Entropy for Random Partitions and Its Applications

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Abstract

Asymptotic properties of partitions of the unit interval are studied through the entropy for random partition

where X1, n < X2, n < … < Xn, n are the order statistics of a random sample {Xi, i ≥ n}, X0, n ≡ −∞, Xn+1, n ≡ +∞ and F(x) is a continuous distribution function. A characterization of continuous distributions based on ℰn(F) is obtained. Namely, a sequence of random observations {Xi, i≥1} comes from a continuous cumulative distribution function (cdf) F(x) if and only if

where γ = 0.577… is Euler's constant. If {Xi, i≥1} come from a density g(x) and F is a cdf with density ƒ(x), some limit theorems for ℰn(F) are established, e.g.,

Statistical estimation as well as a goodness-of-fit test based on ℰn(F) are also discussed.

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