Likelihood-based interval estimation of functionals

Abstract

In this thesis, we are concerned with interval estimation for functions of parameters (or functionals). In particular, we explore topics involving profile likelihood-based interval estimation of functionals in parametric models, as well as models with "missing" data.

Our study is motivated by the inadequacy of intervals based on the large sample normal approximation of test statistics when the normality assumption is not warranted. For example, in parametric settings, interval estimates for functionals based on the delta method have been known to perform poorly in applications. Although many univariate problems admint simple transformations that improve the large sample approximation, analogous approaches do not necessarily carry over to multi-parameter settings in a straightforward manner. For missing data problems, use of the observed information matrix in conjunction with the EM algorithm does not always yield satisfactory interval estimates for essentially the same reasons. While profile likelihood-based approaches to interval estimation are familiar in parametric statistical inference, its use in missing data and semi-parametric settings is not as well-known.

Chapter 1 of this thesis introduces the basic elements of likelihood-based interval estimation, with emphasis on using the profile likelihood to construct interval estimates. We describe the extension of the approach to handle functionals, via Madansky (1965). A few examples serve to round out the discussion. For parametric models, it is well-known that a simple correction factor applied to the likelihood ratio statistic (LRS) improves the quality of the approximation to the reference x^2 distribution. This factor is known as the Bartlett correction and has routinely been applied to tests of hypotheses concerning a parameter vector or a sub-vector of it. In chapter 2, we derive a Bartlett correction to the LRS for testing a parameter function. Unlike the standard approach, our method is based on the basic assumptions and framework of the Lagrange multiplier technique. In many practical situations, we show that this approach can yield a simpler implementation. The improved performance of the corrected LRS is illustrated with examples and evaluation of coverage probabilities.

In chapter 3, we utilize the Lagrange multiplier technique, in conjunction with the EM algorithm, to obtain profile likelihood-based CIs for functionals in "missing" data settings. This yields an alternative interval estimate to those based on the observed information matrix (such as the approach of Louis, 1982). The resulting procedures are computationally intensive, but there is a potential gain in other aspects. To reduce the computational workload, we adapt the EM1 algorithm described by Rai and Matthews (1993) to the parameter function setting.

Chapter 4 is devoted to likelihood-based interval estimation of functionals in failure time models. We first consider Aitkin and Clayton's (1980) "formulation" of the parametric Cox proportional hazards model as a generalized linear model. We adapt their approach in order to derive likelihood-based interval estimates for common failure model functionals, such as quantiles and survival probabilities. Next we consider location-scale failure time models. Therneau (1992) recently fit this class of models via the method of iteratively reweighted least squares (IRLS). We demonstrate that IRLS can be conveniently adapted to constrained maximization. We also show that a Lagrange multiplier argument can be applied to this setting to provide interval estimates of some other useful functionals which are not available via the regular profile likelihood approach.

A concluding chapter provides suggestions for future research.