Empirical likelihood, smoothed likelihood, and their applications

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Nonparametric likelihood methods such as the empirical likelihood and the smoothed likelihood are powerful tools in many areas. They are powerful frameworks for valid statistical inference that do not impose a parametric model on the data. This is particularly useful for data with a complicated structure. However, proposing appropriate nonparametric likelihood methods for data from complex scientific settings and studying the corresponding theoretical properties can be very challenging. There are many open-ended problems in this area. My research is concerned with the empirical likelihood and smoothed likelihood methods, and their application to scientific data of various kinds. This proposal consists of two topics. The first topic is the estimation of species abundance in a closed population. This is a popular problem in many scientific areas, such as biology, ecology, and fishery studies. I will consider the empirical likelihood method in a semiparametric setup. I plan to investigate the asymptotic properties of the maximum empirical likelihood estimator of the abundance and to derive the limiting distribution of the empirical likelihood ratio test, which will be used to construct the confidence interval for the abundance. This confidence interval has at least two advantages: (1) its shape is data-driven and does not need to be symmetric; (2) we do not need to estimate the variance of the estimator for the abundance. This project will make a fundamental contribution to the estimation of species abundance and to the theory and methodology of empirical likelihood. The second topic is the use of the maximum smoothed likelihood to estimate the unknown parameters or functions in mixture models with auxiliary information on the mixing proportions. The motivation for this work comes from scientific problems in many areas such as malaria studies, economics studies, and randomized trials with noncompliance. I will propose an EM-like algorithm to numerically calculate the maximum smoothed likelihood estimates of the unknown parameters and functions. I will investigate the convergence of the algorithm and the asymptotic properties of the maximum smoothed likelihood estimators. This research program will not only solve the problem of interest but is also expected to promote the application of these methods in other research areas. The projects in this proposal aim to address challenging problems in various research areas. They will use advanced statistical tools such as mixture models, the empirical likelihood, and the smoothed likelihood to tackle difficult scientific problems. This research program will in turn benefit theoretical and methodological research into these methods. The outcome of both projects will be applicable to many areas related to NSERC, such as those outlined above.