# Optimal Estimators for Threshold-Based Quality Measures

Authors: Aaron Abrams; Sandy Ganzell; Henry Landau; Zeph Landau; James Pommersheim; Eric Zaslow
Journal: Journal of Probability and Statistics
 (2010)

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## Abstract

We consider a problem in parametric estimation: givenn samples from an unknown distribution, we want to estimate which distribution, from a given one-parameter family, produced the data. Following Schulman and Vazirani (2005), we evaluate an estimator in terms
of the chance of being within a specified tolerance of the correct answer,
in the worst case. We provide optimal estimators for several families of
distributions on ℝ. We prove that for distributions on a compact space,
there is always an optimal estimator that is translation invariant, and
we conjecture that this conclusion also holds for any distribution on ℝ.
By contrast, we give an example showing that, it does not hold for a certain
distribution on an infinite tree.


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## Body

## 1. Introduction

Estimating probability distribution functions is a central problem in statistics. Specifically, beginning with an unknown probability distribution on an underlying spaceX, one wants to be able to do two things: first, given some empirical data sampled from the unknown probability distribution, estimate which one of a presumed set of possible distributions produced the data; and second, obtain bounds on how good this estimate is. For example, the maximum likelihood estimator selects the distribution that maximizes the probability (among those under consideration) of producing the observed data. Depending on what properties of the estimator one is trying to evaluate, this may or may not be optimal. An extensive literature, dating back to the early 20th century, addresses problems of this sort; see for example [1–6].In this paper, we consider one such problem. We presume that samples are coming from an unknown “translate” of a fixed known distribution. The challenge is to guess the translation parameter. More precisely, we are given a distributionμ on a space X, along with an action of a group G on X, which defines a set of translated distributions μθ as follows:
(1.1)μθ(A)=μ({x:θx∈A}),
for A⊂X.Thus, in this context anestimator is a (measurable) function e:Xn→G; the input x=(x1,…,xn) is the list of samples, and the output e(x) is the estimate of θ, the translation parameter. For the majority of the paper, we will study the case of G=ℝ acting by translations (changes in location) on X=ℝ, and the group action will be written additively, as seen beginning from Section 2.We are interested in finding good estimators; thus we need a way of measuring an estimator's quality. A common way to do this is to measure themean squared error, in which case an optimal estimator minimizes this error. Various results are known in this case; for instance, the maximum likelihood estimator (which agrees with the sample mean estimator),
(1.2)e(x)=(1n∑xi)-E(μ),
minimizes the mean squared error if μ is a Gaussian distribution on ℝ.In this paper, we investigate a different and natural measure of quality whereby we consider an estimator to succeed or fail according to whether or not its estimate is within a certain thresholdδ>0 of the correct answer. We then define the quality of the estimator to be the chance of success in the worst case. This notion was introduced in [7] to analyze certain approximation algorithms in computer science. Precisely, the δ-quality of e is defined as
(1.3)Qδ(e)=inf⁡θQδθ(e)=inf⁡θPr{d(e(x),θ)<δ:xiarechosenfromμθ}=inf⁡θμθn({x:d(e(x),θ)<δ}),
where d is a metric on X and μθn is the product measure μθ×⋯×μθ on Xn. (In the case of perverse measures, μ, we must consider the probability as the sup of the intersection of the set {d(e(x),θ)<δ} with all measurable sets. We will ignore this caveat throughout. Indeed, we primarily focus on absolutely continuous measures (as [8, 9] have done, e.g.) and purely atomic measures.) Note that, depending on context, it is sometimes advantageous to define the quality using a closed interval rather than an open one; for example, in the discrete case we could then interpret Q0θ(e) as the probability that e is exactly equal to θ. We write Q(e) when the value of δ is unambiguous. For fixed δ, an (n-sample) estimator e is optimal if Qδ(e)≥Qδ(e′) for all (n-sample) estimators e′. Many authors use the term minimax to describe optimal estimators. Note that much of the literature on this subject uses the notion of loss functions and the associated risk R=1-Q; our point of view is equivalent but more convenient for our purposes.Motivated initially by analyzing an approximate algorithm for determining the average matching size in a graph, Schulman and Vazirani [7] introduce the stronger notion of a majorizing estimator, which is optimal (by the above definition) simultaneously for all δ>0. This was previously studied by Pitman [5], who considered several different optimality criteria and, for each one, constructed optimal “shift-invariant” estimators (defined below). Schulman and Vazirani focus on the Gaussian distribution and prove that the mean estimator is the unique majorizing estimator in this case.In the first part of this paper, we investigate the optimal estimators for several different classes of distributions onℝ. We conjecture that there is always an optimal estimator e that is shift invariant, that is, e satisfies
(1.4)e(x1+c,…,xn+c)=e(x1,…,xn)+c,
for all c,xi∈ℝ. These estimators are typically easier to analyze than general estimators because the quality is the same everywhere, that is, Q(e)=Qθ(e) for every θ. Conditions under which invariant minimax estimators can be obtained have been studied, for example, in [10–12]. Indeed, some of our existence results follow from the quite general Hunt-Stein theorem [12, Theorem 9.5.5], but we give constructions that are very natural and explicit. We obtain general bounds on the quality of shift invariant estimators (Section 2) and general estimators (Section 3), and then we apply these bounds to several families of distributions (Section 4). In each case, we are able to construct an optimal estimator that is shift-invariant. These examples include the Gaussian and exponential distributions, among others.These results motivate our study of shift-invariant estimators on other spaces; these are estimators that are equivariant with respect to the induced diagonal action ofG on either the left or the right on Xn. That is, a left-invariant estimator satisfies
(1.5)e(gx)=ge(x),
where (1.6)g(x1,…,xn)=(gx1,…,gxn).
Right invariance is defined similarly.In Section5, we show that on a compact space X, if e is an estimator for μ, then there is always a shift-invariant estimator with quality at least as high as that of e. The idea is to construct a shift-invariant estimator s as an average of the translates of e; this is essentially a simple proof of a special case of the Hunt-Stein theorem. As there is no invariant probability measure on ℝ, the proof does not extend to the real case.Finally, in the last section, we give an example due to Schulman which shows that (on noncompact spaces) there may be no shift-invariant estimator that is optimal. It continues to be an interesting problem to determine conditions under which one can guarantee the existence of a shift-invariant estimator that is optimal.


## 2. The Real Case: Shift-Invariant Estimators

LetG=X=ℝ, and consider the action of G on X by translations. Because much of this paper is concerned with this context, we spell out once more the parameters of the problem. We assume that δ>0 is fixed throughout. We are given a probability distribution μ on ℝ, and we are to guess which distribution μθ produced a given collection x=(x1,…,xn) of data, where μθ(A)=μ({x:x+θ∈A}). An estimator is a function e:ℝn→ℝ, and we want to maximize its quality, which is given by
(2.1)Q(e)=inf⁡θQθ(e)=inf⁡θPr{|e(x)-θ|<δ:xiarechosenfromμθ}=inf⁡θμθn({x:|e(x)-θ|<δ}).First we present some notation. We will write the group action additively and likewise the induced diagonal action ofG on ℝn; in other words, if x=(x1,…,xn)∈ℝn and a∈ℝ, then x+a denotes the point (x1+a,…,xn+a)∈ℝn. Similarly, if Y⊂ℝn and A⊂ℝ, then Y+A={y+a:y∈Y,a∈A}. We also use the “interval notation” (x+a,x+b) for the set {x+t:a<t<b}; this is a segment of length (b-a)n in ℝn if a and b are finite. If f:ℝn→ℝ is any function, and θ∈ℝ, define fθ(x)=f(x-θ). If f:ℝ→ℝ, then define f[n]:ℝn→ℝ by f[n](x)=f(x1)f(x2)⋯f(xn).We now establish our upper bound on the quality of shift-invariant estimators. Note that a shift invariant estimator has the property thate(x-e(x))=0. Also note that a shift-invariant estimator is determined uniquely by its values on the coordinate hyperplane
(2.2)X0={x∈Rn:x1=0},
and that a shift-invariant estimator exists for any choice of such values on X0. In addition, for e shift-invariant,
(2.3)μθn({x:|e(x)-θ|<δ})=μθn({x:|e(x-θ)|<δ})=μn({x:|e(x)|<δ}),
so the quality can be ascertained by setting θ=0.Definition 2.1.
For fixedn, let 𝒜 denote the collection of all Borel subsets A of the form
(2.4)A=⋃x∈X0x+(f(x)-δ,f(x)+δ),
where f:X0→ℝ is a Borel function. For fixed μ and n, define
(2.5)Sμ,n=Sμ,n(δ)=sup⁡A∈A{μn(A)}.Theorem 2.2.
Letμ and n be given, then any shift-invariant n-sample estimator e satisfies Q(e)≤Sμ,n.Proof.
Due to the observation above, it suffices to bound the quality ofe at θ=0. But this quality is just μn(A), where A=e-1((-δ,δ)). Note that
(2.6)A=⋃x∈X0(x-e(x)-δ,x-e(x)+δ),
and in particular A∈𝒜. Thus, the quality of e is at most Sμ,n.Theorem 2.3.
Letμ and n be given. If the sup in Definition 2.1 is achieved, then there is a shift-invariant n-sample estimator with quality Sμ,n. For any ϵ>0, there is a shift-invariant n-sample estimator e with quality greater than Sμ,n-ϵ.Proof.
For a givenA∈𝒜, let f be the corresponding Borel function (see Definition 2.1). Define the estimator e to be -f(x) on X0 and then extend to all of ℝn to make it shift invariant. Note that A⊂e-1((-δ,δ)), so Q(e)≥μn(A). The theorem now follows from the definition of Sμ,n.


## 3. The Real Case: General Estimators

In this section, we obtain a general upper bound on the quality of randomized estimators, still in the caseG=X=ℝ. The arguments are similar to those of the previous section.Againδ is fixed throughout. A randomized estimator is a function Xn×Ω→ℝ, where Ω is a probability space of estimators; thus for fixed ω∈Ω, e=ẽ(·,ω) is an estimator. The δ-quality of a randomized estimator ẽ is
(3.1)Q(ẽ)=inf⁡θQθ(ẽ),
where
(3.2)Qθ(ẽ)=Eμθn{x∣|ẽ(x)-θ|<δ}.Definition 3.1.
For fixedn, let
(3.3)B={B⊂Rn:B∩(B+2kδ)=∅forallnonzerointegersk}.
For fixed μ and n, define
(3.4)Tμ,n=Tμ,n(δ)=sup⁡B∈B{μn(B)}.Comparing with Definition2.1, we observe that 𝒜⊂ℬ and hence Sμ,n≤Tμ,n.Theorem 3.2.
Letμ and n be given. Any n-sample randomized estimator  satisfies Q(ẽ)≤Tμ,n.Proof.
We will give a complete proof in the case thatμ is defined by a density function f and then indicate the modifications required for the general case. The difference is purely technical; the ideas are the same.
Consider first a nonrandomized estimatore. The performance of e at θ is μθn(e-1((θ-δ,θ+δ))). To simplify notation, we will let Ie,θ denote the set e-1((θ-δ,θ+δ)), and we will suppress the subscript e when no ambiguity exists. Since Q(e) is an infimum, the average performance of e at the k points θi=2δi (i=1,2,…,k) is at least Q(e)as follow:
(3.5)Q(e)≤1k∑i=1kμθin(Iθi).
Now, we use the density functionf. Recall that fθ(x)=f(x-θ). Define f̃ on ℝn by
(3.6)f̃(x)=max⁡i{fθi[n](x)}=max⁡i{fθi(x1)fθi(x2)⋯fθi(xn)}.
Since theIθi are disjoint, we now have
(3.7)Q(e)≤1k∑i=1k∫Iθifθi[n](x)dx≤1k∑i=1k∫Iθif̃(x)dx≤1k∫∪Iθif̃(x)dx≤1k∫Rnf̃(x)dx=1k∫{x1≤0}f̃(x)dx+1k∫{0<x1<2δk}f̃(x)dx+1k∫{x1≥2δk}f̃(x)dx.
We will bound the middle term byTμ,n and show that the first and last terms go to zero (independently of e) as k gets large. The bound on the middle term is a consequence of the following claim.
Claim 3.3.
For anya∈ℝ,
(3.8)∫{a≤x1≤a+2δ}f̃(x)dx≤Tμ,n.
To prove the claim, set Z={x∈ℝn:a≤x1<a+2δ}, and set Zi={x∈Z:iasthesmallestindexsuchthatf̃(x)=fθi[n](x)}. Thus, the Zi are disjoint and cover Z. Now,
(3.9)∫{a≤x1≤a+2δ}f̃(x)dx=∑i∫Zif̃(x)dx=∑i∫Zifθi[n](x)dx=∑i∫Zi-θif[n](x)dx=∫∪(Zi-θi)f[n](x)dx≤Tμ,n.
The last equality follows from the fact that the Zi-θi are disjoint (recall that θi=2δi), and the final step follows because the set B=∪(Zi-θi) is in  ℬ. This proves the claim.
Next, we show that(1/k)∫{x1≤0}f̃(x)dx approaches zero as k grows. Recall that θi=2δi, and set zi=∫{x1≤0}f2δi[n](x)dx=∫{x1≤-2δi}f[n](x)dx. The function f is a probability density function, so f is nonnegative and has total integral 1. The Dominated Convergence Theorem then implies that the sequence {zi} is decreasing to 0. Bounding f̃(x) by ∑i=1kfθi[n](x), we have
(3.10)1k∫{x1≤0}f̃(x)dx≤1k∫{x1≤0}∑i=1kfθi[n](x)dx=1k∑i=1kzi⟶0.
A similar argument shows that the term(1/k)∫{x1≥2δk}f̃(x)dx  goes to 0 as k grows. Since (3.7) holds for all k, we have Q(e)≤Tμ,n for any estimator e.
We have shown that for anyϵ>0, we can find k depending on ϵ and f, such that the average performance of an arbitrary estimator e on the k points θi=2δi is bounded above by Tμ,n+2ϵ. Now, for a randomized estimator ẽ, the quality is bounded above by its average performance on the same k points, and that performance can be no better than the best estimator's performance. We conclude that Q(ẽ)≤Tμ,n+2ϵ, and the theorem follows.
The proof is now complete in the case thatμ has a density f. In general, the argument requires minor technical adjustments. The first step that requires modification is the definition of the function f̃. Let
(3.11)ν=∑i=1kμθin,gi=dμθindμ̃,f̃=max⁡(g1,…gk).
Then f̃·ν=μ̃, and we work with μ̃ rather than f̃. From here, one defines the Zi accordingly, and the remainder of the argument goes through with corresponding changes.


## 4. The Real Case: Examples

We have obtained bounds on quality for general estimators and for shift-invariant ones. In this section, we give several situations where the bounds coincide, and therefore the optimal shift-invariant estimators constructed in Section2 are in fact optimal estimators, as promised by the Hunt-Stein theorem. These examples include many familiar distributions, and they provide evidence for the following conjecture.Conjecture 4.1.
Letμ be a distribution on  ℝ. Then there is an optimal estimator for  μ that is shift invariant.


### 4.1. Warmup: Unimodal Densities—One Sample

Our first class of examples generalizes Gaussian distributions and many others. The argument works only with one sample, but we will refine it in Section4.2. Note that the optimal estimator in this case is the maximum likelihood estimator.We say that a density function isunimodal if for all y, {x:f(x)≥y} is convex.Example 4.2.
Letμ be defined by a unimodal density function f. Then there is a shift-invariant one-sample estimator that is optimal.Proof.
We first show thatTμ,1=Sμ,1. It follows from the definition of ℬ that any set B∈ℬ must have Lebesgue measure less than or equal to 2δ. Since f is unimodal, ∫Bf(x)dx is maximized by concentrating B around the peak of f; thus the best B will be an interval that includes the peak of f. But any interval in ℬ is contained in 𝒜 and thus Tμ,1≤Sμ,1. Since Sμ,1≤Tμ,1 always, we have Tμ,1=Sμ,1.
Now, recalling thatSμ,1 and Tμ,1 are defined as suprema, we observe that the above argument shows that if one is achieved, then so is the other. Therefore, the result follows from Theorems 2.3 and 3.2.


### 4.2. A Sufficient Condition

The next class is more restrictive than the preceding, but with the stronger hypothesis we get a result for arbitraryn. Any Gaussian distribution continues to satisfy the hypothesis.Example 4.3.
Letμ be a distribution defined by a density function of the form f=eλ(x) with λ′(x) continuous and decreasing. Then for any n, there is a shift-invariant n-sample estimator that is optimal.Proof.
For any fixedx∈X0, we define a function hx:ℝ→ℝ by
(4.1)hx(t)=f[n](x+t)=eλ(x1+t)+⋯+λ(xn+t).
Since
(4.2)hx′(t)=eλ(x1+t)+⋯+λ(xn+t)(λ′(x1+t)+⋯+λ′(xn+t))
and λ′ is decreasing, it is clear that for each x, hx′(t)=0 for at most one value of t. Since hx(t)→0 as t→±∞, it follows that for any  x, hx is a unimodal function of t.
Now the argument is similar to Example4.2. We will show that Tμ,n=Sμ,n. Since f[n] restricted to each orbit  x+ℝ is unimodal as we have just shown, a set B∈ℬ on which the integral of f[n] is maximized is obtained by choosing an interval from each orbit. To make this more precise, for each x∈X0, let tx be the center of the length 2δ interval I=(tx-δ,tx+δ) that maximizes ∫Ihxdt. Then let
(4.3)A=⋃x∈X0(x+tx-δ,x+tx+δ).
Now A∈𝒜, and moreover, μn(A)≥μn(B) for any B∈ℬ because ∫A∩(x+ℝ)f[n]≥∫B∩(x+ℝ)f[n] for each  x∈X0.
Thus,supB∈ℬ{μn(B)} is achieved by B=A∈𝒜, and it follows that Sμ,n=Tμ,n and that the best shift-invariant estimator is optimal.


### 4.3. Monotonic Distributions onℝ+

The third class of examples generalizes the exponential distribution, defined by the densityf(x)=λe-λx for x≥0 and f(x)=0 for x<0. The optimal estimator in this case       is not the maximum likelihood estimator. (Note that in a typical estimation problem involving a family of exponential distributions, one is trying to estimate the "scale" parameter λ rather than the "location" θ).Example 4.4.
Letμ be defined by a density function f that is decreasing for x≥0 and identically zero for x<0. Then for any n, there is a shift-invariant n-sample estimator that is optimal.Proof.
We construct the estimator as follows: forx∈ℝn, define e(x)=min{x1,…,xn}-δ. Note that this is shift invariant; therefore Q(e) can be computed at θ=0. That is, it suffices to show that Q0(e)=Tμ,n.
LetB={x∈ℝn:0≤min{x1,…,xn}<2δ}. Note that B=e-1([-δ,δ)), and so μn(B) is the quality of e. Note also that B∈ℬ (in fact B∈𝒜), so certainly μn(B)≤Tμ,n. We will show that any C∈ℬ can be modified to a set C′∈ℬ such that C′⊂B and μn(C)≤μn(C′). It then follows that Tμ,n≤μn(B), and this will complete the proof.
So, letC∈ℬ, and define C′={x∈B:x+2kδ∈Cforsomek∈Z}. Note thatk is determined uniquely by x. Now C′⊂B is in ℬ, and by our hypotheses on f, if x∈B, then f[n](x)≥f[n](x+2kδ) for every integer k. Therefore, μn(C′)-μn(C)=∫C′[f[n](x)-f[n](x+2kδ)]dμn≥0.


### 4.4. Discrete Distributions

Here, we discuss purely atomic distributions on finite sets of points. Because we are only trying to guess withinδ of the correct value of θ, there are many possible choices of estimators with the same quality. Among the optimal ones is the maximum likelihood estimator.Example 4.5.
Letμ be a distribution on a finite set of points Z. There is a shift-invariant one-sample estimator that is optimal. Furthermore, if all of the pairwise distances between points of Z are distinct, then for every n there is a shift-invariant n-sample estimator that is optimal.Proof.
We first treat the casen=1. Since μ is discrete, the supremum defining Sμ,1 is attained; therefore, by Theorems 2.3 and 3.2, it suffices to show that every estimator has quality at most Sμ,1.
LetZ={z1,…,zr}, and for any z∈Z, let pz denote the mass at z. For a finite set, we use |·| to denote the cardinality. Suppose that e is any estimator.
Lemma 4.6.
LetY be any finite subset of ℝ, then
(4.4)Q(e)≤Sμ,1|Y+Z||Y|,
where  Y+Z denotes the set {y+z∣y∈Y,z∈Z}.Proof of Lemma4.6.
To prove the lemma, we estimate the average qualityQθ(e) over θ∈Y. We have
(4.5)∑θ∈YQθ(e)=∑θ∈Yμ(e-1(θ-δ,θ+δ)-θ)=∑θ∈Y∑zpz,
with the inner part of the last sum taken over those z∈Z that lie in e-1(θ-δ,θ+δ)-θ. Using x to denote θ+z, this condition becomes e(x)∈(θ-δ,θ+δ), and the right hand side above may be rewritten as
(4.6)∑θ∈Y∑zpz=∑θ∈Y∑xpx-θ=∑x∈Y+Z∑θpx-θ,
with the inner sum now taken over all θ with e(x)∈(θ-δ,θ+δ). This latter condition implies that z is within δ of x-e(x). But by definition, Sμ,1 is the maximum measure of any interval of length 2δ. Hence, for any fixed x∈Y+Z, the inner sum is at most Sμ,1, and the entire sum is thus bounded above by Sμ,1·|Y+Z|. Dividing by |Y| gives a bound for the average quality over Y, and since Q(e) is defined as an infimum, the lemma follows.
We now apply the lemma to complete the example. Letk be a natural number, and let
(4.7)Yk={h1z1+⋯+hrzr:hi∈Z,0≤hi<k}.
Note that Yk⊆Yk+1 and |Yk|≤kr. It follows that for any ϵ>0, there exists k such that |Yk+1|/|Yk|<1+ϵ, for otherwise, |Yk| would grow at least exponentially in k. Using the fact that Yk+Z⊆Yk+1, the lemma applied to Yk implies that Q(e)≤Sμ,1(1+ϵ). Therefore,Q(e)≤Sμ,1, and this finishes the case n=1.
Lastly, we consider an arbitraryn. If we are given samples x1,…,xn and if any xi≠xj for some i and j, then by our hypothesis the shift θ is uniquely determined. Thus, we may assume that any optimal estimator picks the right θ in these cases, and the only question is what value the estimator returns if all the samples are identical. The above analysis of the one sample case can be used to show that any optimal shift-invariant estimator is optimal.


## 4.1. Warmup: Unimodal Densities—One Sample

Our first class of examples generalizes Gaussian distributions and many others. The argument works only with one sample, but we will refine it in Section4.2. Note that the optimal estimator in this case is the maximum likelihood estimator.We say that a density function isunimodal if for all y, {x:f(x)≥y} is convex.Example 4.2.
Letμ be defined by a unimodal density function f. Then there is a shift-invariant one-sample estimator that is optimal.Proof.
We first show thatTμ,1=Sμ,1. It follows from the definition of ℬ that any set B∈ℬ must have Lebesgue measure less than or equal to 2δ. Since f is unimodal, ∫Bf(x)dx is maximized by concentrating B around the peak of f; thus the best B will be an interval that includes the peak of f. But any interval in ℬ is contained in 𝒜 and thus Tμ,1≤Sμ,1. Since Sμ,1≤Tμ,1 always, we have Tμ,1=Sμ,1.
Now, recalling thatSμ,1 and Tμ,1 are defined as suprema, we observe that the above argument shows that if one is achieved, then so is the other. Therefore, the result follows from Theorems 2.3 and 3.2.


## 4.2. A Sufficient Condition

The next class is more restrictive than the preceding, but with the stronger hypothesis we get a result for arbitraryn. Any Gaussian distribution continues to satisfy the hypothesis.Example 4.3.
Letμ be a distribution defined by a density function of the form f=eλ(x) with λ′(x) continuous and decreasing. Then for any n, there is a shift-invariant n-sample estimator that is optimal.Proof.
For any fixedx∈X0, we define a function hx:ℝ→ℝ by
(4.1)hx(t)=f[n](x+t)=eλ(x1+t)+⋯+λ(xn+t).
Since
(4.2)hx′(t)=eλ(x1+t)+⋯+λ(xn+t)(λ′(x1+t)+⋯+λ′(xn+t))
and λ′ is decreasing, it is clear that for each x, hx′(t)=0 for at most one value of t. Since hx(t)→0 as t→±∞, it follows that for any  x, hx is a unimodal function of t.
Now the argument is similar to Example4.2. We will show that Tμ,n=Sμ,n. Since f[n] restricted to each orbit  x+ℝ is unimodal as we have just shown, a set B∈ℬ on which the integral of f[n] is maximized is obtained by choosing an interval from each orbit. To make this more precise, for each x∈X0, let tx be the center of the length 2δ interval I=(tx-δ,tx+δ) that maximizes ∫Ihxdt. Then let
(4.3)A=⋃x∈X0(x+tx-δ,x+tx+δ).
Now A∈𝒜, and moreover, μn(A)≥μn(B) for any B∈ℬ because ∫A∩(x+ℝ)f[n]≥∫B∩(x+ℝ)f[n] for each  x∈X0.
Thus,supB∈ℬ{μn(B)} is achieved by B=A∈𝒜, and it follows that Sμ,n=Tμ,n and that the best shift-invariant estimator is optimal.


## 4.3. Monotonic Distributions onℝ+

The third class of examples generalizes the exponential distribution, defined by the densityf(x)=λe-λx for x≥0 and f(x)=0 for x<0. The optimal estimator in this case       is not the maximum likelihood estimator. (Note that in a typical estimation problem involving a family of exponential distributions, one is trying to estimate the "scale" parameter λ rather than the "location" θ).Example 4.4.
Letμ be defined by a density function f that is decreasing for x≥0 and identically zero for x<0. Then for any n, there is a shift-invariant n-sample estimator that is optimal.Proof.
We construct the estimator as follows: forx∈ℝn, define e(x)=min{x1,…,xn}-δ. Note that this is shift invariant; therefore Q(e) can be computed at θ=0. That is, it suffices to show that Q0(e)=Tμ,n.
LetB={x∈ℝn:0≤min{x1,…,xn}<2δ}. Note that B=e-1([-δ,δ)), and so μn(B) is the quality of e. Note also that B∈ℬ (in fact B∈𝒜), so certainly μn(B)≤Tμ,n. We will show that any C∈ℬ can be modified to a set C′∈ℬ such that C′⊂B and μn(C)≤μn(C′). It then follows that Tμ,n≤μn(B), and this will complete the proof.
So, letC∈ℬ, and define C′={x∈B:x+2kδ∈Cforsomek∈Z}. Note thatk is determined uniquely by x. Now C′⊂B is in ℬ, and by our hypotheses on f, if x∈B, then f[n](x)≥f[n](x+2kδ) for every integer k. Therefore, μn(C′)-μn(C)=∫C′[f[n](x)-f[n](x+2kδ)]dμn≥0.


## 4.4. Discrete Distributions

Here, we discuss purely atomic distributions on finite sets of points. Because we are only trying to guess withinδ of the correct value of θ, there are many possible choices of estimators with the same quality. Among the optimal ones is the maximum likelihood estimator.Example 4.5.
Letμ be a distribution on a finite set of points Z. There is a shift-invariant one-sample estimator that is optimal. Furthermore, if all of the pairwise distances between points of Z are distinct, then for every n there is a shift-invariant n-sample estimator that is optimal.Proof.
We first treat the casen=1. Since μ is discrete, the supremum defining Sμ,1 is attained; therefore, by Theorems 2.3 and 3.2, it suffices to show that every estimator has quality at most Sμ,1.
LetZ={z1,…,zr}, and for any z∈Z, let pz denote the mass at z. For a finite set, we use |·| to denote the cardinality. Suppose that e is any estimator.
Lemma 4.6.
LetY be any finite subset of ℝ, then
(4.4)Q(e)≤Sμ,1|Y+Z||Y|,
where  Y+Z denotes the set {y+z∣y∈Y,z∈Z}.Proof of Lemma4.6.
To prove the lemma, we estimate the average qualityQθ(e) over θ∈Y. We have
(4.5)∑θ∈YQθ(e)=∑θ∈Yμ(e-1(θ-δ,θ+δ)-θ)=∑θ∈Y∑zpz,
with the inner part of the last sum taken over those z∈Z that lie in e-1(θ-δ,θ+δ)-θ. Using x to denote θ+z, this condition becomes e(x)∈(θ-δ,θ+δ), and the right hand side above may be rewritten as
(4.6)∑θ∈Y∑zpz=∑θ∈Y∑xpx-θ=∑x∈Y+Z∑θpx-θ,
with the inner sum now taken over all θ with e(x)∈(θ-δ,θ+δ). This latter condition implies that z is within δ of x-e(x). But by definition, Sμ,1 is the maximum measure of any interval of length 2δ. Hence, for any fixed x∈Y+Z, the inner sum is at most Sμ,1, and the entire sum is thus bounded above by Sμ,1·|Y+Z|. Dividing by |Y| gives a bound for the average quality over Y, and since Q(e) is defined as an infimum, the lemma follows.
We now apply the lemma to complete the example. Letk be a natural number, and let
(4.7)Yk={h1z1+⋯+hrzr:hi∈Z,0≤hi<k}.
Note that Yk⊆Yk+1 and |Yk|≤kr. It follows that for any ϵ>0, there exists k such that |Yk+1|/|Yk|<1+ϵ, for otherwise, |Yk| would grow at least exponentially in k. Using the fact that Yk+Z⊆Yk+1, the lemma applied to Yk implies that Q(e)≤Sμ,1(1+ϵ). Therefore,Q(e)≤Sμ,1, and this finishes the case n=1.
Lastly, we consider an arbitraryn. If we are given samples x1,…,xn and if any xi≠xj for some i and j, then by our hypothesis the shift θ is uniquely determined. Thus, we may assume that any optimal estimator picks the right θ in these cases, and the only question is what value the estimator returns if all the samples are identical. The above analysis of the one sample case can be used to show that any optimal shift-invariant estimator is optimal.


## 5. The Compact Case

So far, we have dealt only with distributions onℝ, where the shift parameter is a translation. In every specific case that we have analyzed, we have found a shift-invariant estimator among the optimal estimators. In this section, we prove that if G=X is a compact group acting on itself by (left) multiplication, then at least for measures defined by density functions, there is always a shift-invariant estimator as good as any given estimator. In Section 6, we show that the compactness hypothesis cannot be eliminated entirely; we do not know how much it can be weakened, if at all.We will continue to use bothG and X as notations, in order to emphasize the distinction between the two roles played by this object. Eaton [2] discusses estimators in a context in which the group G acts on both the parameter space Θ and the sample space X. In his work, the sample space X is an arbitrary homogeneous space (i.e., a space with a transitive G-action). In this generality, shift-invariant estimators may not exist, since there may not even exist a function from Xn to Θ that preserves the G action. For this reason, we choose to identify the sample space with the group G.As usualG acts diagonally on Xn; we denote the orbit space by Y. An element y of Y is an equivalence class y=[x]={(gx1,…,gxn):g∈G}, which we identify with G via (gx1,…,gxn)↦gx1. For x=(x1,…,xn)∈X, we denote by x0 the point x1-1x; thus, x0 is in the orbit of x and has first coordinate 1. The set X0={x0:x∈X}⊂Xn is naturally identified with Y.EquipG=X with a left- and right-invariant metric d, meaning that d(gx,gy)=d(x,y)=d(xg,yg) for all x,y,g∈G. Let B(g)=Bδ(g) denote the ball of radius  δ around g∈G. If S is a subset of a measure space (T,α), then we denote the measure of S variously by α(S), ∫Sdα, or ∫TχSdα. (The notation χS refers to the characteristic function of the set S.)Fixδ and n, and let μ be an arbitrary measure on X. The following technical lemma says that to evaluate an integral over Xn, we can integrate over each G-orbit and then integrate the result over the orbit space.Lemma 5.1.
There exist measuresν on Y and αy on each orbit y such that for any function F on Xn,
(5.1)∫XnF(x)dμn=∫Y∫GF(gx0)dαydν.Proof.
Letφ:Xn→Xn be defined by φ(x)=(x1,x1-1x2,…,x1-1xn) and let π be the image of μn with respect to φ, that is, π(A)=μn{φ∈A} for all Borel A⊂Xn, then
(5.2)∫F̃(φ(x))dμn(x)=∫F̃(y)dπ(y)
for nonnegative Borel functions F̃ on Xn. Taking F̃(y)=F(y1,y1y2,…,y1yn) yields
(5.3)∫F(x)dμn(x)=∫F(y1,y1y2,…,y1yn)dπ(y)=∫dν̃(y2,…,yn)∫F(y1,y1y2,…,y1yn)dα̃(y2,…,yn)(y1),
for some measures ν̃ and α̃. The right-hand side can then be written as
(5.4)∫X0dν(x0)∫GF(gx0)dαx0(g),
where ν is an image of ν̃ with respect to the function (y2,…,yn)↦(1,y2,…,yn) and αx0=α̃(y2,…,yn) for x0=(1,y2,…,yn), completing the proof.Lemma 5.2.
Ifs is a shift-invariant (n-sample) estimator, then
(5.5)Q(s)=∫Y∫GχB(s(x0)-1)dαx0dν.Proof.
Sinces is shift invariant, its quality can be computed at the identity. Thus, Q(s)=Q1(s)=μn(s-1(B(1)))=∫Xnχs-1(B(1))dμn. By Lemma 5.1, this integral can be decomposed as
(5.6)∫Y∫Gχs-1(B(1))(gx0)dαx0dν.
Now, note that gx0∈s-1(B(1)) if and only if gs(x0)∈B(1) if and only if g∈B(s(x0)-1). Thus, the integral above is the same as the one in the statement of the lemma, and we are done.We are now ready to prove the result. Note that we do not prove that optimal estimators exist—only that if they exist, then one of them is shift invariant.Theorem 5.3.
LetG=X be a compact group, let δ and n be given, and let μ be defined by a density function. If e:Xn→G is any estimator, then there exists a shift-invariant estimator s with Q(s)≥Q(e).Proof.
Lete:Xn→G be any estimator. For each group element γ∈G, we define a shift-invariant estimator sγ that agrees with e on the coset γX0 as follows:
(5.7)sγ(g,gx2,…,gxn)=gγ-1e(γ,γx2,…,γxn).
We will show that there exists γ such that Q(sγ)≥Q(e).
Denote byρ the invariant (Haar) measure on G. Since Q(e) is defined as an infimum, we have
(5.8)Q(e)≤∫θ∈GQθ(e)dρ=∫θ∈G∫Xnχθ-1e-1(B(θ))(x)dμndρ=∫Xn∫θ∈Gχθ-1e-1(B(θ))(x)dρdμn=∫Y∫G∫θ∈Gχθ-1e-1(B(θ))(gx0)dρdαx0dν,
where the last equality comes from Lemma 5.1. The condition that gx0∈θ-1e-1(B(θ)) is equivalent to d(e(θgx0),θ)<δ. Now, we make the substitution γ=θg. Thus, θ=γg-1, and the condition becomes d(e(γx0),γg-1)<δ, or, by invariance of the metric, d(γ-1e(γx0),g-1)<δ. This says that g-1∈B(γ-1e(γx0)), or equivalently, g∈B(e(γx0)-1γ).
This allows us to rewrite the triple integral (5.8), using the measure-preserving transformation θ↦γ=θg, as
(5.9)∫Y∫G∫θ∈Gχθ-1e-1(B(θ))dρdαx0dν=∫Y∫G∫γ∈GχB(e(γx0)-1γ)dρdαx0dν=∫γ∈G(∫Y∫GχB(e(γx0)-1γ)dαx0dν)dρ.
Now, comparing with Lemma 5.2, we see that the inner integral above is exactly the quality of the shift-invariant estimator sγ.
We therefore have(5.10)Q(e)≤∫γQ(sγ)dρ,
or in other words, the average quality of the shift-invariant estimators {sγ} is at least Q(e). Therefore, at least one of the sγ satisfies Q(sγ)≥Q(e).


## 6. A Non-Shift-Invariant Example

In the following example, suggested by  Schulman, the optimal shift-invariant estimator is not optimal. This provides an interesting complement to Conjecture4.1. Lehmann and Casella [11, Section 5.3] also give examples of this phenomenon.LetX be the infinite trivalent tree, which we view as the Cayley graph of the group G=(Z/2Z)*(Z/2Z)*(Z/2Z)=〈a,b,c∣a2=b2=c2=1〉. In words, G is a discrete group generated by three elements a,b, and c, each of order two and with no other relations. Each nonidentity element of G can be written uniquely as a finite word in the letters a, b, and c with no letter appearing twice in a row; we refer to such a word as the reduced form of the group element. (We write 1 for the identity element of G.) Multiplication in the group is performed by concatenating words and then canceling any repeated letters in pairs. Evidently, G is infinite. The Cayley graph X is a graph with one vertex labeled by each group element of G, and with an edge joining vertices w and x if and only if w=xa, w=xb, or w=xc. Note that this relation is symmetric, since a, b, and c each has order 2. Each vertex of X has valence 3, and X is connected and contains no circuits, that is, it is a tree. Finally, X becomes a metric space by declaring each edge to have length one.Because of how we defined the edges ofX, G acts naturally on the left of X: given g∈G, the map g:X→X defined by g(x)=gx is an isometry of X. So if δ>0 is given, μ is a probability distribution, θ∈G, and e is an estimator, then the shift μθ and the quality Qδ(e) are defined as usual by (1.1) and (1.3).We are ready to present the example. Suppose that0<d<1 is fixed, and let μ be the probability distribution with atoms of weight 1/3 at the three vertices a,b,andc. Thus for θ∈G, the distribution μθ has atoms of weight 1/3 at the three neighbors θa,θb,andθc of the vertex θ in X.Example 6.1.
There is an optimal one-sample estimator with quality 2/3, but the quality of any shift-invariant one-sample estimator is at most 1/3.Proof.
Consider the one-sample estimatore that truncates the last letter of the sample (unless the sample is the identity, in which case we arbitrarily assign the value a). That is, for a vertex x of X,
(6.1)e(x)={wifx=wlisreduced,l=a,b,orc,aifx=1.
Geometrically, this estimator takes a sample x and, unless x=1, guesses that the shift is the (unique) neighbor of x that is closer to 1.
We compute the quality ofe. Note that Q1(e)=1, because if θ=1, then the sample will be a,b, or c, and the estimator is guaranteed to guess correctly. In fact, Qa(e)=1 also, as is easily verified. For any other shift θ, the sample is θℓ for ℓ=a,b, or c, and the estimator guesses correctly exactly when ℓ differs from the last letter of θ. So Qθ(e)=2/3, and Q(e)=infθQθ(e)=2/3.
It is easy to see that this estimator is optimal. Suppose thate′ is another estimator and Q(e′)>2/3. Since each local quality Qθ(e′) is either 0, 1/3, 2/3, or 1, we must have Qθ(e′)=1 for all θ. This means that e′ always guesses correctly. But since there are different values of θ that can produce the same sample, this is impossible.
Observe that the estimatore above is neither left nor right invariant. For instance right invariance fails, as e(ba·a)=e(b)=1≠ba=e(ba)·a, and the same example shows the failure of left invariance: e(b·aa)=id≠ba=be(aa).
Indeed, we conclude by showing that the quality of any shift invariant one-sample estimatore′ is at most 1/3. Suppose e′(1)=w. If e′ is left invariant, it follows that e′(x)=xw for all x; if e′ is right invariant, it follows that e′(x)=wx for all x.
Sinceδ<1, the quality of e′ at θ=1 is equal to the probability that e′(x)=1, given that x was sampled from μ. With equal probability x is a, b, or c; since at most one of wa, wb, and wc and one of aw, bw, and cw can equal 1, we conclude that Q(e′)≤Q1(e′)≤1/3.We remark that this example readily generalizes to other non-amenable groups; the key is thate is a two-to-one map but with only one sample, a shift-invariant estimator is necessarily one-to-one.



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*Source: 752750-2010-12-22.xml*