Research on Test and Estimate of Distribution Function and its Functionals

分布函数及其泛函在统计推断中起着重要作用。在参数框架工程中，似然用密度函数来表示。在非参数理论中，几个假设是由分布函数给出的。本研究的目的是发展分布函数及其泛函检验与估计的理论和方法。首席调查员和调查人员单独或共同取得以下结果。在分布函数的估计中，在积分均方误差意义下，核估计优于经验分布函数。在我们的工作中，核估计的优越性在积分平方误差意义下是不成立的。因此，如果我们坚持连续性，我们建议使用核估计，否则使用经验分布函数。2.必须给出核估计的光滑化参数。为了选择参数，我们必须估计误差。结果表明，Bootstrap方法在误差估计中是非常有用的。3.分布函数的一般置信带是基于绝对偏差的上确界。然而，该波段对于极值点并不准确。为了弥补传统方法的不足，给出了一种新的置信度。4.提出了基于数据图形表示和分布刻画的拟合优度检验方法。5.推导了时间序列和非线性回归模型中统计量的二阶和三阶有效性。6.考虑了k阶二项分布和k阶负二项分布的一些推广。推导了两状态马尔可夫链的矩母函数和等待时间的方差。

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.