# Fractional moments of multivariate normal distributions

Is there an analytic formula for fractional moments of multivariate normal distribution? $$E(\prod_{i=1}^k x_i^{\nu_i})={?}$$ where $$X=(x_1,\ldots,x_k) \sim N_k(\mu, \Sigma)$$, $$\nu_i\in \mathcal{R}$$ and $$\nu_i>0$$. I know there is one for univariate normal distributions, but can't find one for the multivariate case.

The formula for the univariate case I know of is $$E[|x-\mu|^{\nu}]=\sigma^{\nu}\frac{2^{\nu/2}\Gamma(\frac{\nu+1}{2})}{\sqrt{\pi}}$$. I understand this is about the central absulate moment, but I am more interested in $$E(\prod_{i=1}^k x_i^{\nu_i})$$ or $$E(\prod_{i=1}^k |x_i|^{\nu_i})$$.

• Perhaps you mean $|x_i|^{\nu_i}$ instead? It would be worth writing out the analytic formula you like for the univariate case, to show what approximations and what special functions you're comfortable with. Apr 15, 2020 at 18:42
• Thanks! I have edited my question. Apr 15, 2020 at 22:36
• This is now asking for a simultaneous generalization from central to noncentral, from unsigned to signed, and from univariate to multivariate — the combination makes it hopeless to provide a nice formula. Apr 16, 2020 at 7:06
• Thanks. What about the central absolute case $E(\prod_{i=1}^k |x_i-\mu_i|^{\nu_i})$ then? And what if $\nu_i$ are integers? Apr 16, 2020 at 14:03

Mathematica gives some results for the bivariate normal distribution with correlation $$\rho$$ and standard marginals. If $$a$$ and $$b$$ are positive integers then
\begin{align} \text{for odd }a+b,\ \ E[x^a y^b] &=0 \\ \text{for odd }a\text{ and }b,\ \ E[x^a y^b] &= \sqrt{\frac{2^{a+1}}\pi}\ \Gamma\!\left(1+\frac a2\right) f(a,b,\rho) \\ \text{for even }a\text{ and }b,\ \ E[x^a y^b] &= \sqrt{\frac{2^{a}}\pi}\ \Gamma\!\left(1+\frac a2\right) f(a,b,\rho) \end{align} where \begin{align} f(a,1,\rho)&=\rho \\ f(a,3,\rho)&=3\rho+(a-1)\rho^3 \\ f(a,5,\rho)&=15\rho+10(a-1)\rho^3+(a-1)(a-3)\rho^5 \\ f(a,2,\rho)&=1+a\rho^2 \\ f(a,4,\rho)&=3+6a\rho^2+a(a-2)\rho^4 \\ f(a,6,\rho)&=15+45a\rho^2+15a(a-2)\rho^4 + a(a-2)(a-4)\rho^6 \end{align} For odd $$a$$ and $$b$$ the coefficients appear to be Ward numbers, and for even $$a$$ and $$b$$ they appear to be the exponential Riordan array.