Nonparametric Families

Let $X_{1}, \ldots , X_{n}$ be iid with distribution $F$.

Let $\mathcal{F}$ be the class of all univariate distribution functions $F$ that have a probability density function $f$ and finite $m$ th moment.

In the preceding problem, show that $1 / \operatorname {var}_{F} X_{i}$ does not have an unbiased estimator for any $n$.

Let $X_{1}, \ldots , X_{n}$ be iid with distribution $F \in \mathcal{F}$ where $\mathcal{F}$ is the class of all symmetric distributions with a probability density. There exists no UMVU estimator of the center of symmetry $\theta$ of $F$ (if unbiasedness is required only for the distributions $F$ for which the expectation of the estimator exists).

If $X_{1}, \ldots , X_{m}$ and $Y_{1}, \ldots , Y_{n}$ are independently distributed according to $F$ and $G \in \mathcal{F}_{0}$, defined in Problem 4.1, the order statistics $X_{(1)}<\cdots <X_{(m)}$ and $Y_{(1)}<\cdots < Y_{(n)}$ are sufficient and complete.

Under the assumptions of the preceding problem, find the UMVU estimator of $\mathbb {P}\left(X_{i}<Y_{j}\right)$.

Under the assumptions of Problem 4.5, let $\xi =\mathbb {E}\left[X_{i}\right]$ and $\eta =\mathbb {E}\left[Y_{j}\right]$. Show that $\xi^{2} \eta^{2}$ possesses an unbiased estimator if and only if $m \geq 2$ and $n \geq 2$.

Let $\left(X_{1}, Y_{1}\right), \ldots ,\left(X_{n}, Y_{n}\right)$ be iid $F \in \mathcal{F}$, where $\mathcal{F}$ is the family of all distributions with probability density and finite second moments. Show that $\delta (X, Y)=\sum \left(X_{i}-\bar{X}\right)\left(Y_{i}-\bar{Y}\right) /(n-1)$ is UMVU for $\operatorname {cov}(X, Y)$.

Under the assumptions of the preceding problem, find the UMVU estimator of

Let $\left(X_{1}, Y_{1}\right), \ldots ,\left(X_{n}, Y_{n}\right)$ be iid with $F \in \mathcal{F}$, where $\mathcal{F}$ is the family of all bivariate densities. Show that the sufficient statistic $T$, which generalizes the order statistics to the bivariate case, is complete.