ON THE PRINCIPLE OF MONOTONE LIKELIHOOD AND LOG-LINEAR MODELS.

Two independent samples from a probability distribution are defined to be activity-equivalent if they contain the same number of independent sample-points and if certain functions of the sample-points take identical values. The Principle of Monotone Likelihood is satisfied for activity-equivalent samples if the sample with lower values of certain cost functions has greater likelihood than the sample with higher values of the cost functions. It is shown that the Principle of Monotone Likelihood is satisfied if and only if the probability distribution is of log linear type (belongs to the exponential family of distributions). Examples are given on distributions, which are implied by (and imply) the Principle of Monotone Likelihood: The exponential and normal distributions, Maxwell-Boltzmann's distribution for the speeds of the molecules of an ideal gas, the Maxwell-Boltzmann energy distribution law, the gravity model in transportation planning.