Admissibility and linear sufficiency in linear model with nuisance parameters

In this paper, we derive characterization of admissible and linearly sufficient estimators of a vector of linear estimable functions of parameters in a partitioned linear model. Then, we compare this class of estimators with the class of admissible and linearly sufficient estimators of the same vector of linear parametric functions in a reduced linear model. We show that these classes are equal with probability one.     

A test of the mean square error criterion for linear admissible estimators

A test for choosing between a linear admissible estimator and the least squares estimator (LSE) is developed. A characterization of linear admissible estimators useful for comparing estimators is presented and necessary and sufficient conditions for superiority of a linear admissible estimator over the LS estimetor is derived for the test. The test is based on the MSE matrix superiority, but also new resl?!ts concerning covariance matrix comparisons of linear estimators are derived. Further,shown that the test of Toro - Vizcarrondo and Wailace applies iioi only the restricted least squares estimators but also to certain estimators outside this class.     

Effective procedures for estimating beta distribution's parameters and Their confidence intervals

First, we compare several methods for computing the ML estimators of the two-parameter beta distribution; the most effective one is identified. Second, a simple way is found to characterize the sampling distribution of the ML estimators; this characterization leads to a practical way of establishing confidence intervals for the ML estimators.     

On the Bayesian Estimation for the Uniform Scale Parameter via a Functional Equation

Bayes uniform model under the squared error loss function is shown to be completely identifiable by the form of the Bayes estimates of the scale parameter. This results in solving a specific functional equation. A complete characterization of differentiable Bayes estimators (BE) and generalized Bayes estimators (GBE) is given as well as relations between degrees of smoothness of the estimators and the priors. Characterizations of strong (generalized Bayes) Bayes sequence (SBS or SGBS) are also investigated. A SBS is a sequence of estimators (one for each sample size) where all its components are BE generated by the same prior measure. A complete solution is given for polynomial Bayesian estimation.     

On Robust and Efficient Estimation of the Center of Symmetry

In this article, a class of estimators of the center of symmetry based on the empirical characteristic function is examined. In the spirit of the Hodges–Lehmann estimator, the resulting procedures are shown to be a function of the pairwise averages. The proposed procedures are also shown to have an equivalent representation as the minimizers of certain distances between two corresponding kernel density estimators. An alternative characterization of the Hodges–Lehmann estimator is established upon the use of a particularly simple choice of kernel.     

A functional law of the iterated logarithm for kernel-type estimators of the tail index

Under von Mises's condition, the kernel tail index estimators of Csörgő et al. (Ann. Statist. 13 (1985) 1050) have been generalized by Groeneboon et al. (Ann. Statist. 31 (2003) 1956) for all real tail indices. Weak consistency and asymptotic normality of their kernel estimators have been established. In this paper, we present a characterization of the almost sure behavior of these estimators and we show their strong consistency.     

Life table estimator revisited

We explore the standard life table (actuarial) estimator for grouped right-censored survival data and its extensions in order to consider its relationship with the Kaplan–Meier estimator, and to investigate the critical properties of the extended life table estimators (ELTEs). We discuss certain conditions for the ELTE to be consistent and develop a characterization of the standard life table estimator using the consistency property under any choice of at least two observation times of a finite interval. We also perform a comparative analysis of the ELTEs with the corresponding maximum likelihood estimators for grouped right-censored survival data.     

A Bayesian approach to non-parametric monotone function estimation

Summary.  The paper proposes two Bayesian approaches to non-parametric monotone function estimation. The first approach uses a hierarchical Bayes framework and a characterization of smooth monotone functions given by Ramsay that allows unconstrained estimation. The second approach uses a Bayesian regression spline model of Smith and Kohn with a mixture distribution of constrained normal distributions as the prior for the regression coefficients to ensure the monotonicity of the resulting function estimate. The small sample properties of the two function estimators across a range of functions are provided via simulation and compared with existing methods. Asymptotic results are also given that show that Bayesian methods provide consistent function estimators for a large class of smooth functions. An example is provided involving economic demand functions that illustrates the application of the constrained regression spline estimator in the context of a multiple-regression model where two functions are constrained to be monotone.     

Characterization of distributions in invariant models

The theorem of Kagan et al. (1967), (Sankhya Ser. A. 27, 405) on the characterization of the normal law is extended to the characterization of a broad class of distribution shapes, also in the linear regression model, and the stability of this characterization is considered. The results enable, among others, to construct an asymptotically efficient estimator from a subclass of equivariant asymptotically linear estimators of θ.     

Invariant methods for estimating variance components in mixed linear models 2

The subject of this contribution is to present a survey on new methods for variance component estimation, which appeared in the literature in recent years. Starting from mixed models treated in analysis of variance research work on this field turned over to a more general approach in which the covariance matrix of the vector of observations is assumed to be a unknown linear combination of known symmetric matrices. Much interest has been shown in developing some kinds op optimal estimators for the unknown parameters and most results were obtained for estimators being invariant with respect to a certain group of translations. Therefore we restrict attention to this class of estimates. We will deal with minimum variance unbiased estimators, least squared errors estimators, maximum likelihood estimators. Bayes quadratic estimators and show some relations to the mimimum norm quadratic unbiased estimation principle (MINQUE) introduced by C. R. Rao [20]. We do not mention the original motivation of MINQUE since the otion of minimum norm depends on a measure that is not accepted by all statisticians. Also we do‘nt deal with other approaches like the BAYEsian and fiducial methods which were successfully applied by S. Portnoy [18], P. Rusolph [22], G. C. Tiao, W. Y. Tan [28], M. J. K. Healy [9] and others, although in very special situations, only. Additionally we add some new results and also new insight in the properties of known estimators. We give a new characterization of MINQUE in the class of all estimators, extend explicite expressions for locally optimal quadratic estimators given by C. R. Rao [22] to a slightly more general situation and prove complete class theorems useful for the computation of BAYES quadratic estimators. We also investigate situations in which BAYES quadratic unbiased estimators do'nt change if the distribution of the error terms differ from the normal distribution.     

Some remarks on a ridge-type-estimator and good prior means

Pliskin (1987) compared modified ridge regression estimators based on prior information with respect to their mean square error matrices. A further characterization of good prior mean is given here, and the case of different ridge parameters is also considered.     

New approach in biased regression

SUMMARY An optimization problem which provides a new characterization for ridge regression is discussed. A variant of this optimization problem leads to a new family of biased estimators that includes the Stein estimation method and principal components regression as particular cases. The whole approach is illustrated on the basis of real data sets.     

Shrinkage Structure of Partial Least Squares

Partial least squares regression (PLS) is one method to estimate parameters in a linear model when predictor variables are nearly collinear. One way to characterize PLS is in terms of the scaling (shrinkage or expansion) along each eigenvector of the predictor correlation matrix. This characterization is useful in providing a link between PLS and other shrinkage estimators, such as principal components regression (PCR) and ridge regression (RR), thus facilitating a direct comparison of PLS with these methods. This paper gives a detailed analysis of the shrinkage structure of PLS, and several new results are presented regarding the nature and extent of shrinkage.     

Simultaneous equivariant estimation of the parameters of matrix scale and matrix location-scale models

Eaton and Olkin (1987) discussed the problem of best equivariant estimator of the matrix scale parameter with respect to different scalar loss functions. Edwin Prabakaran and Chandrasekar (1994) developed simultaneous equivariant estimation approach and illustrated the method with examples. The problems considered in this paper are simultaneous equivariant estimation of the parameters of (i) a matrix scale model and (ii) a multivariate location-scale model. By considering matrix loss function (Klebanov, Linnik and Ruhin, 1971) a characterization of matrix minimum risk equivariant (MMRE) estimator of the matrix parameter is obtained in each case. Illustrative examples are provided in which MMRE estimators are obtained with respect to two matrix loss functions.     

Estimation of mean using double sampling for stratification and multivariate auxiliary information

Several estimators for estimating the mean of a principal variable are proposed based on double sampling for stratification (DSS) and multivariate auxiliary information. The general properties of the proposed estimators are studied, search for optimum estimators is made and the proposed estimators are compared with the corresponding estimators based on unstratified double sampling (USDS).     

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.     

Construction of improved estimators of multinomial proportions

The improved large sample estimation theory for the probabilities of multi¬nomial distribution is developed under uncertain prior information (UPI) that the true proportion is a known quantity. Several estimators based on pretest and the Stein-type shrinkage rules are constructed. The expressions for the bias and risk of the proposed estimators are derived and compared with the maximum likelihood (ml) estimators. It is demonstrated that the shrinkage estimators are superior to the ml estimators. It is also shown that none of the preliminary test and shrinkage estimators dominate each other, though they perform y/ell relative to the ml estimators. The relative dominance picture of the estimators is presented. A simulation study is carried out to assess the performance of the estimators numerically in small samples.     

A New Estimation Method Based on Order Statistics in the Families of Symmetric Location-scale Distributions

In this study, new unbiased and nonlinear estimators based on order statistics are proposed for the family of symmetric location-scale distributions and these estimators can be computed from both uncensored and symmetric doubly Type II censored samples. In addition, other relevant unbiased estimators are proposed to estimate standard deviations of these new estimators. A simulation study has been performed to evaluate the performance of the new estimators compared to BLU estimators for small sample sizes. As a result of the simulation study, the new estimators proposed for the location-scale family in general performed nearly as good as BLU estimators. Furthermore, the computational advantage of the proposed estimators over BLU and ML estimators are worthy of notice. In addition, these new estimators have been applied to real data, and the estimation results obtained have been compatible with those of BLUE methods.     

On invariant quadratic unbiased estimation of variance components

The present study deals with three different invarint quadratic unbiased estimators (IQUE) for variance components namely quadratic least squares estimators (QLSE), weighted quadratic least squares estimators (WQLSE) and Mitra type estimators (MTE). The variance and covariances of these three different estimators are presented for unbalanced one-way random model. The relative performances of these estimators are assessed based on different optimality criteria like, D-optimality, T-optimality and M-optimality together with variances of these estimators. As a result, it has been shown that MTE has optimal properties.     

Bayes estimators in the presence of a guess value

The paper deals with the problem of parameter estimation in the presence of a guess value and attempts to justify the use of Bayes estimators as an alternative to ordinary shrinkage estimators. Finally, certain Bayes estimators of exponential parameters are obtained under type II censoring, and these are compared with the corresponding MLEs and ordinary shrinkage estimators using a Monte Carlo study.