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Estimation and bootstrap for stochastically monotone Markov processes

Author

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  • Michael H. Neumann

    (Friedrich-Schiller-Universität Jena)

Abstract

The Markov property is shared by several popular models for time series such as autoregressive or integer-valued autoregressive processes as well as integer-valued ARCH processes. A natural assumption which is fulfilled by corresponding parametric versions of these models is that the random variable at time t gets stochastically greater conditioned on the past, as the value of the random variable at time $$t-1$$ t - 1 increases. Then the associated family of conditional distribution functions has a certain monotonicity property which allows us to employ a nonparametric antitonic estimator. This estimator does not involve any tuning parameter which controls the degree of smoothing and is therefore easy to apply. Nevertheless, it is shown that it attains a rate of convergence which is known to be optimal in similar cases. This estimator forms the basis for a new method of bootstrapping Markov chains which inherits the properties of simplicity and consistency from the underlying estimator of the conditional distribution function.

Suggested Citation

  • Michael H. Neumann, 2024. "Estimation and bootstrap for stochastically monotone Markov processes," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 87(1), pages 31-59, January.
    • Handle: RePEc:spr:metrik:v:87:y:2024:i:1:d:10.1007_s00184-023-00903-7
    DOI: 10.1007/s00184-023-00903-7
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