Consistency Of Density Kernel Estimator For α Mixing Random Variables

Let X be a random variable with a continuous density function f(x), and {X_t}_t=1ⁿ be a sample drown from the population X. We define the kernel density estimator of f(x) aswhere K(·) is a kernel density function and h is a smoothing bandwidth.Many scholars have done lots of work about consistency of density kernel estimator under independent conditions. But lots of work show that for a huge number of financial and economic time series, independence doesn't hold, and the dependence is an intrinsic feature. As a matter of fact, there is plenty of literature on the discussion ofα-mixing of time series.This paper discusses the strong consistency and uniformly strong consistency of the density kernel estimator Underα-mixing random variable, and gives their convergent rate. We will use the following basic assumptions.A2.1 The process {X_t : t≥1} is strictly stationary andα-mixing with a common density function f(x),and there existsδ> 0 such thatsum from n=1 to∞α^{δ/(2+δ)}(n)<∞A2.2 The kernel function K is a univariate symmetric probability density function, has continuous bounded second derivative and satisfies the following moment conditions:∫_-∞^∞uK(u)du=0 and∫_-∞^∞u²K(u)du=σ_K²<∞．A2.3 The smoothing bandwidth h satisfies h→0, nh→∞. This paper will derive the following results.Theorem 2.1 Let A2.1-A2.3 hold.f has continuous second derivative f"(x) which is bounded in R¹. Ifsum from n=1 to∞(nh^{1+δ/(2+δ)-1/δ}(log₂nlogn)^1/δ<∞． then(?)_n,h(x)-f(x)=O(h²)+o((nh^{1+δ/(2+δ)})^-1/2(log₂nlogn)^1/2) a.s.Bosq.D.(1996, Lemma 2.1 on page 45) has given the following conclusion under geometricα-mixing random variable.Let {X_n,n≥1} is geometricα-mixing random variable with a common density function f(x), in other word,α(n) = c₀ρⁿ where c₀ > 0,ρ∈(0,1). Let K(x) and f(x) satisfy the conditions of Theorem 2.1. If h = c_n(logn/n)^1/5 where c_n→c > 0, then(?)_n,h(x)-f(x)=o((n/logn)^-2/5log₂n) a.s.whenδ→0, from Theorem 2.1 we have the following rateO(h²)+o((nh^{1+δ／(2+δ)})^-1/2(log₂ nlogn)^1/2)=O((n／logn)^-2/5),which is a little faster than the rate of Bosq.D. This shows we can deduce the conclusion of Bosq from our Theorem 2.1.Theorem 2.2 Let A2.1-A2.3 hold.f(x) is uniformly continuous and K(·) satisfies a Lipschitz condition.(1)If exists someγ> 0 such thatn^{1-(1+γ)δ/(2+δ)}h^{1+3δ／(2+δ)}(lognlog₂³n)^{-δ/(2+δ)}／logn→∞,then(2)f has continuous second derivative f″(x) which is bounded in R¹.Ifn^{1/2-(2+γ)δ/(2+δ)}h^{1／2+3δ／(4+2δ)}／log^1／2n→∞,thenUnder geometricα-mixing, if h = c_n(logn/n)^1/5 where c_n→c > 0, we have from Theorem 2.2Hence we have the following corollary. Corollary 2.1 Let A2.1-A2.3 hold, f(x) be uniformly continuous and K(·) satisfy a Lipschitz condition. Assume that the mixed coefficientα(n) is geometric decay. If nh/log n→∞, then for allγ> 0,holds;Corollary 2.2 Let A2.1-A2.3 hold, f(x) be uniformly continuous and K(·) satisfy a Lipschitz condition. Assume that the mixed coefficientα(n) is geometric decay.If h = c_n(log n/n)^1/5 where c_n→c > 0, then for allγ> 0,holds.This is the Bosq's Theorem 2.2. If h = Cn^-1/5, we have the following result.Corollary 2.3 Let A2.1-A2.3 hold, f(x) be uniformly continuous and K(·) satisfy Lipschitz condition. If mixed coefficientα(n) is geometric decay and h = Cn^-1/5, thenholds.Those results greatly improve those obtained by Bosq(1998) in Lemma 2.1 and Theorem 2.2.The arrangement of this thesis is: In chapter 1, some knowledge and research about the density kernel estimator are first introduced .In chapter 2, some basic assumptions are given, meanwhile, the main results in the paper and some remarks about the results are showed.In chapter 3, several lemmas which are needed in the paper are given.In chapter 4, the part is the main proof process.We prove tow results in detail.In chapter 5, we summarize the main results of this paper .