Some Results On Limit Theory For Error Density Estimators In Nonlinear Autoregressive Models

Theory of Probability is a science of quantitatively studying regularity of ran-dom phenomena,which is extensively applied in natural science,technological science,managerial science,economic and finance etc.Hence,it has been developing rapidly since 1930's and many new branches have emerged from time to time.Limit Theory is one of the important branches and also an essential theoretical basis of science of Probability and Statistics.As stated in the classical book“Limit Distributions for Sums of Independent Random Variables"(1954)by Gendenko and Kolmogrov.“The epistemological value of the theory of probability is revealed only by limit theorems.Without limit theorems it is impossible to understand the real content of the primary concept of all our sciences-the concept of probability.”The classical limit theorems of probability theory for independent random variables had been developed successfully in 1930's and 1940 s,and they are the significant achievements in the progress of Prob-ability.The basic results were summed up in Gendenko and Kolmogrov's monograph "Limit Distributions for Sums of Independent Random Variables"(1954)and Petrov's monograph“Sums of Independent Random Variables"(1975).Studying various limiting properties of the time series is one of the orientations of the current study of Limit Theory.The linear models are important in time series analysis and have many applications to other fields,like economics,engineering,and physical science.However,as the movement law of the real world is often nonlinear,and some phenomena essentially display nonlinear behaviors.Realizing this,researchers have proposed numerous nonlinear time series models since the late 1970's.Tong's monograph " Non-Linear time series:A dynamical approach"(1990)and Fan and Yao s monograph " Nonlinear time series:nonparametric and parametric methods"(2003)represents a good account of nonlinear time series.This paper mainly considers the nonlinear autoregressive model of Xi=r?(Xi-1…Xi-s)+?i.let {Xi} be a strictly stationary process,for some ?=(?1,…,?q)'??(?)Rq where r0,???,is a family of known measurable function from Rs? R.We also assume that {?i} are independent and identically distributed random variables with mean zero,finite variance ?a2 and common density f.Moreover,Xi-1,...,Xi-s are assumed that to be independent of {?i}.Next,we need to define the kernel error density based on the true errors.Let the kernel function K(·)be a Borel measurable function on R,hn is a sequence of positive numbers(usually called bandwidth)which tends to zero as n? ?.Then the kernel estimate is defined as Let ?=(?1,...,?q)' be an estimator of ?.Based on the estimator ?,we define the residuals?i=Xi-r?(Xi-1,...,Xi-s),i=1,2,...,n Then the estimator fn(t)of fn(t)is defined as follows:For the proof,we introduce some basic assumptions which will be used throughout the paper.(Al)Let U(?)?(?)Rq be an open neighborhood of ?.We assume that for any y ? Rs,?=(?1,...,?q)?U,j,l=1,...,q where E[M12p(Xi-,....,Xi-s)}<? and E[M22p(Xi-1,...,Xi-s)]<oo for each i? 1.(A2)For j,l=1,...,q,there exists p>2 such that Yij and Zijl satisfying that(A3)Let ?=(?1,...,?q)' be an estimator for ? satisfying that there exists a constant C1(0<C1<?)such that where(?)(A4)K is a function of bounded variation on R,K" is bounded,andIn this paper,the author deals with some results on Limit Theory for error density estimators in nonlinear autoregressive models.In Chapter two,the author discusses the law of the iterated logarithm for error density estimators in nonlinear autoregressive models.Theorem 1 Suppose that at a fixed t ?R,there exists a constant 0<C<?such that f(t)satisfies|f(t)-f(t-y)|?<C|y|(?)y ?R.Assume that hn satisfies hn?0,where a and p satisfy ?>0,p>2 and ?pT>2,T?(0,1).Then,under the assumptions(Al)-(A4),we have that where CK:=?-?+?K2(u)du<?.In Chapter three,the author deals with the almost sure central limit theorem for error density estimators in nonlinear autoregressive models.Theorem 2 Suppose that at a fixed t ? R,there exists a constant 0<C<?such that f(t)satisfies|f(t)-f(t-y)|?|y|,(?)g?R.Assume that hn satisfies hn?0.here a and p satisfy ?>0,p>2 and apT>2,T?(0,1).And dn is a sequence of positive numbers satisfying the following conditions(C1)limsupn?? ndn(log Dn)?/Dn<? for some ?>1,where DN=?n=1N dn.(C2)DN??,DN=o{N?),for any ?>0.Then,under the assumptions(A1)-(A4),for any x ? R,where ?(x)is the standard normal distribution function.In Chapter four,the author obtains the Berry-Esseen bound for error density estimators in nonlinear autoregressive models.Theorem 3 Suppose that at a fixed t ? R,there exists a constant 0<C<?such that f(t)satisfies|f(t)-f(t-y|?C|y|,(?)y?R.Assume that hn satisfies hn?0,then,under the assumptions(Al).(A3)and(A4),for any y?R,where ?(y)is the standard normal distribution function.