Asymptotic equivalence, one of the most important statistical contributions of Lucien Le Cam, is a theory to build the connections among various statistical models. If two models are asymptotically equivalent, all asymptotically optimal statistical estimators can be carried over from one model to the other. A basic principle of establishing asymptotic equivalence is to approximate a complicated statistical model by a more tractable one. The Gaussian location model is a tractable model that captures the essence of a number of statistical settings. The investigator studies explicit and practical procedures to convert a general nonparametric estimation to a Gaussian regression, using improved quantile coupling inequalities and new variance stabilization transformations. Other statistical problems are better understood by relating them to Poisson process models. The investigator studies infinitely divisible approximation to density estimation and its connection to nonparametric edge estimation and classification. The investigator is also proposed to study a long-standing issue in this area -- asymptotic equivalence theory for unbounded loss, and to study the asymptotic equivalence theory for multiple comparisons, functional data analysis and long memory models.
The project would help statisticians in many areas such as robust nonparametric estimation, machine learning, multiple comparison, functional data analysis, long memory models and generalized linear models, to understand and appreciate the simplification of Le Cam's theory and use it as a guidance to produce new theory and methodologies. The models the investigator is studying can be used in signal and image processing, calling data analysis, detection of bioweapons use, Genomic research, disease prevention, etc. The project will integrate research and education by teaching courses on decision theory, by organizing seminars and workshops to disseminate and preserve Le Cam's theory, and by advising graduate students working on this topic. The investigator will serve as the Diversity Coordinator for graduate student admissions in the Yale Statistics Department, and will seek to attract women and minorities to do research on the grant.
Project Outcomes or Findings: One of the most important statistical contributions of Lucien Le Cam is the asymptotic equivalence theory. A basic principle of asymptotic equivalence theory is to approximate general statistical models (also called experiments) by simple ones. There have been several developments in the asymptotic equivalence theory in the past decade. I have attempted to move this area forward by attacking the following problems: (i) Explicit converting procedure;(ii) In?nitely divisible approximation; (iii) Unbounded loss; (iv) High dimensional modelling such as large covariance matrices estimation; (v) Functional data analysis. Here is a detailed discussion on explcit converting procedures, one of our major findings: In the literatures, the procedure of converting a general nonparametric model to a simple Gaussian model was either inexplicit, or explicit but complicated, so that it was hard to for the results of the asymptotic equivalence theory to impact methods of statistical practice. In this project, I worked with several coauthors to propose explicit procedures to convert nonparametric densityestimation (Brown, Cai, Zhang, Zhao and Zhou, 2010) and nonparametric regression in exponential families (Brown, Cai and Zhou, 2008a) to Gaussian regression by mean-matching variance-stabilizating transformations for exponential families withquadratic variance, and to covert robust nonparametric regression (Brown, Cai and Zhou, 2008b) to Gaussian regression by deriving Komlós-Major-Tusnády type resultsfor the median. We had found the framework can be extended to general expo-nential families. The mathematical framework of Lucien Le Cam was often consider incomprehensible by many researchers, while it is appreciated by some leading statisticians. The models we studied are used in signal and image processing, calling data analysis, national security, Genomic research, global temperature. The project had integrated research and education by teaching monograph courses on decision theory and organizing a very sucessful workshop on asymptotic decision theory to disseminate and preserve Le Cam?s theory, and advising graduate students working on this topic.
