The Influence of Assumptions Implicit in a Model on Parametric Inference

Abstract

In choosing a model, there are important considerations beyond the goodness of fit, namely that the subsequent inferences should not rely on questionable assumptions implicit in the model, because otherwise they will be suspect. The two-parameter lognormal distribution provides an example. The symmetry on the log scale leads to the anomaly that relative frequencies of classes in the lower tail can substantially affect inferences on the upper tail. This is contrary to the realities of the alcohol consumption example considered in this article, as well as other practical situations. A mathematical analysis of assumptions is made, using the fact that maximum likelihood estimation is asymptotically equivalent to least squares regression. It is shown that, for inferences regarding the upper tail of the two-parameter lognormal distribution, the introduction of the usual third parameter, or, almost equivalently, censoring the lower tail, can remove the anomaly. The theoretical results are illustrated by a numerical example, using data from an alcohol consumption survey at Busselton, Western Australia, showing how the specification affects the estimates of certain functions of the class probabilities.

Key Words:

• Alcohol consumption
• Contrast
• Linear function
• Lognormal distribution
• Ore evaluation
• Partition of chisquared
• Specification