Comparison of estimators of the location parameter using pitman's closeness criterion

Necessary and sufficient conditions for a linear estimator to dominate another linear estimator of a location parameter under the Pitman's criterion of comparison are discussed. Consequently it is demonstrated that a linear biased estimator can not dominate a linear unbiased estimator under Pitman's criterion and that the sample mean is the Closest Linear Unbiased Estimator (CLUE). It is also shown that the ridge regression estimator with a known biasing constant can not dominate the ordinary least squares estimator. If an estimator δdominates an estimator δin the average loss sense then sufficient conditions are obtained under which δis also preferred over δunder Pitman's criterion. Further we obtain sufficient conditions under which preference under the Pitman's criterion will lead to preference under the mean squared error sense.     

Preliminary test and Bayes Estimation of a Location Parameter Under Blinex Loss

In this article, the preliminary test estimator is considered under the BLINEX loss function. The problem under consideration is the estimation of the location parameter from a normal distribution. The risk under the null hypothesis for the preliminary test estimator, the exact risk function for restricted maximum likelihood and approximated risk function for the unrestricted maximum likelihood estimator, are derived under BLINEX loss and the different risk structures are compared to one another both analytically and computationally. As a motivation on the use of BLINEX rather than LINEX, the risk for the preliminary test estimator under BLINEX loss is compared to the risk of the preliminary test estimator under LINEX loss and it is shown that the LINEX expected loss is higher than BLINEX expected loss. Furthermore, two feasible Bayes estimators are derived under BLINEX loss, and a feasible Bayes preliminary test estimator is defined and compared to the classical preliminary test estimator.     

On Bayes Linear Unbiased Estimator Under the Balanced Loss Function

In this paper, the Bayes linear unbiased estimator (Bayes LUE) is derived under the balanced loss function. Moreover, the superiority of Bayes LUE over ordinary least square estimator is studied under the mean square error matrix criterion and Pitman closeness criterion. Furthermore, we compare Bayes LUE under the balanced loss function with Bayes LUE under the quadratic loss function.     

Limit theorems for intermediate and central order statistics under nonlinear normalization

In this paper the work of Pancheva (1984) for extreme order statistics under nonlinear normalization is extended to order statistics with variable ranks. Two new results are proved. The first is that under nonlinear normalization, the nondegenerate type (family of types) of the distribution functions with two finite growth points is a possible weak limit of any central order statistic with regular rank sequence. The second result is that the possible nondegenerate weak limits of any central order statistic with regular rank under the traditionally linear normalization and under the power normalization are the same. Finally, the class of all possible weak limits for lower and upper intermediate order statistics is derived under power normalization from the corresponding weak limits of extremes under power normalization.     

Empirical Bayes estimation in continuous one-parameter exponential families under associated samples

In this paper, we study the empirical Bayes (EB) estimation in continuous one-parameter exponential families under negatively associated (NA) samples and positively associated (PA) samples. Under certain regularity conditions, it is shown that the convergence rates of proposed EB estimators under NA or PA samples are the same as those of EB estimators under independent observations, which significantly improve the existing results in EB estimation under associated samples.     

Minimax prediction in finite populations

In this paper, minimax predictors are considered for some population quantities of interest, under some specific superpopulation models. Minimax predictors are derived under a binomial superpopulation model. The well known ratio estimator is shown to be minimax under the usual simple regression normal model. Nonparametric minimax predictors are considered under some models. Some of the predictors considered also shown to be admis s ible.     

Single‐arm phase II trial design under parametric cure models

The current practice of designing single‐arm phase II survival trials is limited under the exponential model. Trial design under the exponential model may not be appropriate when a portion of patients are cured. There is no literature available for designing single‐arm phase II trials under the parametric cure model. In this paper, a test statistic is proposed, and a sample size formula is derived for designing single‐arm phase II trials under a class of parametric cure models. Extensive simulations showed that the proposed test and sample size formula perform very well under different scenarios. Copyright © 2015 John Wiley & Sons, Ltd.     

Some efficient classes of estimators of population mean in two-phase successive sampling under random non response

In this paper, an attempt has been made to present some efficient classes of estimators of population mean on current occasion in two-occasion successive sampling under random non response in two-phase setup. Effectiveness of the proposed classes of estimators has been studied under the assumptions that sampling units follow a distribution under the random non response. To check the performances, the proposed classes of estimators are compared with an estimator under the similar situation under the complete response. Results are demonstrated through empirical studies which show the reliable nature of the proposed classes of estimators. Suitable recommendations have been made to the survey practitioners.     

The small sample properties of the restricted principal component regression estimator in linear regression model

In regression analysis, to deal with the problem of multicollinearity, the restricted principal components regression estimator is proposed. In this paper, we compared the restricted principal components regression estimator, the principal components regression estimator, and the ordinary least-squares estimator with each other under the Pitman's closeness criterion. We showed that the restricted principal components regression estimator is always superior to the principal components regression estimator, under certain conditions the restricted principal components regression estimator is superior to the ordinary least-squares estimator under the Pitman's closeness criterion and under certain conditions the principal components regression estimator is superior to the ordinary least-squares estimator under the Pitman's closeness criterion.     

三个多元正态总体在简单半序约束下均值估计

 多元保序回归理论对统计学中研究多维参数在序约束下的估计理论起着关键性作用。本文讨论了当协方差矩阵已知,在简单半序约束下,对三个多元正态总体均值的估计问题,给出了估计的算法。并证明了在多元均方损失条件下,给出的均值估计优于无序约束的均值估计。     

Testing for changes using permutations of U-statistics

The critical values for various tests based on U-statistics to detect a possible change are obtained through permutations of the observations. We obtain the same approximations for the permutated U-statistics under the no change null hypothesis as well as under the exactly one change alternative. The results are used to show that the simulated critical values are asymptotically valid under the null hypothesis and the tests reject with the probability tending to one under the alternative.     

Shrinkage estimator of regression model under asymmetric loss

This article investigates the performance of the shrinkage estimator (SE) of the parameters of a simple linear regression model under the LINEX loss criterion. The risk function of the estimator under the asymmetric LINEX loss is derived and analyzed. The moment-generating functions and the first two moments of the estimators are also obtained. The risks of the SE have been compared numerically with that of pre-test and least-square estimators (LSEs) under the LINEX loss criterion. The numerical comparison reveals that under certain conditions the LSE is inadmissible, and the SE is the best among the three estimators.     

Admissible minimax estimators for the shape parameter of Topp–Leone distribution

Abstract

The shape parameter of Topp–Leone distribution is estimated in this article from the Bayesian viewpoint under the assumption of known scale parameter. Bayes and empirical Bayes estimates of the unknown parameter are proposed under non informative and suitable conjugate priors. These estimates are derived under the assumption of squared and linear-exponential error loss functions. The risk functions of the proposed estimates are derived in analytical forms. It is shown that the proposed estimates are minimax and admissible. The consistency of the proposed estimates under the squared error loss function is also proved. Numerical examples are provided.     

ON THE SIMPLE PROJECTION PREDICTOR IN FINITE POPULATIONS

In this paper, necessary and sufficient conditions for optimality of the simple projection predictor (SPP) are considered under a superpopula-tion model with a not necessarily diagonal covariance matrix. Conditions under which the SPP remains optimal, under possible covariance matrix misspecifications, are also considered.     

Two-stage U-statistics for Hypothesis Testing

Abstract.  A U -statistic is not easy to apply or cannot be applied in hypothesis testing when it is degenerate or has an indeterminate degeneracy under the null hypothesis. A class of two-stage U -statistics (TU-statistics) is proposed to remedy these drawbacks. Both the asymptotic distributions under the null and the alternative of TU-statistics are shown to have simple forms. When the degeneracy is indeterminate, the Pitman asymptotic relative efficiency of a TU-statistic dominates that of the incomplete U -statistics. If the kernel is degenerate under the null hypothesis but non-degenerate under the alternative, a TU-statistic is proved to be more powerful than its corresponding U -statistic. Applications to testing independence of paired angles in ecology and marine biology are given. Finally, a simulation study shows that a TU-statistic is more powerful than its corresponding incomplete U -statistic in almost all cases under two settings.     

The Bayes rule of the variance parameter of the hierarchical normal and inverse gamma model under Stein's loss

For the variance parameter of the hierarchical normal and inverse gamma model, we analytically calculate the Bayes rule (estimator) with respect to a prior distribution IG (alpha, beta) under Stein's loss function. This estimator minimizes the posterior expected Stein's loss (PESL). We also analytically calculate the Bayes rule and the PESL under the squared error loss. Finally, the numerical simulations exemplify that the PESLs depend only on alpha and the number of observations. The Bayes rules and PESLs under Stein's loss are unanimously smaller than those under the squared error loss.     

Estimation in a random effects growth curve model

This paper considers estimation under the growth curve model of Potthoff and Roy (1964) with random effects. Estimation under a multivariate model is also considered. Estimation under incomplete data and estimation of random effects are also discussed. A numerical example of data on bulls is presented to illustrate these techniques.     

Johansen cointegration rank tests under local alternative hypotheses when related drift or linear trend is not dependent upon sample size in VARs

Abstract

This paper discusses Johansen’s likelihood ratio tests to determine the cointegration rank under local alternative hypotheses in the vector autoregressive models (VARs) in which drift or linear trend related to the hypotheses is not dependent upon the sample size, and evaluates related asymptotic properties. We show that the test statistics diverge at the rate of the sample size even under one of local alternative hypotheses, owing to the existence of such a deterministic term. This implies that under our situations, the tests are far more powerful than those under the conventional local alternative hypotheses.     

Fitting a semi-parametric model based on two sources of information

This paper discusses fitting a semi-parametric model based on two sources of information, motivated by a rat sleep dataset. In a recent study, rats were exposed to two different lighting conditions. The first (baseline) condition was a standard 24-hour schedule of 12 hours lights on, 12 hours lights off; the second (test) condition exposed rats to a continuous 3 hours lights on, 3 hours lights off schedule. Rat sleep was believed to be affected mainly by the circadian rhythm under the baseline lighting condition and by both the circadian rhythm and light under the test lighting condition. This paper suggests fitting a non-parametric model for the dataset under the baseline lighting condition. For the dataset under the test lighting condition, a two-part model is suggested. The first part equals an unknown coefficient multiplied by the non-parametric function used for modelling the dataset under the baseline lighting condition, explaining the remnant of the circadian rhythm under the test lighting condition. The second part is a periodic non-parametric function which would explain the effect of the test lighting condition. This modelling procedure can be used to model other physiological parameters affected by both intrinsic and extrinsic factors.     

Uniform stochastic orderings and total positivity

Uniform stochastic orderings of random variables are expressed as total positivity (TP) of density, survival, and distribution functions. The orderings are called uniform because each is a stochastic order that persists under conditioning to a family of intervals—for example, the family consisting of all intervals of the form (-∞,x]. This paper is concerned with the preservation of uniform stochastic ordering under convolution, mixing, and the formation of coherent systems. A general TP₂ result involving preservation of total positivity under integration is presented and applied to convolutions and mixtures of distribution and survival functions. Log-concavity of distribution, survival, and density functions characterizes distributions that preserve the various orderings under convolution. Likewise, distributions that preserve orderings under mixing are characterized by TP₂ distribution and survival functions.