ON ASYMPTOTIC NORMALITY OF PSEUDO LIKELIHOOD ESTIMATES FOR PAIRWISE INTERACTION PROCESSES

JENS LEDET JENSEN^{1} AND HANS R. KÜNSCH^{2}

1Department of Theoretical Statistics, Institute of Mathematics, University of Aarhus, DK-8000 Aarhus C, Denmark 2Seminar für Statistik, ETH, CH-8092 Zürich, Switzerland

(Received February 22, 1993; revised August 26, 1993)

Abstract. We consider point processes defined through a pairwise interaction potential and admitting a two-dimensional sufficient statistic. It is shown that the pseudo maximum likelihood estimate can be stochastically normed so that the limiting distribution is a standard normal distribution. This result is true irrespectively of the possible existence of phase transitions. The work here is an extension of the work Guyon and Künsch (1992, Lecture Notes in Statist., 74, Springer, New York) and is based on viewing a point process interchangeably as a lattice field.

Key words and phrases: Asymptotic normality, Gibbs processes, lattice field, phase transitions, pseudo likelihood, stochastically normed.