Mathematical Sciences: Asymptotically Efficient Semiparametric Estimation

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The proposal is focused on estimation problems involving infinite dimensional parameters, such as cumulative distribution functions, characteristic functions, error distributions in regression models etc. Methodologically, a geometric approach has been successfully developed to solve nonparametric estimations problems from i.i.d. (independent and identically distributed) data. This approach is also applicable to a wide range of nonparametric and semiparametric problems, in which the traditional restrictive i.i.d. assumptions can be relaxed. The main emphasis is on the estimation of differentiable statistical functionals. For a general differentiable transformation of the unknown abstract parameter, the asymptotically efficient estimates will be constructed using the geometric approach. Complex statistical data that are collected in areas such as medical, econometric, and engineering reliability studies often cannot be analyzed using traditional statistical methods. The diversity of people in clinical trials, the multitude of inputs affecting economic outputs, and the intricate interaction of components affecting the performance of sophisticated electronics or equipment do not permit the use of simple statistical modeling techniques. These applications require instead a new methodology that is more flexible and yet does not sacrifice information unnecessarily. The proposed research will develop methods for estimating key features of complex statistical data. These methods do not depend on the validity of traditional assumptions. Results of this work will be applicable to a rich variety of areas, including the estimation of drug efficacy, a population profile of potential patients, proficiency characteristics for industrial output, and the probability of successful operation of electronic equipment.