Concavity of the Log Likelihood

Abstract

For a very general regression model with an ordinal dependent variable, the log likelihood is proved concave if the derivative of the underlying response function has concave logarithm. For a binary dependent variable, a weaker condition suffices, namely, that the response function and its complement each have concave logarithm. The normal, logistic, sine, and extreme-value distributions, among others, satisfy the stronger condition, the t (including Cauchy) distributions only the weaker. Some converses and generalizations are also given. The model is that which arises from an ordinary linear regression model with a continuous dependent variable that is partly unobservable, being either grouped into intervals with unknown endpoints, or censored, or, more generally, grouped in some regions, censored in others, and observed exactly elsewhere.

Key Words:

• Ordinal dependent variables
• Concavity of log likelihood
• Ordered categories
• Strong unimodality
• Grouped dependent variable
• Censored dependent variables