w)du}dwdw'=0.

Then (<ref>) follows from this
result, (<ref>) and Taylor expansion. 

Since {𝒢(·,·;v):v∈ I}
is (uniformly) VC-type with respect to a constant envelope F_𝒢
which satisfies F_𝒢 h_f^-2, it follows from
(<ref>) and a maximal inequality
() that 

    v∈ Isup|1/N· L∑_i,l𝒢_2(B_il;v)|=O_p((log(L)/Lh)^1/2+log(L)/Lh^2).

See the proof of Lemma B.2 of MMS for more details.Now
it follows from this result, (<ref>), (<ref>),
(<ref>) and (<ref>) that

    1/N· L∑_i,l𝕋_il1/h_f^2K_f'(V_il-v/h_f)1/s'(V_il)∫_b^b1/h_rK_r(V_il-ξ(b)/h_r)G(b)-G(b)/g(b)db=O_p((log(L)/Lh)^1/2+log(L)/Lh^2),

uniformly in v∈ I. Then it follows from this result, (<ref>),
(<ref>), (<ref>)
and (<ref>) that 

    1/N· L∑_i,l𝕋_il1/h_f^2K_f'(V_il-v/h_f)s(V_il)-s(V_il)/s'(V_il)
    
    =    1/N-11/N· L∑_i,l𝕋_il1/h_f^2K_f'(V_il-v/h_f)1/s'(V_il)∫_b^b1/h_rK_r(V_il-ξ(b)/h_r)G(b)/g(b)^2(g(b)-g(b))db
        +O_p((log(L)/Lh)^1/2+log(L)/Lh^2+h^3),

where the remainder term is uniform in v∈ I.

By the definition of the MCE, 

    g(b)=1/N· L∑_i,l1/h_gK_g(B_il-b/h_g), for

Therefore, since K_f' and K_r are compactly supported on
[-1,1], it is easy to verify that

    -1/N-11/N· L∑_i,l1/h_f^2𝕋_ilK_f'(V_il-v/h_f)1/s'(V_il)∫_b^b1/h_rK_r(ξ(b)-V_il/h_r)G(b)/g(b)^2(g(b)-g(b))db
    
    =     -1/N-11/N· L∑_i,l1/h_f^2𝕋_ilK_f'(V_il-v/h_f)1/s'(V_il)∫_b^b1/h_rK_r(ξ(b)-V_il/h_r)G(b)/g(b)^2
       ×(1/N· L∑_j,k1/h_gK_g(B_jk-b/h_g)-g(b))db, for all , w.p.a.1.

The conclusion follows from this result, (<ref>),
(<ref>), (<ref>),
(<ref>) and Lemma <ref>. 

Suppose that Assumptions <ref> - <ref>
hold. Let 

    β_g(b)∫{1/h_gK_g(b'-b/h_g)-g(b)}dG(b')

be the bias of the kernel density estimator of g(b).
Then

    μ_ℳ(v)-∫{1/h_f^2K_f'(ξ(u)-v/h_f)1/s'(ξ(u))∫_b^b1/h_rK_r(ξ(b)-ξ(u)/h_r)G(b)β_g(b)/g(b)^2db}dG(u)=o(h^2),

uniformly in v∈ I.

By change of variables,

    ∫{1/h_f^2K_f'(ξ(u)-v/h_f)1/s'(ξ(u))∫_b^b1/h_rK_r(ξ(b)-ξ(u)/h_r)G(b)β_g(b)/g(b)^2db}dG(u)
    
    =    ∫_v-v/h_f^v-v/h_f∫_v-v/h_r^v-v/h_r1/h_fK_f'(w)K_r(z-λ_f/λ_rw)G(s(h_rz+v))β_g(s(h_rz+v))/g(s(h_rz+v))^2s'(h_rz+v)g(s(h_fw+v))dzdw.

Let ψ(z) G(s(z))s'(z)/g(s(z))^2.
By a mean value expansion, the second line of (<ref>)
is equal to

    1/h_f∫_v-v/h_f^v-v/h_f∫_v-v/h_r^v-v/h_rK_f'(w)K_r(z-λ_f/λ_rw){ψ(v)β_g(s(v))+(ψ'(v̇)β_g(s(v̇))+ψ(v̇)β_g'(s(v̇))s'(v̇))h_rz}
       ×{ g(s(v))+g'(s(v̈))s'(v̈)h_fw}dzdw
    
    =    ψ(v)β_g(s(v))g(s(v))1/h_f∫_v-v/h_f^v-v/h_f∫_v-v/h_r^v-v/h_rK_f'(w)K_r(z-λ_f/λ_rw)dzdw
        +ψ(v)β_g(s(v))∫_v-v/h_f^v-v/h_f∫_v-v/h_r^v-v/h_rK_f'(w)K_r(z-λ_f/λ_rw)g'(s(v̈))s'(v̈)wdzdw
        +g(s(v))1/h_f∫_v-v/h_f^v-v/h_f∫_v-v/h_r^v-v/h_rK_f'(w)K_r(z-λ_f/λ_rw)(ψ'(v̇)β_g(s(v̇))+ψ(v̇)β_g'(s(v̇))s'(v̇))h_rzdzdw
        +h_r∫_v-v/h_f^v-v/h_f∫_v-v/h_r^v-v/h_rK_f'(w)K_r(z)(ψ'(v̇)β_g(s(v̇))+ψ(v̇)β_g'(s(v̇))s'(v̇))g'(s(v̈))s'(v̈)zwdzdw,

where v̇ and v̈ are mean values that are dependent
on z and w with |v̇-v|≤ h_r|z|
and |v̈-v|≤ h_f|w|. When h
is small enough,

    ∫_v-v/h_f^v-v/h_f∫_v-v/h_r^v-v/h_rK_f'(w)K_r(z-λ_f/λ_rw)dzdw=0, for all .


By standard arguments for the bias of kernel estimators for the density
(see, e.g., ), since K_g is supported
on [-1,1], for each b∈[s(v_l-δ_0),s(v_u+δ_0)],

    |β_g(b)|≤h_g^3/6b'∈[b-h_g,b+h_g]sup| g”'(b')|∫| u^3K_g(u)|du,

when h_g is small enough. By change of variable and Taylor expansion,
we have

    b∈[s(v_l-δ_0),s(v_u+δ_0)]sup|β_g'(b)|=    b∈[s(v_l-δ_0),s(v_u+δ_0)]sup|∫1/h_gK_g(b'-b/h_g)g'(b')db'-g'(b)|
    ≤   b∈[s(v_l-δ_0),s(v_u+δ_0)]suph_g^2/2|∫ K_g(u)u^2(g”'(ḃ)-g”'(b))du|,

when h_g is small enough, where ḃ is the mean value
depending on u with |ḃ-b|≤ h_g|u|.
Since g”' is uniformly continuous on any inner closed subset of
[b,b], the assumption that K_g
is supported on [-1,1] and (<ref>)
imply that β_g'(b)=o(h^2) uniformly
in b∈[s(v_l-δ_0),s(v_u+δ_0)].
The conclusion follows from these results, (<ref>)
and (<ref>). 

Suppose that Assumptions <ref> - <ref>
hold. Then 

    f_RGPV(v)-f(v)     =    1/(N-1)1/(N· L)^2∑_i,l(ℳ_2(B_il;v)-μ_ℳ(v))+1/2f”(v)(∫ K_f(u)u^2du)h_f^2
           +1/2(s”'(v)f(v)+s”(v)f'(v))s'(v)-s”(v)^2f(v)/s'(v)^2(∫ K_r(u)u^2du)h_r^2
           +O_p(log(L)/Lh^3+(log(L)/Lh)^1/2+h^2).


Hoeffding decomposition
gives

    1/(N· L)^2∑_i,l∑_j,kℳ(B_il,B_jk;v)     =    μ_ℳ(v)+{1/N· L∑_i,lℳ_1(B_il;v)-μ_ℳ(v)} +{1/N· L∑_i,lℳ_2(B_il;v)-μ_ℳ(v)}
           +1/(N· L)_2∑_(2){ℳ(B_il,B_jk;v)-ℳ_1(B_il;v)-ℳ_2(B_jk;v)+μ_ℳ(v)}
           +1/(N· L)^2∑_i,lℳ(B_il,B_il;v)-1/(N· L)(N· L)_2∑_(2)ℳ(B_il,B_jk;v).

By standard arguments used in the proofs of Lemma B.2 and Lemma B.3
of MMS, it can be easily verified that the class {ℳ(·,·;v):v∈ I}
is (uniformly) VC-type with respect to a constant envelope F_ℳ
that satisfies F_ℳ=O(h^-3). Therefore,
it is easy to check that 

    |1/(N· L)^2∑_i,lℳ(B_il,B_il;v)|=O_p((Lh^3)^-1) and |1/(N· L)(N· L)_2∑_(2)ℳ(B_il,B_jk;v)|=O_p((Lh^3)^-1),

uniformly in v∈ I, and 

    v∈ Isup|1/(N· L)_2∑_(2){ℳ(B_il,B_jk;v)-ℳ_1(B_il;v)-ℳ_2(B_jk;v)+μ_ℳ(v)}|=O_p((Lh^3)^-1)

follows from the maximal inequality <cit.>
and Markov's inequality.

By change of variables,

    ℳ_1(b;v)     =    1/h_f^2K_f(ξ(b)-v/h_f)ξ'(b)∫_b^b1/h_rK_r(ξ(b)-ξ(z)/h_r)G(z)/g(z)^2β_g(z)dz
        =    1/h_f^2K_f(ξ(b)-v/h_f)ξ'(b)∫_v-ξ(b)/h_r^v-ξ(b)/h_rK_r(u)G(s(h_ru+ξ(b)))β_g(s(h_ru+ξ(b)))s'(h_ru+ξ(b))/g(s(h_ru+ξ(b)))^2du.

By the arguments used in the proof of Lemma B.3, it can be verified
that {ℳ_1(·;v):v∈ I}
is (uniformly) VC-type with respect to a constant envelope F_ℳ_1
that satisfies

    F_ℳ_1 h_f^-2b∈[s(v_l-δ_0),s(v_u+δ_0)]sup|β_g(b)|,

when h is small enough. The maximal inequality <cit.>
yields 

    E[v∈ Isup|1/N· L∑_i,lℳ_1(B_il;v)-μ_ℳ(v)|] L^-1/2F_ℳ_1=O(L^-1/2h).

The conclusion follows from this result, Markov's inequality, (<ref>),
(<ref>), (<ref>) and Lemma
<ref>.
