Research on Test and Estimate of Distribution Function and its Functionals

分布函数及其泛函的检验与估计研究

• 批准号: 03630014
• 负责人: SHIRAHATA Shingo
• 金额: $ 0.96万
• 依托单位: Osaka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (C)
• 财政年份: 1991
• 资助国家: 日本
• 起止时间: 1991 至 1992
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-03630014/
• 关键词: Distribution Function; Empirical Distribution; Kernel Estimate; Goodness of Fit Test; Estimate; Confidence Band; Bootstrap; Mean Integrated Square Error; 核関数; U-統計量; 検定; ブ-ツトラップ

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.

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.

项目成果

SHIRAHATA,S.(SHIRAHATA,S.): "Estimate of variance of U-statistics" Communications in Statistics;Theory&Methods. 21. 2969-2981 (1992)

SHIRAHATA,S.（SHIRAHATA,S.）：“U 统计方差估计”统计通讯；理论

Isogai, T. (Hiratsuka, A.): "Study on sludge dewatering techniques using a statistical model-in view of both quantity of filtrate and filtrate quality" Res. Env. System. 20. 231-241 (1992)

Isogai, T.（Hiratsuka, A.）：“使用统计模型研究污泥脱水技术 - 考虑滤液量和滤液质量”Res。

Taniguchi, M. (Kondo, M.): "Two sample problem in time series analysis" Proc. 3rd Pacific Area Statist. Conf. (1993)

Taniguchi, M.（Kondo, M.）：“时间序列分析中的两个样本问题”Proc。

Aki, S. (Hirano, K.): "On number of occurrences of success runs of specified length in a two-state Markov chain" Statist. Sinica. (1993)

Aki, S.（Hirano, K.）：“关于二态马尔可夫链中指定长度的成功运行的出现次数”统计者。

ISOGAI,T.(ISOGAI,T.): "Some measurement models velated to an inverse Gaussian distribution" Reports of Statistical Application Research. 36. 1-11 (1991)

ISOGAI,T.(ISOGAI,T.)：“一些与逆高斯分布相关的测量模型”统计应用研究报告。