The Three-Parameter Lognormal Distribution and Bayesian Analysis of a Point-Source Epidemic

Some unusual features of the likelihood function of the three-parameter lognormal distribution ln(t – γ)∼N(μ, σ2) are explored. In particular, it is shown that there exist paths along which the likelihood function of any sample t 1, …, tn tends to ∞ as (γ, μ, σ2) approaches (t (1), – ∞, + ∞), where t (1) is the smallest of the ti , and hence that in a meaningful sense this is the maximum-likelihood estimate. Estimation is then considered from a Bayesian point of view, and some natural posterior distributions are explored. A statistical model for a point-source epidemic is presented, and the theory developed is used in estimating the time of onset and other parameters.