Renyi information, loglikelihood and an intrinsic distribution measure

Renyi information of order lambda for a continuous random variable is considered and its connection to the loglikelihood is established. From this relation, sin intrinsic loglikelihood-based distribution measure S-f is proposed. It is derived from the gradient of the spectrum of Renyi information and it can be represented, in the case of important parametric families, by a single function the trigamma function. Serving as a measure of the shape of a distribution, S-f can be used to compare the tails and shapes of various frequently used densities, whereas the traditional kurtosis measure is not applicable, including many heavy-tailed distributions such as Student's t with any degree of freedom, Cauchy, Cramer, Levy and Pareto distributions. A nonparametric procedure of estimating the measure S-f is also developed and its strong consistency is established. (C) 2001 Elsevier Science B.V. All rights reserved. MSG: primary 62B10; 62A10; secondary 62G05; 62G20.