论文标题
马尔可夫链的无偏见估计器
Unbiased time-average estimators for Markov chains
论文作者
论文摘要
我们考虑了马尔可夫链功能的时间平均值估算器$ f_ {k} $。在耦合假设下，我们表明$ f_ {k} $的期望具有限制$μ$，因为时间步长的数量流向无穷大。我们描述了$ f_ {k} $的修改，该修改产生了$μ$的无偏估计器$ \ hat f_ {k} $。结果表明，$ \ hat f_ {k} $是正方形的，并且预期运行时间有限。在某些条件下，$ \ hat f_ {k} $无需任何预先计算就可以构建，并且渐近至少与$ f_ {k {k} $一样高，最多可任意接近$ 1 $的乘法常数。我们的方法为$ f_ {k} $的偏置提供了无偏的估计器。我们研究了对波动性预测，队列和高维高斯载体的模拟的应用。我们的数值实验与我们的理论发现一致。
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.
