Expected maximum given population size, mean, and variance

Question

How would one estimate the maximum given population size, a few moments, and perhaps some additional assumption on the distribution?

Something like "I'm going to do $N_s≫1$ measurements out of population of size $N_p≫N_s$; will record mean $μ_s$, standard deviation $σ_s$, and maximal value in the sample $X_s$; I am willing to assume binomial (or Poisson, etc) distribution; what is the expected maximal value $X_p$ of the entire population?"

Related question: does one need to make the assumptions on the nature of the population distribution, or the sample statistics would be enough to estimate $X_p$?

Edit: the background I just added in the comments may not be clear enough. So here it is:

The end purpose it to print a set of shapes (wires, gates, etc) on a VLSI circuit that matches the designed shapes (a.k.a. targets) as well as possible. The measure of fitness of the manufactured set of shapes is the MAXIMAL difference from the target, rather than the $\sigma$ along the $~10^9$ location. The reason for evaluating the maximum difference is clear: a single short circuit is bad enough to bring down the entire chip, and then it wouldn't matter how close you were to the target in the remaining 99.999999% of the chip's location.

The problem is that it's very costly to measure the printed shape in too many locations: you literally need to look though an electron microscope at the half-manufactured chip (that's going to get trashed after the destructive measurements), adjust for metrology errors, etc. Therefore more than $10^4$ measurements is hardly ever being done. The result of those measurement is the maximal target difference $X_s$ of the SAMPLE, as well as any other sample statistics you may wish for.

And now one needs to estimate the maximal difference $X_p$ for the entire population... And now one wishes that he paid more attention in the statistics class back in college...

Answer 1

Try 1:

If $X \sim U[a,b]$ (uniform, either discrete or continuous), then the MLE estimator for b (which is $\max_{x \in [a,b]} X$) is essentially $\max_{i=1,...,N_s}x_i$.

I chose uniform distribution because it is the worst case distribution in terms of entropy. This is in line with the MaxEnt (maximum entropy) principle. I also assumed a linear order in the values of the random variable.

We can make the following claim about the estimator $\max_{i=1,...,N_s}x_i$ to its mean using Hoeffdings inequality (without assuming that $X \sim U[a,b]$). Assuming $x_i$ are i.i.d from some distribution with bounded support $[a,b]$, we have \begin{align*} \mathbb{P}_{x_1,...,x_{N_s}}\left(|\max_{i=1,...,N_s}x_i - \mathbb{E}[\max_{i=1,...,N_s}x_i]| \geq \epsilon\right) \leq 2\exp\left(\frac{-2\epsilon^2}{N_s(b-a)}\right) \end{align*} Here we do not need to know $b$ exactly, any rough or crude upper bound will suffice. The above concentration is only saying that the estimator is close to the expected value of the estimator which is not the same as being close to the unknown $\max_{x \in [a,b]}X = b$.

Additional comment: I would make the measurements uniformly at random over the plane/chip so that hopefully no region with high $X$ values is missed. This observation is independent of the above.

Answer 2

Try 2:

This is a heuristic and I don't know of any statistical guarantees. The procedure is as follows:

• construct the empirical distribution function. If it looks exponential, convert the values to log scale to see a power-law tail.
• Fit a curve on this modified histogram. That is, do a 1-D regression. Hopefully the curve mimics the tail of a well-behaved distribution.
• Pick the point where the line intersects the x-axis in the interval $[\max_{i=1,...,N_s}x_i,\infty)$.

This is another estimator of the max value of the support of the population.