An Existence Problem from Statistical Estimation Theory

The Hájek–LeCam convolution theorem and the Cramér–Rao inequality are the two best-known results from statistical estimation theory that bound the performance of estimators. The Hájek–LeCam convolution theorem is an asymptotic result and the Cramér–Rao inequality is valid for finite sample sizes. Another finite sample result is the spread inequality.
Let $T$ be an estimator of $\theta\in\mathbb R$. Let $G$ be the distribution function obtained by averaging over $\theta$ the distribution function $G_\theta$ of $T-\theta$ under $\theta$. The spread inequality states that this distribution function $G$ is more spread out than a well-defined distribution function $K$, in the sense that all pairs of quantiles of $G$ are at least as far apart as the corresponding quantiles of $K$. Here, the distribution function $K$ depends on the model for the observations on which the estimator is based, and on the averaging density used to define $G$. Of course, $K$ does not depend on the estimator $T$.
For multivariate $\theta\in\mathbb R^k$ this univariate spread inequality may be used to get for every estimator $T$ of $\theta$ a bound on its performance as follows. For every $a\in\mathbb R^k$ the average distribution of the estimator $a^\top(T-\theta)$ is at least as spread out as the distribution function $K_a$, which is the bound from the univariate spread inequality applied to estimation of $a^\top\theta$.
The problem now is to determine conditions under which there exists a $k$-dimensional random vector $Z$ with distribution function $K$ such that for every $a\in\mathbb R^k$ the random variable $a^\top Z$ has distribution function $K_a$. In the lecture we will present the spread inequality, its proof, and its consequences in some detail. We will also discuss the open problem of existence of a multivariate distribution $K$.