The Edge Frequency Polygons Estimator Of Density For α-mixing Random Fields

In the unknown population distribution,the less known overall information,the not neces-sarily independent sample,the nonparametric density estimators is a very good choice.Due to various reasons,we can not arbitrarily assume the overall distribution of the data in the actual data analysis,so we need to use non parametric model to estimate the density function f(x).Nonpara-metric density estimators has been widely used in the fields of environmental science,electronic physics,biomedicine,geology,economics and finance,regional economy,etc.There are many nonparametric density estimators methods,which includes histogram,Rosen-blatt estimators,Kernel density estimators,likelihood estimators,frequency polygon,etc.and in which the most widely used is the frequency polygon,and the estimators effect is well.The frequency polygon that will exceed the n-4/5 integral mean square error(IMSE)convergence rate of the non-negative Kernel estimator and is faster than the histogram estimation’s n-2/3.In the numerical calculation,the calculation of the estimated frequency is relatively small,so it has more advantages than the Kernel density estimator.Therefore,it is of great significance to study the estimator of the frequency polygon.Since Scott(1985)proposed the frequency polygon estimator,it has attracted many scholars to study it.Then Jones(1998)improved the frequency polygon estimator,and the edge frequency interpolation estimation was proposed,and its’ the asymptotic mean square error(AMSE)of the estimator is smaller than the traditional frequency polygon.At the same time he proved that this new estimator method.the edge frequency polygon estimator,has a better theoretical performance than the traditional midpoint frequency polygon.Therefore,this paper choose a better method of the edge frequency polygon density who has a better theoretical performance.But so far,the frequency polygon estimator is very little research in the spatial data,which is the random field,only Carbon et al,Bensaid and Dabo-Niang and Machkouri El were integral studied IMSE,the optimal bin width,the asymptotic variance,the asymptotic normality and the uniformly strong consistency under the random field.However,at present,there is still no scholar to study the asymptotic performances of the edge frequency polygon density estimator in random field.So.this paper will study the performances of the edge frequency polygon in α-mixing random field.Firstly,we will prove that the α-mixing random field under some certain conditions,and the edge frequency polygons estimator of density has asymptotic variance.Secondly,we will prove that the random field samples meet certain conditions,when n→∞,((?)bn)1/2[fn(x)-Efn(x)]σ-1(x)has a standard normal distribution.And,we will proved the strong consistency of the edge frequency polygons estimator of density for α-mixing random fields.Finally,different sample sizes and window widths are discussed by means of numerical simulation,further verify and illustrate the rationality and correctness of the conclusion.