The Annals of Statistics

Residual empirical processes for long and short memory time series

Ngai Hang Chan and Shiqing Ling

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This paper studies the residual empirical process of long- and short-memory time series regression models and establishes its uniform expansion under a general framework. The results are applied to the stochastic regression models and unstable autoregressive models. For the long-memory noise, it is shown that the limit distribution of the Kolmogorov–Smirnov test statistic studied in Ho and Hsing [Ann. Statist. 24 (1996) 992–1024] does not hold when the stochastic regression model includes an unknown intercept or when the characteristic polynomial of the unstable autoregressive model has a unit root. To this end, two new statistics are proposed to test for the distribution of the long-memory noises of stochastic regression models and unstable autoregressive models.

Article information

Ann. Statist., Volume 36, Number 5 (2008), 2453-2470.

First available in Project Euclid: 13 October 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G30: Order statistics; empirical distribution functions
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Empirical process long-memory time series residuals unit root weak convergence


Chan, Ngai Hang; Ling, Shiqing. Residual empirical processes for long and short memory time series. Ann. Statist. 36 (2008), no. 5, 2453--2470. doi:10.1214/07-AOS543.

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