Browsing by Author "John F. Monahan, Committee Chair"

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    An Optimization Approach for the Parameter Estimation of the Nonlinear Mixed Effects Models
    (2004-07-29) Wang, Jing; Bhattacharyya, Bibhuti, Committee Member; John F. Monahan, Committee Chair; Marie Davidian, Committee Member; Dickey, David, Committee Member
    Nonlinear mixed-effects models (NLMM) have received a great deal of attention in the statistical literature in recent years because of the flexibility they offer in handling the unbalanced repeated-measurements data that arise in different areas of investigation, such as pharmacokinetics. We concentrate here on maximum likelihood estimation for the parameters in nonlinear mixed-effects models. A rather complex numerical issue for maximum likelihood estimation in nonlinear mixed-effects models is the evaluation of a multiple integral that, in most cases, does not have a closed-form expression. We restrict our attention in this article on numerical methods that are based on approximation for the likelihood. Several numerical approximations for the likelihood have been proposed such as first-order linearization (FOL), Laplace approximation, Importance Sampling, and Gaussian Quadrature (GQ). In addition, for a general optimization problem, iterative methods are usually required to update the parameter estimates iteratively. A large number of parameter updating methods have been developed such as Newton-Raphson, Steepest Descent, Stochastic optimization, etc. Many current optimization algorithms implement a Newton iterative method to update the parameter estimates in NLMM. The objective of this thesis is to propose an optimization approach for the parameter estimation in nonlinear mixed-effects models. This optimization method implements Importance Sampling for approximating likelihood and a stochastic recursive procedure for updating parameter estimates in NLMM. In Chapter 1, we describe the model and introduce several likelihood approximations and parameter updating procedures for these models. The proposed optimization approach is illustrated in Chapter 2. In order to compare this approach to the other optimization methods, simulations are performed and conclusions are drawn based on simulation results in Chapter 3. Some future work is presented in Chapter 4.
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    A Stationary Stochastic Approximation Algorithm for Estimation in the GLMM
    (2008-05-18) Chang, Sheng-Mao; Daowen Zhang, Committee Member; John F. Monahan, Committee Chair; Bibhuti Bhattacharyya, Committee Member; Dennis D. Boos, Committee Member
    Estimation in generalized linear mixed models is challenging because the marginal likelihood is an integral without closed form. In many of the leading approaches such as Laplace approximation and Monte Carlo integration, the marginal likelihood is approximated, and the maximum likelihood estimate (MLE) can only be reached with error. An alternative, the simultaneous perturbation stochastic approximation (SPSA) algorithm is designed to maximize an integral and can be employed to find the exact MLE under the same circumstances. However, the SPSA does not directly provide an error estimate if the algorithm is stopped in a number of finite steps. In order to estimate the MLE properly with an statistical error bound, we propose the stationary SPSA (SSPSA) algorithm. Assuming that the marginal likelihood, objective function, is quadratic around the MLE, the SSPSA takes the form of a random coefficient vector autoregressive process. Under mild conditions, the algorithm yields a strictly stationary sequence where the mean of this sequence is asymptotically unbiased to the MLE and has a closed-form variance. Also, the SSPSA sequence is ergodic providing certain constraints on the step size, a parameter of the algorithm, and the mechanism that directs the algorithm to search the parameter space. Sufficient conditions for the stationarity and ergodicity are provided as a guideline for choosing the step size. Several implementation issues are addressed in the thesis: pairing numerical derivative, scaling, and importance sampling. Following the simulation study, we apply the SSPSA on several GLMMs: Epilepsy seizure data, lung cancer data, and salamander mating data. For the first two cases, SSPSA estimates are similar to published results whereas, for the salamander data, our solution greatly differs from others.