A Multivariate Student's T-Distribution
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Open Journal of Statistics, 2016, 6, 443-450 Published Online June 2016 in SciRes. http://www.scirp.org/journal/ojs http://dx.doi.org/10.4236/ojs.2016.63040 A Multivariate Student’s t-Distribution Daniel T. Cassidy Department of Engineering Physics, McMaster University, Hamilton, ON, Canada Received 29 March 2016; accepted 14 June 2016; published 17 June 2016 Copyright © 2016 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract A multivariate Student’s t-distribution is derived by analogy to the derivation of a multivariate normal (Gaussian) probability density function. This multivariate Student’s t-distribution can have different shape parameters νi for the marginal probability density functions of the multi- variate distribution. Expressions for the probability density function, for the variances, and for the covariances of the multivariate t-distribution with arbitrary shape parameters for the marginals are given. Keywords Multivariate Student’s t, Variance, Covariance, Arbitrary Shape Parameters 1. Introduction An expression for a multivariate Student’s t-distribution is presented. This expression, which is different in form than the form that is commonly used, allows the shape parameter ν for each marginal probability density function (pdf) of the multivariate pdf to be different. The form that is typically used is [1] −+ν Γ+((ν n) 2) T ( n) 2 +Σ−1 n 2 (1.[xx] [ ]) (1) ΓΣ(νν2)(π ) This “typical” form attempts to generalize the univariate Student’s t-distribution and is valid when the n marginal distributions have the same shape parameter ν . The shape of this multivariate t-distribution arises from the observation that the pdf for [ xy] = [ ] σ is given by Equation (1) when [ y] is distributed as a multivariate normal distribution with covariance matrix [Σ] and σ 2 is distributed as chi-squared. The multivariate Student’s t-distribution put forth here is derived from a Cholesky decomposition of the scale matrix by analogy to the multivariate normal (Gaussian) pdf. The derivation of the multivariate normal pdf is How to cite this paper: Cassidy, D.T. (2016) A Multivariate Student’s t-Distribution. Open Journal of Statistics, 6, 443-450. http://dx.doi.org/10.4236/ojs.2016.63040 D. T. Cassidy given in Section 2 to provide background. The multivariate Student’s t-distribution and the variances and covariances for the multivariate t-distribution are given in Section 3. Section 4 is a conclusion. 2. Background Information 2.1. Cholesky Decomposition A method to produce a multivariate pdf with known scale matrix [Σs ] is presented in this section. For nor- mally distributed variables, the covariance matrix [Σ=Σ] [ s ] since the scale factor for a normal distribution is the standard deviation of the distribution. An example with n = 4 is used to provide concrete examples. Consider the transformation [ yx] = [M ][ ] where [ y] and [ x] are 41× column matrices, [M ] is 44× square matrix, and the elements of [ x] are independent random variables. The off-diagonal elements of [M ] introduce correlations between the elements of [ y] . yx11m1,1 000 mm 00 yx222,1 2,2 = (2) yxmmm3,1 3,2 3,3 0 33 yx44mmmm4,1 4,2 4,3 4,4 T The scale matrix [Σ=s ] [MM][ ] . The covariance matrix [Σ] has elements Σ=ij, E{yyij} where E{yyij} is the expectation of yyij and Σ=Σi,, j ji. If the [ x] are normally distributed, then T [Σ=Σ] [ s ] =[MM][ ] , where the superscript T indicates a transpose of the matrix. If [Σs ] is known, then [M ] is the Cholesky decomposition of the matrix [Σs ] [2]. For the 44× example of Equation (2), 2 m1,1 mm1,1 2,1 mm1,1 3,1 mm1,1 4,1 22 mm1,1 2,1 m2,1++ m 2,2 mm2,1 3,1 mm 2,2 3,2 mm2,1 4,1 + mm 2,2 4,2 Σ= . (3) [ s ] 22 2 mmmmmm1,1 3,1 2,1 3,1+ 2,2 3,2 mmm3,1++ 3,2 3,3 mmmmmm3,1 4,1+ 3,2 4,2 + 3,3 4,3 22 22 mmmmmmmmmmm1,1 4,1 2,1 4,1+ 2,2 4,2 3,1 4,1 ++ 3,2 4,2 3,3m4,3 mmmm 4,1+++ 4,2 4,3 4,4 T T 2 From linear algebra, det ([MM]) = det[ M] det[ M] = ( det [ M]) . For [M ] as defined in Equation (2), 2 det M= mm mm and det Σ=mm22 mm 2 2=det M whereas Σ=E yy =σ 2 is the va- [ ] 1,1 2,2 3,3 4,4 [ s ] 1,1 2,2 3,3 4,4 ( [ ]) ii, { ii} yi Σ=E yy =σ 2 riance of the zero-mean random variable yi and ij, { i j} yyij is the covariance of the zero-mean random variables yi and y j . 2.2. Multivariate Normal Probability Density Function To create a multivariate normal pdf, start with the joint pdf fxN ([ ]) for n unit normal, zero mean, independent random variables [ x]: n 11 1 1T f([ x]) =exp −=x2 exp −[xx] [ ] (4) N ∏ i n i=1 2π 22(2π) T where [x] is an n-row column matrix: [x] = [ xx12,,, xn ] . fN ([ x])dd xx12 d xn gives the probability that the random variables [ x] lie in the interval [x] <[ x] ≤+[ xx] [d ]. The requirement for zero mean random variables is not a restriction. If E{xi } = µi , then xxii′ = − µi is a zero mean random variable with the same shape and scale parameters as xi . Use Equation (2) to transform the variables. The Jacobian determinant of the transformation relates the products of the infinitesimals of integration such that ∂ ( xx12,,, xn ) ddxxin2 d x= ddyy12 d. yn (5) ∂ ( yy12,,, yn ) 444 D. T. Cassidy The magnitude of the Jacobian determinant of the transformation [ y] = [ Mx][ ] is (Appendix) ∂ ( xx,,, x) 11 12 n = = (6) ∂ yy, , , y det M ( 12 n ) [ ] det[Σs ] 2 where the equality det[Σ=s ] ( det [M ]) has been used. −1 − Σ= T Σ=−1 T1− = 1 Since [ s ] [MM][ ] , [ s ] ([MM] ) [ ] , and since [xMy] [ ] [ ] , the multivariate “z-score” −1 − T TT− 1= T Σ −1 T1Σ Σ=Σ [xx] [ ] becomes [ yM] ([ ] ) [ My] [ ] [ y] [ s ] [ y] , which equals [ yy] [ ] [ ] since [ s ] [ ] for normally distributed variables. The result is that the unit normal, independent, multivariate pdf, Equation (4), becomes under the trans- formation Equation (2) 11T −1 fyN ([ ]) = exp−Σ[ y] [ s ] [ y] (7) n 2 (2π) det [Σs ] T where [ y] is a n-row column matrix: [ y] = [ yy12,,, yn ] and [Σ=Σs ] [ ] . For the 44× example, 1 0 00 m 1,1 m 1 − 2,1 00 −1 mm1,1 2,2 m2,2 [M ] = , (8) mm− mm m 1 3,2 2,1 3,1 2,2 − 3,2 0 mmm1,1 2,2 3,3 mm2,2 3,3 m3,3 − mm− mm m 1 1 4,3 3,2 4,2 3,3 − 4,3 m1,4 mmm2,2 3,3 4,4 mm3,3 4,4 m 4,4 −1 Σ=−1 T1− from which [ s ] ([MM] ) [ ] can be calculated. In Equation (8), −1 mmm2,1 3,2 4,3−−+ mmm 4,2 2,1 3,3 mmm 2,2 3,1 4,3 mmm 4,1 2,2 3,3 m1,4 = − . (9) mm1,1 2,2 mm 3,3 4,4 −1 The denominator in the expression for m1,4 is det [M ] . 3. Multivariate Student’s t Probability Density Function A similar approach can be used to create a multivariate Student’s t pdf. Assume truncated or effectively truncated t-distributions, so that moments exist [3] [4]. For simplicity, assume that support is [µ−+bb βµ, β] where b is a positive, large number, β is the scale factor for the distribution, and µ is the location parameter for the distribution. If b is a large number, then a significant portion of the tails of the distribution are included. If b = ∞ then all of the tails are included. Start with the joint pdf for n independent, zero-mean (location parameters [µ] = 0 ) Student’s t pdfs with shape parameters [ν ] , and scale parameters [β ] = 1 : −+(ν 12) nn2 i Γ+((ν i 12) ) x f([ x];[νν]) = 1; +=i gx( ) (10) t ∏∏ν i ii ii=11Γ (ννii2) π i = with −∞ ≤xi ≤ +∞ . ft ([ x];[ν ]) dd xx12 d xn gives the probability that a random draw of the column matrix [ x] from the joint Student’s t-distribution lies in the interval [x] <[ x] ≤+[ xx] [d ]. The pdf gxi( ii;ν ) is a function of only xi and the shape parameter ν i , and thus is independent of any other gxj( jj;ν ) , ji≠ . 445 D. T. Cassidy Use the transformation of Equation (2) to create a multivariate pdf −+ν 2 ( i 12) i − my1 n ∑ ij, j 1 Γ+((ν i 12) ) j=1 fyt ([ ];[ν ]) = ∏ 1.+ (11) = Γ ν det [M ] i 1 (ννii2) π i = i −1 The solution xi ∑ j=1 myij, j of the transformation Equation (2) was used. The elements of the inverse −1 −1 matrix [M ] , mij, , are given in terms of the mij, by Equation (8) for the n = 4 example. Note that the shape parameters ν i of the constituent distributions need not be the same in the multivariate t-distribution given by fyt ([ ];[ν ]) . ft ([ y];[ν ]) dd yy12 d yn gives the probability that a random draw of the column matrix [ y] from the multivariate Student’s t-distribution with shape parameters [ν ] lies in the interval [ y] <≤+[ y] [ yy] [d ] .