Re: Question relating to order statistics of normal random variables


David Jones wrote:
Reef Fish wrote:
shrishri@xxxxxxxxx wrote:
Hi all,

I am looking for a solution / direction to solve the following
problem.

Any pointers will be greatly appreciated!

The problem I am seeking a solution for is as follows.
Let X_1, X_2,... X_N be N i.i.d Gaussian random variables with unit
variance and zero mean.
Suppose we arrange the realizations in increasing values of their
absolute values
(similar to order statistics, except that we are using the absolute
values to sort them), and let the corresponding sequence of
random variables be Y_1, Y_2,...,Y_N.

So far so good.

I am looking for the expected value of sum of squares of the first
K
terms of the sequence Y_i,

Because of the symmetry of N(0,1) around 0, you distribution of !X!
is just the half unit normal, but when you're considering the sum of
SQUARES of !X!, how is that different from the sum of the first K
order statistics from a Chi-square distribution with 1 d.f.? (an
exponential distribution).


Close ... An exponential is a Chi-square distribution with 2 d.f..

Thanks for the correction. Indeed I slipped one d.f. in my
recollection.
I remembered the chi-square being an exponential because in the
exponential distribution form, most people would not recognize it to
be a special case of the chi-square (with TWO d.f.).

Another reason for that slip was the ease one could work with the
exponential distribution, but as you pointed out, the special Gamma
distribution is not quite as easy to work with.

On the other hand, the r.v. in question seems to be related to the
Rayleigh distribution in 1 dimension, and the Continuous Univariate
Distribution book by Johnson, Kotz, and Balakrishnan has quite a
bit of information about the Rayleigh distributions and their order
statistics.

-- Reef Fish Bob.


The special case of the Gamma distribution for 1 d.f. doesn't look
helpful directly.

I think it may be best to ignore the assumption of
"Normal/half-Normal" distribution and first work out the asymptotics
for a general distribution function of the absolute values or squares,
or probably better, use the quantile function to give values in terms
of an underlying uniform distribution and work from the CDF of order
statistics of the uniforms. This might yield a tractable integral of
some function of the CDF or quantile function.

David Jones

.

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