Unbiased time-average estimators for Markov chains
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866918204264153088 |
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| author | Kahale, Nabil |
| author_facet | Kahale, Nabil |
| contents | We consider a time-average estimator $f_{k}$ of a functional of a Markov chain. Under a coupling assumption, we show that the expectation of $f_{k}$ has a limit $μ$ as the number of time-steps goes to infinity. We describe a modification of $f_{k}$ that yields an unbiased estimator $\hat f_{k}$ of $μ$. It is shown that $\hat f_{k}$ is square-integrable and has finite expected running time. Under certain conditions, $\hat f_{k}$ can be built without any precomputations, and is asymptotically at least as efficient as $f_{k}$, up to a multiplicative constant arbitrarily close to $1$. Our approach provides an unbiased estimator for the bias of $f_{k}$. We study applications to volatility forecasting, queues, and the simulation of high-dimensional Gaussian vectors. Our numerical experiments are consistent with our theoretical findings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_09581 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Unbiased time-average estimators for Markov chains Kahale, Nabil Statistics Theory Computation We consider a time-average estimator $f_{k}$ of a functional of a Markov chain. Under a coupling assumption, we show that the expectation of $f_{k}$ has a limit $μ$ as the number of time-steps goes to infinity. We describe a modification of $f_{k}$ that yields an unbiased estimator $\hat f_{k}$ of $μ$. It is shown that $\hat f_{k}$ is square-integrable and has finite expected running time. Under certain conditions, $\hat f_{k}$ can be built without any precomputations, and is asymptotically at least as efficient as $f_{k}$, up to a multiplicative constant arbitrarily close to $1$. Our approach provides an unbiased estimator for the bias of $f_{k}$. We study applications to volatility forecasting, queues, and the simulation of high-dimensional Gaussian vectors. Our numerical experiments are consistent with our theoretical findings. |
| title | Unbiased time-average estimators for Markov chains |
| topic | Statistics Theory Computation |
| url | https://arxiv.org/abs/2209.09581 |