The Annals of Statistics

Nonparametric estimation in null recurrent time series

Hans Arnfinn Karlsen and Dag Tjøstheim

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We develop a nonparametric estimation theory in a nonstationary environment, more precisely in the framework of null recurrent Markov chains. An essential tool is the split chain, which makes it possible to decompose the times series under consideration into independent and identical parts. A tail condition on the distribution of the recurrence time is introduced. This condition makes it possible to prove weak convergence results for sums of functions of the process depending on a smoothing parameter. These limit results are subsequently used to obtain consistency and asymptotic normality for local density estimators and for estimators of the conditional mean and the conditional variance. In contradistinction to the parametric case, the convergence rate is slower than in the stationary case, and it is directly linked to the tail behavior of the recurrence time. Applications to econometric, and in particular to cointegration models, are indicated.

Article information

Ann. Statist., Volume 29, Number 2 (2001), 372-416.

First available in Project Euclid: 24 December 2001

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62G07: Density estimation
Secondary: 60J05: Discrete-time Markov processes on general state spaces

Nonstationary time series models null recurrent Markov chain nonparametric kernel estimators split chain


Karlsen, Hans Arnfinn; Tjøstheim, Dag. Nonparametric estimation in null recurrent time series. Ann. Statist. 29 (2001), no. 2, 372--416. doi:10.1214/aos/1009210546.

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