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

The vector of medians Mₙ and the vector of medians of averages of pairs Wₙ are investigated as competitors of the vector mean Ẋₙ 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 Mₙ and Wₙ are given. For Wₙ, 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. Wₙ and Mₙ when compared to Ẋₙ in terms of the Wilks generalized variance are robust in the case of asymptotically independent coordinates. But for p 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, Wₙ is highly efficient, although Mₙ can have arbitrarily small efficiency. However, Wₙ 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, Wₙ becomes infinitely more efficient than Ẋₙ 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.