Two-stage james-stein estimators of the mean based on prior knowledge

We suggest five types of two-stage James-Stein type estimators of the mean vector μ based on prior knowledge about μ and two-stage sampling scheme proposed by Waikar and Katti(1971) Their risks are evaluated and calculated to compare with two-stage estimator suggested by Waikar and Katti(1971) when the prior form of an initial estimate of μ is 0. We find that the five estimators suggested here all have high efficiencies in large dimensions and/or in large value of ratio of two sample sizes at each stage.     

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.     

Improved estimation of the mean vector for student-t model

Improved James-Stein type estimation of the mean vector μ of a multovaroate Student-t population of dimension p with ν degrees of freedom is considered. In addition to the sample data, uncertain prior information on the value of the mean vector, in the form of a null hypothesis, is used for the estiamtion. The usual maximum liklihood estimator((mle) of μ is obtained and a test statistic for testing H₀:μ=μ₀ is derived. Based on the mle of μ and the tes statistic the preliminary test estimator (PTE), Stein-type shrinkage estimator (SE) and positive-rule shrinkage esiimator (PRSE) are defined. The bias and the quadratic risk of the estimators are evaiuated. The relative performances of the estimators are mvestigated by analyzing the risks under different condltlons It is observed that the FRSE dommates over he other three estimators, regardless of the vaiidity of the null hypothesis and the value ν.     

A Note on the Comparison of the Stein Estimator and the James-Stein Estimator

The seminal work of Stein (1956 Stein, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. Proc. Third Berkeley Symp. Mathemat. Statist. Probab., University of California Press, 1:197–206. [Google Scholar]) showed that the maximum likelihood estimator (MLE) of the mean vector of a p-dimensional multivariate normal distribution is inadmissible under the squared error loss function when p ? 3 and proposed the Stein estimator that dominates the MLE. Later, James and Stein (1961 James, W., Stein, C. (1961). Estimation with quadratic loss. Proc. Fourth Berkeley Symp. Mathemat. Statist. Probab., University of California Press, 1:361–379. [Google Scholar]) proposed the James-Stein estimator for the same problem and received much more attention than the original Stein estimator. We re-examined the Stein estimator and conducted an analytic comparison with the James-Stein estimator. We found that the Stein estimator outperforms the James-Stein estimator under certain scenarios and derived the sufficient conditions.     

On a generalized stein estimator of regression coefficients

This paper studies a generalized Stein estimator of regression coefficients. The small disturbance approximations for the bias and mean square error matrix of the estimator are derived and a necessary and sufficient condition is obtained for the estimator to dominate the ordinary least squares estimator under the mean square error criterion.     

Comparison of the Stein and the usual estimators for the regression error variance under the Pitman nearness criterion when variables are omitted

This paper compares the Stein and the usual estimators of the error variance under the Pitman nearness (PN) criterion in a regression model which is mis-specified due to missing relevant explanatory variables. The exact expression of the PN-probability is derived and numerically evaluated. Contrary to the well-known result under mean squared errors (MSE), with the PN criterion the Stein variance estimator is uniformly dominated by the usual estimator when no relevant variables are excluded from the model. With an increased degree of model mis-specification, neither estimator strictly dominates the other. The authors are grateful to two anonymous referees for their valuable comments. Also, the first author is grateful to the Japan Society for the Promotion of Science for partial financial support.     

Small confidence sets for the mean of a spherically symmetric distribution

Summary.  Suppose that X has a k -variate spherically symmetric distribution with mean vector θ and identity covariance matrix. We present two spherical confidence sets for θ , both centred at a positive part Stein estimator     . In the first, we obtain the radius by approximating the upper α -point of the sampling distribution of     by the first two non-zero terms of its Taylor series about the origin. We can analyse some of the properties of this confidence set and see that it performs well in terms of coverage probability, volume and conditional behaviour. In the second method, we find the radius by using a parametric bootstrap procedure. Here, even greater improvement in terms of volume over the usual confidence set is possible, at the expense of having a less explicit radius function. A real data example is provided, and extensions to the unknown covariance matrix and elliptically symmetric cases are discussed.     

Nonparametric Shrinkage Estimation for Aalen's Additive Hazards Model

Aalen's nonparametric additive model in which the regression coefficients are assumed to be unspecified functions of time is a flexible alternative to Cox's proportional hazards model when the proportionality assumption is in doubt. In this paper, we incorporate a general linear hypothesis into the estimation of the time‐varying regression coefficients. We combine unrestricted least squares estimators and estimators that are restricted by the linear hypothesis and produce James‐Stein‐type shrinkage estimators of the regression coefficients. We develop the asymptotic joint distribution of such restricted and unrestricted estimators and use this to study the relative performance of the proposed estimators via their integrated asymptotic distributional risks. We conduct Monte Carlo simulations to examine the relative performance of the estimators in terms of their integrated mean square errors. We also compare the performance of the proposed estimators with a recently devised LASSO estimator as well as with ridge‐type estimators both via simulations and data on the survival of primary billiary cirhosis patients.     

Estimation of the mean of a spherically symmetric distribution with constraints on the norm

The authors consider the problem of estimating, under quadratic loss, the mean of a spherically symmetric distribution when its norm is supposed to be known and when a residual vector is available. They give a necessary and sufficient condition for the optimal James‐Stein estimator to dominate the usual estimator. Various examples are given that are not necessarily variance mixtures of normal distributions. Consideration is also given to an alternative class of robust James‐Stein type estimators that take into account the residual vector. A more general domination condition is given for this class.     

Stein-type estimation using ranked set sampling

In this study, we consider the application of the James–Stein estimator for population means from a class of arbitrary populations based on ranked set sample (RSS). We consider a basis for optimally combining sample information from several data sources. We succinctly develop the asymptotic theory of simultaneous estimation of several means for differing replications based on the well-defined shrinkage principle. We showcase that a shrinkage-type estimator will have, under quadratic loss, a substantial risk reduction relative to the classical estimator based on simple random sample and RSS. Asymptotic distributional quadratic biases and risks of the shrinkage estimators are derived and compared with those of the classical estimator. A simulation study is used to support the asymptotic result. An over-riding theme of this study is that the shrinkage estimation method provides a powerful extension of its traditional counterpart for non-normal populations. Finally, we will use a real data set to illustrate the computation of the proposed estimators.     

Semiparametric Stein estimators

Stein [Stein, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. In Proc. 3rd Berkeley symp. math. statist. and pro. (pp. 197–206). University of California Press], in his seminal paper, came up with the surprising discovery that the sample mean is an inadmissible estimator of the population mean in three or higher dimensions under squared error loss. The past five decades have witnessed multiple extensions and variations of Stein’s results. In this paper we develop Stein-type estimators in a semiparametric framework and prove their coordinatewise asymptotic dominance over the sample mean in terms of Bayes risks.     

A two-step robust estimation of the process mean using M-estimator

Parameter estimation is the first step in constructing control charts. One of these parameters is the process mean. The classical estimators of the process mean are sensitive to the presence of outlying data and subgroups which contaminate the whole data. In existing robust estimators for the process mean, the effects of the presence of the individual outliers are being considered, while, in this paper, a robust estimator is being proposed to reduce the effect of outlying subgroups as well as the individual outliers within a subgroup. The proposed estimator was compared with some classical and robust estimators of the process mean. Although, its relative efficiency is fourth among the estimators tested, its robustness and efficiency are large when the outlying subgroups are present. Evaluation of the results indicated that the proposed estimator is less sensitive to the presence of outliers and the process mean performs well when there are no individual outliers or outlying subgroups.     

On the mean squared error of nonparametric quantile estimators under random right-censorship

For randomly right-censored data, new asymptotic expressions for the mean squared errors of the product-limit quantile estimator and a kernel-type quantile estimator are presented in this paper. From these results a comparison of the two quantile estimators with respect to their mean squared errors is given.     

An explicit formula for the risk of James-Stein estimators

Expressions for the risk of a Stein estimator and its principal derivatives involve a well-defined. but nonevaluated. expectation term. Stein (1966) has suggested a second-order approximation. In this paper we present an alternative exact expression.     

The occurrence of fingerprint characteristics as a two-dimensional poisson process

The individuality of n fingerprint is based on the configuration of occurences of the ten Galton characteristics ( ridge endings, forks, etc. ). A model ( Osterburg, Parthasarthy, Raghavan, Sclove, 1977 ) for the occurence of these characteristics, in terms of a grid of cells, is further developed. The occurence of the characteristics is modelled as a two-dimensional multivariate Poisson process. This approach allows one to treat multiple occurrences in a more satisfying way than in Osterburg, Parthasarathy, Raghavan and Sclove ( 1977 ) or Sclove ( 1978 )     

Comparison of estimation methods for the finite population mean in simple random sampling: symmetric super-populations

In this paper, a new estimator combined estimator (CE) is proposed for estimating the finite population mean ¯ Y _N in simple random sampling assuming a long-tailed symmetric super-population model. The efficiency and robustness properties of the CE is compared with the widely used and well-known estimators of the finite population mean ¯ Y _N by Monte Carlo simulation. The parameter estimators considered in this study are the classical least squares estimator, trimmed mean, winsorized mean, trimmed L-mean, modified maximum-likelihood estimator, Huber estimator (W24) and the non-parametric Hodges–Lehmann estimator. The mean square error criteria are used to compare the performance of the estimators. We show that the CE is overall more efficient than the other estimators. The CE is also shown to be more robust for estimating the finite population mean ¯ Y _N , since it is insensitive to outliers and to misspecification of the distribution. We give a real life example.     

A comparison of various estimators of the mean of an inverse gaussian distribution

In this paper we consider the Inverse Gaussian distribution whose variance is proportional to the mean. Assuming that the data are available from IGD(,μ,c,μ ²), and also from its length biased version, simulation studies are presented to compare the MVUE and MLE in terms of their variances and mean square errors from both kinds of data. Some tables and graphs are provided to analyze the comparisons. Finally, some recommendations and conclusions are given when one or both kinds of data are available.     

Estimating a positive normal mean

Let X ₁, X ₂, ..., X _n be a random sample from a normal population with mean μ and variance σ ². In many real life situations, specially in lifetime or reliability estimation, the parameter μ is known a priori to lie in an interval [a, ∞). This makes the usual maximum likelihood estimator (MLE) ̄ an inadmissible estimator of μ with respect to the squared error loss. This is due to the fact that it may take values outside the parameter space. Katz (1961) and Gupta and Rohatgi (1980) proposed estimators which lie completely in the given interval. In this paper we derive some new estimators for μ and present a comparative study of the risk performance of these estimators. Both the known and unknown variance cases have been explored. The new estimators are shown to have superior risk performance over the existing ones over large portions of the parameter space.     

Improved estimation of a finite population mean based on paired observations

This paper considers an improvement of the customary estimator of a finite population mean under a single stage sampling design when paired data, are available on each unit of the sample. Guided by the well known problem of “corninon mean”, a mixture i.e. a weighted combination of the mean of the principal characteristic and that of the auxiliary (possibly transformed) characteristic is proposed. It is shown that, under some conditions, improveinent (with respect to MSE) over the traditional estimator is possible for a broad range of the values of the mixing constant. An estimator of the MSE of the proposed estimator is also provided.     

General procedure for estimating the population mean using ranked set sampling

In this paper, we suggest a class of estimators for estimating the population mean ? of the study variable Y using information on X?, the population mean of the auxiliary variable X using ranked set sampling envisaged by McIntyre [A method of unbiased selective sampling using ranked sets, Aust. J. Agric. Res. 3 (1952), pp. 385–390] and developed by Takahasi and Wakimoto [On unbiased estimates of the population mean based on the sample stratified by means of ordering, Ann. Inst. Statist. Math. 20 (1968), pp. 1–31]. The estimator reported by Kadilar et al. [Ratio estimator for the population mean using ranked set sampling, Statist. Papers 50 (2009), pp. 301–309] is identified as a member of the proposed class of estimators. The bias and the mean-squared error (MSE) of the proposed class of estimators are obtained. An asymptotically optimum estimator in the class is identified with its MSE formulae. To judge the merits of the suggested class of estimators over others, an empirical study is carried out.