TESTING DEPARTURES FROM GAMMA, RAYLEIGH AND TRUNCATED NORMAL DISTRIBUTIONS

JOAN DEL CASTILLO AND PEDRO PUIG

Unitat d’Estadistica i Investigació Operativa, Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

(Received May 15, 1995; revised June 27, 1996)

Abstract. This paper provides necessary and sufficient conditions for a solution to likelihood equations for an exponential family of distributions, which includes Gamma, Rayleigh and singly truncated normal distributions. Furthermore, the maximum likelihood estimator is obtained as a limit case when the equations have no solution. These results provide a way to test departures from Rayleigh and singly truncated normal distributions using the likelihood ratio test. A new easy way to test departures from a Gamma distribution is also introduced.

Key words and phrases: Exponential families, Gamma, Rayleigh and singly truncated normal distributions.