# Estimation After Selection on Non-central F Random Variables

Suppose that you observe $F_1,F_2,\ldots,F_k$ each independently. drawn from non-central F distributions with common, known, d.f. $\nu_1, \nu_2$, and with (unknown) non-centrality parameters $\lambda_1,\lambda_2,\ldots,\lambda_k$. Suppose that the sample is ordered by $F_{(1)} \le F_{(2)} \le \ldots \le F_{(k)}$. I would like to somehow estimate $\lambda_{(k)}$. That is, estimate the non-centrality parameter associated with the largest of the $F_i$. (Actually, procedures for estimating arbitrary $\lambda_{(i)}$ would be nice too.)

There is a large body of literature on estimation after selection that typically assumes the RVs are normally distributed (Gupta and Miescke inter alios), or exponential, uniform, etc. I looked into the normalizing transforms of the non-central F, but these require you to know the $\lambda_i$ (they seem to exist to construct tables), and don't work well with estimates $\hat{\lambda_i}.$

Milton & Rizvi (1989) is a related reference.

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I am both puzzled and curious as to why you would be interested in solving such a problem. I think the term "largest sample" is not what you mean. Don't you mean that you want to estimate the noncentrality parameter of the maximum of the Fs? Also if you really mean i.i.d. then you are assuming all the noncentrality parameters are the same. If you are allowing them to be different then maybe you mean that the noncentral Fs are indpendent but not necessarily identically distributed. In the case where you have k different lambdas, I see no way to identify any of the noncentrality parameters. – Michael Chernick Jul 20 '12 at 22:30
If you assume all of them are the same (the iid case) there is hope and you could apply maximum likelihood. Perhaps in the non iid case if you assume some are the same and hence reduce the number of parameters the likelihood approach might still work. – Michael Chernick Jul 20 '12 at 22:33
You can always write down the cumulative distribution for the maximum. But it will involve not only the lambda that you want to estimate but also the other k-1 that are nuisance parameters. – Michael Chernick Jul 20 '12 at 22:35
You can find some estimators in this paper. Maximum likelihood estimators do not seem to be too difficult to calculate numerically for moderate $k$ (at least that is my impression after some numerical experiments). I would not spend lots of time looking for closed estimators of such cumbersome distributions. – user10525 Jul 21 '12 at 13:45
@Procrastinator thanks for the link! I had been using the unbiased estimator, but the KRS estimator helps me somewhat. The MLE is also easy to compute by brute force. – shabbychef Jul 24 '12 at 5:07