Is there any theoretical results about the determinants of a Non-Central Wishart matrix?

Question

As we know that a Non-Central Wishart matrix is defined as
$W:=XX^T$, where $X \in \mathbb{R}^{p \times N}$, and
$X:= M + E$, with $M \in \mathbb{R}^{p \times N}$ a deterministic and non-zero matrix, and each element of $E$ obeying the normal distribution $\mathcal{N}(0,\sigma^2)$.
While on the contrary, the Central Wishart matrix can be viewed as a special case of the Non-Central one when $M=0$ as shown above.
My question is, is there any theoretical result about the determinant of this Non-Central Wishart matrix $W$? i.e., what do we know about
$det(W)=det((M+E)(M+E)^T)$
e.g. do we know about the expectation, the moment or even the concentration inequalily about this random determinant?
And what is the relationship of the determinant between Non-Central Wishart matrices and central Wishart matrices?

Answer 1

For more on the determinants Noncentral Wishart Matrices see Muirhead (1982,2005) Chapter 10. In particular see Theorem 10.3.7. Muirhead also argues that the moments of the determinant can be used to obtain asymptotic distributions in builds towards this in his problems 10.2 10.3 and 10.4.

Answer 2

upper and lower bounds for the expectation of $\log\det W$ --- with complex matrix elements, but I presume readily generalized to real matrix elements --- have been derived in On the log-determinant of noncentral Wishart matrices (2003):

if the real eigenvalues of $\sigma^{-2}MM^{\ast}$ are ordered from large to small, $\lambda_1\geq\lambda_2\cdots\geq\lambda_p$, then

$$g_N(\lambda_1)+\sum_{k=2}^{p}\psi(N-k+1)\leq E(\log\det W)-p\log\sigma^2\leq\sum_{k=1}^p g_N(\lambda_k)$$

$${\rm with}\;\;g_N(\lambda)=\log\lambda-{\rm Ei}\,(-\lambda)+\sum_{k=1}^{N-1}(-1/\lambda)^k\left[e^{-\lambda}(k-1)!-\frac{(N-1)!}{k(N-1-k)!}\right]$$

and $\psi$ the digamma function, Ei the exponential integral. Equality in the lower bound iff $\lambda_k=0$ for $k>1$.