On the bounded regret of empirical bayes estimators

Let (θ₁,x₁),…,(θ_n,x_n) be independent and identically distributed random vectors with E(xθ) = θ and Var(x|θ) = a + bθ + cθ². Let t_i be the linear Bayes estimator of θ_i and θ~_i be the linear empirical Bayes estimator of θ_i as proposed in Robbins (1983). When Ex and Var x are unknown to the statistician. The regret of using θ~_i instead of t_i because of ignorance of the mean and the variance is r_i = E(θ_i ? θ_i)² ?E(t_i ?θ_i)². Under appropriate conditions cumulative regret R_n = r₁+…r_n is shown to have a finite limit even when n tends to infinity. The limit can be explicitly computed in terms of a,b,c and the first four moments of x.     

Empirical bayes interval estimates involving uniform distributions

Bayesian and empirical Bayesian decision rules are exhibited for the interval estimation of the parameter 0 of a Uniform (0,θ) distribution. The estimate ?,δ>resulting in the interval [?,?+δ]suffers loss given by L(?,δ>,θ)=1-[?≦e≦?+δ]+c₁((?-θ)²+(?+δ?θ)²))+c₂δ. The solution is presented for prior distributions G which have bounded support, no point masses,∫θ^?mdG(θ)<∞ and for some integer m. An example is presented involving a particular parametric form for G and rates of risk convergence in the empirical Bayes problem for this example are calculated.     

Empirical Bayes testing for equivalence

For a fixed point θ₀ and a positive value c₀, this paper studies the problem of testing the hypotheses H₀:|θ−θ₀|≤c₀ against H₁:|θ−θ₀|>c₀ for the normal mean parameter θ using the empirical Bayes approach. With the accumulated past data, a monotone empirical Bayes test is constructed by mimicking the behavior of a monotone Bayes test. Such an empirical Bayes test is shown to be asymptotically optimal and its regret converges to zero at a rate (lnn)^2.5/n where n is the number of past data available, when the current testing problem is considered. A simulation study is also given, and the results show that the proposed empirical Bayes procedure has good performance for small to moderately large sample sizes. Our proposed method can be applied for testing close to a control problem or testing the therapeutic equivalence of one standard treatment compared to another in clinical trials.     

On the admissibility of c[Xbar] + d with respect to the linex loss function

Let X₁, …,X_n be a random sample from a normal distribution with mean θ and variance σ² The problem is to estimate θ with loss function L(θ,e) = v(e-θ) where v(x) = b(exp(ax)-ax-l) and where a, b are constants with b>0, a¦0. Zellner (1986), showed that [Xbar] ? σ²a/2n dominates [Xbar] and hence [Xbar] is inadmissible. The question of what values of c and d render c[Xbar]+ d admissible is studied here.     

MINIMUM MSE ESTIMATION BY LINEAR COMBINATIONS OF ORDER STATISTICS1

In this paper we assume that in a random sample of size ndrawn from a population having the pdf f(x; θ) the smallest r₁ observations and the largest r₂ observations are censored (r₁0, r₂0). We consider the problem of estimating θ on the basis of the middle n-r₁-r₂ observations when either f(x;θ)=θ^-1f(x/θ) or f(x;θ) = (aθ)¹f(x-θ)/aθ) where f(·) is a known pdf, a (<0) is known and θ (>0) is unknown. The minimum mean square error (MSE) linear estimator of θ proposed in this paper is a “shrinkage” of the minimum variance linear unbiased estimator of θ. We obtain explicit expressions of these estimators and their mean square errors when (i) f(·) is the uniform pdf defined on an interval of length one and (ii) f(·) is the standard exponential pdf, i.e., f(x) = exp(–x), x0. Various special cases of censoring from the left (right) and no censoring are considered.     

Empirical Bayes Testing for the Mean Lifetime of Exponential Distributions: Unequal Sample Sizes Case

This paper deals with an empirical Bayes testing problem for the mean lifetimes of exponential distributions with unequal sample sizes. We study a method to construct empirical Bayes tests {δ* _nl + 1,n }^∞ _n = 1 for the sequence of the testing problems. The asymptotic optimality of {δ* _nl + 1,n }^∞ _n = 1 is studied. It is shown that the sequence of empirical Bayes tests {δ* _nl + 1,n }^∞ _n = 1 is asymptotically optimal, and its associated sequence of regrets converges to zero at a rate (ln n)^4M?1/n, where M is an upper bound of sample sizes.     

Asymptotic properties of Bayes risk for one-sided tests

Consider a given sequence {T_n} of estimators for a real-valued parameter θ. This paper studies asymptotic properties of restricted Bayes tests of the following form: reject H₀:θ ≤ θ₀ in favour of the alternative θ > θ₀ if T_n ≤ C_n, where the critical point C_n is determined to minimize among all tests of this form the expected probability of error with respect to the prior distribution. Such tests may or may not be fully Bayes tests, and so are called T_n-Bayes. Under fairly broad conditions it is shown that and the T_n-Bayes risk where a_n is the order of the standard error of T_n, - is the prior density, and μ is the median of F, the limit distribution of (T_n – θ)/a_nb(θ). Several examples are given.     

Estimators of shift based on statistics of the Kolmogorov-Smirnov type

This paper is concerned with the estimation of a shift parameter δ_o, based on some nonnegative functional Hg₁ of the pair (D^δ_N(x), f?^δ_N(x)), where D^δ_N(x) = K_N/b {F_2,n(x)—F_1,m (x + δ)}, +^δ_N(x) = {mF_1,m (x + δ) + nF_2,n(x)}/N, where F_1,m and F_2,n are the empirical distribution functions of two independent random samples (N = m + n), and where K²_N = mn/N. First an estimator δ_N, is defined as a value of δ minimizing a functional H of the type of H₁. A second estimator δ¹_N is also defined which is a linearized version of the first. Finite and asymptotic properties of these estimators are considered. It is also shown that most well-known test statistics of the Kolmogorov-Smirnov type are particular cases of such functionals H₁. The asymptotic distribution and the asymptotic efficiency of some estimators are given.     

Local asymptotic normality of intermediate and central order statistics

Consider n independent random variables Z_i,…, Z_n on R with common distribution function F, whose upper tail belongs to a parametric family F(t) = F_θ(t),t ≥ x₀, where θ ∈ ? ? R ^d. A necessary and sufficient condition for the family F_θ, θ ∈ ?, is established such that the k-th largest order statistic Z_n?k+1:n alone constitutes the central sequence yielding local asymptotic normality ( LAN ) of the loglikelihood ratio of the vector (Z_n?i+1:n)¹ _i=kof the k largest order statistics. This is achieved for k = k(n)→_n→∞∞ with k/n→_n→∞ 0.

In the case of vectors of central order statistics ( Z_r:n, Z_r+1:n,…, Z_s:n ), with r/n and s/n both converging to q ∈ ( 0,1 ), it turns out that under fairly general conditions any order statistic Z_m:n with r ≤ m ≤s builds the central sequence in a pertaining LAN expansion.These results lead to asymptotically optimal tests and estimators of the underlying parameter, which depend on single order statistics only     

Non-parametric estimation of an acceleration parameter

Let X₁,… X_m be a random sample of m failure times under normal conditions with the underlying distribution F(x) and Y₁,…,Y_n a random sample of n failure times under accelerated condititons with underlying distribution G(x);G(x)=1?[1?F(x)]^θ with θ being the unknown parameter under study.Define:U_ij=1 otherwise.The joint distribution of _ijdoes not involve the distribution F and thus can be used to estimate the acceleration parameter θ.The second approach for estimating θ is to use the ranks of the Y-observations in the combined X- and Y-samples.In this paper we establish that the rank of the Y-observations in the pooled sample form a sufficient statistic for the information contained in the U_ii 's about the parameter θ and that there does not exist an unbiassed estimator for the parameter θ.We also construct several estimators and confidence interavals for the parameter θ.     

On empirical Bayes simultaneous selection procedures for comparing normal populations with a standard

In this paper, we derive statistical selection procedures to partition k normal populations into ‘good’ or ‘bad’ ones, respectively, using the nonparametric empirical Bayes approach. The relative regret risk of a selection procedure is used as a measure of its performance. We establish the asymptotic optimality of the proposed empirical Bayes selection procedures and investigate the associated rates of convergence. Under a very mild condition, the proposed empirical Bayes selection procedures are shown to have rates of convergence of order close to O(k^−1/2) where k is the number of populations involved in the selection problem. With further strong assumptions, the empirical Bayes selection procedures have rates of convergence of order O(k^{−α(r−1)/(2r+1)}), where 1<α<2 and r is an integer greater than 2.     

Minimax estimation of a lower-bounded scale parameter of a gamma distribution for scale-invariant squared-error loss

Let X have a gamma distribution with known shape parameter θr;aL and unknown scale parameter θ. Suppose it is known that θ ≥ a for some known a > 0. An admissible minimax estimator for scale-invariant squared-error loss is presented. This estimator is the pointwise limit of a sequence of Bayes estimators. Further, the class of truncated linear estimators C = {θ_ρ|θ_ρ(x) = max(a, ρ), ρ > 0} is studied. It is shown that each θ_ρ is inadmissible and that exactly one of them is minimax. Finally, it is shown that Katz's [Ann. Math. Statist., 32, 136–142 (1961)] estimator of θ is not minimax for our loss function. Some further properties of and comparisons among these estimators are also presented.     

An inclusion-consistent solution to the problem of absurd confidence statements: 2. Consistent nonexact-confidence-interval estimation

Two consistent nonexact-confidence-interval estimation methods, both derived from the consistency-equivalence theorem in Plante (1991), are suggested for estimation of problematic parametric functions with no consistent exact solution and for which standard optimal confidence procedures are inadequate or even absurd, i.e., can provide confidence statements with a 95% empty or all-inclusive confidence set. A belt C(·) from a consistent nonexact-belt family, used with two confidence coefficients (γ = inf_θ P_θ [ θ ? C(X)] and γ⁺ = sup_θ P_θ[θ ? C(X)], is shown to provide a consistent nonexact-belt solution for estimating μ₂ -μ₁ in the Behrens-Fisher problem. A rule for consistent behaviour enables any confidence belt to be used consistently by providing each sample point with best upper and lower confidence levels [δ⁺(x) ≥ γ⁺, δ(x) ≤ γ], which give least-conservative consistent confidence statements ranging from practically exact through informative to noninformative. The rule also provides a consistency correction L(x) = δ⁺(x)-δ(X) enabling alternative confidence solutions to be compared on grounds of adequacy; this is demonstrated by comparing consistent conservative sample-point-wise solutions with inconsistent standard solutions for estimating μ₂/μ₁ (Creasy-Fieller-Neyman problem) and $\sqrt {\mu _1^2 + \mu _2^2 }$, a distance-estimation problem closely related to Stein's 1959 example     

Bayes and Empirical Bayes Two-Stage Procedures for Sampling Inspection

A batch of M items is inspected for defectives. Suppose there are d defective items in the batch. Let d ₀ be a given standard used to evaluate the quality of the population where 0 < d ₀ < M. The problem of testing H ₀: d < d ₀ versus H ₁: d ≥ d ₀ is considered. It is assumed that past observations are available when the current testing problem is considered. Accordingly, the empirical Bayes approach is employed. By using information obtained from the past data, an empirical Bayes two-stage testing procedure is developed. The associated asymptotic optimality is investigated. It is proved that the rate of convergence of the empirical Bayes two-stage testing procedure is of order O (exp(? c? n)), for some constant c? > 0, where n is the number of past observations at hand.     

Convergence Rates of Empirical Bayes Estimation for the Parameter of the Uniform Distribution U(0, θ) Under Random Censorship

ABSTRACT

This article considers the empirical Bayes estimation problem in the uniform distribution U(0, θ) with censored data. For the parameter θ, using the empirical Bayes (EB) approach, we propose an EB estimation of θ which possesses a rate of convergence can be arbitrarily close to O(n ^?1/2) when the historical samples are randomly censored from the right, where n is the number of historical sample. A sample and some simulation results are also presented.     

Empirical Bayes testing for guarantee lifetime: Non identical components case

This article considers an empirical Bayes testing problem for the guarantee lifetime in the two-parameter exponential distributions with non identical components. We study a method of constructing empirical Bayes tests under a class of unknown prior distributions for the sequence of the component testing problems. The asymptotic optimality of the sequence of empirical Bayes tests is studied. Under certain regularity conditions on the prior distributions, it is shown that the sequence of the constructed empirical Bayes tests is asymptotically optimal, and the associated sequence of regrets converges to zero at a rate O(n^{? 1 + 1/[2(r + α) + 1]}) for some integer r ? 0 and 0 ? α ? 1 depending on the unknown prior distributions, where n is the number of past data available when the (n + 1)st component testing problem is considered.     

Exponential and polynomial tailbounds for change-point estimators

Let X_1n,…,X_nn be independent random elements with an unknown change point θ∈(0,1), that is X_in has a distribution ν₁ or ν₂, respectively, according to i⩽[nθ] or i>[nθ]. We propose an estimator θ_n of θ, which is defined as the maximizer of a weighted empirical process on (0,1). Finding upper bounds of polynomial and exponential type for the tails of nθ_n−[nθ], we are able to derive rates of almost sure convergence, of distributional convergence, of L_p-convergence and of convergence in the Ky-Fan- and in the Prokhorov-metric.     

Sequential Fixed-Width Confidence Interval for a Function of the Parameters

Let X₁…, X_m and Y₁…, Y_n be two independent sequences of i.i.d. random variables with distribution functions F_x(.|θ) and F_y(. | φ) respectively. Let g(θ, φ) be a real-valued function of the unknown parameters θ and φ. The purpose of this paper is to suggest a sequential procedure which gives a fixed-width confidence interval for g(θ, φ) so that the coverage probability is approximately α (preas-signed). Certain asymptotic optimality properties of the sequential procedure are established. A Monte Carlo study is presented.