Wishart Moments Calculator

This is a SAGE based moment calculator for Wishart distributed random variables.

Among others, we compute the expectation of $W^k$ for $k\in \mathbb{N} $ and $W^{-k}$ for $n > 2k + (r-1)$, for $W \sim \mathrm{Wishart} (n, \Sigma)$, with parameters $n \in \mathbb{N}$ and $\Sigma $ (positive definite matrix of dimension $r$).

More specifically, following Letac G, Massam H (2004), we compute the moments of \(\frac{1}{k }L_{(i)} (W)\), where, for every portrait \( (i)\in I_k = \{ (i_1,\ldots,i_k)\in\mathbb{N}^k \,:\, \sum_{j=1}^k i_j = k \} , \) \[ L_{(i)} (W) = r_{(i)} (W) \sum_{j=1}^k j i_j \frac{W^j}{\hbox{tr}\,(W^j)} \] with $$r_{(i)}(W) = \prod_{j=1}^k \left(\hbox{tr} (W^j) \right)^{i_j}.$$ Here $\hbox{tr}(A)$ indicates the trace of a square matrix.

We also compute the expectation of \( \frac{1}{k}L_{(i)} (W^{-1}) \) for every portrait $(i) \in I_k$ for $n > 2k + (p-1)$.

The index k should be entered as a parameter for the computation; then press return. The slider is used to specify the different invariant moments. The leftmost position of the slider is $(i) = (k)$, which results in the moments $ \mathbb{E}(W^{\pm k}) $.

Other capabilities:

• The \( \rm\LaTeX \) commands for every expression can be obtained by right-clicking the formula and then selecting Show math as > TeX commands
• Computations can also be made for a specific matrix.