ASYMPTOTICS AND BOOTSTRAP FOR INVERSE GAUSSIAN REGRESSION

GUTTI JOGESH BABU^{1} AND YOGENDRA P. CHAUBEY^{2}

1Department of Statistics, 319 Classroom Building, Pennsylvania State University,

University Park, PA 16802, U.S.A.

2Department of Mathematics and Statistics, Concordia University, Loyola Campus, 7141 Sherbrooks Street West, Montreal, Quebec, Canada H4B 1R6

(Received January 13, 1995; revised May 24, 1995)

Abstract. This paper studies regression, where the reciprocal of the mean of a dependent variable is considered to be a linear function of the regressor variables, and the observations on the dependent variable are assumed to have an inverse Gaussian distribution. The large sample theory for the pseudo maximum likelihood estimators is available in the literature, only when the number of replications increase at a fixed rate. This is inadequate for many practical applications. This paper establishes consistency and derives the asymptotic distribution for the pseudo maximum likelihood estimators under very general conditions on the design points. This includes the case where the number of replications do not grow large, as well as the one where there are no replications. The bootstrap procedure for inference on the regression parameters is also investigated.

Key words and phrases: Chi-square distribution, inverse Gaussian distribution, pseudo maximum likelihood estimator, strong consistency, weak convergence.