Ross Kindermann

On the relation between Markov random fields and social networks

Holland and Leinhardt (1977a) introduced a continuous time Markov chain to model changes in a social network. Their model was further studied by Wasserman (1977). Holland and Leinhardt (1977b) have also proposed a general class of probability measures on random networks. The purpose of this paper is to show that these probability measures may be viewed as Gibbs measures induced by a nearest neighbor potential. As such they have a Markov field property which is a natural generalization of Markov chains to spatial situations. The dynamic models are shown to fit into a general class of Markov processes suggested by the study of interacting particle systems. A related “voter model” is discussed.

An asymptotically efficient closed form estimator for the three - parameter lognormal distribution

The asymptotic distribution theory of the closed form estimator proposed by Munro and Wixley (1970) for the three-parameter lognormal distribution is developed. This estimator is shown to be consistent and asymptotically normally distributed with co-variance matrix equal to the inverse of the information matrix. Methods of adjusting the estimator for type II censored data sets are also presented, and small sample properties of the estimator are briefly explored.

Closed form asymptotically efficient estimators based upon order statistics

An estimation procedure, which is related to Q-Q plotting techniques, is proposed. The technique is easily programmed for distributions such as the three parameter lognormal and Weibull, and can be adapted to cases involving multiple type II censoring.

A condition for best asymptotic normality in a regular family of distributions

A theorem is presented which provides a simple sufficient condition for a weakly consistent estimator of a parameter in a regular family of distributions to be best asymptotically normal (B.A.N.). As a corollary the B.A.N. property of a maximum likelihood estimator is established under weaker conditions than those of Zacks (1971). Two examples are provided to illustrate the technique.