Linear Combination of multivariate t distribution

Question

I am looking for a resource where i can find derivation of the linear combination of multivariate t distribution. Does anyone here know any good site or place (s)he can point me to? I am trying to see if the linear combination of multivariate t distribution will give a multivariate t distribution. In other words, what is the distribution of the linear combination of two or more multivariate t distribution?

Does anyone here have any idea on this?

thanks.

Answer 1

I am trying to see if the linear combination of multivariate t distribution will give a multivariate t distribution.

In general, no, this is not the case, even with univariate t's (see here and here for example; note that the difference of two t-random variables is the sum of two t-random variables, but with the second component having its mean that of the original random variable multiplied by -1)

In some very particular cases, yes. Consider:

(i) the limiting case of infinite degrees of freedom, linear combinations of multivariate normals are multivariate normal;

(ii) if the component t-variables are perfectly dependent their sums will be multivariate-t;

(iii) in the univariate case, sums of independent Cauchy random variables will be Cauchy. I haven't checked, but this may well extend more to subsets of the multivariate case than vectors of independent Cauchy (and the perfectly-dependent case mentioned above);

(iv) in the limit of very large numbers of components, where none of the components dominates variance-wise (that is, where the coefficient of each component times the variance of that component doesn't become too large), you may be able to invoke a version of the central limit theorem.

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In the case where the weights on the components are equal (effectively converting it to a scaled sum) and you're dealing with standard t (rather than ones with general means and variances), this paper has some information. Extending it to the case of a general mean is straightforward but it doesn't deal with the general case of arbitrary scales, or equivalently arbitrary linear combinations.

Answer 2

Please have a look at Walker, Glenn A., and John G. Saw. "The distribution of linear combinations of t-variables." Journal of the American Statistical Association 73.364 (1978): 876-878.

The resulting PDF is described as a weighted sum of student-t distribution, and the paper shows how to obtain the weight. The author started from observing for odd degrees of freedom the characteristic function of a student-t r.v. is expressible in closed form, i.e. proportional to the modified Bessel function of the third kind. It seems to me that the paper proposes the solution for all odd t-distribution cases, and no even degree-of-freedoms should be involved.

Answer 3

P15 of Multivariate T distribution and their application (Kotz and Nadarajah) says " If X has the p-variate t distribution with degrees of freedom v, mean vector p, and correlation matrix R, then, for any nonsingular scalar matrix C and for any a, CX + a has the p-variate t distribution with degrees of freedom v, mean vector Cp+a, and correlation matrix CRC'. This result is of importance in applications and is similar to the corresponding result for the multivariate normal distribution. "

edit:

In a univariate language, e.g., considering $aT_1+bT_2$ where $T_i$ is t distributed with mean $m_1,m_2$, scale (in same order as variance) $S_1,S_2$ and with the same degree of freedom. Then define $T:=[T1, T2]'$ which is a bivariate T distribution with zero covariance by stacking the $T_1$ and $T_2$ into a vector. Then $aT_1+bT_2=[a, b]T$ . Applying above conclusion, $aT_1+bT_2$ is univariate t distributed with the same degree of freedom, whose mean is $am_1+bm_2$ and scale is $a^2S_1+b^2S_2$.

An R simulation that supports this theory is:

require('mas3321')
n=10000
sample=c()
mean1=.6
mean2=1.2
scale1=.5
scale2=1
p1=10
p2=20
samples_combt=p1*rgt(n, 10, mean1, scale1)+p2*rgt(n, 10,mean2 , scale2)
hist(samples_combt,probability = T)    
mean_comb= p1*mean1+p2*mean2
scale_comb=p1^2*scale1+p2^2*scale2
curve(dgt(x, 10,mean_comb, scale_comb), add=TRUE)

Answer 4

If $X$ follows a multivariate t-distribution, then any linear combination of $X$ also follows a multivariate t-distribution with the same degrees of freedom:

$$ X \sim t(\mu, \Sigma, \nu) \quad \Rightarrow \quad Y = AX + b \sim t(A\mu + b, A\Sigma A^\mathrm{T}, \nu) \; . $$

Thus, the intended combination only works, if the two multivariate t-distributions have the same dimensions and degrees of freedom. Let us assume that

$$ \begin{split} X_1 &\sim t(\mu_1, \Sigma_1, \nu) \\ X_2 &\sim t(\mu_2, \Sigma_2, \nu) \end{split} $$

where $X_1$ and $X_2$ are independent $n \times 1$ random vectors. If that is the case, we have:

$$ X = \left[ \begin{array}{c} X_1 \\ X_2 \end{array} \right] \sim t\left( \left[ \begin{array}{c} \mu_1 \\ \mu_2 \end{array} \right], \; \left[ \begin{array}{cc} \Sigma_1 & 0_{nn} \\ 0_{nn} & \Sigma_2 \end{array} \right], \; \nu \right) \; . $$

The random variable you seem to have in mind is probably this:

$$ Y = c_1 X_1 + c_2 X_2 \; . $$

Note that $Y$ can be emulated from $X$ by specifying an appropriate linear combination:

$$ A = \left[ \begin{array}{c} c_1 I_n & c_2 I_n \end{array} \right], \; b = 0_{n} \quad \Rightarrow \quad Y = AX + b = c_1 X_1 + c_2 X_2 \; . $$

Thus, we can apply the linear transformation theorem from above:

$$ Y \sim t\left( \left[ \begin{array}{c} c_1 I_n & c_2 I_n \end{array} \right] \left[ \begin{array}{c} \mu_1 \\ \mu_2 \end{array} \right] + 0_{n}, \; \left[ \begin{array}{c} c_1 I_n & c_2 I_n \end{array} \right] \left[ \begin{array}{cc} \Sigma_1 & 0_{nn} \\ 0_{nn} & \Sigma_2 \end{array} \right] \left[ \begin{array}{c} c_1 I_n & c_2 I_n \end{array} \right]^\mathrm{T}, \; \nu \right) \; . $$

This gives:

$$ Y \sim t\left( c_1 \mu_1 + c_2 \mu_2, \; c_1^2 \Sigma_1 + c_2^2 \Sigma_2, \; \nu \right) \; . $$

Fun Fact: The above theorem can also be used to prove the relationship between the multivariate t-distribution and the F-distribution.

Answer 5

Student-t distribution is a special case of the Generalised Hyperbolic Distribution which is closed under affine transform according to wikipedia page (all linear transforms are affine transforms). Hence I would think all linear transformations of student-t random variables (with same degree of freedom) are student-t distributed.

I believe if $$X \sim t_d(\nu, \mathbf{\mu}, \Sigma)$$ then $$\mathbf{w}^T X + c \sim t_1(\nu, \mathbf{w}^T \mathbf{\mu} + c, \mathbf{w}^T \Sigma \mathbf{w})$$

This is just my understanding after reading the below references and adds to answer posted by @user31575.

https://en.wikipedia.org/wiki/Generalised_hyperbolic_distribution

Also in "Quantitative risk management: Concepts, techniques and tools" section 2.3.1 equation 2.31

Also referenced in this paper Hu, W. and Kercheval, A.N., 2010. Portfolio optimization for student t and skewed t returns.