Asymptotically Nearly Efficient Estimators of Multivariate Location Parameters

Abstract

Common estimates of multivariate location parameters have the property that each component of the parameter is estimated using only the corresponding component of the observations. This is true of the sample mean, sample median and the vector of medians of averages (studied in [1]), as well as of the rank-order statistics often applied to testing for location. In some cases, particularly the multivariate normal, such estimators achieve asymptotic efficiency, but in general information is lost. This paper presents three methods of estimating multivariate location parameters which use more information than is available in the marginal distributions. These classes of estimators are asymptotically nearly efficient (ANE), in the sense that for every $\epsilon > 0$ there is an estimator in the class with asymptotic efficiency $> 1 - \epsilon$ (if efficiency is measured by a comparison of the asymptotic covariance matrix to the inverse of the information matrix). Our ANE estimators are motivated by those of Ogawa [6] for univariate location parameters. Ogawa obtained the asymptotically minimum-variance asymptotically unbiased estimator (ABLUE) for location or scale from a chosen set of sample quantiles. It was soon observed (Tischendorf [10]) that the reciprocal of the asymptotic variance of Ogawa's estimator (properly normalized) is essentially a Riemann sum for the information integral for the parameter being estimated. Thus under mild regularity conditions the ABLUE approaches asymptotic efficiency as larger sets of more closely spaced quantiles are chosen for use. Ogawa's estimators are therefore ANE for univariate location parameters. In the present paper we describe three classes of ANE estimators for multivariate location parameters. The first two consist of linear estimators, and represent multivariate generalizations of Ogawa's ANE class. Our three classes are as follows: (1) Choose a set of marginal sample quantiles in each direction from a continuous $r$-variate location parameter distribution. These quantiles generate a random partition of Euclidean $r$-space $R_r$, and for $r > 1$ the observed cell frequencies contain additional information. We obtain the ABLUE's in terms of the sample quantiles and the observed cell frequencies for $r = 2$ and show that they are ANE. (2) For all $r > 1$, linear ANE estimators are obtained by choosing a single sample quantile in each direction and partitioning $R_r$ by marking off fixed distances from these. The ABLUE's in terms of the $r$ chosen quantiles and the observed cell frequencies are ANE. These estimators have much simpler coefficients than do those of class (1). (3) Finally, ANE estimators can be obtained by exploiting analytic properties of RBAN estimators (Neyman [5]) for a sequence of multinomial problems related to the given location parameter family. These estimators are usually not expressed in closed form. Ogawa proceeded by applying least squares theory to the asymptotic distribution of his chosen set of sample quantiles. The ABLUE's of classes (1) and (2) are here derived by the same method, but establishing the joint asymptotic distribution of the marginal sample quantiles and the observed cell frequencies is non-trivial. Our method is to reduce the problem to one involving the multinomial distribution. A similar idea was used by Weiss [11] to obtain the joint asymptotic distribution of the quantiles alone, but the present problem requires more elaborate arguments. Section 2 contains a preliminary result for the multinomial distribution. The ANE classes (1) and (2) are discussed in Section 3, while Section 4 presents the third class. Estimators of all three classes will require use of a computer if the distribution function cannot be expressed in closed form, and may therefore be of limited practical usefulness. Section 5, however, contains an example for which estimators of classes (1) and (2) can be computed with relative ease. For this example, a bivariate logistic distribution, the performance of our estimators is compared with that of the sample mean and median and of Ogawa's univariate ANE estimators. Throughout, $K$ denotes a generic positive constant, $\mathscr{L}\{X\}$ is the probability law of the random variable $X$, and $\mathscr{L}\{X_n\} \rightarrow \mathscr{L}\{X\}$ designates convergence in law. $N(\mu, \Sigma)$ is the normal law with mean $\mu$ and covariance matrix $\Sigma$ (which may be $1 \times 1$).