The Asymptotic Properties Of Edge Frequency Polygons Density Estimator For Strong Mixing Samples

Nonparametric density estimation is a kind of important density estimation, it has widely used in real life. In nature, Most general still unknown distribution and the sample is not necessarily independent, so the nonparametric density function estimation has attracted much attention in the research of mathematical statistics. From the initial histogram density estimation, nonparamet-ric density function through constantly exploring gradually has some density estimation, such as Rosenblatt density estimation, Parzen kernel estimation, kernel density estimation, nearest neigh-bor density estimates, moving average histogram density estimation and frequency interpolation density estimation. Jones (1998) introduced The Edge Frequency Polygons Density Estimator on the basis of histogram, and under the independent samples showed that the estimation of mean square error (mse) is smaller than the frequency interpolation density estimation mean square error (mse). In later studies found that the edge frequency interpolation density estimation and his-togram are basically the same on the amount calculation, but its convergence speed is faster than the histogram. In addition, the alpha blending sequence have a wide range of applications in time series model, reliability theory, ecological system research and other fields, so statisticians have interesting in studying alpha blending sequences.In statistics, the density function is one of the most basic concepts. The density function estimation is widely used for the research of the non-parametric density estimation. In nature, most of the distributions of the general are unknown and the samples are not necessarily independent. The problem of the nonparametric density estima-tion has attracted a widely attention in the research of mathematical statistics. The nonparametric estimation of density function started from the most primitive histogram. After the continuous explorations in the later years, Rosenblatt estimation, Parzen kernel estimation, kernel density es-timation, the nearest-neighbor density estimation, the moving average histogram estimation and the frequency interpolation density estimation and so on have come into beings gradually. Jones,et al.(1998)introduced he concept of edge frequency interpolation density estimation based on the histogram of non-parametric estimation and proved that the mean square error was less than the one in the reaserch of the front. In addition, the mixing sequence has a widely application in reliability theory, time series model, ecosystem research and other fields, so the statisticians show a widely attention and interest in the study of mixed sequence. This article is to discuss the asymptotic property of the mixed sequence frequency interpolation.In1998, Jones proposed that the edge frequency interpolation density estimation, and under the independent sample got the mean square error (mse) of the estimates. However, there is no literature has more research on the nature of the estimate. Therefore, it is meaningful to research the edge density estimation frequency interpolation asymptotic properties under the alpha hybrid dependent samples.Under the alpha blend samples, this article was mainly research edge density estimation fre-quency interpolation asymptotic properties. First of all, according to the definition of asymptotic unbiasedness, this article proved asymptotic unbiasedness of the edge frequency interpolation den-sity estimation. Secondly, Using method of frequency interpolation density estimation to study the properties of the fringe frequency interpolation density estimation, got the optimum window width, asymptotic variance, consistency, and the corresponding rate of convergence of the fringe frequency interpolation on the edge of the alpha blend sample. Finally, by using matrix inequality and index inequality to prove that edge frequency interpolation asymptotic normality on the edge of the alpha blend sample.