Truncation, Information, and the Coefficient of Variation

The Fisher information in a random sample from the truncated version of a distribution that belongs to an exponential family is compared with the Fisher information in a random sample from the un- truncated distribution. Conditions under which there is more information in the selection sample are given. Examples involving the normal and gamma distributions with various selection sets, and the zero-truncated binomial, Poisson, and negative binomial distributions are discussed. A property pertaining to the coefficient of variation of certain discrete distributions on the non-negative integers is introduced and shown to be satisfied by all binomial, Poisson, and negative binomial distributions.