As we know that a NonCentral 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 nonzero 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 NonCentral one when $M=0$ as shown above.
My question is, is there any theoretical result about the determinant of this NonCentral 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 NonCentral Wishart matrices and central Wishart matrices?
2 Answers
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 logdeterminant 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
$$g_N(\lambda_1)+\sum_{k=2}^{p}\psi(Nk+1)\leq E(\log\det W)p\log\sigma^2\leq\sum_{k=1}^p g_N(\lambda_k)$$
$${\rm with}\;\;g_N(\lambda)=\log\lambda{\rm Ei}\,(\lambda)+\sum_{k=1}^{N1}(1/\lambda)^k\left[e^{\lambda}(k1)!\frac{(N1)!}{k(N1k)!}\right]$$
and $\psi$ the digamma function, Ei the exponential integral. Equality in the lower bound iff $\lambda_k=0$ for $k>1$.

$\begingroup$ Thank you for your answer! I'm still wondering if there is any further result about this log determinant random variable $\log\det W$? e.g., the distribution, or something like the confidence interval about this log determinant? $\endgroup$– HelmholzMar 19, 2014 at 2:45
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.