My interest in statistics is both on the theoretical end of things and on the applied. As far as theory goes, I have been working for the last few years on generalizations of classical likelihood ratio statistics and likelihood ratio tests. These tests form the basis of much common statistical methodology, such as the t-test and the F-test. However, one needs to make quite strong assumptions about the form of the underlying distributions in order for these tests to be valid. Some more modern techniques aim to make fewer assumptions, even going so far as to be nonparametric, or to allow for different types of structure in the data, such as some kinds of dependence which can be modeled by martingales. Questions then focus on how well tests based on these new likelihood functions perform when compared to the standard likelihood theory tests. There are several criteria which can be used to make this evaluation, relating to the accuracy and the power of the tests. Furthermore, many of these methods are computationally intense; we might want to look at the cost in time and resources involved in using these alternative tests - how much do we gain by using these methods (in that we make fewer assumptions but still obtain accurate or powerful tests) and how much do we pay (in terms of having a more complicated procedure, whose performance might not be that different from a standard test)? Since so-called "artificial likelihoods" are becoming more prevalent in statistics, these questions are taking on a new importance.
One of the great things about applied statistics is that it is possible to become interested in a new field and start exploring it! In the past, my area of application has been centered on psychology; specifically, I worked in psychometrics and surveys. There are many interesting questions that arise in these two fields, including how to maintain the validity and reliability of a selection system based on written tests, interviews and peer ratings, or how to design an effective questionnaire. I have investigated some of these issues in populations as diverse as officer candidates in the Israeli army and recently hospitalized patients. In the past year, I have renewed a long-standing interest in anthropology. I have been fascinated with history and different cultures for most of my life, but now I have been thinking more in terms of physical anthropology. This involves, among other things, measurements of various bone fragments and skeletal remains, in order to investigate the physical relations between them. One project for which I acted as a statistical consultant attempted to make statements about the size of a skull based on the lengths of the skull sutures. In any given specimen, some of these suture measurements might be present and others absent; how then can we use the available information? If we want to incorporate other factors such as race and sex, the problem becomes even more complex. It seems to me that there is room to explore methodologies for spatial analysis that have not yet been taken advantage of.
Some related publications:
Lazar, N.A., Eddy, W.F., Genovese, C.R., and Welling, J. Statistical issues in fMRI for brain imaging. To appear: International Statistical Review.
Lazar, N. and Mykland, P.A. (1998). An evaluation of the power and conditionality properties of empirical likelihood. Biometrika, 85, 3, 523-534.
Lazar, N. and Mykland, P.A. (1999). Empirical likelihood in the presence of nuisance parameters. Biometrika, 86, 1, 203-211.