Let $k < p$ be positive integers and $g: \mathbb R^k \rightarrow \mathbb R^p$ be a smooth Lipschitz continuous function. Let $y_1,\ldots, y_N \in \mathbb R^p$ and $a = (a_1,\ldots,a_N) \in \mathbb R^N$ be fixed. For each $i=1,\ldots,N$, define $e_i(y, a) := \|y-y_i\|^2_2 - a_i$, for $y \in \mathbb R^p$. Now, for positive real numbers $\lambda_1,\ldots,\lambda_N$ and for a standard $k$-dimensional Gaussian random variable $Z$, and for each $i=1,\ldots,N$, consider the transformed random variable $$W_i := \frac{\lambda_i\exp(-e_i(g(Z),a))}{\sum_{j=1}^N\lambda_j\exp(-e_j(g(Z), a))}. $$
Question:
- (A) What is a "good estimator" for expected value of $W$ ?
- (B) For which choices of the parameters $\lambda_1,\ldots,\lambda_N$ is the variance of $W_i$ minimized ?
Setting: I want to estimate the expectations of the $W_i$'s to good precision. Indeed, if possible I'd prefer to minimize all the $\operatorname{Var}(W_i)$ at once, i.e solve $$\underset{\lambda_1,\ldots,\lambda_N>0}{\operatorname{minimize}}\;\max_{i} \operatorname{Var}(W_i). $$ I considered the decoupled problems for convenience (presumably much easier).
