Central limit theorem for the empirical process of a linear sequence with long memory

We discuss the functional central limit theorem (FCLT) for the empirical process of a moving-average stationary sequence with long memory. The cases of one-sided and double-sided moving averages are discussed. In the case of one-sided (causal) moving average, the FCLT is obtained under weak conditions of smoothness of the distribution and the existence of (2+δ)-moment of i.i.d. innovations, by using the martingale difference decomposition due to Ho and Hsing (1996, Ann. Statist. 24, 992–1014). In the case of double-sided moving average, the proof of the FCLT is based on an asymptotic expansion of the bivariate probability density.     

Testing of a sub-hypothesis in linear regression models with long memory errors and deterministic design

This paper considers the problem of testing a sub-hypothesis in homoscedastic linear regression models where errors form long memory moving average processes and designs are non-random. Unlike in the random design case, asymptotic null distribution of the likelihood ratio type test based on the Whittle quadratic form is shown to be non-standard and non-chi-square. Moreover, the rate of consistency of the minimum Whittle dispersion estimator of the slope parameter vector is shown to be n^-(1-α)/2$n^{-(1-\alpha )/2}$, different from the rate n^-1/2$n^{-1/2}$ obtained in the random design case, where α$\alpha$ is the rate at which the error spectral density explodes at the origin. The proposed test is shown to be consistent against fixed alternatives and has non-trivial asymptotic power against local alternatives that converge to null hypothesis at the rate n^-(1-α)/2$n^{-(1-\alpha )/2}$.     

Goodness-of-fit testing under long memory

This paper discusses the problem of fitting a distribution function to the marginal distribution of a long memory moving average process. Because of the uniform reduction principle, unlike in the i.i.d. set up, classical tests based on empirical process are relatively easy to implement. More importantly, we discuss fitting the marginal distribution of the error process in location, scale, location–scale and linear regression models. An interesting observation is that in the location model, location–scale model, or more generally in the linear regression models with non-zero intercept parameter, the null weak limit of the first order difference between the residual empirical process and the null model is degenerate at zero, and hence it cannot be used to fit an error distribution in these models for the large samples. This finding is in sharp contrast to a recent claim of Chan and Ling (2008) that the null weak limit of such a process is a continuous Gaussian process. This note also proposes some tests based on the second order difference for the location case. Another finding is that residual empirical process tests in the scale problem are robust against not knowing the scale parameter.