Asymptotic Properties Of Error Density Estimator In Autoregressive Modles Under Strong Mixing Assumptions

In this paper,we mainly consider the following problem:Suppose that{Xi,i=-p+1,-p+2,...}is a sequence of strictly stationary real random variables satisfying the nonlinear autoregressive model of orderp Xi=gθ(Xi-1,…,Xi-p)+εi, i≥1. for someθ=(θ1,…,θq)'∈(?) Rq,where gθ,θ∈(?),is a family of known measurable func-tions from Rq→R.Also the{εi} are strong mixing random variables with mean zero,finite varianceσ2,common distribution F and common density f.In time series we do not observeε1,ε2,…,εn,we can observe X1,X2,…,Xn.We will first compute an estimatorθn,=(θn1,…,θnq)'forθ=(θ1,…,θq)'.Letθn=(θnl,…,B,q)'be an estimator forθ=(θ1,…,θq)',and letεi=Xi-gθn(Xi-1,…,Xi-P)i≥1. denote the residuals.Based on these residual,we construct a histogram-type error density estimator of the error density f as follows:In this paper,we have following main result:Theorem 1(Asymptotic normality)Suppose f that satisties the local Lipschitz condi-tion of order 1 at x∈R with f(x)＞0.Letαk=O(k-r)for some r>2,andThen,under some assumptions,we have Theorem 2 (law of the iterated logarithm) Supposef that satisfies the local Lipschitz condition of order 1 atx∈R withf(x)>0.Letαk-O(k-r) for somer>3,andThen,under some assumptions,we have...