Open Access
November 2013 Marked empirical processes for non-stationary time series
Ngai Hang Chan, Rongmao Zhang
Bernoulli 19(5A): 2098-2119 (November 2013). DOI: 10.3150/12-BEJ444


Consider a first-order autoregressive process $X_{i}=\beta X_{i-1}+\varepsilon_{i}$, where $\varepsilon_{i}=G(\eta_{i},\eta_{i-1},\ldots)$ and $\eta_{i}$, $i\in\mathbb{Z}$ are i.i.d. random variables. Motivated by two important issues for the inference of this model, namely, the quantile inference for $H_{0}\colon\ \beta=1$, and the goodness-of-fit for the unit root model, the notion of the marked empirical process $\alpha_{n}(x)=\frac{1}{n}\sum_{i=1}^{n}g(X_{i}/a_{n})I(\varepsilon_{i}\leq x)$, $x\in\mathbb{R}$ is investigated in this paper. Herein, $g(\cdot)$ is a continuous function on $\mathbb{R}$ and $\{a_{n}\}$ is a sequence of self-normalizing constants. As the innovation $\{\varepsilon_{i}\}$ is usually not observable, the residual marked empirical process $\hat{\alpha}_{n}(x)=\frac{1}{n}\sum_{i=1}^{n}g(X_{i}/a_{n})I(\hat{\varepsilon}_{i}\leq x)$, $x\in\mathbb{R}$, is considered instead, where $\hat{\varepsilon}_{i}=X_{i}-\hat{\beta}X_{i-1}$ and $\hat{\beta}$ is a consistent estimate of $\beta$. In particular, via the martingale decomposition of stationary process and the stochastic integral result of Jakubowski (Ann. Probab. 24 (1996) 2141–2153), the limit distributions of $\alpha_{n}(x)$ and $\hat{\alpha}_{n}(x)$ are established when $\{\varepsilon_{i}\}$ is a short-memory process. Furthermore, by virtue of the results of Wu (Bernoulli 95 (2003) 809–831) and Ho and Hsing (Ann. Statist. 24 (1996) 992–1024) of empirical process and the integral result of Mikosch and Norvaiša (Bernoulli 6 (2000) 401–434) and Young (Acta Math. 67 (1936) 251–282), the limit distributions of $\alpha_{n}(x)$ and $\hat{\alpha}_{n}(x)$ are also derived when $\{\varepsilon_{i}\}$ is a long-memory process.


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Ngai Hang Chan. Rongmao Zhang. "Marked empirical processes for non-stationary time series." Bernoulli 19 (5A) 2098 - 2119, November 2013.


Published: November 2013
First available in Project Euclid: 5 November 2013

zbMATH: 06254555
MathSciNet: MR3129045
Digital Object Identifier: 10.3150/12-BEJ444

Keywords: Goodness-of-fit , long-memory , Marked empirical process , Quantile regression , unit root

Rights: Copyright © 2013 Bernoulli Society for Mathematical Statistics and Probability

Vol.19 • No. 5A • November 2013
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