Some remarks on an admissibility condition

Summary In this note we deal with some admissibility conditions proved by G. B. Tranquilli to be sufficient in the class of unbiased estimators of finite population parameters and with respect to (w.r.t.) a quadratic loss function. We show that the same conditions:i) are sufficient for the admissibility of an unbiased estimator with any loss function;ii) imply hyperadmissibility with reference to a particular (critical) population of the. From this fact we deduce that, for a fixed critical population, there is at most one estimator, in the class of all unbiased estimator of a finite population parameter, which satisfies Tranquilli condition. This research was partially supported by a M.U.R.S.T. grant ?Metodi inferenziali basati sul ricampionamento?.     

Admissible unbiased estimation of finite population variance under a randomized response model

We consider the problem of estimation of a finite population variance related to a sensitive character under a randomized response model and prove (i) the admissibility of an estimator for a given sampling design in a class of quadratic unbiased estimators and (ii) the admissibility of a sampling strategy in a class of comparable quadratic unbiased strategies.     

Une condition nécessaire d'admissibilité et ses conséquences sur les estimateurs à rétrécisseur de la moyenne d'un vecteur normal

Considering exponential families of distributions, we estimate parameters which are not the natural parameters. We prove that the admissible estimators of these parameters are limits of Bayes estimators and can be expressed through a given functional form. An important particular case of this model pertains to the estimation of the mean of a multidimensional normal distribution when the variance is known up to a multiplicative factor. We deduce from the main result a necessry condition for the admissibility of matricial shrinkage estimators.     

Non-minimaxity of linear combinations of restricted location estimators and related problems

The estimation of a linear combination of several restricted location parameters is addressed from a decision-theoretic point of view. Although the corresponding linear combination of the unbiased estimators is minimax under the restricted problem, it has a drawback of taking values outside the restricted parameter space. Thus, it is reasonable to use the linear combination of the restricted estimators such as maximum likelihood or truncated estimators. In this paper, a necessary and sufficient condition for such restricted estimators to be minimax is derived, and it is shown that the restricted estimators are not minimax when the number of the location parameters is large. The condition for minimaxity is examined for some specific distributions. Finally, similar problems of estimating the product and sum of the restricted scale parameters are studied, and it is shown that analogous non-dominance properties appear when the number of the scale parameters is large.     

Estimation of the parameter of a dynamically selected population for two subclasses of the exponential family

We introduce the problem of estimation of the parameters of a dynamically selected population in an infinite sequence of random variables and provide its application in the statistical inference based on record values from a non stationary scheme. We develop unbiased estimation of the parameters of the dynamically selected population and evaluate the risk of the estimators. We provide comparisons with natural estimators and obtain asymptotic results. Finally, we illustrate the applicability of the results using real data.     

Admissible estimators in finite population sampling employing various types of prior information

We consider some estimators of the total and variance of a finite population from Bayesian and pseudo-Bayesian perspectives. Recently, Meeden and Ghosh (1982a, 1982b) have provided quite simple but powerful tools for proving admissibility of estimators and estimator-design pairs is finite population sampling problems. We consider what these techniques yield in the way of admissibility results for the estimators discussed.     

A Class of Estimators of Population Variance Using Auxiliary Information in Sample Surveys

This article advocates the problem of estimating the population variance of the study variable using information on certain known parameters of an auxiliary variable. A class of estimators for population variance using information on an auxiliary variable has been defined. In addition to many estimators, usual unbiased estimator, Isaki's (1983), Upadhyaya and Singh's (1999), and Kadilar and Cingi's (2006) estimators are shown as members of the proposed class of estimators. Asymptotic expressions for bias and mean square error of the proposed class of estimators have been obtained. An empirical study has been carried out to judge the performance of the various estimators of population variance generated from the proposed class of estimators over usual unbiased estimator, Isaki's (1983), Upadhyaya and Singh's (1999) and Kadilar and Cingi's (2006) estimators.     

ON NON-NEGATIVE UNBIASED ESTIMATION OF QUADRATIC FORMS IN FINITE POPULATION SAMPLING

The paper investigates non-negative quadratic unbiased (NnQU) estimators of positive semi-definite quadratic forms, for use during the survey sampling of finite population values. It examines several different NnQU estimators of the variance of estimators of population total, under various sampling designs. It identifies an optimal quadratic unbiased estimator of the variance of the Horvitz-Thompson estimator of population total.     

Estimating the exponential reliability function for a system in series with type II censored data

The problem of estimating the one parameter exponential reliability function for a system composed of l componentes in series is considered. Under the type II censoring scheme, the Bayes nature of the minimum variance unbiased estimator is demonstrated and the admissibility of related generalized Bayes estimators is established. For the one component case, the best unbiased estimator is admissible.     

Unbiased Estimation of the Distribution Function of a Two-Parameter Exponential Population Using Order Statistics

In this article, we consider the problem of unbiased estimation of the distribution function of a two-parameter exponential population using order statistics based on a random sample from the population. We give necessary and sufficient conditions for the existence of an unbiased estimator based on an arbitrary set of order statistics and suggest unbiased estimators in some situations where unbiased estimators exist. A few properties of the suggested estimators for some special cases have also been discussed.     

Re-parameterization of multinomial distributions and diversity indices

It is shown in this paper that the parameters of a multinomial distribution may be re-parameterized as a set of generalized Simpson's diversity indices. There are two important elements in the generalization: (1) Simpson's diversity index is extended to populations with infinite species; (2) weighting schemes are incorporated. A class of unbiased estimators for the generalized Simpson's biodiversity indices is proposed. Asymptotic normality is established for the estimators. Both the unbiasedness and the asymptotic normality of the estimators hold for all three cases of the number of species in the population: infinite, finite and known, and finite but unknown. In the case of a population with a finite number of species, known or unknown, it is also established that the proposed estimators are uniformly minimum variance unbiased and are asymptotically efficient.     

Estimation of the Mean and Standard Deviation of the Logistic Distribution Based on Multiply Type-II Censored Samples

In this paper, we discuss the problem of estimating the mean and standard deviation of a logistic population based on multiply Type-II censored samples. First, we discuss the best linear unbiased estimation and the maximum likelihood estimation methods. Next, by appropriately approximating the likelihood equations we derive approximate maximum likelihood estimators for the two parameters and show that these estimators are quite useful as they do not need the construction of any special tables (as required for the best linear unbiased estimators) and are explicit estimators (unlike the maximum likelihood estimators which need to be determined by numerical methods). We show that these estimators are also quite efficient, and derive the asymptotic variances and covariance of the estimators. Finally, we present an example to illustrate the methods of estimation discussed in this paper.     

Admissible linear estimation in singular linear models

The admissibility results of Rao (1976), proved in the context of a nonsingular covariance matrix, are exteneded to the situation where the covariance matrix is singular. Admi.s s Lb Le linear estimators in the Gauss-Markoff model are characterized and admis-sibility of the best linear unbiased estimator is investigated.     

Discriminant analysis of survey data

We consider the problem of the effect of sample designs on discriminant analysis. The selection of the learning sample is assumed to depend on the population values of auxiliary variables. Under a superpopulation model with a multivariate normal distribution, unbiasedness and consistency are examined for the conventional estimators (derived under the assumptions of simple random sampling), maximum likelihood estimators, probability-weighted estimators and conditionally unbiased estimators of parameters. Four corresponding sampled linear discriminant functions are examined. The rates of misclassification of these four discriminant functions and the effect of sample design on these four rates of misclassification are discussed. The performances of these four discriminant functions are assessed in a simulation study.     

Optimal sequential estimation for semi-markov and markov renewal processes

We consider the problem of obtaining efficient estimators and sampling plans for semi-Markov and Markov-renewal processes. A lower bound for the variance of an unbiased estimator of a function of the parameters is obtained under a sequential scheme and we characterize the parametric functions and sampling plans which admit minimum variance unbiased estimators.     

Examples of Differing Weighted and Unweighted Estimates from a Sample Survey

Unweighted estimators using data collected in a sample survey can be badly biased, whereas weighted estimators are approximately unbiased for population parameters. We present four examples using data from the 1988 National Maternal and Infant Health Survey to demonstrate that weighted and unweighted estimators can be quite different, and to show the underlying causes of such differences.     

Second-order properties of intraclass correlation estimators for a symmetric normal distribution

Consider the problem of estimating the intraclass correlation coefficient of a symmetric normal distribution under the squared error loss function. The general admissibility of the standard estimators of the intraclass correlation coefficient is hard to check due to their complicated sampling distributions. We follow the asymptotic decision-theoretic approach of Ghosh and Sinha (1981) and prove that the three standard intraclass correlation estimators (the maximum-likelihood estimator, the method-of-moments estimator and the first-order unbiased estimator) are second-order admissible for all p ≥ 2, p being the dimension of the distribution.     

Estimation of parameters and the mean life of a mixed failure time distribution

This paper deals with estimation of parameters and the mean life of a mixed failure time distribution that has a discrete probability mass at zero and an exponential distribution with mean O for positive values. A new sampling scheme similar to Jayade and Prasad (1990) is proposed for estimation of parameters. We derive expressions for biases and mean square errors (MSEs) of the maximum likelihood estimators (MLEs). We also obtain the uniformly minimum variance unbiased estimators (UMVUEs) of the parameters. We compare the estimator of O and mean life fj based on the proposed sampling scheme with the estimators obtained by using the sampling scheme of Jayade and Prasad (1990).     

Estimation of parameters of exponential distribution in the truncated space using asymmetric loss function

This paper is concerned with estimation of location and scale parameters of an exponential distribution when the location parameter is bounded above by a known constant. We propose estimators which are better than the standard estimators in the unrestricted case with respect to the suitable choice of LINEX loss. The admissibility of the modified Pitman estimators with respect to the LINEX loss is proved. Finally the theory developed is applied to the problem of estimating the location and scale parameters of two exponential distributions when the location parameters are ordered.     

Separate Ratio Estimators for the Population Variance in Stratified Random Sampling

We propose separate ratio estimators for population variance in stratified random sampling. We obtain mean square error equations and compare proposed estimators about efficiency with each other. By these comparisons, we find the conditions which make proposed estimators more efficient than others. It has been shown that proposed classes of estimators are more efficient than usual unbiased estimator. We find that separate ratio estimators are more efficient than combined ratio estimators for population variance. The theoretical results are supported by a numerical illustration with original data. A simulation study is also carried out to investigate empirical performance of estimators.