Multivariate t-distribution

In statistics, the multivariate t-distribution (or multivariate Student distribution) is a multivariate probability distribution. It is a generalization to random vectors of the Student's t-distribution, which is a distribution applicable to univariate random variables. While the case of a random matrix could be treated within this structure, the matrix t-distribution is distinct and makes particular use of the matrix structure. In statistics, the multivariate t-distribution (or multivariate Student distribution) is a multivariate probability distribution. It is a generalization to random vectors of the Student's t-distribution, which is a distribution applicable to univariate random variables. While the case of a random matrix could be treated within this structure, the matrix t-distribution is distinct and makes particular use of the matrix structure. One common method of construction of a multivariate t-distribution, for the case of p {displaystyle p} dimensions, is based on the observation that if y {displaystyle mathbf {y} } and u {displaystyle u} are independent and distributed as N ( 0 , Σ ) {displaystyle {mathcal {N}}({mathbf {0} },{oldsymbol {Sigma }})} and χ ν 2 {displaystyle chi _{ u }^{2}} (i.e. multivariate normal and chi-squared distributions) respectively, the matrix Σ {displaystyle mathbf {Sigma } ,} is a p × p matrix, and y / u / ν = x − μ {displaystyle {mathbf {y} }/{sqrt {u/ u }}={mathbf {x} }-{oldsymbol {mu }}} , then x {displaystyle {mathbf {x} }} has the density and is said to be distributed as a multivariate t-distribution with parameters Σ , μ , ν {displaystyle {oldsymbol {Sigma }},{oldsymbol {mu }}, u } . Note that Σ {displaystyle mathbf {Sigma } } is not the covariance matrix since the covariance is given by ν / ( ν − 2 ) Σ {displaystyle u /( u -2)mathbf {Sigma } } (for ν > 2 {displaystyle u >2} ). In the special case ν = 1 {displaystyle u =1} , the distribution is a multivariate Cauchy distribution. There are in fact many candidates for the multivariate generalization of Student's t-distribution. An extensive survey of the field has been given by Kotz and Nadarajah (2004). The essential issue is to define a probability density function of several variables that is the appropriate generalization of the formula for the univariate case. In one dimension ( p = 1 {displaystyle p=1} ), with t = x − μ {displaystyle t=x-mu } and Σ = 1 {displaystyle Sigma =1} , we have the probability density function and one approach is to write down a corresponding function of several variables. This is the basic idea of elliptical distribution theory, where one writes down a corresponding function of p {displaystyle p} variables t i {displaystyle t_{i}} that replaces t 2 {displaystyle t^{2}} by a quadratic function of all the t i {displaystyle t_{i}} . It is clear that this only makes sense when all the marginal distributions have the same degrees of freedom ν {displaystyle u } . With A = Σ − 1 {displaystyle mathbf {A} ={oldsymbol {Sigma }}^{-1}} , one has a simple choice of multivariate density function