Are there any properties of commonly encountered probability distributions that cannot be effectively estimated by sampling them?(adsbygoogle = window.adsbygoogle || []).push({});

Searching for "inestimable" lead to irrelevant links. Those links discussed not being able to estimate some parameters of a model when certain types of data are missing.

Searching for "estimable parameter" lead to links about using statistical software packages, which aren't relevant either. My question is theoretical.

I want to know about things that can (or cannot) be estimated in the sense that for each given [itex] \epsilon > 0 [/itex] , the probability that the estimate is within [itex] \epsilon [/itex] of the actual value approaches 1 as the number of independent random samples approaches infinity. (This brings up the the technical question of whether the term "estimator" denotes a function of a fixed number of variables. If I want to talk about letting the number of samples approach infinity, should I talk about asequenceof estimators instead of speaking of a single estimator? )

The "properties" of a distribution are more general than the "parameters" of it. I'll define a "property" of a distribution to be some function of its parameters. For example, a (wierd) example of a property of a Normal distribution is whether it's variance is rational number. You can express this kind of property as a function of the parameters. A similar example is:

On the family of Normal distributions, parameterized by their mean [itex] \mu [/itex] and the variance [itex] \sigma^2 [/itex], define the function [itex] g(k,\mu,\sigma^2) [/itex] by

[itex] g(k, \mu,\sigma^2) = 1 [/itex] if the k-th moment of the normal distribution wiith those parameters is irrational.

[itex] g(k,\mu,\sigma^2) = 0 [/itex] otherwise.

We can also define more complicated functions, such as

[itex] \zeta(\mu,\sigma^2) = \sum_{k=1}^{\infty} \frac {g(k,\mu,\sigma^2)}{2^k} [/itex]

Can such things be effectively estimated?.

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# Un-estimatable properties of distributions

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