Statistical Inference Based on Pseudo-Maximum Likelihood Estimators in Elliptical Populations

Abstract

In this article we develop statistical inference based on the maximum likelihood method in elliptical populations with an unknown density function. The method assuming the multivariate normal distribution, using the sample mean and the sample covariance matrix, is basically correct even for elliptical populations under a certain kurtosis adjustment, but is not statistically efficient, especially when the kurtosis of the population distribution has higher than moderate values. On the other hand, several methods of statistical inference assuming a particular family (e.g., multivariate T distribution) of elliptical distributions have been recommended as a robust procedure against outliers or distributions with heavy tails. Such inference also will be important to maintain a high efficiency of statistical inference in elliptical populations. In practice, however, it is very difficult to choose an appropriate family of elliptical distributions, and one may misspecify the family. Furthermore, extra parameters (i.e., other than means and covariances) may make computation heavy. Here we investigate the maximum likelihood method assuming a particular family of elliptical distributions with extra parameters replaced by inexpensive estimators when the assumed family may be misspecified. Consistency and asymptotic normality of the estimators are proved, and the asymptotic equivalence among the likelihood ratio, Wald and Score test statistics, and their chi-squaredness under a constant correction are shown. Two easy methods of estimating extra parameters are proposed. A criterion for choosing a family among competing elliptical families is also provided.

KEY WORDS:

• Likelihood ratio test
• Misspecification
• Multivariate kurtosis
• Nuisance parameter
• Robustness
• Score test
• Wald test