Text

theses

This thesis contains two parts. The first part, in Chapter 2-4, addresses three connected issues in penalized least-square estimation for high-dimensional data. The second part, in Chapter 5, concerns nonparametric maximum likelihood methods for mixture models. In the fi rst part, we prove the estimation, prediction and selection properties of concave penalized least-square estimation (PLSE) under fully observed and noisy/missing design, and validate an essential condition for PLSE: the restricted-eigenvalue condition. In Chapter 2, we prove that the concave PLSE matches the oracle inequalities for prediction and coefficients estimation of the Lasso, based only on the restricted eigenvalue condition, one of the mildest condition imposed on the design matrix. Furthermore, under a uniform signal strength assumption, the selection consistency does not require any additional conditions for proper concave penalties such as the SCAD penalty and MCP. A scaled version of the concave PLSE is also proposed to jointly estimate the regression coefficients and noise level. Chapter 3 concerns high-dimensional regression when the design matrix are subject to missingness or noise. We extend the PLSE for fully observed design to noisy or missing design and prove that the same scale of coefficients estimation error can be obtained, while requiring no additional condition. Moreover, we show that a linear combination of the `2 norm of regression coefficients and the noise level is large enough as penalty level when noise or missingness exists. This sharpens the commonly understood results where l-1 norm of coefficients is required. Chapter 4 validates the restricted eigenvalue (RE) type conditions required in Chapter 2 and Chapter 3 and considers a more general groupwise version. We prove that the population version of the groupwise RE condition implies its sample version under a low moment condition given usual sample size requirement. Our results include the ordinary RE condition as a special case. In the second part, we consider nonparametric maximum likelihood (NPML) methods for mixture models, a nonparametric empirical Bayes approach. We provide concrete guidance on implementing multivariate NPML methods for mixture models, with theoretical and empirical support; topics covered include identifying the support set of the mixing distribution, and comparing algorithms (across a variety of metrics) for solving the simple convex optimization problem at the core of the approximate NPML problem. In addition, three diverse real data applications are provided to illustrate the performance of nonparametric maximum likelihood methods.

ETD_8263

Ph.D.

Includes bibliographical references

by Long Feng

doi:10.7282/T3NZ8BRP

ETD doctoral

The author owns the copyright to this work.