TITLE: Estimation of a function of a random vector
QUESTION [2 upvotes]: The joint probability distribution of the random variables $X_1$, $X_2$, ..., $X_n$ is available. The objective is to find the distribution of $\epsilon = g(X_1, X_2, ...,X_n)$. In other words, $g$ is unknown. A sample of size $m$ is drawn from the population: $([X_{i1}, X_{i2},..., X_{in}], \epsilon_i)$ is available for $i=1,...,m$. How to find $\epsilon^*$ such that $P(\epsilon>\epsilon^*)<0.05$. On a more general note, how to find an optimized estimator of $g$, namely $\hat{g}$?
Probably a common problem for professionals in this field but I'm a beginner, so your help is highly appreciated.

REPLY [1 votes]: I have a suggestion for what method you can use. As I said in the comment, you have no assumption about the function $g$, so this sollution is more likely heuristic, assuming that the function $g$ is nice with respect to this method.
When you are given a point $[X_1, X_2, X_3, \ldots]$ and you need to calculate $g(X_1, X_2, X_3, \ldots)$, you take the measurements within radius $R$ and estimate the 0.95-quantile of them. You just need to fit the radius $R$ to your data, so it gives satisfactory results. Or more generally, you weighten all measurements by a kernel function of your choice (a non-increasing function of distance between two points). Again, you have to choose a fitting kernel function.
If you choose a kernel function which is infinite for zero distance (for example inverted square distance), it will have a special effect. If you are given a measurement at a point and then you ask for the same point, the measurement at that point will outweight all other points, so you get the measurement which you got.
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