## Bernoulli

• Bernoulli
• Volume 19, Number 5A (2013), 2098-2119.

### Marked empirical processes for non-stationary time series

#### Abstract

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.

#### Article information

Source
Bernoulli, Volume 19, Number 5A (2013), 2098-2119.

Dates
First available in Project Euclid: 5 November 2013

https://projecteuclid.org/euclid.bj/1383661215

Digital Object Identifier
doi:10.3150/12-BEJ444

Mathematical Reviews number (MathSciNet)
MR3129045

Zentralblatt MATH identifier
06254555

#### Citation

Chan, Ngai Hang; Zhang, Rongmao. Marked empirical processes for non-stationary time series. Bernoulli 19 (2013), no. 5A, 2098--2119. doi:10.3150/12-BEJ444. https://projecteuclid.org/euclid.bj/1383661215

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