Abstract
This paper offers sufficient conditions for the Miller–Madow estimator and the jackknife estimator of entropy to have respective asymptotic normalities on countably infinite alphabets.
Highlights
Let X = {`k ; k ≥ 1} be a finite or countably infinite alphabet, let p = { pk ; k ≥ 1} be a probability distribution on X, and define K = ∑k≥1 1[ pk > 0], where 1[·] is the indicator function, to be the effective cardinality of X under p
This paper offers sufficient conditions for the Miller–Madow estimator and the jackknife estimator of entropy to have respective asymptotic normalities on countably infinite alphabets
The problem of statistical estimation of entropy has a long history. It is well-known that no unbiased estimators of entropy exist, and, for this reason, much energy has been focused on deriving estimators with relatively little bias (see [5] and the references therein for a discussion of some of these)
Summary
Its theoretical properties have been studied going back, at least, to [6], where conditions for consistency and asymptotic normality, in the case of finite alphabets, were derived. It would be almost fifty years before corresponding conditions for the countabe case would appear in the literature. The jackknife entropy estimator is another commonly used estimator designed to reduce the bias of the plug-in. It is calculated in three steps: 2.
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