Statistics & Biostatistics

The research of this group covers areas such as sequential analysis, resampling techniques, permutation methods, time series, reliability theory, design of experiments, applied probability, and mathematical statistics.

Benzion Boukai

Prof. Boukai is working in the general area of mathematical statistics where he develops parametric and non-parametric methodological frameworks for various statistical inference problems. In sequential analysis, which is his primary area of expertise, he studies the properties of statistical procedures involving inference or estimation based on sequential sampling schemes and associated stopping rules. In recent years he has a growing interest in investigating problems arising from applications involving random effects in nonlinear models. These types of models are often encountered in real-life applications ranging from statistical pharmacokinetics modeling to econometric studies involving auctions of financial assets. This topic often links areas of hierarchical Bayesian modeling with structured parametric modeling and their applications. In this general context, he studies how parameters of such models are affected by random shocks and other such effects and is working on developing novel resampling and recycling schemes that would allow a more accurate evaluation of the small-sample and the large-sample properties of estimates obtained to various models' parameters.

Fang Li

Prof. Li's principal research interest focuses on inference of stochastic processes including long-range dependence and time series, in which nonparametric kernel estimations are explored to estimate regression, autoregressive, density, and heteroschadastic error variance functions. Some of her ongoing projects involve comparing multiple time series, which has diverse applications in economics and finance. In collaboration with research groups from biological fields, she is applying nonparametric and general linear mixed models to real life data. Additionally, she has strong interest in extending the nonparametric approaches to survival and longitudinal data analysis.

Ryan Martin

Prof. Martin's primary research focus is on nonparametric estimation of mixing distributions in mixture or latent-variable models. These are notoriously difficult problems, both computationally and theoretically, but recent progress has been made using a fast recursive algorithm called Predictive Recursion (PR). Dr. Martin and his collaborators have used martingale methods to derive first- and second-order asymptotics for PR estimates for a wide class of mixture models, even when the mixture model is mis-specified. PR, it turns out, is closely related to stochastic approximation -- a stochastic root-finding procedure popular in the engineering literature -- and this connection is expected to lead to improved bounds on the PR rate of convergence.

The technological development of high-throughput devices, such as DNA microarrays, has lead to a surge of interest in empirical Bayes methods for detecting sparse signals. Indeed, empirical Bayes methodology is now being used in a wide range of new and exciting applications, including gene expression profiling and genetic association mapping. Using his work on estimating prior/mixing distributions, Dr. Martin and his collaborators have proposed a general nonparametric empirical Bayes framework for high-dimensional inference. This new methodology is based on a semiparametric extension of the PR algorithm that is more flexible than the original PR. The results are quite promising, and efforts to derive its theoretical properties and to speed up computation are ongoing.

Jyoti Sarkar

Prof. Sarkar's major research activities are in statistical reliability theory. The general class of problems he and his collaborators study can be described as follows: "A monitored system experiences a cycle of lifetimes and repair times lasting for random 27 durations. After failure, the system undergoes a repair and is restored to operation. What is the probability that the system will be in the functioning state at any given time?" Dr. Sarkar has studied the availability of repairable systems under different models arising from the implementation of different inspection policies, different repair policies, and the presence of one or more spare units to support the main operating unit. Many other practical variations of the system availability problem remain to be solved. For example, the aspect of cost in any maintenance policy and the effect of a random environment need to be properly addressed. Dr. Sarkar is engaged in extending the frontiers of availability theory research.

Sarkar's research in statistical inference focuses on the performance of the Graybill-Deal estimator of the common mean of two populations under a two-stage or a fully sequential sampling scheme. He has also studied the optimal allocation of a fixed sampling budget in the context of estimating the common mean of a bivariate normal population when the component variables have different sampling costs. He is now working on the companion problem of optimal sampling when the objective is hypothesis testing.

Aside from his primary research areas in Statistics, Dr. Sarkar also collaborates with several economists to conduct research in location theory. They study location equilibrium for firms in various models of competition arising from different types of competition (in price or in quantity), different number of competitors (duopolists or oligopolists) and different number of stores per firm (single-store or multi-store), different types of markets (network, linear, or circular) and different entry times into the market (simultaneous or sequential).