Estimating a restricted normal mean

Let X ₁, X ₂, ..., X _n be a random sample from a normal distribution with unknown mean μ and known variance σ ². In many practical situations, μ is known a priori to be restricted to a bounded interval, say [−m, m] for some m > 0. The sample mean , then, becomes an inadmissible estimator for μ. It is also not minimax with respect to the squared error loss function. Minimax and other estimators for this problem have been studied by Casella and Strawderman (Ann Stat 9:870–878, 1981), Bickel (Ann Stat 9:1301–1309, 1981) and Gatsonis et al. (Stat Prob Lett 6:21–30, 1987) etc. In this paper, we obtain some new estimators for μ. The case when the variance σ ² is unknown is also studied and various estimators for μ are proposed. Risk performance of all estimators is numerically compared for both the cases when σ ² may be known and unknown.     

Gamma-minimax estimation of a multivariate normal mean

Summary The mean vector of a multivariate normal distribution is to be estimated. A class Γ of priors is considered which consists of all priors whose vector of first moments and matrix of second moments satisfy some given restrictions. The Γ-minimax estimator under arbitrary squared error loss is characterized. The characterization follows from an application of a result of Browder and Karamardian published in Ichiishi (1983) which is a special version of a minimax inequality due to Ky Fan (1972). In particular, it is shown that within the set of all estimators a linear estimator is Γ-minimax. The authors would like to thank the Deutsche Forschungsgemeinschaft for financial support.     

Minimax estimation for the bounded mean of a bivariate normal distribution

A bivariate normal distribution is considered whose mean lies in an equilateral triangle. We show by a convexity argument that the three point prior having mass 1/3 at each of the edges is least favourable if the length of a side of the equilateral triangle is less than or equal to . Thus the corresponding Bayes estimator is minimax in that case. Numerical studies are given as well.     

The moments of products of quadratic forms in normal variables*

Abstract The expectation of the product of an arbitrary number of quadratic forms in normally distributed variables is derived.     

On estimating the reliability after sequentially estimating the mean: The exponential case

Sequential estimation problems for the mean parameter of an exponential distribution has received much attention over the years. Purely sequential and accelerated sequential estimators and their asymptotic second-order characteristics have been laid out in the existing literature, both for minimum risk point as well as bounded length confidence interval estimation of the mean parameter. Having obtained a data set from such sequentially designed experiments, the paper investigates estimation problems for the associatedreliability function. Second-order approximations are provided for the bias and mean squared error of the proposed estimator of the reliability function, first under a general setup. An ad hoc bias-corrected version is also introduced. Then, the proposed estimator is investigated further under some specific sequential sampling strategies, already available in the literature. In the end, simulation results are presented for comparing the proposed estimators of the reliability function for moderate sample sizes and various sequential sampling strategies.     

A property of Stein's estimator for the mean of a multivariate normal distribution

A uniform bound on the risk (under squared error loss) of Stein's estimator Ψ₁ for the mean of the multivariate normal distribution is given. Using the bound, the asymptotic behaviour of the risk of Ψ₁ under a Bayesian assumption is obtained.     

A note on the estimation of mean in normal population

An estimator for the mean of a Normal population is presented and its properties are analyzed. The efficiency with respect to the conventional sample mean is also examined.     

Remarks on sequential estimation of a linear function of two means: The normal case

Asymptotic normality of the stopping time ofMukhopadhyay [1976] relating to the point estimation problem is proved. Also moderate sample size behaviour of this stopping time has been studied by Monte-Carlo methods.     

Subsampling inference for the mean in the heavy-tailed case

In this article, asymptotic inference for the mean of i.i.d. observations in the context of heavy-tailed distributions is discussed. While both the standard asymptotic method based on the normal approximation and Efron's bootstrap are inconsistent when the underlying distribution does not possess a second moment, we propose two approaches based on the subsampling idea of Politis and Romano (1994) which will give correct answers. The first approach uses the fact that the sample mean, properly standardized, will under some regularity conditions have a limiting stable distribution. The second approach consists of subsampling the usual t-statistic and is somewhat more general. A simulation study compares the small sample performance of the two methods. Received: December 1998     

Further developments in estimation of the largest mean ofK normal populations

We revisit the bounded maximal risk point estimation problem as well as the fixed-width confidence interval estimation problem for the largest mean amongk(≥2) independent normal populations having unknown means and unknown but equal variance. In the point estimation setup, we devise appropriate two-stage and modified two-stage methodologies so that the associatedmaximal risk can bebounded from aboveexactly by a preassigned positive number. Kuo and Mukhopadhyay (1990), however, emphasized only the asymptotics in this context. We have also introduced, in both point and interval estimation problems,accelerated sequential methodologies thereby saving sampling operations tremendously over the purely sequential schemes considered in Kuo and Mukhopadhyay (1990), but enjoying at the same time asymptotic second-order characteristics, fairly similar to those of the purely sequential ones.     

Consistent maximum-likelihood estimation with dependent observations: The general (non-normal) case and the normal case

In this paper we aim to establish intuitively appealing and verifiable conditions for the existence and weak consistency of ML estimators in a multi-parameter framework, assuming neither the independence nor the identical distribution of the observations. The paper has two parts. In the first part (Theorems 1 and 2) we assume that the joint density of the observations is known (except for the values of a finite number of parameters to be estimated), but we do not specify this distribution. In the second part (Theorems 3–6), we do specify the distribution and assume joint normality (but not independence) of the observations. Some examples are also provided.     

Estimation of conditional moment restrictions without assuming parameter identifiability in the implied unconditional moments

A well-known difficulty in estimating conditional moment restrictions is that the parameters of interest need not be globally identified by the implied unconditional moments. In this paper, we propose an approach to constructing a continuum of unconditional moments that can ensure parameter identifiability. These unconditional moments depend on the “instruments” generated from a “generically comprehensively revealing” function, and they are further projected along the exponential Fourier series. The objective function is based on the resulting Fourier coefficients, from which an estimator can be easily computed. A novel feature of our method is that the full continuum of unconditional moments is incorporated into each Fourier coefficient. We show that, when the number of Fourier coefficients in the objective function grows at a proper rate, the proposed estimator is consistent and asymptotically normally distributed. An efficient estimator is also readily obtained via the conventional two-step GMM method. Our simulations confirm that the proposed estimator compares favorably with that of Domínguez and Lobato (2004, Econometrica) in terms of bias, standard error, and mean squared error.     

Sequential estimation of normal mean under asymmetric loss function with a shrinkage stopping rule

The problem of estimating a normal mean with unknown variance is considered under an asymmetric loss function such that the associated risk is bounded from above by a known quantity. In the absence of a fixed sample size rule, a sequential stopping rule and two sequential estimators of the mean are proposed and second-order asymptotic expansions of their risk functions are derived. It is demonstrated that the sample mean becomes asymptotically inadmissible, being dominated by a shrinkage-type estimator. Also a shrinkage factor is incorporated in the stopping rule and similar inadmissibility results are established. Received September 1997     

Estimators of long-memory: Fourier versus wavelets

Semi-parametric estimation methods of the long-memory exponent of a time series have been studied in several papers, some applied, others theoretical, some using Fourier methods, others using a wavelet-based technique. In this paper, we compare the Fourier and wavelet approaches to the local regression method and to the local Whittle method. We provide an overview of these methods, describe what has been done and indicate the available results and the conditions under which they hold. We discuss their relative strengths and weaknesses both from a practical and a theoretical perspective. We also include a simulation-based comparison. The software written to support this work is available on demand and we illustrate its use at the end of the paper.     

The informativeness of estimation moments

This paper introduces measures for how each moment contributes to the precision of parameter estimates in generalized method of moments settings. For example, one of the measures asks what would happen to the variance of the parameter estimates if a particular moment was dropped from the estimation. The measures are all easy to compute. We illustrate the usefulness of the measures through two simple examples as well as an application to a model of joint retirement planning of couples. We estimate the model using the British Household Panel Survey, and we find evidence of complementarities in leisure. Our sensitivity measures illustrate that the estimate of the complementarity is primarily informed by the distribution of differences in planned retirement dates. The estimated econometric model can be interpreted as a bivariate ordered-choice model that allows for simultaneity. This makes the model potentially useful in other applications.     

On estimating the mean of the selected normal population under the LINEX loss function

Following Parsian and Farsipour (1999), we consider the problem of estimating the mean of the selected normal population, from two normal populations with unknown means and common known variance, under the LINEX loss function. Some admissibility results for a subclass of equivariant estimators are derived and a sufficient condition for the inadmissibility of an arbitrary equivariant estimator is provided. As a consequence, several of the estimators proposed by Parsian and Farsipour (1999) are shown to be inadmissible and better estimators are obtained. Received January 2001/Revised May 2002     

Sequential point estimation of normal mean under LINEX loss function

A sequential point estimation of the mean of a normal distribution is considered under LINEX loss function. The regret of sequential procedures are obtained. Furthermore, it is shown that a sequential procedure with the sample mean as an estimate is asymptotically inadmissible. An accerelated stopping time is also considered. Received: December 1999     

On the stopping boundary for sequential selection of the normal population with the largest mean

This paper provides an improved stopping boundary for open sequential selection of the normal population with the largest mean when all populations have common known variance. The proof relies on the theory of Brownian motion processes with drift.     

Utility Theory and reciprocal pairwise comparisons: The Eigenvector Method

The criticisms of Utility Theory focus on either its axioms or the construction of utility functions. Here we present a method which avoids the problems of uniqueness encountered in the construction of utility functions when using either the certainty equivalence method or the probability equivalence method. The method is based on the construction of ratio scale value functions from reciprocal pairwise comparisons and Saaty's Eigenvector Method. We show that under the assumption of cardinal consistency utility functions are a particular case of these ratio scales. Reciprocal pairwise comparisons allow decision makers to relax the transitivity assumption and help to derive a unique scaling of preferences.