基于残差的非线性自回归模型的拟合优度检验

时间序列模型中，考虑误差分布的拟合优度检验是很重要的．Lee和Na（2002）考虑了在线性自回归模型下，基于残差的Bickel—Rosenblatt检验问题．他们指出了在原假设条件下，检验统计量的极限分布与利用独立同分布观测值的经典Bickel—Rosenblatt检验相同．本文主要讨论无限阶的非线性自回归模型的基于残差的Bickel—Rosenblatt检验统计量的渐近性质．我们证明了在自回归函数未知的情况下，当满足一定条件时，检验统计量的渐近性质与基于真实误差的统计量的性质相同．

GOODNESS-OF-FIT TEST USING RESIDUALS IN INFINITE-ORDER NONLINEAR AUTOREGRESSIVE MODELS

It is of importance to consider the goodness-of-fit tests of the error distribution in time series models. Recently, Lee and Na（2002） considered the Bickel-Rosenblatt test based on the integrated squared error of the true error density function and kernel density estimate from the residuals in linear autoregressive models. They derived the asymptotic distribution under the null-hypothesis, which is the same as that for the classical Bickel-Rosenblatt test for the distribution of i.i.d observations. In this paper, we extend the results from the set up of linear autoregressive models to the case of infinite-order nonlinear autoregressive time series models. We study the asymptotic behavior of the Bickel-Rosenblatt test statistic. It is proved that without knowing the nonlinear autoregressive function, and under some conditions, the test statistic behaves asymptotically the same as the one based on the true errors.