We extend the above results to the case when each of the input
RVs have a different distribution
.
We restrict the analysis, however,
to the case when
(the largest RV). We would also like to include
the indexes of the order statistics in the joint distribution.
This is because when the input RVs have different distributions, the
indexes of the order statistics are no longer independent.
Said another way, our expectation of the amplitudes of the
top RVs would change if we knew the indexes. Whereas,
if the input RVs were iid, knowledge of the
indexes would not change our expectations of their amplitudes.
Let
, be the set of indexes
corresponding to the top RVs.
We define the complete feature vector as
The combined probability density and discrete probability function
is defined as the limit as
of
times the probability that the largest RV
had index and is between
and
, AND
the next largest RV had index and is between
and
,
etc, AND the sum of the remaining RVs is between
and
.
The exact solution is given by [44]

(7.5) 
where is the set of indexes between and that do not include
through , is the unit step function, is the contour in the
MGF domain , and
where

(7.6) 
The best numerical solution is obtained by
finding the saddlepoint (the real
value of for which the integrand achieves the minimum value),
then integrating from to vertically in the complex plane
at that real value of .
A saddlepoint approximation to this integral is also available
[45].
Baggenstoss
20170519