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.

Theory of best integer equivariant estimation for contaminated normal and multivariate t-distribution with applications

2021 ◽  
Author(s):  
Peter Teunissen
Keyword(s):  
Normal Distribution ◽  
Least Squares ◽  
Unbiased Estimation ◽  
Multivariate T Distribution ◽  
Integer Least Squares ◽  
Wide Range ◽  
Best Linear Unbiased ◽  
Best Linear Unbiased Estimators ◽  
T Distribution ◽  
Multivariate T

<p>Best integer equivariant (BIE) estimators provide minimum mean squared error (MMSE) solutions to the problem of GNSS carrier-phase ambiguity resolution for a wide range of distributions. The associated BIE estimators are universally optimal in the sense that they have an accuracy which is never poorer than that of any integer estimator and any linear unbiased estimator. Their accuracy is therefore always better or the same as that of Integer Least-Squares (ILS) estimators and Best Linear Unbiased Estimators (BLUEs).</p><p>Current theory is based on using BIE for the multivariate normal distribution. In this contribution this will be generalized to the contaminated normal distribution and the multivariate t-distribution, both of which have heavier tails than the normal. Their computational formulae are presented and discussed in relation to that of the normal distribution. In addition a GNSS real-data based analysis is carried out to demonstrate the universal MMSE properties of the BIE estimators for GNSS-baselines and associated parameters.</p><p> </p><p><strong>Keywords: </strong>Integer equivariant (IE) estimation · Best integer equivariant (BIE) · Integer Least-Squares (ILS) . Best linear unbiased estimation (BLUE) · Multivariate contaminated normal · Multivariate t-distribution . Global Navigation Satellite Systems (GNSSs)</p>

scholarly journals BAYESIAN ANALYSIS OF DYNAMIC FACTOR MODELS USING MULTIVARIATE T DISTRIBUTION

2018 ◽  
Vol 36 (1) ◽  
pp. 140
Author(s):  
Larissa Ribeiro de ANDRADE ◽  
Daniel Furtado FERREIRA ◽  
Thelma SÁFADI ◽  
Lúcia Pereira BARROSO
Keyword(s):  
Normal Distribution ◽  
Factor Model ◽  
Graphical Analysis ◽  
Dynamic Factor ◽  
Multivariate Normal ◽  
Chi Square ◽  
Multivariate T Distribution ◽  
Convergence Diagnostics ◽  
T Distribution ◽  
Multivariate T

The multivariate t models are symmetric and have heavier tail than the normal distribution and produce robust inference procedures for applications. In this paper, the Bayesian estimation of a dynamic factor model is presented, where the factors follow a multivariate autoregressive model, using the multivariate t distribution. Since the multivariate t distribution is complex, it was represented in this work as a mix of the multivariate normal distribution and a square root of a chi-square distribution. This method allowed the complete dene of all the posterior distributions. The inference on the parameters was made taking a sample of the posterior distribution through a Gibbs Sampler. The convergence was veried through graphical analysis and the convergence diagnostics of Geweke (1992) and Raftery and Lewis (1992).

scholarly journals Calculations Involving the Multivariate Normal and Multivariate t Distributions with and without Truncation

2018 ◽  
Vol 18 (4) ◽  
pp. 826-843
Author(s):  
Michael J. Grayling ◽  
Adrian P. Mander
Keyword(s):  
Normal Distribution ◽  
Multivariate Normal Distribution ◽  
Distribution Functions ◽  
Multivariate Normal ◽  
Random Vectors ◽  
Multivariate T Distribution ◽  
T Distribution ◽  
Multivariate T

In this article, we present a set of commands and Mata functions to evaluate different distributional quantities of the multivariate normal distribution and a particular type of noncentral multivariate t distribution. Specifically, their densities, distribution functions, equicoordinate quantiles, and pseudo–random vectors can be computed efficiently, in either the absence or the presence of variable truncation.

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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