# Variance of the sum of elements of a Wishart distributed matrix

Looking for the variance of $$S=\sigma _{1,3}-\sigma _{1,4}-\sigma _{2,3}+\sigma _{2,4}$$, where $$\sigma_{i,j}$$ are Wishart-distributed elements of the random matrix

$$\Sigma =\left( \begin{array}{cccc} \sigma _1^2 & \sigma _{1,2} & \sigma _{1,3} & \sigma _{1,4} \\ \sigma _{1,2} & \sigma _2^2 & \sigma _{2,3} & \sigma _{2,4} \\ \sigma _{1,3} & \sigma _{2,3} & \sigma _3^2 & \sigma _{3,4} \\ \sigma _{1,4} & \sigma _{2,4} & \sigma _{3,4} & \sigma _4^2 \\ \end{array} \right)$$ the $$m$$-sample estimation of the covariance matrix of 4 multivariate Gaussian distributed random variables with $$n$$ observations each.

(I tried Math Stack Exchange with no result).

Let $$a = (\begin{matrix} 1 & -1 & -1 &1\end{matrix})$$ and $$\sigma = (\begin{matrix} \sigma_{1,3} & \sigma_{1,4} &\sigma_{2,3} & \sigma_{2,4}\end{matrix})'$$.

Then $$S=a\sigma$$. Let $$V$$ be the variance-covariance matrix of $$\sigma$$. We have $$\mathrm{Var}(S) = aVa'$$

About $$V$$, the variance-covariance matrix of 4 multivariate Gaussian distributed random variables is needed. Then referring to eqn (1) on page 4 of http://www.math.unm.edu/~fletcher/Wishart.pdf, you can get the answer.

• When you say "About V, the variance-covariance matrix of 4 multivariate Gaussian distributed random variables is needed " do you mean $\Sigma$ in the OP? Or V shd be the Kronecker product of $\Sigma$ by $\Sigma$ ?
– Nero
Nov 6, 2018 at 21:09
• I mean the Kronecker product of Σ by Σ in eqn (1) on page 4 linked paper. Nov 6, 2018 at 21:16
• Thanks! Then a should have 16 elemts, s.a. (0,0,0,1,-1....), no?
– Nero
Nov 6, 2018 at 21:32
• depend on you. If you want to use that 16 $\sigma$s, yes, need 16 dimension vector $a$. If you want to get a specific 4x4 matrix from that 16x16 matrix, you can keep a=(1,-1,-1,1). Both ways will generate the exact same results. Nov 6, 2018 at 21:39
• What I am getting from $a V a'$ is $S^2$. I am looking for $E(S^2)$. Any glitch?
– Nero
Nov 8, 2018 at 5:45