Research on Test and Estimate of Distribution Function and its Functionals

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Distribution function and its functionals play important roles in statistical inferences. In parametric frame works, the likelihood is represented by density functions. In nonparametric theory, several hypotheses are given by distribution functions. The aim of this research project is to develop theory and method on test and estimate of distribution function and its functionals. The head investigator and the investigators singly or jointly obtained the following results.1. In the estimate of distribution function, the kernel estimators are better than the empirical distribution function in the sense of integrated mean squared error. In our work, the superiority of the kernel estimator is shown to be not true in the sense of integrated squared error. Hence, we recommend to use kernel estimator if we adhere the continuity, and to use empirical distribution function if otherwise. 2. We must give the smoothing parameter in the kernel estimators. In order to select the parameter, we must estimate the errors. It is shown that the bootstrap method is very useful in the estimate of the errors. 3. The usual confidence band of distribution function is based on the supremum of the absolute deviation. However, the band is not accurate for extreme points. A new confidence band is given in order to recover the defect of the usual method. 4. Goodness of fit tests based on graphical representation of data and a characterization of distribut ions are proposed. 5. The second order and the third order efficiency of statistics in time series and non-linear regression models are derived. 6. Some extensions of binomial distribution of order k and negative binomial distribution of order k are considered. Moment generating functions and variances of waiting time occurred in two-state Markov chain are derived.