Statistical Inference For The AR Models With Skew Normal Distributions

In the recent years, there has been increasing interest in a scheme for the genesisof families of distributions whose main features are in the review papers by Azzalini.The more emblematic and in a sense simplest representative of these distributions isso-called skew-normal (SN) distribution. The density function is, for the onedimensional case,whereφ(?) andΦ(?) denote the N (0,1) density and distribution function ,respectively,andξ,σandλare location, scale and shape parameters, respectively, andξ,λ∈R ,σ∈Z~+ .The SN family forms a superset of the normal distribution, which corresponds tothe choice ofλ= 0 . Whenλ= 0 , thenThe density (1.1) reduce to the N (ξ,σ2 ) . Obviously, the SN distribution extends thefamily of normal distribution by the addition of a shape parameter which regulatesskewness.The interest in skew-normal distribution comes from two directions. On thetheoretical side, it enjoys a number of formal properties which reproduce or resemblethose of the normal distribution and appear to justify its name skew-normal distribution,if ( , , ) , then For scale case, .From the applied viewpoint, there is general tendency in the statistical literaturetowards more flexible methods to represent features of the data as adequately aspossible and to reduce unrealistic assumptions. In practical problems, we usuallyshould deal with many data with skewness, multimodality and heavily tail, but stillretains some broad similarity with the normal distribution. Although the normalityassumption leads to simple, mathematically tractable and powerful test, normaldistribution is not fit for dealing these problems here. For the reasons mentioned above, in this paper, we mainly introduce the SNdistribution and its statistical properties in section 2. Then we consider the aspects ofparameter estimation including moment estimation and maximum likelihood estimation(MLE) in section 3, and we give the main results as follow. Firstly, the momentestimations of parameters is,where .Then, the ECM-algorithm for MLE is given as follow. We give two propositionsabout parameters estimation at first.Proposition 1.1 If random variable (τ|Y=y) whose density defined as (3.2.7),thenProposition 1.2 If random variable (τ|Y=y) whose density defined similarabove, thenThen we will get the Q function with complete data,For , the steps of ECM-algorithm are,E-step: Given , computewhere is the iteration.CM-step 1: Update by maximum Q function overξ, then we have iteration Namely , .CM-step 2: Fixed , update by maximum Q function overξσ~2 , then we have iteration ,Namely,CM-step 3:Fixed , and , update by maximum Qfunction overδ, then we have iteration by the solution of Namely,For , then , so we only should choose one of rootsfrom the equation, after that , we can get .Noting that CM-step 3 requires a one-dimensional search for the root ofδfroma complex equation, so this method is very slow for searching the root. We update by optimizing the following constrained actual log-likelihood function, Specifically,Differentiating the equation overλand let it equals to zero,Namely,After that, we discuss the asymptotic behavior of the MLE. The asymptoticcovariance of MLE can be estimated by the inverse of the observed informationmatrix, , where is the score vector corres-ponding to the single observation y evaluated at .Expressions for the elements of the score vector with respect to , , aregiven by setting . Then we havewher C is constant. ThenAt last, we put application SN distribution to AR models in section 4, and give theconditional least estimation (CLS) for autoregression coefficients,Namely,Then, we introduce the tests of hypothesis of AR models.