On Some Alternative Estimates for Shift in the $P$-Variate One Sample Problem

Abstract

The vector of medians $\mathbf{M}_n$ and the vector of medians of averages of pairs $\mathbf{W}_n$ are investigated as competitors of the vector mean $\dot{\mathbf{X}}_n$ in estimating the location parameters in the $p$-variate one-sample problem. These estimates are found to be asymptotically normal and unbiased. Necessary and sufficient conditions for the degeneracy of the asymptotic distribution of $\mathbf{M}_n$ and $\mathbf{W}_n$ are given. For $\mathbf{W}_n$, in the case $p = 2$, these reduce to the condition that one coordinate variable be a monotone function of the other. Sufficient symmetry conditions are given for the asymptotic independence of the coordinates of these estimates. $\mathbf{W}_n$ and $\mathbf{M}_n$ when compared to $\dot{\mathbf{X}}_n$ in terms of the Wilks generalized variance are robust in the case of asymptotically independent coordinates. But for $p \geqq 3$ they can have arbitrarily small efficiency even in the non-singular $p$-variate normal case, if the underlying distribution is permitted to approach a suitable degenerate distribution arbitrarily closely. For $p = 2$, in the normal case, $\mathbf{W}_n$ is highly efficient, although $\mathbf{M}_n$ can have arbitrarily small efficiency. However, $\mathbf{W}_n$ is also shown to have arbitrarily small efficiency for a suitable highly correlated family of distributions even in the case $p = 2$. On the other hand, $\mathbf{W}_n$ becomes infinitely more efficient than $\dot{\mathbf{X}}_n$ as a given fixed distribution is mixed with an increasingly heavy gross error distribution. The behavior of these estimates is also considered for other non-normal families.