Entropy for Random Partitions and Its Applications

Abstract

Asymptotic properties of partitions of the unit interval are studied through the entropy for random partition $$E_n (F) \equiv - \sum\limits_{j = 1}^{n + 1} {[F(X_{j,n} ) - F(X_{j - 1,n} )]\log \{ [F(X_{j,n} ) - F(X_{j - 1,n} )](n + 1)\} }$$ where \(X_{1,n} < X_{2,n} < \cdot \cdot \cdot < X_{n,n}\) 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 \(E_n (F)\) 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 $$\mathop {\lim }\limits_{n \to \infty } E_n (F) = \gamma - 1{\text{ a}}{\text{.s}}{\text{.}}$$ 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 f(x), some limit theorems for \(E_n (F)\) are established, e.g., $$\mathop {\lim }\limits_{n \to \infty } E_n (F) = - \int_{\{ x:g(x) > 0\} } {f(x)\log \frac{{f(x)}}{{g(x)}}dx + \gamma - 1{\text{ in probability}}}$$ Statistical estimation as well as a goodness-of-fit test based on \(E_n (F)\) are also discussed.