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I'd like to have some statistical/probabilistic formalisations (solutions..) of the following concrete case I have heard :

"Imagine you have a set of candidates to be interviewed for a job. You split randomly this set into two populations, distributed in 1/3 and 2/3. Then you interview all the candidates of the 1/3-set (first third), and then select the best (with the higher score according to your criteria..) of this set, denoted Cm.

Then you begin to interview candidates of the remaining 2/3-set.. And as soon as you find a better candidate than Cm, denoted CM for example, you stop the procedure and definitely choose your candidate CM. (If such a candidate doesn't exist in the 2/3-set we take the best of this set)

We can say that "on average" CM is our best global candidate (i.e over the whole candidates population), and this despite we did not look necessarily at the whole 2/3-set. How to show that?"

I know this statement could be unclear (what does "on average" exactly mean?) or not complete, but this is part of my question, how can we understand it ?

Thanks a lot for your lights !

Have a nice day.

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  • $\begingroup$ "On average" usually refers to an expectation value. Can your question be rephrased as follows: "What is the probability to pick the best candidate with this procedure"? And "on everage" would then mean that this probability is about 50%? Note that this problem can be easily simulated, so you can guesstimate the probability by randomly trying it out a million times with the assumption of a normally distributed "quality" of applicants. $\endgroup$
    – cdalitz
    Commented Mar 18, 2021 at 12:26
  • $\begingroup$ Thanks for your comment, yes exactly I think it's a good way to rephrase, by finding the probability to choose the best with this procedure, or at least find an interval/estimation for this probability... And I also agree that assuming that the intrinsic scores of the applicants are normally distributed is a good standard assumption. And yes I know this can be easily simulated, but I wanted to have some short and right theoretical rationals to give a frame to this problem (and so to forecast a simulation on this for example) $\endgroup$
    – GregP
    Commented Mar 19, 2021 at 11:03

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Formally, I think your model is acombination of an order stastitic (maxiumum $m_1$) followed by a geometric distribution (repeat until a value greater than $m_1$ occurs). The distribution of the maximum of normal distributions is derived both exactly and asymptotically in

G. Elfving: "The asymptotical distribution of range in samples from a normal population." Biometrika 34, pp. 111-119, 1949

Let us call the first maximum $m_1$ and the second maximum $m_2$ occurring at $n_2$. Then you are looking for the probability $$P(X_i < m_2 \mbox{ for all } i>n_2) = P(X<m_2)^{n-n_2}$$ The problem is that both $m_2$ and $n_2$ are random variables: $n_2$ is geometrically distributed and $m_2$ according to the distrubution given by Elfving. You must thus use the formula for the total probability and integrate/sum over all possible values of $m_2$ and $n_2$ weighted with their respective probabilites, and as $P(n_2)$ depends on $m_1$ you must recursively compute its total probability by integrating over the values of $m_1$. You thus get a double integral and a sum, which might be numerically evaluated by some quadrature (numeric integration) algorithm.

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  • $\begingroup$ Ok ! Thanks very much for your detailed answer. I think your reasoning fits my expectations, it's a good roadmap to follow! Have a nice day ! $\endgroup$
    – GregP
    Commented Mar 19, 2021 at 15:11

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