Nonparametric Kernel Regression Estimation With Random Left-truncation Responses For Functional Stationary Ergodic Data

With the approach of big data,the data we processing in our real life is diversified. A lot of data obtained in scientific experiments is function data. This is one of the hot spots in recent years of scholars. There is a difference between the traditional data research and function data research. Traditional data research is confined to the case of finite dimensionality. However,function data of the research is carried on the case of infinite dimensionality. Needing to define a new measure. In electricity consumption data and spectrometric data research,the data we obtained is function data. Research on the theory of function data, is of great significance in the real scientific research.However,in our real life,the data we processing is incomplete. For various reasons, the data may have missing phenomenon. We get only part of the data. There are three kinds of the data missing:left-truncation,missing data,censored data.Left-truncation in our real life, such as in medicine, earthquake, biostatistics, econometrics are concerned, it has important research value. This paper is to explore nonparametric kernel regression estimation with random left-truncation responses for function data in stationary ergodic. And they have achieved some results, the contents are as follows:Firstly,in the left-truncation condition, by using the Nadarage-Watson kernel estimation of regression function by ergodic function under the condition of the left-truncation data to estimate the convergence in probability, asymptotic normality.Secondly, by the asymptotic normality we obtain some lemmas and as applications the approximate (1-α) confidence interval of the regression function is also presented.Thirdly, in addition,at the end of the paper, we present a study about the bias of the nonparametric hazard function in stationary ergodic condition.