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I believe your two empirical approximations are the same: aren't they both the total fraction of observations less than or equal to $x_0$? As for the question of asymptotic normality, Donsker's Theor …

If you're only interested in estimating $F$ at a single point $x_0$, then you are really just estimating the probability $p_0 = \mathbb{P}\left(X\in (-\infty,x_0]\right)$. The indicator variables $B_ …

My question concerns an operation on probability distributions which has arisen in some applied research. It is well-defined mathematically (at least in a limited context), but I don't know how to in …

This follows (modulo any minor technical details I haven't checked) from the theory of exponential families. The main result there says that the distribution which maximizes entropy subject to constr …

Here is my original answer (see below for a better one): Writing down \[ g(p) = \sum_{k=0}^n h(k)\binom{n}{k}p^k(1-p)^{n-k} \] and differentiating twice gives \[ g''(p) = \sum_{k=0}^n h(k)\binom{n} …

The hit-and-run algorithm and variants are popular choices. These are Monte Carlo methods but should be much better than rejection sampling. Unfortunately I don't know of a canonical reference.

It is a standard result that the matrices of the form $\mu^{\otimes 2}$ for nonzero $\mu$ are the extreme rays of the positive semidefinite cone. That is to say, your condition on the second moments …