I am trying to find the $k$:th, ($:=S^k$) order statistic for a collection "affinely bernoulli" trials $\mathcal{C}=\{b_1 + c_1X_{1},...,b_I+c_IX_{I}\}$. In particular, we are given that $X_i\,{\buildrel d \over =}\, Bernoulli_{0,1}(p_i)$ and the $X_i$ are all independent with the $b_i>0,c_i>0$. Moreover,also assume $p_1<p_2,...,<p_I$, $b_1>b_2>...>b_I$ and that if $j>i$ then $b_j+c_j>b_i+c_i$.

I think we should think of this last property as "$i>j$ if both are flipped low (0) and $j>i$ if both are flipped high (1)".

Using this, it is fairly easy to find the max and minimum statistics. For $1<i<I$, $S_i=\max \mathcal{C}$ iff all $X_j=0$ for $j>i$ and $X_i=1$. So by independence this probability is $p_i\prod_{j=i+1}^I(1-p_{j})$. The extremal cases are easily handled. Namely, for $i=1$ this is just the probability that all others are all tails: $\prod_{j=2}^I(1-p_{j})$ and for $i=I$ it is just the probability of heads: $p_I$. A similar argument holds for the minimum by symmetry.

Now, I am particularly interested in the distribution of the second largest value statistic but the $k$:th statistic in general for this problem is of great interest to me. Is there any way to find nice formulas like for the maximum above? I know about the Bapat-Beg formula but I'd much rather have a less complex expression. Any help to this end would be much appreciated.

EDIT: If there is no nice formula, I would also appreciate references to suitable monotonicity properties of these distributions!

  • $\begingroup$ I wonder if there is a better description in the title than "almost Bernoulli trials", but I don't have any suggestions at the moment. $\endgroup$ – Silverfish Sep 28 '16 at 13:49
  • $\begingroup$ Yeah, I agree. I used affinely bernoulli at one point which might avoid the confusion which could arise with regard to "almost surely". $\endgroup$ – Winston Sep 28 '16 at 13:51
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    $\begingroup$ The statistics are trivial to compute: sort the values. This question reads like you might be asking about the distributions of the order statistics. There will be no "nice" formulas unless there are some special relationships among the $b_i$ and $c_i$. Incidentally, you seem to assume all the $c_i$ are positive, for otherwise your formula for the distribution of the maximum is incorrect. $\endgroup$ – whuber Sep 28 '16 at 14:19
  • $\begingroup$ You are correct in both regards, thanks! I'm looking for the distribution and indeed both $b_i$ and $c_i$ are positive for all $i$. I've edited the title and question. I'll take it you dont think the ordering of the constants is sufficiently "nice"? $\endgroup$ – Winston Sep 28 '16 at 14:29
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    $\begingroup$ Thank you. It might be worth noting that the positivity assumption loses no generality: any negative $c_i$ can be made positive upon replacing $X_i$ by $1-X_i$ (which adds $c_i$ to $b_i$ and turns $p_i$ into $1-p_i$) and the resulting $b_i$ can be made positive by subtracting $-\min(b_i)$ from each, which merely shifts all order statistics by the same amount. $\endgroup$ – whuber Sep 28 '16 at 14:33

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