Consider the following expression xHAx, where x is a known vector, i.e. a determinisitc parameter, and xH its hermitian, and A is a hermitian, positive semi-definite matrix constructed as the summation of D rank-one matrices of the form didiH for i=1,...,D. In this case, the vectors di are distributed as a standard complex Gaussian (zero mean and unit variance).
The question is which is the statistical model of xHAx?
Thanks in advance.
Data presented in abstract well is not good for statistical modelling, because the phenomena itself must be related with the explanation and the model. Probably a Bootsrap approach could be a better option for your model, but is necessary an approximation of the size of the matrices and an empirical explanation about what the data is related on. Good luck.
Dear Jordi Borràs Pino,
Focussing on the case of say n-dimensional vectors, the question requires an explanation what should be understood under the statement
"vectors di are distributed as a standard complex Gaussian (zero mean and unit variance)."
1. Are these vectors with deterministic (in general - different) directions multiplied by standard complex Gaussian random variables (independent - not independent)? Then: what are the mutual orientations of the vectors?
OR
2. Do all the coordinates of the vectors are independent and possess joint Gaussian distribution? Then: What is the standard (zero mean and unit variance) distribution - of each single coordinate or . . . ?
The general case of joint normal distribution of all coordinates of all vectors d_i-s obviously will lead to the Wishart distribution of the square of the modulus of the complex number
∑_k x_k^{⋆} ⋅ d_{i,k}
where x_k and d_{i,k} denote the k-th coordinates, k=1,2,...,n. (as it correctly has been pointed at by the former answers by Bo and Valerio). However, without more precise information about the joint distribution of the vectors d_i nothing can be stated about the parameters of the resulting distribution of the modulus.
Suggestion: consider the most general case of distribution for low number of vectors d_i and low dimension n of the vectors. E.g. for i=1,2 and n=3 the real dimension of the joint distribution of the coordinates is 12 , which can be managed by hand-writing (hopefully:). As a starting analysis one can pose suitable strong assumptions about the distribution (cf. subquestions 1 and 2 above).
Regards
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