Estimating risk reduction in stein estimation

It is shown that the unbiased estimator of the risk reduction in Stein estimation is unsatisfactory from a mean-squared-error point of view. A truncated form of the unbiased estimator and various empirical Bayes estimators of the risk reduction are shown to perform much better than the unbiased estimator. A simple practical estimator is proposed whose performance is a compromise between that of the truncated and empirical Bayes estimators.     

A formal bayes multiple shrinkage estimator

A new minimax multiple shrinkage estimator is constructed. This estimator which can adaptively shrink towards many subspace targets, is formal Bayes with respect to a mixture of harmonic priors. Unbiased estimates of risk and simulation results suggest that the risk properties of this estimator are very similar to those of the multiple shrinkage Stein estimator proposed by George (1986a). A special case is seen to be admissible.     

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.     

A note on estimating the common mean of k normal distributions and the stein problem

An unbiased estimator for the common mean of k normal distributions is suggested. A necessary and sufficient condition for the estimator Lo have a smaller variance than each sample mean is given. In the case of estimating the common mean vector of k p-variate (p ≤ 3) normal distributions a combined unbiased estimator may be used. We give a class of estimators which are better than the combined estimator when the loss is quadratic and the restriction of unbiasedness is removed.     

Limit expressions for the risk of james-stein estimators

Limit expressions (as the dimension p ← ∞ ) are derived for the relative risk of the James-Stein estimator and its positive-part version. The limit is simple to evaluate, and gives the amount of improvement in risk that is possible. The technique used is to bound the risk, both above and below. with bounds that converge to the same limit. For the James-Stein estimator these bounds are simple to calculate, and are quite accurate even for moderate dimensions.     

Group decision analysis and its application to combining opinions

This paper is about the graphical depiction of the set of feasible gains-in-utilities accruing to three Bayesians involved in the joint estimation of a multivatiate normal mean vector. The basic theory for this problem is sketched. Then a suitable parametrization of surface contours is developed. These contours allow the surface to be mapped and graphically displayed. This is done with the LIG language for interactive graphics. As the opinions of the three Bayesians diverge, the illustrations contained in the paper show how the initially smooth balloon shaped structure develops and ‘clicks’ through a flat spot and eventually becomes highly irregular with a central indentation. The result provides insight into the nature of disensus where explicit mathematical analysis is extremely difficult.     

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.     

Confidence intervals for the largest mean from k correlated normal populations

Let X= (X₁,…, X_k)’ be a k-variate (k ≥ 2) normal random vector with unknown population mean vector μ = (μ₁ ,…, μ_k)’ and covariance matrix Σ of order k and let μ_[1] ≤ … ≤ μ_[k] be the ordered values of the μ ’ s. No prior knowledge of the pairing of the μ_[i] with the X_j. (or μ_[i] with the σ_j ²) is assumed for any i and j (1 ≤ i, j ≤ k). Based on a random sample of N independent vector observations on X, this paper considers both upper and lower (one-sided) and two-sided 100γ% (0 < γ < 1) confidence intervals for μ_[k] and μ_[1], the largest and the smallest mean, respectively, when Σ is known and when Σ is equal to σ²R with common unknown variance σ² > 0 and correlation matrix R known, respectively. An optimum two-sided confidence interval via finding the shortest length from this class is also considered. Necessary tables and computer program to actually apply these procedures are provided.     

Confidence sets of fixed size with predetermined confidence level

In this paper a three stage method to obtain a confidence sets of given form and size, with a preassigned confidence coefficient, is suggested for the case of normally distributed observations from populations with different means and common unknown variance. The method is quite general, and it allows for most common forms of confidence sets. Asymptotic properties of the random sample sizes of the method are studied and compared to the same properties of other methods with the same aim.     

Confidence sets in a linear regression model for interval data

The construction of confidence sets for the parameters of a flexible simple linear regression model for interval-valued random sets is addressed. For that purpose, the asymptotic distribution of the least-squares estimators is analyzed. A simulation study is conducted to investigate the performance of those confidence sets. In particular, the empirical coverages are examined for various interval linear models. The applicability of the procedure is illustrated by means of a real-life case study.     

Simultaneous Confidence Tubes in Multivariate Linear Regression

Simultaneous confidence bands have been shown in the statistical literature as powerful inferential tools in univariate linear regression. While the methodology of simultaneous confidence bands for univariate linear regression has been extensively researched and well developed, no published work seems available for multivariate linear regression. This paper fills this gap by studying one particular simultaneous confidence band for multivariate linear regression. Because of the shape of the band, the word ‘tube’ is more pertinent and so will be used to replace the word ‘band’. It is shown that the construction of the tube is related to the distribution of the largest eigenvalue. A simulation‐based method is proposed to compute the 1 ? α quantile of this eigenvalue. With the computation power of modern computers, the simultaneous confidence tube can be computed fast and accurately. A real‐data example is used to illustrate the method, and many potential research problems have been pointed out.     

Estimation of parameters and reliability from generalized life models

Simultaneous estimation of parameters with p (≥ 2) components, where each component has a generalized life distribution, is considered under a sum of squared error loss function. Improved estimators are obtained which dominate the maximum likelihood and the niinimum mean square estimators. Robustness of the improved estimators is shown even when the component distributions are dependent. The result is extended to the estimation of the system reliability when the components are connected in series. Several numerical studies are performed to demonstrate the risk improvement and the Pitman closeness of the new estimators.     

Confidence Intervals for Reliability of Stress-Strength Models in the Normal Case

In this article, we propose some procedures to get confidence intervals for the reliability in stress-strength models. The confidence intervals are obtained either through a parametric bootstrap procedure or using asymptotic results, and are applied to the particular context of two independent normal random variables. The performance of these estimators and other known approximate estimators are empirically checked through a simulation study which considers several scenarios.     

Simultaneous estimation of several CDF’s: homogeneity constraint

Let {x_ij(1 ? j ? n_i)|i = 1, 2, …, k} be k independent samples of size n_j from respective distributions of functions F_j(x)(1 ? j ? k). A classical statistical problem is to test whether these k samples came from a common distribution function, F(x) whose form may or may not be known. In this paper, we consider the complementary problem of estimating the distribution functions suspected to be homogeneous in order to improve the basic estimator known as “empirical distribution function” (edf), in an asymptotic setup. Accordingly, we consider four additional estimators, namely, the restricted estimator (RE), the preliminary test estimator (PTE), the shrinkage estimator (SE), and the positive rule shrinkage estimator (PRSE) and study their characteristic properties based on the mean squared error (MSE) and relative risk efficiency (RRE) with tables and graphs. We observed that for k ? 4, the positive rule SE performs uniformly better than both shrinkage and the unrestricted estimator, while PTEs works reasonably well for k < 4.     

A note on james-stein and bayes empiricl bayes estimators

In an empirical Bayes decision problem, a simple class of estimators is constructed that dominate the James-Stein

estimator, A prior distribution A is placed on a restricted (normal) class G of priors to produce a Bayes empirical Bayes estimator, The Bayes empirical Bayes estimator is smooth, admissible, and asymptotically optimal. For certain A rate of convergence to minimum Bayes risk is 0(n^-1)uniformly on G. The results of a Monte Carlo study are presented to demonstrate the favorable risk bebhavior of the Bayes estimator In comparison with other competitors including the James-Stein estimator.     

Profiles identification on hierarchical tree structure data sets

In this work we study a way to explore and extract more information from data sets with a hierarchical tree structure. We propose that any statistical study on this type of data should be made by group, after clustering. In this sense, the most adequate approach is to use the Mahalanobis–Wasserstein distance as a measure of similarity between the cases, to carry out clustering or unsupervised classification. This methodology allows for the clustering of cases, as well as the identification of their profiles, based on the distribution of all the variables that characterises each subject associated with each case. An application to a set of teenagers' interviews regarding their habits of communication is described. The interviewees answered several questions about the kind of contacts they had on their phone, Facebook, email or messenger as well as the frequency of communication between them. The results indicate that the methodology is adequate to cluster this kind of data sets, since it allows us to identify and characterise different profiles from the data. We compare the results obtained with this methodology with the ones obtained using the entire database, and we conclude that they may lead to different findings.     

Non-parametric Estimation of the Death Rate in Branching Diffusions

We consider finite systems of diffusing particles in with branching and immigration. Branching of particles occurs at position dependent rate. Under ergodicity assumptions, we estimate the position-dependent branching rate based on the observation of the particle process over a time interval [0, t ]. Asymptotics are taken as t  → ∞. We introduce a kernel-type procedure and discuss its asymptotic properties with the help of the local time for the particle configuration. We compute the minimax rate of convergence in squared-error loss over a range of Hölder classes and show that our estimator is asymptotically optimal.     

Estimating a multivariate treatment effect under a biased allocation rule

In a clinical trial with a biased allocation rule whereby all and only those patients at risk are given the new treatment, Robbins and Zhang (1989) derived an asymptotically normal and efficient estimator of the mean difference between the new and old treatments on those at risk. This paper with the use of a well known identity of Stein (1981) generalizes the result to the multivariate situation.     

Incorporating historical controls using a random-effects model with a normal prior

Employing a variance stabilization transformation, a random-effects model with a normal prior is proposed for incorporating historical controls in the estimation of dose-response relationships. The advantages and disadvantages of maximum likelihood and empirical Bayes estimators are discussed. A two-stage method is introduced for the prudent use of historical controls.