Strawderman-type estimators for a scale parameter with application to the exponential distribution

It is shown that Strawderman's [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190–198] technique for estimating the variance of a normal distribution can be extended to estimating a general scale parameter in the presence of a nuisance parameter. Employing standard monotone likelihood ratio-type conditions, a new class of improved estimators for this scale parameter is derived under quadratic loss. By imposing an additional condition, a broader class of improved estimators is obtained. The dominating procedures are in form analogous to those in Strawderman [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190–198]. Application of the general results to the exponential distribution yields new sufficient conditions, other than those of Brewster and Zidek [1974. Improving on equivariant estimators. Ann. Statist. 2, 21–38] and Kubokawa [1994. A unified approach to improving equivariant estimators. Ann. Statist. 22, 290–299], for improving the best affine equivariant estimator of the scale parameter. A class of estimators satisfying the new conditions is constructed. The results shed new light on Strawderman's [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190–198] technique.     

A class of modified stein estimators with easily computable risk functions

Consider the problem of estimating the mean of a p (≥3)-variate multi-normal distribution with identity variance-covariance matrix and with unweighted sum of squared error loss. A class of minimax, noncomparable (i.e. no estimate in the class dominates any other estimate in the class) estimates is proposed; the class contains rules dominating the simple James-Stein estimates. The estimates are essentially smoothed versions of the scaled, truncated James-Stein estimates studied by Efron and Morris. Explicit and analytically tractable expressions for their risks are obtained and are used to give guidelines for selecting estimates within the class.     

On the comparison of the pre-test and shrinkage estimators for the univariate normal mean

The estimation of the mean of an univariate normal population with unknown variance is considered when uncertain non-sample prior information is available. Alternative estimators are defined to incorporate both the sample as well as the non-sample information in the estimation process. Some of the important statistical properties of the restricted, preliminary test, and shrinkage estimators are investigated. The performances of the estimators are compared based on the criteria of unbiasedness and mean square error in order to search for a ‘best’ estimator. Both analytical and graphical methods are explored. There is no superior estimator that uniformly dominates the others. However, if the non-sample information regarding the value of the mean is close to its true value, the shrinkage estimator over performs the rest of the estimators. Received: June 19, 1999; revised version: March 23, 2000     

The superharmonic condition for simultaneous estimation of means in exponential familles

The mean vector associated with several independent variates from the exponential subclass of Hudson (1978) is estimated under weighted squared error loss. In particular, the formal Bayes and “Stein-like” estimators of the mean vector are given. Conditions are also given under which these estimators dominate any of the “natural estimators”. Our conditions for dominance are motivated by a result of Stein (1981), who treated the N_p (θ, I) case with p ≥ 3. Stein showed that formal Bayes estimators dominate the usual estimator if the marginal density of the data is superharmonic. Our present exponential class generalization entails an elliptic differential inequality in some natural variables. Actually, we assume that each component of the data vector has a probability density function which satisfies a certain differential equation. While the densities of Hudson (1978) are particular solutions of this equation, other solutions are not of the exponential class if certain parameters are unknown. Our approach allows for the possibility of extending the parametric Stein-theory to useful nonexponential cases, but the problem of nuisance parameters is not treated here.     

Sequential estimators and the Cramér-Rao lower bound

While all nonsequential unbiased estimators of the normal mean have variances which must obey the Cramér-Rao inequality, it is shown that some sequential unbiased estimators do not.     

Back-to-back mixtures of discrete distributions

A method for expanding the choices for fits of discrete data is given. The method is very simple: a breakpoint is chosen for the data set on either side of which two separate discrete distributions are fit. Thus, the method is a mixture of two discrete distributions. The method is appealing in light of the ease with which the likelihood equations simplify. For illustrative purposes, the method is used on the data set that motivated its conception.     

Antipodally symmetric distributions for orientation statistics

The conventional antipodally symmetric Bingham matrix distribution on the Stiefel manifold is generalized. Large sample maximum likelihood estimation and uniformity tests are discussed, and a parametric model for axial orientations (X-shapes) is suggested. A generalization of the Khatri-Mardia matrix distribution is developed to provide a model suitable for hybrids (T-shapes).     

Simple bounds on availability in a model with unknown life and repair distributions

We take a fresh look at the classic model of a device supported by a single statistically identical spare and provision for repairs, with system failure resulting whenever the currently operating unit fails before the repair of the previously failed unit is completed to allow it to become a spare. The limiting availability A(F,G) of this system depends on the life distribution F and repair time distribution G through α=∫GdF and the expected downtime. In this paper we derive several computable and sharp bounds on A(F,G) when F,G have suitable life distribution characteristics in the sense of reliability theory but are otherwise unknown except for at most two moments. Among other results, we find a sharp bound which involves the MTBF, MTTR and the second moment of the life-distribution of the device through its coefficient of variation. This leads to a maximin result for DFR repairs and DMRL lives.     

On the estimation of restricted mean

In sampling from a continuous distribution with unknown mean μ and variance σ² the problem of estimation of μ, when it is known that μ∈(a, ∞) (or μ∈(-∞, b)), is considered. The estimators proposed here lie in the interval (a, ∞) (or (-∞, b)) almost surely. The performance of these estimators is compared to that of some known estimators in the case of sampling from a normal, exponential and a weighted difference of two independent chi-square distributions.     

Characterization of minimax linear estimators in linear regression

Two characterization theorems of the minimax linear estimator (Mile) are proven for the case, where the regression parameter varies only in an arbitrary ellipsoid. Furthermore, the existence, uniqueness and admissibility of Mile are shown. The explicit determination of Mile is carried out for a special case.     

Combining coordinates in simultaneous estimation of normal means

The problem of combining coordinates in Stein-type estimators, when simultaneously estimating normal means, is considered. The question of deciding whether to use all coordinates in one combined shrinkage estimator or to separate into groups and use separate shrinkage estimators on each group is considered. A Bayesian viewpoint is (of necessity) taken, and it is shown that the ‘combined’ estimator is, somewhat surprisingly, often superior.     

On regressing regression coefficients

This paper considers a regression model in which coefficients obtained from a previous regression are themselves the object of analysis. It is shown that the parameters of interest can be obtained in two ways: pooling across observations and subsamples, or a two-stage process of first estimating the coefficients within each subsample, and then using these coefficients as dependent variables in a second stage regression. The relative properties of these estimators are analyzed, and the conditions under which the two estimators are equivalent are derived.     

A comparison of the independent-samples t-test and the paired-samples t-test when the observations are nonnegatively correlated pairs

An account of the behavior of the independent-samples t-test when applied to homoschedastic bivariate normal data is presented, and a comparison is made with the paired-samples t-test. Since the significance level is not violated when applying the independent-samples t-test to data which consist of positively correlated pairs and since the estimate of the variance is based on a larger number of ‘degrees of freedom’, the results suggest that when the sample size is small, one should not worry much about the possible existence of weak positive correlation. One may do better, powerwise, to ignore such correlation and use the independent-samples t-test, as though the samples were independent.     

On the robustness properties for maximum likelihood estimators of parameters in exponential power and generalized T distributions*

Abstract

Examining the robustness properties of maximum likelihood (ML) estimators of parameters in exponential power and generalized t distributions has been considered together. The well-known asymptotic properties of ML estimators of location, scale and added skewness parameters in these distributions are studied. The ML estimators for location, scale and scale variant (skewness) parameters are represented as an iterative reweighting algorithm (IRA) to compute the estimates of these parameters simultaneously. The artificial data are generated to examine performance of IRA for ML estimators of parameters simultaneously. We make a comparison between these two distributions to test the fitting performance on real data sets. The goodness of fit test and information criteria approve that robustness and fitting performance should be considered together as a key for modeling issue to have the best information from real data sets.     

Confidence regions for the common mean vector of several multivariate normal populations

Suppose that there are independent samples available from several multivariate normal populations with the same mean vector m? but possibly different covariance matrices. The problem of developing a confidence region for the common mean vector based on all the samples is considered. An exact confidence region centered at a generalized version of the well-known Graybill-Deal estimator of m? is developed, and a multiple comparison procedure based on this confidence region is outlined. Necessary percentile points for constructing the confidence region are given for the two-sample case. For more than two samples, a convenient method of approximating the percentile points is suggested. Also, a numerical example is presented to illustrate the methods. Further, for the bivariate case, the proposed confidence region and the ones based on individual samples are compared numerically with respect to their expected areas. The numerical results indicate that the new confidence region is preferable to the single-sample versions for practical use.     

On location and scale parameters of exponential distributions with censored observations

In this paper we first address the problem of estimating the common scale of several exponential distributions with unknown location parameters when censored samples are observed. The improved estimators are basically Stein type testimators. These testimators are then used to construct improved estimators of location parameters.     

Some bounds on balanced block designs

Bounds on the latest root of the C-matrix and the number of blocks for a variance-balanced block design are given. These results contain the well-known results as special cases.     

Burden of bioinformatics in medical research: Statistical perspectives and controversies

The ongoing evolution of genomics and bioinformatics has an overwhelming impact on medical and clinical research, albeit this development is often marked by genuine controversies as well as lack of scientific clarities and acumen. The search for disease genes and the gene–environment interaction has drawn considerable interdisciplinary scientific attention: environmental health, clinical and medical sciences, biological as well as computational and statistical sciences are most noteworthy. Statistical reasoning (quantitative modeling and analysis perspectives) has a focal stand in this respect while data mining resolutions are far from being statistically fully understood or interpretable. The use of human subjects, though unavoidable, under various extraneous restraints, medical ethics perspectives, and human rights undercurrents, has raised concern all over the world, especially in the developing countries. In the genomics context, clinical trials may be designed on chips and yet there are greater challenges due to the curse of dimensionality perspectives. Some of these challenging statistical issues in medical and clinical research (with emphasis on clinical trials) are appraised in the light of existing statistical tools, which are available for less complex clinical research problems.     

On a random series in a hilbert space

Let (S_n) be partial sums of a non-degenerate sequence of Identically and independently distributed random variables taking values in a separable Hilbert space. Then for 0 ≤ β ≤ 3/2, the series converges almost nowhere. For β > 3/2 this may not be true.     

Note on the Cramér-Rao inequality in the nonregular case: the family of uniform distributions

We consider the family of uniform distributions with range of unit length. The main result of this note asserts that the average variance of any unbiased estimator of the midpoint of the range is not less than (2(n+1))(n+2))^-1 and this lower bound is sharp. The proof is based upon a nonregular version of the Cramér-Rao inequality.