Thank you very much, Ray !
You mentioned Pearson's result. Could you post
a pointer to a book/paper which I can use and cite.
It is likely that I can not find the original from 1901,
but any recent discussion of his or related result would do.
Also, correct me if I'm wrong, but I think,
if N -> infinity, then P -> 1 (for any K=2,3,4,...), given that p>q,
and P -> 0, if p<q.
Thank you again.
[EMAIL PROTECTED] (Ray Koopman) wrote in message news:<[EMAIL PROTECTED]>...
> [EMAIL PROTECTED] (Dan Esperantos) wrote in message
> news:<[EMAIL PROTECTED]>...
> > Suppose we are given a multinomial probability distribution
> > N!
> > P(X_1=N_1, X2=N_2,..., X_K=N_K) = ---------------- p^N_1 * q^(N-N_1),
> > N_1! N_2!...N_K!
> >
> > where p + q*(K-1) = 1, and N_1 + N_2 + N_K = N.
> >
> > How to compute an approximation of the following
> > probability, where {X_i}s are distributed as above:
> >
> > P( X_1 = max{X_1, X_2,..., X_K} )?
> >
> > if p > q or if p > q*(K-1)
> >
> > For example, when K = 2 we simply have (assume N is even):
> > N N!
> > P( X_1 = max{X_1, X_2} ) = P( X_1 >= X_2 ) = Sum -------- p^i*(1-p)^(N-i)
> > i=N/2 i!(N-i)!
> >
> > Are any approximations known for such sums?
> > What about the case of general K?
> > In my applications N is on order of several hundreds,
> > so exact computation is not feasible.
>
> Let Y_i = X_1 - X_i, i = 2,...,K. Then
>
> P(X_1 >= max{X_2,...,X_K}) = P(Y_2 >= 0 & ... & Y_K >= 0)
> = P(min{Y_2,...,Y_K} >= 0).
>
> The {Y_i} are exchangeable with means = N(p-q),
>
> variances = Var(X_1) + Var(X_i) - 2 Cov(X_1,X_i)
> = Np(1-p) + Nq(1-q) - 2(-Npq),
> = N(p + q - (p-q)^2),
>
> covariances = Var(X_1) - Cov(X_1,X_i) - Cov(X_1,X_j) + Cov(X_i,X_j)
> = Np(1-p) - 2(-Npq) + (-Nq^2)
> = N(p - (p-q)^2).
>
> If Nq is sufficiently large -- say 10 or so? -- then the distribution
> of {X_i}, and hence of {Y_i}, is approximately multivariate normal,
> and the desired probability is approximately equal to the probability
> that K-1 exchangeable multinormal variables whose means, variances,
> and covariances are given above, are all positive (or, equivalently,
> that their minimum is positive).
>
> K = 2: P = F(z), where F is the standard normal cdf,
> and z = Mean(Y_i)/SD(Y_i) = N(p-q)/Sqrt(N(p + q - (p-q)^2))).
>
> K = 3: Pearson (1901), working on a more general problem, reduced
> the obvious two-dimensional integral to a one-dimensional integral:
> R
> P = (F(z))^2 + Int f(z,z;r)dr, where z is as defined above,
> 0
> f(x,y;r) = exp((-1/2)(x^2 + y^2 - 2rxy)/(1-r^2))/(2Pi Sqrt(1-r^2))
>
> is the bivariate standard normal pdf with correlation r, and
>
> R = Cov(Y_i,Y_j)/Var(Y_i) = (p-(p-q)^2)/(p+q-(p-q)^2)
>
> is the correlation of Y_i and Y_j. The integrand simplifies a little
> when we set x = y = z, but numeric integration is still necessary.
>
> K = 4: I vaguely remember seeing a paper about the three-dimensional
> case, but I have no idea whose or where or when.
>
> K > 4: ???
.
.
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