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

Jean Sun comm at 263.net
Wed Apr 16 13:54:40 CEST 2003

```A doubt:
Multiplying together the characteristics functions is available only if all a(i)|R(i)| is independent,i think.However, obviously, that is not true herein.

I will try the methods you mentioned.Thanks a lot!

2003-04-14 08:10:00 Spencer Graves wrote£º
>The standard asymptotic theory would start by deriving the
>characteristic funciton of |R(i)|, then of a(i)|R(i)|, then multiplying
>together the characteristics functions.  Then invert the characteristic
>function with liberal use of Taylor's theorem.
>
>Any good book on asymptotics and approximation theory in Statistics
>(especially Edgeworth expansions) will discuss this.  The modern theory
>of saddlepoint approximations may do something different, but I'm not
>familiar with that.
>
>Hope this helps.
>
>Spencer Graves
>
>Thomas W Blackwell wrote:
>> 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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