Research Based On Asymptotic Properties Of Nonparameter Kernel Mode Estimator For Dependent Left-truncated Data

With the development of society and technology, statistics has been paid moreand more attention. An important part of the statistical research is to collect data,but some cannot be collected in the process of data collection for unknown reasons,which is called truncated data (or censored data). Truncated data containsleft-truncated data, right-censored data and interval truncated data, whose researchderives from such fields as astronomy, economics, epidemiology, biostatistics andso on. The main researching methods of truncated data include parametric,semi-parametric and nonparametric estimation, and the last one is a hot topic ofinternational research. In the nonparametric methods, the kernel estimation isfavored for its good smoothness, small amount of calculation and wideserviceability. In order to study the general level and concentration trend oftruncated data, scholars usually choose the mode, because it is gotten simply andhardly influenced by extreme data.In recent years, the theory of truncated data has been rapidly developed andwidely applied. Meanwhile there are also many problems to be solved. In thisdissertation, we mainly study the asymptotic behaviors of nonparametric kernelestimation of the mode based on the α-mixing dependent left-truncated data, andget some better results. The details are given as follows:Firstly, we investigate the nonparametric kernel estimation of the first andsecond derivative of the density function under left-truncated and stationaryα-mixing conditions. Moreover, we establish the strong uniform convergence of theestimators with the Fuk-Nagaev inequality under certain assumptions.Secondly, we study the nonparametric kernel estimation of the mode functionof α-mixing left-truncated data. Then we get the estimate model with the method ofTaylor’s expansion, as well as the asymptotic normality of estimation with theBernstein’s big-block and small-block method. Further-more, the confidenceinterval for mode estimation is constructed.Finally, we generate several groups of α-mixing data from the AR (1) model,and get the mean square error table, histogram and normal probability plot of their mode estimation with the Matlab software, which show the rationality of mainconclusions.