Statistical inference for location and scale of elliptically contoured models with monotone missing data

In this paper statistical inference is developed for the estimation and testing problems of the location and scale parameters of the elliptically contoured family of distributions. The data matrix is of a monotone missing pattern. The analytic form of the maximum likelihood estimators of location and scale are derived, and based on them, the likelihood ratio test statistics are obtained for testing the following: (i) the location and scale parameters are, separately, equal to a specified vector and matrix, (ii) the location and scale parameters are, simultaneously, equal to a specified vector and matrix, and (iii) the hypothesis of lack of correlation between sets of variates that jointly described by the elliptically contoured family of distributions. The test of sphericity is also derived in the particular case of the multivariate normal distribution. The asymptotic null distributions of the resulting test statistics are derived for k=2, as well as, for k>2 steps of monotone missing data. The results are illustratively applied in the Appendix A, to specific elliptically contoured models like the multivariate t-distribution. The results are also illustrated using simulated data from a multivariate t-distribution.