Estimation of parametric functions for multivariate normal distributions with incomplete observations

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The theory of statistical inference in multivariate normal distributions with complete samples can be seen in many textbooks of multivariate analysis. Some monographs deal with the case when incomplete or missing observations are given. In this research we mainly studied statistical estimation for multivariate normal distributions with incomplete observations. The following are the main results.(1) Measures of niche overlap are used to asses the similarity of two populations. The problem of estimating Matusita's measure when samples from multivariate normal distributions with unknown mean vectors and covariance matrices was considered for the case of complete samples (Minami and Shimizu, 1999). Asymptotic variances and biases of Matusita's measure estimates were derived and three bias reduction methods were compared. The case for incomplete observations should be studied in the future.(2) Threshold methods for estimating area rain-rate first product moment and covariance were proposed … More (Hossain and Shimizu, 1999). Empirical study shows a strong correlation between the first product moment and the, probability of exceeding two threshold values for two geographically different locations, land and island, for Tokyo Metropolitan rain-fall data. Theoretical optimal thresholds under the bivariate lognormal distribution as a model were chosen by minimizing the asymptotic normalized variance. A good agreement between the empirical and theoretical results was observed.(3) The problem of estimating Matusita's measure when the niches are bivariate normal distributions with missing observations was discussed (Minami, Shimizu and Mishra, to appear).The plug-in estimates of Matusita's measure by the Maximum Likelihood (ML) estimates and the Restricted Maximum Likelihood (REML) estimates for dispersion parameters were considered. Simulation study shows that the plug-in estimate by the REML estimation tends to have smaller mean square error than that by the ML estimation.(4) The ground-based lidar data (June 1996-March 1999) continuously observed in Tsukuba with the National Institute for Environmental Studies compact lidar were analyzed (Takagiwa, et al., to appear). The vertical distribution and seasonal variation were studied. In the ground-based lidar measurement, clouds above thick lower clouds are not detected. The cloud overlapping model will be validated by combining the statistical analysis of ground-based lidars and the space lidar.(5) Other are in Miyaoka and Tazaki (1999) and in Smith and Miyaoka (1999). Less