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# Exact bounds on the truncated-tilted mean, with applications

May 28, 2020
###### Abstract

Exact upper bounds on the Winsorised-tilted mean, , of a random variable in terms of its first two moments are given. Such results are needed in work on nonuniform Berry–Esseen-type bounds for general nonlinear statistics. As another application, optimal upper bounds on the Bayes posterior mean are provided. Certain monotonicity properties of the tilted mean are also presented.

{frontmatter}\runtitle

Bounds on the truncated-tilted mean

{aug}\runauthor

Iosif Pinelis

\affiliation

Michigan Technological University

Department of Mathematical Sciences
Michigan Technological University
Houghton, Michigan 49931, USA

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[class=AMS] \kwd[Primary ]60E15 \kwd[; secondary ]60E10 \kwd60F10 \kwd60F05 {keyword} \kwdexact upper bounds \kwdWinsorization \kwdtruncation \kwdlarge deviations \kwdnonuniform Berry-Esseen bounds \kwdCramér tilt transform \kwdmonotonicity \kwdBayes posterior mean

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chapter

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subsubsection

## 1 Introduction

Cramér’s tilt transform of a random variable (r.v.) is a r.v.  such that

 Ef(Xh)=Ef(X)ehXEehX

for all nonnegative Borel functions , where is a real parameter. This transform is an important tool in the theory of large deviation probabilities , where is a large number; then the appropriate value of the parameter is positive. Unfortunately, if the right tail of the distribution of decreases slower than exponentially, then for all and thus the tilt transform is not applicable. The usual recourse then is to replace in the exponent by its truncated counterpart, say or , where is a real number. As shown in [12, 13], of the two mentioned kinds of truncation, it is the so-called Winsorization, , of the r.v.  that is more useful in the applications considered there.

In particular, in [13] one needs a good upper bound on the mean

 Eh,wX:=EXeh(X∧w)Eeh(X∧w). (1.1)

of the Winsorised-tilted distribution of . Note that is well defined and finite for any , any , and any r.v.  such that .

In [12], exact upper bounds on the denominator of the ratio in (1.1) were provided, which effected a significant improvement on previously obtained upper bounds on (also, [12] contained applications to pricing of certain financial derivatives). However, before the present study, the numerator of the ratio in (1.1) was bounded separately from the denominator, which entailed a serious loss in accuracy. In this paper, exact upper bounds on will be provided, in terms of the first two moments of the r.v. . In fact, without loss of generality one may assume that , since

 Eh,w(X+m)=m+Eh,w−mX (1.2)

for any (and, again, any and ).

We shall show (Theorem 2.1) that the maximum of is attained at a r.v.  with taking one just two values. This allows for further analysis leading to a rather easily computable expression for the maximum of , as well as simple and explicit, but at the same time optimal, upper bounds on this maximum; these latter results are provided in Theorem 2.4. We shall also present various monotonicity properties of the maximum (part (II) of Theorem 2.1, and Proposition 2.6), and also demonstrate uniqueness of the maximizer (part (I) of Theorem 2.4). In addition, we shall apply some of these results to obtain optimal upper bounds on the Bayes posterior mean.

## 2 Summary and discussion

Take any and in and any . It will sometimes be more convenient to state the results, not in terms of r.v.’s , but in terms of the corresponding probability distributions , so that we shall use the notation

 Mh,wP:=∫Rxeh(x∧w)P(dx)∫Reh(x∧w)P(dx)

instead of — with standing for “the (Winsorised-tilted) mean”.

Let denote the set of all probability distributions (that is, Borel probability measures) on . Let then

 Pσ :={P∈P:∫RxP(dx)=0,∫Rx2P(dx)=σ2}, (2.1) P⩽σ :={P∈P:∫RxP(dx)=0,∫Rx2P(dx)∈(0,σ2]}, P(2)w,σ :={Pu,v:−∞<−u

where, for any positive real numbers and , the symbol stands for the unique zero-mean probability distribution on the two-point set . Note here that (i) the conditions and imply that and ; and (ii) ; so, and .

Let denote the class of all r.v.’s whose probability distributions belong to the set ; similarly define the classes , , and .

Define now the corresponding suprema:

 \SSh,w,σ:= sup{Mh,wP:P∈Pσ} =sup{Eh,wX:X∈Xσ}, (2.2) \SSh,w,⩽σ:= sup{Mh,wP:P∈P⩽σ} =sup{Eh,wX:X∈X⩽σ}, \SS(2)h,w,σ:= sup{Mh,wP:P∈P(2)w,σ} =sup{Eh,wX:X∈X(2)w,σ}, \SS(2)h,w,⩽σ:= sup{Mh,wP:P∈P(2)w,⩽σ} =sup{Eh,wX:X∈X(2)w,⩽σ}.

Consider also the attainment sets for these suprema:

 Ah,w,σ:= {P∈Pσ:Mh,wP=\SSh,w,σ}, (2.3) Ah,w,⩽σ:= {P∈P⩽σ:Mh,wP=\SSh,w,⩽σ}, A(2)h,w,σ:= {P∈P(2)w,σ:Mh,wP=\SS(2)h,w,σ}, A(2)h,w,⩽σ:= {P∈P(2)w,⩽σ:Mh,wP=\SS(2)h,w,⩽σ}.

We shall say, interchangeably, that some or all of the four suprema in (2.2) (and the related suprema in (3.1) below) are attained at a r.v.  or at a probability distribution , assuming that is the distribution of .

###### Theorem 2.1.

The following statements hold:

1. the four suprema in (2.2) are all the same:

 \SSh,w,⩽σ=\SSh,w,σ=\SS(2)h,w,σ=\SS(2)h,w,⩽σ; (2.4)
2. each of the four suprema in (2.4) is (strictly) increasing in ;

3. each of these suprema is attained, and

 Ah,w,⩽σ=Ah,w,σ=A(2)h,w,σ=A(2)h,w,⩽σ. (2.5)

The proofs will be given in Section 3.

Let us now show how to compute effectively the four equal suprema in Theorem 2.1; in particular, we shall see that each of the four attainment sets in (2.5) is a singleton one. We shall also provide simple (and, in a sense, optimal) upper bounds on these suprema; such bounds are what was needed in [13].

###### Remark 2.2.

The shift-transformation formula (1.2) allowed us to reduce the consideration to zero-mean distributions. One can also do rescaling, to reduce the set of all possible values of the Winsorization level from to . Indeed, observe that

 Eh,wX=|w|Eh|w|,w/|w|X|w|

for any real (and any ), which implies

 \SSh,w,σ=|w|\SSh|w|,w/|w|,σ/|w|

and the similar formulas for the other three suprema in (2.4). So, without loss of generality let us assume that , which will allow us to simplify the writing.

To state Theorem 2.4 below, more notation is needed. For any , let

 u∗(h,ε):=ε2e(1+ε)h−1−εh1+εh−e−(1+ε)h. (2.6)

Let mean “equals in sign to”.

The following proposition allows one to define terms used in the statement of Theorem 2.4.

###### Proposition 2.3.
1. There is a unique root of the equation

 u∗(h,σ2h)=σ2h.
2. If , then there is a unique root of the equation

 u∗(h,~εh,σ)=σ2; (2.7)

moreover, for all .

3. For each , there is a unique root of the equation

 rw,1(εh,w,σ) =0, where (2.8) rw,1(ε) :=e(ε+w)h(1+εh)(ε2+σ2)−ε2e2(ε+w)h−σ2.

Also, let us recall that stands for the unique zero-mean probability distribution on the two-point set .

###### Theorem 2.4.

Take any . Then the following statements hold.

1. Each of the four attainment sets in (2.5) coincides with the singleton set

 Ah,w,σ={Pεh,w,σ,σ2/εh,w,σ}, (2.9)

where is defined by (2.8) for , and

 εh,1,σ:= {σ2 if σ⩽σh,~εh,σ if σ>σh, (2.10)

with defined by equation (2.7).

2. Moreover,

 \SSh,w,σ=Mh,wPεh,w,σ,σ2/εh,w,σ

where

 Kw(h):=⎧⎪⎨⎪⎩eh−1 if% w=1,h−L−1(−ehw−1) if w∈{−1,0},

is a branch of the Lambert product-log function such that for each the value is the only root of the equation (see e.g. [1] concerning properties of the Lambert function). One may observe that (i) and (ii) as and as .

3. The constant factor in (2.11) is the best possible.

Of course, in view of (2.4), the supremum can be replaced in (2.11) by any of the other three suprema.

###### Remark 2.5.

By Remark 2.2 and (2.9), for any given the four attainment sets in (2.5) coincide with the same singleton set; that is, each of the four suprema in (2.4) is attained at the same unique maximizer.

Let us now propose a complement to part (II) of Theorem 2.1; in fact, this proposition is a corollary to Theorems 2.1 and 2.4. To state it, let denote, as usual, the support (set) of any Borel measure on ; so, is the complement of the union of all open sets with ; equivalently, is the set of all points such that for all open sets containing the point . For any r.v. , let denote the support of the measure that is the probability distribution of ; also, let

 iX:=infsuppXandsX:=supsuppX;

note that one may have and/or .

###### Proposition 2.6.

The following statements hold:

1. for any r.v. , is nondecreasing in and in ;

2. for any r.v.  with and any , is increasing in ;

3. for any r.v.  and any , is increasing in ;

4. each of the four equal suprema in (2.4) is increasing in and in .

The case of a positive Winsorization level is the one most important in applications. In accordance with Remark 2.2, this case is represented in Theorem 2.4 by . Although the corresponding upper bound on is very simple and the constant factor in it is optimal, the relative error of this bound is small only if or is small, as illustrated in (the right panel of) Figure 1, showing the graphs of the ratios of to . However, it is small values of that are of particular interest in the application of Theorem 2.4 in [13].

For , the relative errors are seen to be rather small even for as large as and as large as . Also, the relative errors appear to be monotonic in , but not in or in .

It is obvious that the upper bounds on will hold if the factor of in the numerator of the ratio in (1.1) is replaced by any r.v. that is no greater than . Thus, by Theorem 2.4 and Remark 2.2, one has

###### Corollary 2.7.

Take any , , and in . Take then any r.v.  and let be any r.v. such that with probability . Then

 EYeh(X∧w)Eeh(X∧w)⩽\SSh,w,σ

In particular, one can take here or .

###### Remark 2.8.

Suppose that , , , and are as in Corollary 2.7 and, in addition, . Then, by Corollary 2.7,

 EXehXEehX⩽\SSh,w,σ

Moreover, according to Theorem 2.4, the bound on is still exact (even under the additional condition ) — provided that ; cf. (2.10). As pointed out before, in the applications that motivated this study, the values of are typically small and thus will satisfy the condition . Also, the proof of Theorem 2.4 shows (see especially the paragraph containing formula (3.21)) that the factor in the second upper bound in (2.12) will still be optimal, even under the additional restriction — because the supremum of in is “attained” in the limit as , and, in turn, is a supremum in , which is attained at , which latter goes to as .

At this point, one is ready to present

### 2.1 Application: Optimal prior bounds on the Bayes posterior mean for exponential families

Consider a so-called exponential family of probability densities on some measurable “data” space with respect to some measure on ; that is,

 p(x|θ)=eθT(x)c(θ)q(x)

for some positive Borel-measurable function on some nonempty Borel-measurable “parameter space” , some nonnegative measurable functions and on , all , and all ; one also needs to require that be measurable in . Thus, is what is usually referred as the natural parameter. For instance, for the family of the Poisson distributions with parameter , the natural parameter is and .

Suppose that . Further, let be a Borel measure on , which will play the role of a so-called prior distribution. Note that does not have to be a probability distribution. In fact, it will be convenient here to normalize and/or the function so that . Let us exclude the trivial case when . Then, clearly, . Finally, suppose that the variance of the (renormalized by the factor ) prior distribution is known to be bounded: for some ; such an assumption appears especially reasonable in empirical Bayes settings, when accumulated prior knowledge may greatly reduce the uncertainty about the value of .

Consider now the posterior mean

 ^θ(x):=∫Θθp(x|θ)π(dθ)∫Θp(x|θ)π(dθ)=∫Θθeθtc(θ)π(dθ)∫Θθeθtc(θ)π(dθ)

given some observable “data” (with ), where . Then, by formula (1.2) and Remarks 2.8 and 2.2, one has

 ^θ(x)−m⩽\SSt,θmax−m,σ

again for and any (with ). If is small enough, then the upper bounds in (2.13) on may be much smaller than the trivial bound . Also, by Remark 2.8, the bound on is exact, and the factor in the second upper bound in (2.13) is optimal. Recall that the main concern with the Bayesian approach is uncertainty about the choice of the prior distribution. So, the bounds in (2.13) may be of help, as they rely only on the first two moments and an upper bound of the support of such a distribution.

## 3 Proofs

An approach one could try to use to establish an exact upper bound on the ratio , defined in (1.1), is to fix — besides the first two moments of — a value of the denominator, , and then maximize the numerator, , subject to these three (affine) restrictions on the measure that is the distribution of . In fact, here one has one more, less explicit restriction on this measure, which can be written as , of course meaning that the measure is a probability one. Then one can use some of well-known results such as those in [4, 6] to reduce the optimization problem to the case when consists of at most four points, corresponding to the four restrictions on the measure. Then the problem will be reduced to calculus with variables ( variables for the four support points of the measure and the four corresponding masses; one variable for the previously fixed value of ; and also the parameters , , and ). Of these variables, four can be eliminated using the four restrictions on the measure, and one of the parameters can be eliminated by rescaling. Yet, this would leave variables and a highly nonlinear function to maximize, with a number of restrictions on the variables. Such a problem appears too difficult, in terms of the amount of required calculations, especially symbolic ones.

Here this difficulty is overcome mainly by a thorough exploitation of the duality principle, the idea of which goes back to Chebyshev; see e.g. [krein59, 5, 9, 7, pin-games-transl, 11]. A general expression of this duality is the so-called minimax duality, which goes back to von Neumann [15]; see also e.g. [3, 8, 14, pin-games-transl]; in particular, a necessary and sufficient condition for minimax duality for concave-convex functions was given in [pin-games-transl]. However, more convenient in a number of problems in probability and statistics turn out to be sufficient conditions for duality given by Kemperman [7], which will be used in the present paper as well. Another significant idea in the proof of the basic Theorem 2.1 in this paper is a reduction of the maximization of the ratio of two affine functions (of the distribution of ) to the maximization of a linear combination of these two functions, with an appropriately chosen value of the constant . As the result, we show that the maximum of is attained at a r.v.  with consisting just of two (rather than four) points. This allows for further analysis, to be presented in the proof of Theorem 2.4, resulting in a rather easily computable expression for the maximum of , as well as simple and explicit, but at the same time optimal, upper bounds on this maximum.

In accordance to some of the above discussion, Theorem 2.1 is obtained as a rather easy corollary of Theorem 3.1 below. To state the latter, let us introduce, for all ,

 \SSk;h,w,σ :=sup{∫R(x−k)eh(x∧w)P(dx):P∈Pσ} (3.1) =sup{E(X−k)eh(X∧w):X∈Xσ}, \SSk;h,w,⩽σ :=sup{∫R(x−k)eh(x∧w)P(dx):P∈P⩽σ} =sup{E(X−k)eh(X∧w):X∈X⩽σ}, \SS(2)k;h,w,σ :=sup{∫R(x−k)eh(x∧w)P(dx):P∈P(2)w,σ} =sup{E(X−k)eh(X∧w):X∈X(2)w,σ}, \SS(2)k;h,w,⩽σ :=sup{∫R(x−k)eh(x∧w)P(dx):P∈P(2)w,⩽σ} =sup{E(X−k)eh(X∧w):X∈X(2)w,⩽σ}.

Similarly to (2.3), define now the attainment sets , , , and (as subsets of ) for the corresponding suprema in (3.1).

###### Theorem 3.1.

For any

1. the four suprema in (3.1) are all the same:

 \SSk;h,w,⩽σ=\SSk;h,w,σ=\SS(2)k;h,w,σ=\SS(2)k;h,w,⩽σ; (3.2)
2. each of the four suprema in (3.2) is (strictly) increasing in ;

3. each of these suprema is attained, and

 Ak;h,w,⩽σ=Ak;h,w,σ=A(2)k;h,w,σ=A(2)k;h,w,⩽σ. (3.3)
###### Proof of Theorem 2.1 (modulo Theorem 3.1).

Note first that all the four suprema in (2.4) are finite, because and, for any r.v. , one has , , and hence . If now is chosen to coincide with , then

 E(X−k)eh(X∧w)=(Eh,wX−k)Eeh(X∧w)⩽0 (3.4)

for all , so that ; in fact, , since the factor stays between and , and hence is bounded. So, by Theorem 3.1, . Therefore,

 EXeh(X∧w)⩽kEeh(X∧w) (3.5)

for all , which implies that . The reverse inequality, , is trivial. So, the second equality in (2.4) is verified. Moreover, by Theorem 3.1, the supremum is attained at some r.v. , for which inequality (3.5) must then turn into the equality , which is equivalent to . So, the suprema and are attained.

Let us now verify the monotonicity of in (which will also yield the first and third equalities in (2.4), as well as the attainment of the suprema and ). Here the reasoning is similar to that in the previous paragraph, again with , which, as was shown, equas to . Then relations (3.4) hold for all , so that . Take now any . Then, by part (II) of Theorem 3.1, . Again using the equality in (3.4) (with its two sides interchanged), one has for all , which implies — since, by what has been already proved, the supremum is attained at some .

It remains to observe that (2.5) follows from (3.3). Indeed, for any , one has if and only if for and at that ; and the same is true for each of the pairs of the subscript/superscript attributes , , in place of the pair . ∎

###### Proof of Theorem 3.1.

Introduce more notation. First, let stand for the set , equipped with the natural topology and the corresponding Borel sigma-algebra; then is compact. Next, take indeed any and define the real-valued functions on by the formulas

 aj(s):=sj−11+s2andb(s):=(s−k)eh(s∧w)1+s2

for all and (assuming the convention ), with the values of these functions on the set defined by continuity, so that for and .

Further, let stand for the set of all (nonnegative) Borel measures on . Introduce now the sets

 A:= {(∫a1dμ,∫a2dμ,∫a2dμ)∈R3:μ∈M, ∫(|a1|+|a2|+|a3|)dμ<∞}, Mσ:= {μ∈M:(∫a1dμ,∫a2dμ,∫a2dμ)=(1,0,σ2)}, C:= {(c1,c2,c3)∈R3:c1a1(s)+c2a2(s)+c3a3(s)⩾b(s) for all s∈S}, (3.6)

where the integrals are over . Also, let

 B(c1,c2,c3):= {s∈S:c1a1(s)+c2a2(s)+c3a3(s)=b(s)}; (3.7)

for , the set is sometimes referred to as the contact set — compare (3.6) and (3.7).

Observe the following.

• For all , one has , since . For the same reason, . Moreover, the strict inequality holds (on ) for some large enough (depending on and ).

• The point lies in the interior of the set . This follows because the condition implies that (i) there is a measure with and (ii) the restrictions of the functions to the three-point set are linearly independent.

(As usual, denotes the cardinality.) Therefore, by Theorems 3 and 4 in [7] (see also comments in the penultimate paragraph of [7, Section 3]), there exist and such that

 suppμ∘⊆B(c∘1,c∘2,c∘3)andsup{∫bdμ:μ∈Mσ}=∫bdμ∘. (3.8)

Moreover, the conditions and imply that . So, by Lemma 3.2 below,

 suppμ∘={−u,v} (3.9)

for some real numbers and such that ; in particular, .

Next, observe that the formula

 P(ds)=μ(ds)1+s2 (3.10)

defines a one-to-one correspondence between the set and the set of zero-mean probability measures on , defined in (2.1) (the formal meaning of (3.10) is of course that the Radon–Nikodym derivative of relative to is the function ). Moreover, for any so-related measures and , one has . Now, in view of (3.8) and (3.1),

 \SSk;h,w,σ= sup{∫R(x−k)eh(x∧w)P(dx):P∈Pσ} (3.11) = sup{∫Sbdμ:μ∈Mσ,suppμ⊆R} ⩽ sup{∫Sbdμ:μ∈Mσ} = ∫Sbdμ∘ = ∫R(x−k)eh(x∧w)P∘(dx)⩽\SS(2)k;h,w,σ⩽\SSk;h,w,σ,

where is the measure corresponding to by means of (3.10), so that, by (3.9), . Thus, both suprema and are attained at and are equal to each other, so that the second equality in (3.2) holds.

Next, let us prove the monotonicity of in , which will also yield the first and third equalities in (3.2), as well as the attainment of the suprema and . Take any and then take any .

Let now the measure with be defined by the formula , in accordance with the correspondence (3.10). It follows that

 ∫R(x−k)eh(x∧w)P(dx)=∫bdμ⩽∫(c∘1a1+c∘2a2+c∘3a3)dμ=c∘1+c∘3σ21

here,

• the first equality follows by the definition of ;

• the first inequality, by the condition ;

• the second equality, because ;

• the second inequality — because, by Lemma 3.2 below, the condition implies , while ;

• the third equality, because ;

• the last equality, by (3.11).

Now one can see that ; moreover, this latter inequality is strict, since the last inequality in (3.12) is strict and, by what was already proved, the supremum is attained. This concludes the proof of parts (I) and (II) of Theorem 3.1 — modulo Lemma 3.2.

Next, let us prove (3.3). First here, note that the obvious relation
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