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I was wondering if the No Free Lunch (NFL) theorem applies to even the estimation problem. Suppose there are $N$ points in the input. We are trying to estimate the mean value say weights associated with the $N$ points. By randomly selecting $n$ of the $N$ points, we obtain an unbiased estimate of the population mean weight.

However, the NFL states that the knowledge of the $n$ doesn't imply the goodness of the estimate of the remaining $N-n$ points. Does this mean there is no best estimator of the population mean in statistics?

I will also be very grateful if someone could share some papers on NFL in sampling and statistics since I haven't been able to find them yet.

Clarification: I asked this question in maths forum but didn't get any answers, so I am re-posting it here.

learner
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  • I meant an estimator whose error is minimum. In other words, I would evaluate a sampler by its mean weight estimate of the population with the actual mean weight of the population. – learner Apr 03 '19 at 21:38
  • I don't have an exact error formulation but I am okay with assuming that all weights are between 0 and 1. I am thereafter interested in mean squared error for example. I am alright with making other simplifying assumptions if needed. I am just interested in understanding the performance across different problem versions (i.e. weight distributions). In other words, if I propose another sampling policy say step based sampling over the points, will it lead to the same average performance (error) over all possible problems. – learner Apr 03 '19 at 21:44
  • @Tim Does it mean that I am interpreting the NFL theorem wrongly. Does it mean that it is not applicable for estimators? I was trying to think of estimators as predictors, wherein the estimator estimates from $n$ samples and tries to predict for the remaining $N-n$ points. – learner Apr 03 '19 at 22:01
  • Not sure exactly what NFL ought to mean, particularly for statistics. It is certainly true that the best information would result from exactly correct measurements of all $N$ population elements, and so any estimate based on a random sample of $n < N$ elements is bound to be somehow less than perfect. – BruceET Apr 03 '19 at 22:31

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