Minimax Estimation of the Bounded Parameter of Some Discrete Distributions Under LINEX Loss Function

For a class of discrete distributions, including Poisson(θ), Generalized Poisson(θ), Borel(m, θ), etc., we consider minimax estimation of the parameter θ under the assumption it lies in a bounded interval of the form [0, m] and a LINEX loss function. Explicit conditions for the minimax estimator to be Bayes with respect to a boundary supported prior are given. Also for Bernoulli(θ)-distribution, which is not in the mentioned class of discrete distributions, we give conditions for which the Bayes estimator of θ ∈ [0, m], m < 1 with respect to a boundary supported prior is minimax under LINEX loss function. Numerical values are given for the largest values of m for which the corresponding Bayes estimators of θ are minimax.     

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.     

Estimation of the location parameter and the average worth of the selected subset of two parameter exponential populations under LINEX loss function

Suppose a subset of populations is selected from k exponential populations with unknown location parameters θ₁, θ₂, …, θ_k and common known scale parameter σ. We consider the estimation of the location parameter of the selected population and the average worth of the selected subset under an asymmetric LINEX loss function. We show that the natural estimator of these parameters is biased and find the uniformly minimum risk-unbiased (UMRU) estimator of these parameters. In the case of k = 2, we find the minimax estimator of the location parameter of the smallest selected population. Furthermore, we compare numerically the risk of UMRU, minimax, and the natural estimators.     

A Note on the Kuks-Olman Estimator

In the linear regression model the unknown parameter vector θ is supposed to vary in a known ellipsoid. Under this parameter constraint Kuks and Olman derived an estimator by demanding a minimax property. Since sometimes the Kuks-Olman estimator takes values outside of the ellipsoid a modification is proposed in the paper. It is shown that this modified variant is a least squares estimator in the restricted model.     

A note on a modified double stage shrinkage estimator

A modified double stage shrinkage estimator has been proposed for the single parameter θ of a distribution function _Fθ. It is shown to be locally better in comparison to the usual double stage shrinkage estimator in the sense of smaller mean squared error in a certain neighbourhood of prior estimate θ_o of θ.     

ESTIMATING A PARAMETER WHEN IT IS KNOWN THAT THE PARAMETER EXCEEDS A GIVEN VALUE

In some statistical problems a degree of explicit, prior information is available about the value taken by the parameter of interest, θ say, although the information is much less than would be needed to place a prior density on the parameter's distribution. Often the prior information takes the form of a simple bound, ‘θ > θ₁ ’ or ‘θ < θ₁ ’, where θ₁ is determined by physical considerations or mathematical theory, such as positivity of a variance. A conventional approach to accommodating the requirement that θ > θ₁ is to replace an estimator, , of θ by the maximum of and θ₁. However, this technique is generally inadequate. For one thing, it does not respect the strictness of the inequality θ > θ₁ , which can be critical in interpreting results. For another, it produces an estimator that does not respond in a natural way to perturbations of the data. In this paper we suggest an alternative approach, in which bootstrap aggregation, or bagging, is used to overcome these difficulties. Bagging gives estimators that, when subjected to the constraint θ > θ₁ , strictly exceed θ₁ except in extreme settings in which the empirical evidence strongly contradicts the constraint. Bagging also reduces estimator variability in the important case for which is close to θ₁, and more generally produces estimators that respect the constraint in a smooth, realistic fashion.     

Estimation of a multivariate mean with constraints on the norm

Estimation of the mean θ of a spherical distribution with prior knowledge concerning the norm ||θ|| is considered. The best equivariant estimator is obtained for the local problem ||θ|| = λ₀, and its risk is evaluated. This yields a sharp lower bound for the risk functions of a large class of estimators. The risk functions of the best equivariant estimator and the best linear estimator are compared under departures from the assumption ||θ|| = λ₀.     

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.     

BAYESIAN SMOOTHING WITH OCCASIONAL JUMPS

Consider a sequence of independent random variables X ₁, X ₂,…,X _n observed at n equally spaced time points where X _i has a probability distribution which is known apart from the values of a parameter θ _i ∈ R which may change from observation to observation. We consider the problem of estimating θ = (θ₁, θ₂,…,θ _n ) given the observed values of X ₁, X ₂,…,X _n . The paper proposes a prior distribution for the parameters θ for which sets of parameter values exhibiting no change, or no change apart from a few sudden large changes, or lots of small changes, all have positive prior probability. Markov chain sampling may be used to calculate Bayes estimates of the parameters. We report the results of a Monte Carlo study based on Poisson distributed data which compares the Bayes estimator with estimators obtained using cubic splines and with estimators derived from the Schwarz criterion. We conclude that the Bayes method is preferable in a minimax sense since it never produces the disastrously large errors of the other methods and pays only a modest price for this degree of safety. All three methods are used to smooth mortality rates for oesophageal cancer in Irish males aged 65–69 over the period 1955 through 1994.     

Improving on truncated linear estimates of exponential and gamma scale parameters

We consider the problem of estimating the scale parameter of an exponential or a gamma distribution under squared error loss when the scale parameter θ is known to be greater than some fixed value θ₀. Natural estimators in this setting include truncated linear functions of the sufficient statistic. Such estimators are typically inadmissible, but explicit improvements seem difficult to find. Some are presented here. A particularly interesting finding is that estimators which are admissible in the untruncated problem which take values only in the interior of the truncated parameter space are found to be inadmissible for the truncated problem.     

Simultaneous selection of extreme populations:a bayesian approach

The problem of simultaneously selecting two non-empty subsets, S_Land S_U, of k populations which contain the lower extreme population (LEP) and the upper extreme population (UEP), respectively, is considered. Unknown parameters θ₁,…,θ_kcharacterize the populations π₁,…,π_kand the populations associated with θ_[1]=min θ_i. and θ_[k]= max θ_i. are called the LEP and the UEP, respectively. It is assumed that the underlying distributions possess the monotone likelihood ratio property and that the prior distribution of θ= (θ₁,…,θ_k) is exchangeable. The Bayes rule with respect to a general loss function is obtained. Bayes rule with respect to a semi-additive and non-negative loss function is also determined and it is shown that it is minimax and admissible. When the selected subsets are required to be disjoint, it shown that the Bayes rule with respect to a specific loss function can be obtained by comparing certain computable integrals, Application to normal distributions with unknown means θ₁,…,θ_kand a common known variance is also considered.     

Optimality of the quasi-score estimator in a mean–variance model with applications to measurement error models

We consider a regression of y$y$ on x$x$ given by a pair of mean and variance functions with a parameter vector θ$\theta$ to be estimated that also appears in the distribution of the regressor variable x$x$. The estimation of θ$\theta$ is based on an extended quasi-score (QS) function. We show that the QS estimator is optimal within a wide class of estimators based on linear-in-y$y$ unbiased estimating functions. Of special interest is the case where the distribution of x$x$ depends only on a subvector α$\alpha$ of θ$\theta$, which may be considered a nuisance parameter. In general, α$\alpha$ must be estimated simultaneously together with the rest of θ$\theta$, but there are cases where α$\alpha$ can be pre-estimated. A major application of this model is the classical measurement error model, where the corrected score (CS) estimator is an alternative to the QS estimator. We derive conditions under which the QS estimator is strictly more efficient than the CS estimator.     

Some aspects of the theory of estimating equations

An estimating equation for a parameter θ, based on an observation ?, is an equation g(x,θ)=0 which can be solved for θ in terms of x. An estimating equation is unbiased if the funaction g has 0 mean for every θ. For the case when the form of the frequency function p(x,θ) is completely specified up to the unknown real parameter θ, the optimality of the m.1 equation $\text{?}\text{log}\text{p}\text{?θ}\text{=0}$ in the class of all unbiased estimating equations was established by Godambe (1960). In this paper we allow the form of the frequency function p to vary assuming that $\text{x=(x}{}_1\text{,…,x}{}_{\mathrm{n}}\text{)?}\text{R}{}^{\mathrm{n}}$ and that under p, E(x_i)=θ. x₁,…, x_n are independent observations on a variate x, it is shown that among all the unbiased estimating equations for θ, $\text{x}\text{?}\text{?θ=0}$ is uniquely optimum up to a constant multiple.     

A note on the admissibility of hammersley's estimator of an integer mean

Let X₁, X₂, …, X_n be identically, independently distributed N(i,1) random variables, where i = 0, ±1, ±2, … Hammersley (1950) showed that d = [X?_n], the nearest integer to the sample mean, is the maximum likelihood estimator of i. Khan (1973) showed that d is minimax and admissible with respect to zero-one loss. This note now proves a conjecture of Stein to the effect that in the class of integer-valued estimators d is minimax and admissible under squared-error loss.     

Empirical bayes tests in a positive exponential family with partial information on priors

We investigate an empirical Bayes testing problem in a positive exponential family having pdf f{x/θ)=c(θ)u(x) exp(?x/θ), x>0, θ>0. It is assumed that θ is in some known compact interval [C₁, C₂]. The value C₁ is used in the construction of the proposed empirical Bayes test δ^* _n. The asymptotic optimality and rate of convergence of its associated Bayes risk is studied. It is shown that under the assumption that θ is in [C₁, C₂] δ^* _n is asymptotically optimal at a rate of convergence of order O(n^?1/n n). Also, δ^* _n is robust in the sense that δ^* _n still possesses the asymptotic optimality even the assumption that "C₁≦θ≦C₂ may not hold.     

Estimating the parameter of selected uniform population under the squared log error loss function

Let π₁, …, π_k be k (? 2) independent populations, where π_i denotes the uniform distribution over the interval (0, θ_i) and θ_i > 0 (i = 1, …, k) is an unknown scale parameter. The population associated with the largest scale parameter is called the best population. For selecting the best population, We use a selection rule based on the natural estimators of θ_i, i = 1, …, k, for the case of unequal sample sizes. Consider the problem of estimating the scale parameter θ_L of the selected uniform population when sample sizes are unequal and the loss is measured by the squared log error (SLE) loss function. We derive the uniformly minimum risk unbiased (UMRU) estimator of θ_L under the SLE loss function and two natural estimators of θ_L are also studied. For k = 2, we derive a sufficient condition for inadmissibility of an estimator of θ_L. Using these condition, we conclude that the UMRU estimator and natural estimator are inadmissible. Finally, the risk functions of various competing estimators of θ_L are compared through simulation.     

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.     

Minimax property of stein's estimator

Stein's estimator and some other estimators of the mean of a K-variate normal distribution are known to dominate the maximum likelihood estimator under quadratic loss for K > 3, and are therefore minimax. In this paper it is shown that the minimax property of Stein's rule is preserved with respect to a generalized loss function.     

some properties of posterior pitman closeness

This article presents some results on a Bayesian notion of Pitman Closeness, defined in Ghosh and Sen (1991) and called Posterior Pitman Closeness (PPC). Their criterion avoids some of the drawbacks of the well-known (frequentist) Pitman closeness criterion, as introduced by Pitman (1937). It is shown that, if two estimators have the same posterior distribution of the distance from θ, the posterior distribution of θ has to be symmetric. This implies, in particular, that the estimators are Posterior Pitman equivalent. It is also shown that the PPC criterion does not suffer from another paradoxical property illustrated by Blyth and Pathak (1985) - that of an estimator δ₁ being stochastically closer to a parameter θ than another estimator δ₂ and yet being Pitman closer to θ than δ₁. It turns out that, if δ₁ is stochastically closer to θ than δ₂, conditional on x, then it is also Posterior Pitman closer.

We show that the original multivariate concept of PPC is no longer transitive. We provide necessary and sufficient conditions for a Posterior Pitman closest estimator to exist, thus generalizing Theorems 2 and 3 of Ghosh and Sen (1991). We show that a Posterior Pitman closest estimator does not always exist in several dimensions.     

THE LIMITING FORM OF THE NON-CENTRAL WISHART DISTRIBUTION1,2

Let S (p×p) have a Wishart distribution -with v degrees of freedom and non-centrality matrix θ= [θ_jK] (p×p). Define θ₀= min {| θ_jk |}, let θ₀→∞, and suppose that | θ_jK | = 0(θ_o). Then the limiting form of the standardized non-central distribution, as θ while n? remains fixed, is a multivariate Gaussian distribution. This result in turn is used to obtain known asymptotic properties of multivariate chi-square and Rayleigh distributions under somewhat weaker conditions.