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You say the X1...Xn are independent. Are they also identically distributed?
If not, you will have some very cumbersome expressions.If we use f(Xk) as
the density and F(Xk) as the cdf of the k'th r.v., then the density for the
largest (which we call U) is n*F(U)^(n-1)f(U).
That is, the size of the sample times the (n-1) power of the CDF times the
density at U. The most complete reference on such issues is Sahran and
Greenbergs' Contributions to Order Statistics, about 1960 from John Wiley
and Sons.
Fabio Ulisse Pardi wrote:
> Can anybody give me a hint about this problem?
>
> Let the random variables X1,...,Xn be independent and let M be the index
>
> of the maximum among them (i.e. M=i implies Xi >= X1,...,Xn).
> We want to find nice formulas that calculate the distribution of M from
> the distributions of X1...Xn, that we suppose belonging to the same
> class
> of distributions:
> for instance if we assume that all of X1...Xn are normally distributed,
> with parameters (m1,v1),...,(mn,vn), we would like to obtain a formula
> of the kind
> Pr[M=i] = Fi(m1,...,mn,v1,...,vn)
> for every i=1..n.
>
> The problem is that the integral that calculates Pr[M=i] is quite
> complicated, and I haven't figured out how to express its value as a
> simple function of the parameters.
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You say the X1...Xn are independent. Are they also identically distributed?
If not, you will have some very cumbersome expressions.If we use f(Xk)
as the density and F(Xk) as the cdf of the k'th r.v., then the density
for the largest (which we call U) is n*F(U)^(n-1)f(U).
<br>That is, the size of the sample times the (n-1) power of the CDF times
the density at U. The most complete reference on such issues is Sahran
and Greenbergs' <u>Contributions to Order Statistics,</u> about 1960 from
John Wiley and Sons.
<p>Fabio Ulisse Pardi wrote:
<blockquote TYPE=CITE>Can anybody give me a hint about this problem?
<p>Let the random variables X1,...,Xn be independent and let M be the index
<p>of the maximum among them (i.e. M=i implies Xi >= X1,...,Xn).
<br>We want to find nice formulas that calculate the distribution of M
from
<br>the distributions of X1...Xn, that we suppose belonging to the same
<br>class
<br>of distributions:
<br>for instance if we assume that all of X1...Xn are normally distributed,
<br>with parameters (m1,v1),...,(mn,vn), we would like to obtain a formula
<br>of the kind
<br> Pr[M=i] = Fi(m1,...,mn,v1,...,vn)
<br>for every i=1..n.
<p>The problem is that the integral that calculates Pr[M=i] is quite
<br>complicated, and I haven't figured out how to express its value as
a
<br>simple function of the parameters.</blockquote>
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