A theorem on invariance of estimability of linear parametric functions in linear models

Consider the general linear model Y = Xβ + ? , where E[??'] = σ²I and rank of X is less than or equal to the number of columns of X. It is well known that the linear parametric function λ'β is estimable if and only if λ' is in the row space of X. This paper characterizes all orthogonal matrices P such that the row space of XP is equal to the row space of X, i.e. the estimability of λ'β is invariant under P. An additional property of these matrices is the invariance of the spectrum of the information matrix X'X. An application of the results is also given.     

Admissibility of Test Procedures for Linear Hypotheses in General Linear Model

This article shows that an F-test procedure is admissible for testing a linear hypothesis concerning one of the split mean vectors in a general linear model and an F-test procedure is also admissible for testing a linear hypothesis concerning another of the split mean vectors in the same model. These results are proved by showing that the critical functions of the tests are unique Bayes procedures with respect to proper prior distributions set in common for the null hypotheses and for the alternative ones, respectively.     

Accent on Teaching Materials

Estimation of covariance components in the multivariate random-effect model with nested covariance structure is discussed. There are two covariance matrices to be estimated, namely, the between-group and the within-group covariance matrices. These two covariance matrices are most often estimated by forming a multivariate analysis of variance and equating mean square matrices to their expectations. Such a procedure involves taking the difference between the between-group mean square and the within-group mean square matrices, and often produces an estimated between-group covariance matrix that is not nonnegative definite. We present estimators of the two covariance matrices that are always proper covariance matrices. The estimators are the restricted maximum likelihood estimators if the random effects are normally distributed. The estimation procedure is extended to more complicated models, including the twofold nested and the mixed-effect models. A numerical example is presented to illustrate the use of the estimation procedure.     

On multiple constraint skewed models

The Azzalini [A. Azzalini, A class of distributions which includes the normal ones, Scandi. J. Statist. 12 (1985), pp. 171–178.] skew normal model can be viewed as one involving normal components subject to a single linear constraint. As a natural extension of this model, we discuss skewed models involving multiple linear and nonlinear constraints and possibly non-normal components. Particular attention is devoted to a distribution called the extended two-piece normal (ETN) distribution. This model is a two-constraint extension of the two-piece normal model introduced by Kim [H.J. Kim, On a class of two-piece skew normal distributions, Statistics 39(6) (2005), pp. 537–553.]. Likelihood inference for the ETN distribution is developed and illustrated using two data sets.     

On minimaxity of the normal precision matrix estimator of Krishnamoorthy and Gupta

We consider the orthogonally invariant estimation problem of the inverse of the scale matrix of Wishart distribution using Stein's loss (entropy loss). In this problem Krishnamoorthy and Gupta [2] Krishnamoorthy, K. and Gupta, A. K. (1989). Improved minimax estimation of a normal precision matrix. Canad. J. Statist., 17: 91–102. [Crossref], [Web of Science ®] , [Google Scholar] proposed an estimator and showed its good performance in a Monte Carlo simulation. They conjectured their estimator is minimax. Perron [3] Perron, F. (1997). On a conjecture of Krishnamoorthy and Gupta. J. Multivariate Anal., 62: 110–120.  [Google Scholar] proved its minimaxity for p?=?2. In this paper we prove it for p?=?3 by using a new method.     

Analysis of hybrid censored competing risks data

In this paper, we consider the analysis of hybrid censored competing risks data, based on Cox's latent failure time model assumptions. It is assumed that lifetime distributions of latent causes of failure follow Weibull distribution with the same shape parameter, but different scale parameters. Maximum likelihood estimators (MLEs) of the unknown parameters can be obtained by solving a one-dimensional optimization problem, and we propose a fixed-point type algorithm to solve this optimization problem. Approximate MLEs have been proposed based on Taylor series expansion, and they have explicit expressions. Bayesian inference of the unknown parameters are obtained based on the assumption that the shape parameter has a log-concave prior density function, and for the given shape parameter, the scale parameters have Beta–Gamma priors. We propose to use Markov Chain Monte Carlo samples to compute Bayes estimates and also to construct highest posterior density credible intervals. Monte Carlo simulations are performed to investigate the performances of the different estimators, and two data sets have been analysed for illustrative purposes.     

The beta generalized gamma distribution

For the first time, a new five-parameter distribution, called the beta generalized gamma distribution, is introduced and studied. It contains at least 25 special sub-models such as the beta gamma, beta Weibull, beta exponential, generalized gamma (GG), Weibull and gamma distributions and thus could be a better model for analysing positive skewed data. The new density function can be expressed as a linear combination of GG densities. We derive explicit expressions for moments, generating function and other statistical measures. The elements of the expected information matrix are provided. The usefulness of the new model is illustrated by means of a real data set.     

Design-based random permutation models with auxiliary information

We extend the random permutation model to obtain the best linear unbiased estimator of a finite population mean accounting for auxiliary variables under simple random sampling without replacement (SRS) or stratified SRS. The proposed method provides a systematic design-based justification for well-known results involving common estimators derived under minimal assumptions that do not require specification of a functional relationship between the response and the auxiliary variables.     

Bayes designs for multiple linear regression on the unit sphere

We deal with experimental designs minimizing the mean square error of the linear BAYES estimator for the parameter vector of a multiple linear regression model where the experimental region is the k-dimensional unit sphere. After computing the uniquely determined optimum information matrix, we construct, separately for the homogeneous and the inhomogeneous model, both approximate and exact designs having such an information matrix.     

A Comparison of Multiple Outlier Detection Methods for Regression Data

The problem of outliers in statistical data has attracted many researchers for a long time. Consequently, numerous outlier detection methods have been proposed in the statistical literature. However, no consensus has emerged as to which method is uniformly better than the others or which one is recommended for use in practical situations. In this article, we perform an extensive comparative Monte Carlo simulation study to assess the performance of the multiple outlier detection methods that are either recently proposed or frequently cited in the outlier detection literature. Our simulation experiments include a wide variety of realistic and challenging regression scenarios. We give recommendations on which method is superior to others under what conditions.