The research program primarily concerns statistical inference using likelihood based methods and especially, likelihood ratios in nonparametric monotone function estimation problems. A distinguishing feature of the monotone function models is a slower (cube root of n) pointwise rate of convergence of maximum likelihood estimators of the monotone function of interest, with a non-Gaussian limit distribution; this property is referred to as ``non-regularity''. While some progress in likelihood based inference for these problems has been achieved over the past few decades, the behavior of likelihood ratios is by and large unknown. In this project, the P.I. seeks to develop a theory of likelihood ratio inference for these ``non-regular'' monotone function models. This is motivated by the wide applicability of likelihood ratio based inference in regular parametric, semiparametric and nonparametric problems. The emergence of a chi-squared distribution as the limit of log-likelihood ratios allows the construction of test procedures and confidence regions for the parameters of interest, based on the known chi-squared distributions and circumvents the need to estimate nuisance parameters. It is thus natural to ask whether the advantages of the likelihood ratio paradigm carry over to the domain of shape-restricted (and more particularly, monotone) function estimation. The current research program investigates this for various models and applications of interest. More specifically, the main components of the proposed research program are: (i) Investigation of the universality of the limit, D (ii) Studying monotone function models with measured covariates on the individuals, which is typically the case in applications, from both nonparametric and semiparametric angles (iii) Developing methods of constructing pointwise confidence sets and confidence bands for monotone functions of interest using likelihood based methods and comparison of these procedures to currently existing methods. Also on the agenda are related research issues, like the study of competing likelihood ratio statistics and the computational and analytical characterization of the associated limit distributions.
The study of shape--restricted functions arises in a wide variety of problems. In particular, monotonicity, which is a very natural shape-constraint appears in many different areas of application, such as reliability, renewal theory, survival analysis, epidemiology, biomedical studies and astronomy. Through its use of attractive statistical concepts like likelihood and likelihood ratios, for estimating monotone functions, this project is expected to have a broad impact on the theory and practice of nonparametric statistics. It will lead to significantly improved methods for analyzing data using likelihood ratio based methods in medicine, public health, reliability and numerous other application areas and will trigger the development of analogous methods of statistical inference in related fields. The ideas and results of this project will also be fruitful in the training and development of future statisticians through inclusion in the curriculum of advanced courses.
