# [R] A statistical problem.Anybody can help me?

Thomas W Blackwell tblackw at umich.edu
Mon Apr 14 15:56:04 CEST 2003

```If this were my problem, I would try to separate real and imaginary
parts - write everything out in polar coordinates - and recognize
chi-squared random variables where they occur.  But that's very much
a beginner's approach.

HTH  -  tom blackwell  -  u michigan medical school  -  ann arbor  -

On Mon, 14 Apr 2003, comm wrote:

> Sorry for the contents not relating to R.
>
> Assume there are N i.i.d zero-mean complex gaussian random
> variables(RVs),as w(i),0<=i<N} with known variance,from which one
> can generate another N RVs,as
>
>     R(0)=sum over i {w(i)*w'(i)}
>     R(1)=sum over i {w(i+1)*w'(i)}
>     ...
> up to
> 	R(N-1)= w(N-1)w'(i)
>
> where w'(i) is the complex conjugate of w(i).
> (from viewpoint of signal processing, R(i) are serial correlation of time series w(i))
>
> If one defines a new random variable using {R(k)} as
>
> Z=a(0)R(0)+a(1)|R(1)|+...  a(N-1)|R(N-1)|,
>
> with {a(k)} are known and |.| is modulus operation.It's a decision
> statistic encountered in my work. I wish to find its approximated(using
> Central Limit Theorem) statistical characteristics in close-form.Mean and
> variance are enough.
>
> Does anybody have any ideas or references which can solve this problem?
>
> (below is my previous thoughts and now it is tested not work because RVs appear to be Rician distributed)
> Mean of Z is easy to get. However its variance is troublesome. I think it can be calculated by
>
>     Var=alpha*C*alpha',
>
> where alpha=[a(0) a(1) ... a(N-1)],C is covariance matrix of vector [R(0),|R(1)|,...,|R(N-1)|].
>
> Besides the diagonal and first row and first column, the other elements is
> small that can be ignored,which can be shown by simulations.Namely weak
> cross-correlation is hold between any two RVs of set
> {|R(1)|,|R(2)|,R(N-1)},while the crosss-correlation between R(0) and each
> RV of set {|R(1)|,|R(2)|, |R(N-1)|} and self-correlation of set
> {|R(0)|,|R(1)|,|R(N-1)|} is large and should not be ignored. The former is
> what i seek. I almost exhausted,so i came here for help.
>
> Any suggestion will be appreciated.
>
> Regards,
> Jeans Sun

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