Metrika Metrika (1994) 41:109-119 An Information Theoretic Argument for the Validity of the Exponential Model KONSTANTINOSZOGRAFOS AND KOSMAS FERENTINOS University of loannina, Department of Mathematics, Section of Probability--Statistics and Operational Research, 45110 loannina, Greece Abstract: Based on the Cram6r-Rao inequality (in the multiparameter case) the lower bound of Fisher information matrix is achieved if and only if the underlying distribution is the r-parameter exponential family. This family and the lower bound of Fisher information matrix are characterized when some constraints in the form of expected values of some statistics are available. If we combine the previous results we can find the class of parametric functions and the corresponding UMVU estimators via Cramer-Rao inequality. Key Words and Phrases: Fisher information, Cramer-Rao inequality, UMVU estimators, exponen- tial family. I Introduction The estimation of a probability distribution, using the information contained in a random sample, is of great interest in statistics and many other fields including image restoration, quantum mechanics, diffraction theory, reliability estimation, pattern recognition, queuing theory and computer system modelling (cf. Shore and Johnson (1980), Holevo (1982), Frieden (1988)). There are several approaches to this matter. In statistical information theory the principle of maximum entropy is well known and provides a general method of determining an unknown probability distribution based on the knowledge of the mean values of some random variables called hereafter constraints (cf. Jaynes (1957), Kapur (1989)). The minimum discrimination information principle (cf. Kullback (1959), Shore and Johnson (1980)) is a generalization of maximum entropy, that applies in cases where there exist a prior estimate of the distribu- tion in addition to the constraints. These concepts are also equivalent to Gauss's principle for the derivation of an unknown distribution (cf. Campbell (1970)). Recently, Frieden (1988) has proposed a critirion for estimating the unknown probability distribution according to which the distribution is required to have the maximum Cram6r-Rao lower bound. As it is known the Cram6r-Rao lower bound depends only on the distribution involved. This method can be consid- 0026-1335/94/2/109-119 $2.50 9 1994 Physica-Verlag, Heidelberg