Matricvariate and matrix multivariate T distributions and associated distributions

Several distributions are studied, simultaneously in the real, complex, quaternion and octonion cases. Specifically, these are the central, nonsingular matricvariate and matrix multivariate T and beta type II distributions and the joint density of the singular values are obtained for real normed division algebras.     

Generalised shape theory via SV decomposition I

This work finds in terms of zonal polynomials, the non isotropic noncentral elliptical shape distributions via singular value decomposition; it avoids the invariant polynomials and the open problems for their computation. The new shape distributions are easily computable and then the inference procedure is based on exact densities, instead of the published approximations and asymptotic distribution of isotropic models. An application of the technique is illustrated with a classical landmark data of Biology, for this, three Kotz type models are proposed (including Gaussian); then the best one is chosen by using a modified BIC criterion.     

Robustness of some non-uniform random variate generators

Often the usefulness of a random variate generator for a distribution is based on the successful generation of independent variates from the uniform distribution. However, no pseudo-random number generator is capable of generating a truly random uniform sequence. As the cost of computations is now dramatically reduced, the computational efficiency should not be the primary criterion in choosing a good generator. Proposed here is a robustness criterion which is designed to study the effect of an imperfect pseudo-random number generator on the accuracy of a random variate generator. In particular, the beta and binomial variate generators are examined for their robustness to the non-uniformity of a pseudo-random number generator.     

Local power functions of tests for double unit roots

The purpose of this paper is to characterize three commonly used double unit root tests in terms of their asymptotic local power. To this end we study a class of nearly doubly integrated processes which in the limit will behave as a weighted integral of a double indexed Ornstein-Uhlenbeck process. Based on a numerical examination of the analytical distributions, a comparison of the tests is made via their asymptotic local power functions.     

Distribution of personal income: Development of a new model and its application to U.S. income data

This paper proposes a four-parameter statistical model of the personalized distribution of income using the ‘income share elasticity’ approach suggested by Esteban (1986). Our proposed model includes the Singh–Maddala (1976) and Dagum (1977) distributions as special cases. The generalized beta II distribution of McDonald (1984) is also a variant of this model. It appears to give an excellent fit to US income data and its empirical performance turns out to be superior to those of the Singh–Maddala (1976), Dagum (1977), the five-parameter Champernowne (1953) distributions and the generalized beta II distribution of McDonald (1984) on some data.     

On Families of Distributions with Shape Parameters

Univariate continuous distributions are one of the fundamental components on which statistical modelling, ancient and modern, frequentist and Bayesian, multi‐dimensional and complex, is based. In this article, I review and compare some of the main general techniques for providing families of typically unimodal distributions on with one or two, or possibly even three, shape parameters, controlling skewness and/or tailweight, in addition to their all‐important location and scale parameters. One important and useful family is comprised of the ‘skew‐symmetric’ distributions brought to prominence by Azzalini. As these are covered in considerable detail elsewhere in the literature, I focus more on their complements and competitors. Principal among these are distributions formed by transforming random variables, by what I call ‘transformation of scale’—including two‐piece distributions—and by probability integral transformation of non‐uniform random variables. I also treat briefly the issues of multi‐variate extension, of distributions on subsets of and of distributions on the circle. The review and comparison is not comprehensive, necessarily being selective and therefore somewhat personal. © 2014 The Authors. International Statistical Review © 2014 International Statistical Institute     

On a noncentral chi-square distribution in the analysis of weapon systems effectiveness

Summary A natural conjugate prior distribution for the parameters involved in the noncentral chi-square leads to many known distributions. The applications of the distributions thus obtained are briefly pointed out in evaluating the ‘kill’ probability in the analysis of weapon systems effectiveness. The ‘kill’ probabilities or the expected coverage are obtained associated with a gamma prior distribution and compared with those obtained byMcnolty. This paper is read in a symposium on Mathematical Sciences held under the auspices of Delhi University, Delhi im January 1966.     

Control variate method for stationary processes

The sample mean is one of the most natural estimators of the population mean based on independent identically distributed sample. However, if some control variate is available, it is known that the control variate method reduces the variance of the sample mean. The control variate method often assumes that the variable of interest and the control variable are i.i.d. Here we assume that these variables are stationary processes with spectral density matrices, i.e. dependent. Then we propose an estimator of the mean of the stationary process of interest by using control variate method based on nonparametric spectral estimator. It is shown that this estimator improves the sample mean in the sense of mean square error. Also this analysis is extended to the case when the mean dynamics is of the form of regression. Then we propose a control variate estimator for the regression coefficients which improves the least squares estimator (LSE). Numerical studies will be given to see how our estimator improves the LSE.     

Recurrence relations for moment and conditional moment generating functions of generalized order statistics

In this paper, we obtain recurrence relations for moment and conditional moment generating functions of generalized order statistics (gos) based on random samples drawn from a population whose distribution is a member of a doubly truncated class of distributions denoted by . Members of the class are characterized in Section (2) based on recurrence relations for moment generating functions (moments) of gos. In Section (3), we shall characterize members of the class based on recurrence relations for conditional moment generating functions (conditional moments) of gos. These results are specialized to the left, right and non-truncated cases. Ordinary order statistics and ordinary record values are also obtained as special cases of the gos. Characterizations of some members of class such as the Weibull, compound Weibull, Pareto, power function (beta is a special case), Gompertz and compound Gompertz distributions are given as illustrative examples.     

Instrumental values

This paper studies identification of partial differences of nonseparable structural functions. A model is defined which admits structural functions exhibiting a degree of monotonicity with respect to a latent variate. The model identifies partial differences when there are instrumental values of covariates over which the latent variate exhibits a local quantile invariance, and a local order condition holds. The result is useful when covariates exhibit discrete variation, as arises often in practice, and when restricting latent variates and covariates to be statistically independent is unpalatable. The results are illustrated with data from the returns-to-schooling study of Angrist and Krueger [1991. Does compulsory schooling attendance affect schooling and earnings? Quarterly Journal of Economics 106, 979–1014].     

A Tutorial on Palm Distributions for Spatial Point Processes

This tutorial provides an introduction to Palm distributions for spatial point processes. Initially, in the context of finite point processes, we give an explicit definition of Palm distributions in terms of their density functions. Then we review Palm distributions in the general case. Finally, we discuss some examples of Palm distributions for specific models and some applications.     

Statistical treatment choice based on asymmetric minimax regret criteria

This paper studies the problem of treatment choice between a status quo treatment with a known outcome distribution and an innovation whose outcomes are observed only in a finite sample. I evaluate statistical decision rules, which are functions that map sample outcomes into the planner’s treatment choice for the population, based on regret, which is the expected welfare loss due to assigning inferior treatments. I extend previous work started by Manski (2004) that applied the minimax regret criterion to treatment choice problems by considering decision criteria that asymmetrically treat Type I regret (due to mistakenly choosing an inferior new treatment) and Type II regret (due to mistakenly rejecting a superior innovation) and derive exact finite sample solutions to these problems for experiments with normal, Bernoulli and bounded distributions of outcomes. The paper also evaluates the properties of treatment choice and sample size selection based on classical hypothesis tests and power calculations in terms of regret.     

Optimal Estimation of the Parameters of the Gompertz Distribution Based on the Doubly Type II Censored Sample

In this paper, we develop exact confidence intervals and exactjoint confidence regions for the parameters of the Gompertzdistribution under the doubly type II censored sample. We alsoprovide optimal criteria for finding a best exact confidenceinterval and a best exact joint confidence region among theseinterval estimations. Finally, we give a numerical example toillustrate our proposed method. Furthermore,when compared to estimation of Chen (1997), our proposedmethod can get a better parameter estimation.     

Bayesian Analysis in the <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Norm of the Mixing Proportion Using Discriminant Analysis

We consider the mixing proportion π in a mixture of two independent distributions, and establish the expression of its posterior density, in closed form and in terms of L ¹-norms of various related functions, using a prior beta and the optimal classification rule for the two populations provided by Discriminant analysis. A numerical example fully illustrates the concepts presented.Research partially supported by CRSNG 9249 (Canada). The authors wish to thank the Faculty of Science and the Department of Statistics of UNISA for their generous support that has led to this joint work. Also, thanks to Ms. Jeannette LeBlanc for her excellent technical support, and to an anonymous referee for very helpful comments that have helped to improve the presentation of the paper.     

Lognormal Distributions and Geometric Averages of Symmetric Positive Definite Matrices

This article gives a formal definition of a lognormal family of probability distributions on the set of symmetric positive definite (SPD) matrices, seen as a matrix‐variate extension of the univariate lognormal family of distributions. Two forms of this distribution are obtained as the large sample limiting distribution via the central limit theorem of two types of geometric averages of i.i.d. SPD matrices: the log‐Euclidean average and the canonical geometric average. These averages correspond to two different geometries imposed on the set of SPD matrices. The limiting distributions of these averages are used to provide large‐sample confidence regions and two‐sample tests for the corresponding population means. The methods are illustrated on a voxelwise analysis of diffusion tensor imaging data, permitting a comparison between the various average types from the point of view of their sampling variability.     

Generalized trapezoidal distributions

We present a construction and basic properties of a class of continuous distributions of an arbitrary form defined on a compact (bounded) set by concatenating in a continuous manner three probability density functions with bounded support using a modified mixture technique. These three distributions may represent growth, stability and decline stages of a physical or mental phenomenon. Revised: April 2002     

Testing outliers from an exponential population

Summary Likes obtains the distributions ofDixon’s statistics for an exponential population and tabulates upper 100α% points (α=0.1, 0.05, 0.01) of some of these distributions. The distribution of these statistics can be expressed in terms of finite series of beta functions and hence the probabilities of the rejection of suspected observed outliers may be easily calculated on a desk calculator. Thus we may avoid the difficult task of tabulating 100α% values of these statistics.     

On approximating the distributions ofFriedman's 75-175-175-1and related statistics

Error bounds depending explicitly on parameters of the problem are given for large-sample approximations to the central and noncentral distributions ofFriedman's [1937]x _r ² statistic,Steel's [1959] statistics for comparingr treatments with a control, andNemenyi's [1963] statistics for pair-wise comparisons among all treatments. These bounds are of ordero(N ^–1/2); an improved bound of ordero (N ^–r/(r+1)) is given forFriedman's statistic in the central case. Applications yield bounds on the actual type 1 and type 2 error probabilities in terms of their normal-theory approximations.Supported in part by ONR Contract N00014-72A-0136-003 from the U.S. Office of Naval Research.     

Goodness-of-fit tests for long memory moving average marginal density

This paper addresses the problem of fitting a known density to the marginal error density of a stationary long memory moving average process when its mean is known and unknown. In the case of unknown mean, when mean is estimated by the sample mean, the first order difference between the residual empirical and null distribution functions is known to be asymptotically degenerate at zero, and hence can not be used to fit a distribution up to an unknown mean. In this paper we show that by using a suitable class of estimators of the mean, this first order degeneracy does not occur. We also investigate the large sample behavior of tests based on an integrated square difference between kernel type error density estimators and the expected value of the error density estimator based on errors. The asymptotic null distributions of suitably standardized test statistics are shown to be chi-square with one degree of freedom in both cases of the known and unknown mean. In addition, we discuss the consistency and asymptotic power against local alternatives of the density estimator based test in the case of known mean. A finite sample simulation study of the test based on residual empirical process is also included.     

Relative information functions and their type (α, β) generalizations

The concept of relative information functions by means of a functional equation under suitable boundary conditions is introduced and discussed. Further the relative information is defined in terms of these functions and this idea is extended to the new concept of relative information functions of type (α, β) (α, β>0, α≠β) and relative information of type (α,β) which are the generalizations of the earlier ones. Finally the different types of relative informations of type (α, β) are defined for bivariate distributions and the relations between them are established.