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Problem

I have an array with $r$ (some relatively small number (say, 5 or 10)) rows and $c$ (some very large number) columns. The $i^{th}$ row of the array is populated with elements randomly drawn from a Gaussian distribution with mean $0$ and standard deviation $\sigma_i$. I slide an $r \times q$ (with $q \ll c$) window across this array. The window starts with its leftmost column aligned with the first column of the array, and move across the array in increments of 1 column. Every time the window moves, I do the following:

  1. If at least $r_{thresh} < r$ rows within the window contain at least one element exceeding some threshold $t$, I record the value of the largest sample of each row, and divide it by the standard deviation of the Gaussian distribution associated with that row in the array. I then record the $k^{th}$ largest of these values (call it $x$). The process continues.
  2. If fewer than $r_{thresh}$ rows within the window contain at least one element exceeding $t$, the process continues.

My goal is to determine the expectation value of $\langle x \rangle$ after some very large number of iterations.


What I've tried

I began by implementing a slightly simplified version of the algorithm in code. If I assume that all $i$ Gaussian distributions are the same (i.e. $\sigma_1=\sigma_2=\ldots \sigma_i$), I find that the mean value of $x$ after some large number of iterations (independent of the sizes of $r,q,c$ and $r_{thresh}$) is roughly linear in $\sigma$.

I'd like to instead solve for (or approximate) this value analytically, as it is highly computationally expensive to actually run the algorithm. I had originally thought I could simply compute the $n^{th}$ Gaussian order statistic, and divide this by the standard deviation, however it seems that all approximations for the Gaussian order statistic are linear in $\sigma$, hence dividing by $\sigma$ yields a constant value (which is not what I see when I actually do this).

How can I solve for (or approximate) this, analytically?

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