The Asymptotic Properties Of Frequency Polygons Density Estimator For Weakly Dependent Processes

The parameter density estimation is always a hot issue of the density estimation of mathe-matical statistics, its application is also more and more widely in ecological, biology, medicine, economic and other fields. The parameter estimation of the density function has a lot of kinds, such as the histogram estimate, the frequency polygons estimate, Rosenblatt estimate, Parzen kernel estimate and the most nearest-neighbor estimate.Scott, D.W (1985) proposed the frequency polygons estimate based on the histogram technol-ogy. The people later found that the frequency polygon estimate has the same rate of convergence as those of modem non-negative kernel estimators on the integral mean square error. In the nu-merical calculation, the frequency polygons estimate has the same calculated amount as histogram estimate basically, but its rate of convergence is faster than that of histogram estimate.Scott, D. W(1985) point out that for large bivariate data sets,the computational simplicity of the frequency polygon estimate and the ease of determining exact equiprobable contours may outweigh the in-creased accuracy of a kernel estimate. Bivariate contour plots based on millions of observations are increasingly required in applications including high-energy physics simulation experiments, cell sorters, and geographical data representation. Furthermore, such data are usually collected in binned form.The frequency polygon estimate can be a valuable tool for examination and presenta-tion of data. Thus in-depth study of the frequency polygons estimate has very important practical value.Some scholars have done theoretical research on the frequency polygons estimate now. Basing on method of histogram estimate under weakly dependent sequence by Lanh Tat Tran (1994), Michel carbon (1997) have studied integral mean square error and the optimal window, asymptotic variance, as well as the strong consistency of frequency interpolation density estimation under weakly dependent sequence, attained the corresponding convergence rate. Michel carbon (2010) investigated asymptotic normality of the multi-dimension frequency polygon estimator under a stationary α-mixing process, but didn’t give corresponding convergence rate. Nadia Bensaid (2010) discussed integral mean square error and convergence speed of strong consistency of multi-dimension frequency polygon estimate under α-mixing condition.These research conclusion is a preliminary conclusion, because some conditions are too complicated to verify, some conclusions have more demand on α-mixing rate, and some conclusions are given no convergence rate, so the further theory research on frequency polygons estimate is necessary.Firstly, This paper adopt exponent inequality and moment inequalities to prove the asymp-totic properties of frequency polygons estimate for weakly dependent processes, not only get consistency and asymptotic normality of frequency polygons estimate under the condition which is weaker than that in Michel carbon (1997,2010), but also attain the corresponding convergence rate. We also attain consistency of frequency polygons estimate under ψ-mix and NA sequence, and attain the corresponding convergence rate. Secondly, we choose two time series model AR (1) and AR (2) to make numerical simulation for frequency polygons estimate. Basing on this, we in-vestigate the influence on frequency polygons estimate under weakly dependent processes by the sample size, dependent coefficient, dependency structure complexity. Finally we choose textile clothing plate index and the Shanghai index logarithmic return rate to make empirical study, and the result shows that the frequency polygons estimate has better practicability for the α-mixing sequence of AR (1) model structure.