The theory and application of the Cox (1972) failure time regression model has, almost without exception, assumed an exponential form for the dependence of the hazard function on regression variables. Other regression forms may be more natural or descriptive in some applications. For example, a linear relative risk regression model provides a convenient framework for studying epidemiologic risk factor interactions. This note uses the counting process formulation of Andersen and Gill (1982) to develop asymptotic distribution theory for a class of intensity function regression models in which the usual exponential regression form is relaxed to an arbitrary non-negative twice differentiable form. Some stability and regularity conditions, beyond those of Andersen and Gill, are required to show the consistency of the observed information matrix, which in general need not be positive semidefinite.