A unified minorization-maximization approach for fast and accurate estimation in high-dimensional parametric and semiparametric models

Abstract

In this thesis, a unified assembly and decomposition (AD) approach is introduced for constructing separable minorizing functions in developing MM algorithms. This AD technique works effectively for fast and accurate estimation in high-dimensional parametric and semiparametric models.

The MM algorithm provides a powerful tool for optimization in statistical applications. A challenging and subjective issue in developing an MM algorithm is to construct an appropriate minorizing function. For numerical convenience, this thesis first proposes a new assembly and decomposition (AD) approach to constructing the minorizing function as the sum of separable univariate functions in constructing MM algorithms. We employ the assembly technique (A-technique) and the decomposition technique (D-technique). The A-technique introduces a bank of complemental assembly functions which are often the building blocks of various MM algorithms. The D-technique decomposes the objective function into three parts and separately minorizes them. We illustrate the utility of the proposed approach in multiple applications and investigate the theoretical behaviors of the AD-based MM algorithms such as the local convergence, global convergences and convergence rate at the same time. Furthermore, some numerical experiments are conducted to demonstrate its advantages.

Secondly, this thesis proposes three profile MM algorithms for the semiparametric shared gamma frailty models. Gamma frailty survival models have been extensively used for the analysis of multivariate failure time data such as clustered failure time data and recurrent event data. Estimation and inference procedures in these models often center on the nonparametric maximum likelihood method and its numerical implementation via the EM algorithm. Despite its popularity and well celebrated success in dealing with incomplete data problems, the EM algorithm uses Newton's method and involves matrix inversion and hence may not fare well in high-dimensional situations. To address this problem, this thesis proposes a class of profile MM algorithms with good convergence properties. As a key step in constructing minorizing functions, the high-dimensional objective function is decomposed into a sum of separable low-dimensional functions. This allows the algorithm to bypass the difficulty of inverting large matrix and facilitates its pertinent use in high-dimensional problems. Simulation studies show that the proposed algorithms perform well in various situations and converge reliably with practical sample sizes. The method is illustrated using data from a colorectal cancer study.