Likelihood Ratio Bounds as Constrained Nonlinear Optimization Problems

The standard method for finding likelihood ratio confidence bounds for a parameter of a lifetime model is the profile likelihood method. This method entails eliminating all but the parameter of interest by effectively projecting the graph of the likelihood function onto a plane and solving the resulting single-variable optimization problem. Confidence bounds on functions of parameters (such as reliability at a particular time or mean life) can be found by inverting the function of interest to create a reparameterization of the lifetime model and then proceeding as before. Another, less well-known method, instead finds the likelihood ratio bounds on a parameter or function of parameters as the maximum and minimum of a constrained, multivariable nonlinear optimization problem. The likelihood ratio is used to form the constraint set for this problem. In this paper we discuss these two methods and indicate some practical advantages that the constrained nonlinear optimization method has over the profile likelihood method. We also extend the theory on the constrained nonlinear optimization method. Finally, we present numerical results indicating the asymptotic accuracy of the constrained nonlinear optimization method, as well as its improved speed versus the profile likelihood method.