The multivariate Behrens–Fisher distribution

Applied Mathematics, 2012

This paper deals with a direct derivation of Fisher's information matrix for bivariate Bessel distribution of type I. Some tools for the numerical computation and some tabulations of the Fisher's information matrix are provided.

This paper makes an attempt to justify a multivariate t -model and provides a modest review of most important results of this model developed in recent years. Essential properties and applications of the model in various fields are discussed. Special attention is given to pre-test and shrinkage estimation for regression parameters under certain restrictions. The predictive distributions under the multivariate t -distribution are also discussed. It is observed that the multivariate t -distribition is more convincing to model multivariate data than multivariate normal distribution because of its fat tail.

Journal of Multivariate Analysis, 2013

Birnbaum and Saunders introduced in 1969 a two-parameter lifetime distribution which has been used quite successfully to model a wide variety of univariate positively skewed data. Diaz-Garcia and Leiva-Sanchez [9] proposed a generalized Birnbaum-Saunders distribution by using an elliptically symmetric distribution in place of the normal distribution. Recently, Kundu et al.

This paper reviews most important properties of a location-scale multivariate t-distribution. A conditional representation of the distribution is exploited to outline moments, characteristic function, marginal and conditional distributions, distribution of linear combinations and quadratic forms. Stochastic representation is also used to determine the covariance matrix of the distribution. It also makes an attempt to justify an uncorrelated tmodel and overviews distribution of the sum of products matrix and correlation matrix. Estimation strategies for parameters of the model is briefly discussed. Finally the recent trend of linear regression with the uncorrelated tmodel is discussed.

Communications for Statistical Applications and Methods, 2014

This study derives the characteristic functions of (multivariate/generalized) t distributions without contour integration. We extended Hursts method (1995) to (multivariate/generalized) t distributions based on the principle of randomization and mixtures. The derivation methods are relatively straightforward and are appropriate for graduate level statistics theory courses.

Filomat, 2019

This research paper stands for an extension to the multivariate t?distribution introduced in 1954 by Cornish, Dunnett and Sobel, namely the multiparameter t?distribution. This distribution is expressed in two different ways. The first way invests the mixture of a normal vector with a natural extension to the Wishart distribution, that is the Riesz distribution on symmetric matrices. The second one rests upon the Cholesky decomposition of the Riesz matrix. An algorithm for generating this distribution is investigated using the Riesz distribution arising obtained through not only the distribution of the empirical normal covariance matrix for samples with monotone missing data but also through Cholesky decomposition. In addition, Some fundamentals properties of the multiparameter t?distribution such as the infinite divisibility are identified. Besides, the Expectation Maximization algorithm is used to estimate its parameters. Finally, the performance of these estimators is assessed by ...

Mathematics

The Behrens–Fisher problem occurs when testing the equality of means of two normal distributions without the assumption that the two variances are equal. This paper presents approaches based on the exact and near-exact distributions for the test statistic of the Behrens–Fisher problem, depending on different combinations of even or odd sample sizes. We present the exact distribution when both sample sizes are odd and the near-exact distribution when one or both sample sizes are even. The near-exact distributions are based on a finite mixture of generalized integer gamma (GIG) distributions, used as an approximation to the exact distribution, which consists of an infinite series. The proposed tests, based on the exact and the near-exact distributions, are compared with Welch’s t-test through Monte Carlo simulations, in particular for small and unbalanced sample sizes. The results show that the proposed approaches are competent solutions to the Behrens–Fisher problem, exhibiting preci...

2022

Assumption of normality in statistical analysis had been a common practice in many literature, but in the event where small sample is obtainable, then normality assumption will lead to erroneous conclusion in the statistical analysis. Taking a large sample had been a serious concern in practice due to various factors. In this paper, we further derived some inferential properties for log student’s t-distribution (simply log-t distribution) which makes it more suitable as substitute to log-normal when carrying out analysis on right-skewed small sample data. Mathematical and Statistical properties such as the moments, cumulative distribution function, survival function, hazard function and log-concavity are derived. We further extend the results to case of multivariate log-t distribution; we obtained the marginal and conditional distributions. The parameters estimation was done via maximum likelihood estimation method, consequently its best critical region and information matrix were d...

Journal of Statistical Research, 2006

This paper reviews most important properties of a location-scale multivariate t-distribution. A conditional representation of the distribution is exploited to outline moments, characteristic function, marginal and conditional distributions, distribution of linear combinations and quadratic forms. Stochastic representation is also used to determine the covariance matrix of the distribution. It also makes an attempt to justify an uncorrelated tmodel and overviews distribution of the sum of products matrix and correlation matrix. Estimation strategies for parameters of the model is briefly discussed. Finally the recent trend of linear regression with the uncorrelated tmodel is discussed.

Computational Statistics & Data Analysis, 2013

In this paper, we consider the Bayesian analysis of the Marshall-Olkin bivariate Weibull distribution. It is a singular distribution whose marginals are Weibull distributions. This is a generalization of the Marshall-Olkin bivariate exponential distribution. It is well known that the maximum likelihood estimators of the unknown parameters do not always exist. The Bayes estimators are obtained with respect to the squared error loss function and the prior distributions allow for prior dependence among the components of the parameter vector. If the shape parameter is known, the Bayes estimators of the unknown parameters can be obtained in explicit forms under the assumptions of independent priors. If the shape parameter is unknown, the Bayes estimators cannot be obtained in explicit forms. We propose to use importance sampling method to compute the Bayes estimators and also to construct associated credible intervals of the unknown parameters. The analysis of one data set is performed for illustrative purposes. Finally we indicate the analysis of data sets obtained from series and parallel systems.

Revstat-statistical Journal, 2019

The traditional Behrens–Fisher (B-F) problem is to test the equality of the means µ1 and µ2 of two normal populations using two independent samples, when the quotient of the population variances is unknown. Welch [43] developed a frequentist approximate solution using a fractional number of degrees of freedom t-distribution. We make a a comprehensive review of the existing procedures, propose new procedures, evaluate these for size and power, and make recommendation for the B-F and its analogous problems for non-normal populations. On the other hand, we investigate and answer a question: does the same size fit all all, i.e. is the t-test with Welch’s degree of freedom correction robust enough for the B-F problem analogs, and what sample size is appropriate to use a normal approximation to the Welch statistic.

Journal of Taibah University for Science, 2020

In this article, we propose a new family of distributions, namely, a modified T-X family of distributions. The case of the newly proposed family is advocated with respect to three most attractive features: flexibility, efficiency and parsimony. The statistical features are established through simulation studies. The applicability of the scheme is further assessed by using two diverse data sets. The performance evaluation study is conducted with respect to five existing distributions while considering various goodness-of-fit criteria. Moreover, the estimation of the parameters is delineated under the classic approach and Bayesian framework. Based on the findings of this research, we see our proposition as a more efficient and parsimonious alternative to the existing well-known cubic transmutation family of distributions. Finally, the flexible nature of the devised family of distributions may project as a useful candidate in survival analysis and reliability theory.

2002

Weerahandi (1995b) suggested a generalization of the Fisher’s solution to the Behrens-Fisher problem to the problem of multiple comparisons with unequal variances by the method of generalized p-values. In this paper we present a brief outline of the Fisher’s solution and its generalization as well as the methods to calculate the p-values required for deriving the conservative joint confidence interval estimates for the pairwise mean differences, refered to as the generalized Scheff é int rvals. Further, we present the corresponding tables with critical values for simultaneous comparisons of the mean differences of up to k = 6 normal populations with unequal variances based on independent random samples with very small sample sizes. MSC: primary 62F04; secondary 62E15

Journal of Mathematical Sciences & Computational Mathematics, 2021

Multivariate Behrens-Fisher Problem is a problem that deals with testing the equality of two means from multivariate normal distribution when the covariance matrices are unequal and unknown. However, there is no single procedure served as a better performing solution to this problem, Adebayo (2018). In this study effort is made in selecting five different existing procedures and examined their power and rate to which they control type I error using a different setting and conditions observed from previous studies. To overcome this problem a code was designed via R Statistical Software, to simulate random normal data and independently run 1000 times using MASS package in other to estimate the power and rate at which each procedure control type I error. The simulation result depicts that, in a setting when variance covariance matrices S1 > S2 associated with a sample sizes (n1 > n2) in Table 4.1, 4.2, 4.5, and 4.6, shows that, Adebayos’ procedure performed better but at a sample...

Journal of Multivariate Analysis, 2004

In this paper, we introduce a new family of multivariate distributions as the scale mixture of the multivariate power exponential distribution introduced by Go´mez et al. (Comm. Statist. Theory Methods 27(3) (1998) 589) and the inverse generalized gamma distribution. Since the resulting family includes the multivariate t distribution and the multivariate generalization of the univariate GT distribution introduced by McDonald and Newey (Econometric Theory 18 (11) (1988) 4039) we call this family as the ''multivariate generalized t-distributions family'', or MGT for short. We show that this family of distributions belongs to the elliptically contoured distributions family, and investigate the properties. We give the stochastic representation of a random variable distributed as a multivariate generalized t distribution. We give the marginal distribution, the conditional distribution and the distribution of the quadratic forms. We also investigate the other properties, such as, asymmetry, kurtosis and the characteristic function. r