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Entropy estimators with almost sure convergence and an O(n-1) variance

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  • The problem of the estimation of the entropy rate of a stationary ergodic process $\mu$ is considered. A new nonparametric entropy rate estimator is constructed for a sample of n sequences $(X_1$(1)$,\ldots, (X_m$(1)$),\ldots, (X_1$(n) $,\ldots, (X_m$(n)$)$ independently generated by $\mu$. It is shown that, for $m=O(\log n)$, the estimator converges almost surely and its variance is upper-bounded by $O(n$−1$)$ for a large class of stationary ergodic processes with a finite state space. As the order $O(n$−1$)$ of the variance growth on $n$ is the same as that of the optimal Cramer-Rao lower bound, presented is the first near-optimal estimator in the sense of the variance convergence.
    Mathematics Subject Classification: Primary: 94A15, 94A17; Secondary: 37M25.


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