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We consider the problem of improving the efficiency of the nonparametric entropy estimation for a stationary ergodic process. Our approach is based on the nearest-neighbor distances. We propose a broad class of metrics on the space of right-sided infinite sequences drawn from a finite alphabet. The new metric has a parameter that is a nonincreasing function. We prove that, under certain conditions, our estimators have a small variance and show that a special selection of the metric parameters reduces the estimator's bias.

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Abstract. New invariants of measures, called the β-statentropy, are described. They are similar to the entropy and the HP -spectrum for dimensions. The β-statentropy admits construction of a statistical estimator calculated by n independent points distributed in accordance with a given measure. The accuracy of this estimator is O(n −c ), where c is some constant, and the complexity of calculation is O(n 2 ).It is shown that for an exact dimensional measure the 0-statentropy coincides with the Hausdorff dimension, and for a Markov measure the β-statentropy coincides with the HP -spectrum for dimensions.An application of the β-statentropy to finding the entropy and dimensional characteristics of dynamical systems is described.