scholarly journals Objective priors for the number of degrees of freedom of a multivariate t distribution and the t-copula

2018 ◽  
Vol 124 ◽  
pp. 197-219 ◽  
Author(s):  
Cristiano Villa ◽  
Francisco J. Rubio
Keyword(s):  
Degrees Of Freedom ◽  
Multivariate T Distribution ◽  
T Copula ◽  
T Distribution ◽  
Multivariate T

Some Extensions of the Multivariate t-Distribution and the Multivariate Generalization of the Distribution of the Regression Coefficient

1961 ◽  
Vol 57 (1) ◽  
pp. 80-85 ◽  
Author(s):  
A. M. Kshirsagar
Keyword(s):  
Normal Distribution ◽  
Degrees Of Freedom ◽  
Regression Coefficient ◽  
Bivariate Normal Distribution ◽  
Practical Applications ◽  
Multivariate T Distribution ◽  
The Matrix ◽  
Pre Treatment ◽  
T Distribution ◽  
Multivariate T

If the components x1, x2,…, xk of a vector X have a non-singular multivariate normal distribution having a null vector of means and variance-covariance matrix Σ= σ2, the matrix R=[ρij] (where ρii = 1) is known in certain cases but σ2 is unknown. If s2 is an estimate of σ2 based on ƒ degrees of freedom and is distributed independently of X, the distribution of the vector t=x/s is known as the multivariate t-distribution. This distribution was first obtained by Dunnett and Sobel (6) and independently by Cornish (3). Dunnett, Sobel and Bechhofer(2) have discussed some practical applications of this distribution. Cornish (3) obtained this distribution while considering the pre-treatment to be given to certain types of replicated experiments. This distribution possesses some useful properties and makes it suitable as a basis for exact tests of significance in various problems, and Dunnett and Sobel (6), by providing tables of the probability integral, have taken the first step towards its use in practice. Cornish, in a later paper (4) considered the sampling distribution of statistics derived from the multivariate t-distribution and using this he obtained the well-known ((7), (8)) distribution of the sample regression coefficient of one variate with respect to another, when both have a bivariate normal distribution.

Missing data imputation in multivariate t distribution with unknown degrees of freedom using expectation maximization algorithm and its stochastic variants

2020 ◽  
Vol 15 (3) ◽  
pp. 263-272
Author(s):  
Paul Kimani Kinyanjui ◽  
Cox Lwaka Tamba ◽  
Luke Akong’o Orawo ◽  
Justin Obwoge Okenye
Keyword(s):  
Parameter Estimation ◽  
Missing Data ◽  
Expectation Maximization ◽  
Degrees Of Freedom ◽  
Data Imputation ◽  
Missing Data Imputation ◽  
Multivariate T Distribution ◽  
Monte Carlo Em ◽  
T Distribution ◽  
Multivariate T

Many researchers encounter the missing data problem. The phenomenon may be occasioned by data omission, non-response, death of respondents, recording errors, among others. It is important to find an appropriate data imputation technique to fill in the missing positions. In this study, the Expectation Maximization (EM) algorithm and two of its stochastic variants, stochastic EM (SEM) and Monte Carlo EM (MCEM), are employed in missing data imputation and parameter estimation in multivariate t distribution with unknown degrees of freedom. The imputation efficiencies of the three methods are then compared using mean square error (MSE) criterion. SEM yields the lowest MSE, making it the most efficient method in data imputation when the data assumes the multivariate t distribution. The algorithm’s stochastic nature enables it to avoid local saddle points and achieve global maxima; ultimately increasing its efficiency. The EM and MCEM techniques yield almost similar results. Large sample draws in the MCEM’s E-step yield more or less the same results as the deterministic EM. In parameter estimation, it is observed that the parameter estimates for EM and MCEM are relatively close to the simulated data’s maximum likelihood (ML) estimates. This is not the case in SEM, owing to the random nature of the algorithm.

Robust Curve Clustering Based on a Multivariate $t$-Distribution Model

2010 ◽  
Vol 21 (12) ◽  
pp. 1976-1984 ◽  
Author(s):  
Zhi Min Wang ◽  
Qing Song ◽  
Yeng Chai Soh ◽  
Kang Sim
Keyword(s):  
Distribution Model ◽  
Multivariate T Distribution ◽  
Curve Clustering ◽  
T Distribution ◽  
Multivariate T

scholarly journals Cumulants of Multivariate Symmetric and Skew Symmetric Distributions

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1383
Author(s):  
Sreenivasa Rao Jammalamadaka ◽  
Emanuele Taufer ◽  
Gyorgy H. Terdik
Keyword(s):  
Comprehensive Treatment ◽  
Multivariate Distributions ◽  
Symmetric Distributions ◽  
Multivariate T Distribution ◽  
T Distribution ◽  
Multivariate T

This paper provides a systematic and comprehensive treatment for obtaining general expressions of any order, for the moments and cumulants of spherically and elliptically symmetric multivariate distributions; results for the case of multivariate t-distribution and related skew-t distribution are discussed in some detail.

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