<p>We first present a Complex Singular Value Decomposition (CSVD)analysis of credit cyle and explore the lead-lag relation betweencredit cycle and business cycle, then propose a GeneralizedLinear Model (GLM) of credit rating transition probabilitiesunder the impact of business conditions.To detect the cyclic trend existence of credit condition in U.S.economy, all credit variables and business variables aretransformed to complex values and the transformed data matrix isapproximated by first order of CSVD analysis. We show that theeconomy, represented by both credit conditions and businessconditions, is changing recurrently but with different frequenciesfor different time periods. Credit variables making the greatestlinear contribution to first Principal Component can be identifiedas credit cycle indicators. The result of leading businessvariables to credit variables in an economy provides the basis topredict credit condition by business cycle indicators.The credit rating system is a publicly available measure of theriskiness of financial securities and a rating transition matrixquantifies the risk, by permitting calculation of the probabilityof downgrade or default. Credit migration is observed to beinfluenced both by business conditions and by an issuer's owncredit status. We assume the rating history for a particularinstitution is Markovian, and histories for differentinstitutions are assumed to be statistically independent, in bothcases the history of market conditions are known. With a simpleGLM, we investigate the significance of business conditions andtheir two major impacts - creditworthinessdeterioration/improvement and credit stability. We propose amodel of transition probability in discrete time and a model ofinstantaneous transition rates in continuous time, and fit themby maximum likelihood. Business conditions are shown to have asignificant effect: higher likelihood for credit qualityimprovement and stability under good business conditions whilehigher likelihood for credit quality deterioration and driftunder severe business conditions. The two business impacts aresignificant and business deterioration/improvement impact isgreater than its stability impact on credit rating transitions.Investment-grade rating transitions are more sensitive to longrate risk while speculative-grade rating transitions are moresensitive to short rate risk. Compared to a discrete model, thecontinuous transition model has much greater over-dispersion butis more practical.<P>

<p>LUNCEFORD, JARED KENNETH. Estimating Causal Treatment Effects Via thePropensity Score and Estimating Survival Distributions in Clinical TrialsThat Follow Two-stage Randomization Designs. (Under the direction ofProfessor Marie Davidian)Estimation of treatment effects with causalinterpretation from obervational data is complicated by the fact thatexposure to treatment is confounded with subject characteristics. Thepropensity score, the probability of exposure to treatment conditionalon covariates, is the basis for two competing classes of approachesfor adjusting for confounding: methods based on stratification ofobservations by quantiles of estimated propensity scores, and methodsbased on weighting individual observations by weights depending onestimated propensity scores. We review these approaches andinvestigate their relative performance.Some clinical trials follow a design in which patientsare randomized to a primary therapy upon entry followed by anotherrandomization to maintenance therapy contingent upon diseaseremission. Ideally, analysis would allow different treatmentpolicies, i.e. combinations of primary and maintenance therapy ifspecified up-front, to be compared. Standard practice is to conductseparate analyses for the primary and follow-up treatments, which doesnot address this issue directly. We propose consistent estimators ofthe survival distribution and mean survival time for each treatmentpolicy in such two-stage studies and derive large sampleproperties. The methods are demonstrated on a leukemia clinical trialdata set and through simulation.<P>

<p>There has been much interest in testing nonstationarity of panel data in the econometric literature. In the last decade, several tests based on the ordinary least squares and Lagrange multiplier methodhave been developed. In contrast to a unit root test in the univariate case,test statistics in panel data have Gaussian limiting distributions.This dissertation considers weighted symmetric estimation and maximum likelihood estimation in the autoregressive model with individual effects.The asymptotic distributions have been derived as the number of individuals and time periods become large. The power study from Monte Carloexperiments shows that the proposed test statistics perform substantiallybetter than those in previous studies even for small samples.As an example, we consider the real Gross Domestic Product per Capita for 12 countries.<P>

<p>The primary objectives of this research are to develop andstudy estimators for generalized linear measurement errormodels when the mean function contains error-free predictorsas well as predictors measured with error and interactions between error-free and error-prone predictors. Attention is restricted to generalized linear models in canonical form with independent additive Gaussian measurement error in the error-prone predictors.Estimators appropriate for the functional (Fuller, 1987, Ch.1) version of the measurement error model are derived and studied. The estimators are also appropriate in the structural version of the model and thus the methods developed in this research are functional in the sense of Carroll, Ruppert and Stefanski (1995, Ch. 6).The primary approach to the development of estimators in this research is the conditional-score method proposed byStefanski and Carroll (1987) and described by Carroll et al.(1995, Ch. 6). Sufficient statistics for the unobserved predictors are obtained and the conditional distribution of the observed data given these sufficient statistics is derived. The latter admits unbiased score functions that arefree of the nuisance parameters (the unobserved predictors) and are used to construct unbiased estimating equations for model parameters.Estimators for the parameters of the model of interest are also derived using the corrected approach proposed by Nakamura (1990) and Stefanski (1989). These are also functional estimators in the sense of Carroll et al. (1995, Ch. 6) that are less dependent on the exponential-family model assumptions and thus provide a benchmark against whichto compare the conditional-score estimators.Large-sample distribution approximations for both theconditional-score and corrected-score estimators are derivedand the performance of the estimators and the adequacy of the large-sample distribution theory are studied via Monte Carlo simulation.<P>

<p>CHEN, JUNLIANG. A Monte Carlo EM algorithm for generalized linear mixed modelswith flexible random effects distribution. (Under the direction of DaowenZhang and Marie Davidian)A popular way to model correlated binary, count, or other data arising inclinical trials and epidemiological studies of cancer and other diseases is byusing generalized linear mixed models (GLMMs), which acknowledge correlationthrough incorporation of random effects. A standard model assumption is thatthe random effects follow a parametric family such as the normal distribution.However, this may be unrealistic or too restrictive to represent the data,raising concern over the validity of inferences both on fixed and randomeffects if it is violated.Here we use the seminonparametric (SNP) approach (Davidian and Gallant 1992,1993) to model the random effects, which relaxes the normality assumption andjust requires that the distribution of random effects belong to a class of``smooth'' densities given by Gallant and Nychka (1987). This representation allows the density of random effects to be very flexible, including densitiesthat are skewed, multi--modal, fat-- or thin--tailed relative to the normal, andthe normal as a special case. We also provide a reparameterization of thisrepresentation to avoid numerical instability in estimating the polynomialcoefficients.Because an efficient algorithm to sample from a SNP density is available, wepropose a Monte Carlo expectation maximization (MCEM) algorithm using arejection sampling scheme (Booth and Hobert, 1999) to estimate the fixedparameters of the linear predictor, variance components and the SNP density. Astrategy of choosing the degree of flexibility required for the SNP density isalso proposed. We illustrate the methods by application to two data sets fromthe Framingham and Six Cities Studies, and present simulations demonstratingperformance of the approach.<P>

<p>Monte Carlo importance sampling is a useful numerical integration technique, particularly in Bayesian analysis. A successful importance sampler will mimic the behavior of the posterior distribution, not only in the center, where most of the mass lies, but also in the tails (Geweke, 1989). Typically, the Hessian of the importance sampler is set equal to the Hessian of the posterior distribution evaluated at the mode. Since the importance sampling estimates are weighted averages, their accuracy is assessed by assuming a normal limiting distribution. However, if this scaling of the Hessian leads to a poor match in the tails of the posterior, this assumption may be false (Geweke, 1989). Additionally, in practice, two commonly used importance samplers, the Multivariate Normal Distribution and the Multivariate Student-t Distribution, do not perform well for a number of posterior distributions (Monahan, 2000). A generalization of the Multivariate Logistic Distribution (the Elliptical Multivariate Logistic Distribution) is described and its properties explored. This distribution outperforms the Multivariate Normal distribution and the Multivariate Student-t distribution as an importance sampler for several posterior distributions chosen from the literature. A modification of the scaling by Hessians of the importance sampler and the posterior distribution is explained. Employing this alternate relationship increases the number of posterior distributions for which the Multivariate Normal, the Multivariate Student-t, and the Elliptical Multivariate Logistic Distribution can serve as importance samplers. <P>

<p>In stochastic volatility models, the unit root test on the time series of theunobserved log-volatilities may be performed by applying the commonly usedfrequentist unit root tests. For instance, augmented Dickey Fuller tests based on the log-squared meancorrected returns can be used. The log-squared meancorrected returns have the same second order properties as that of an autoregressive moving average process. However, we observed that the moving average parameter of the resulting process (based on the log-squared meancorrected returns) is typically close to the autoregressive parameter. For this reason,the unit root tests applied to stochastic volatility models tend to reject theunit root in finite samples. We propose a method for performing thefrequentist unit root tests in stochastic volatility models based on the finite sampling distribution of the well known test statistics. In addition to the frequentist testing procedures, Bayesian unit root testscan be used to test for a unit root in stochastic volatility models as well.A Bayesian test based on the Bayes factor has been suggested by So and Li (1999).In this approach, they work with the mean corrected returns instead of thelog-squared mean corrected returns. They treat the unobserved log-volatilitiesas missing observations. The prior densities they use for the autoregressiveparameter are continuous densities defined on an interval that does not include the value beingtested. Such prior densities for the autoregressive parameterare not suitable where one's main concern is to test for a unit root inlog-volatilities. We introduce a new prior density for this parameter that puts a positive mass on thepoint being tested. We also consider continuous prior densities defined on an interval that includes thepoint one. These prior densities allow us to use the posterior interval ofthe autoregressive parameter as a testing criterion.The advantage of our method is that it is simple and useful.The performance of these tests are demonstrated by a simulation study. We illustrate thetesting procedures by applying them to four sets of exchange rates.<P>

<p>In many clinical studies, researchers are interested in theeffects of a set of prognostic factors on the hazard of death from a specific disease even though patients may die from other competing causes. Often the time to relapse is right-censored for some individuals due to incomplete follow-up. In some circumstances, it may also be the case that patients are known to die but the cause of death is unavailable. When cause of failure is missing, excluding the missing observations from the analysis or treating them as censored may yield biased estimates and erroneous inferences. Under the assumption that cause of failure is missing at random, we propose three approaches to estimate the regression coefficients. The imputation approach isstraightforward to implement and allows for the inclusion ofauxiliary covariates, which are not of inherent interest formodeling the cause-specific hazard of interest but may be related to the missing data mechanism. The partial likelihood approach we propose is semiparametric efficient and allows for more general relationships between the two cause-specific hazards and more general missingness mechanism than the partial likelihood approach used by others. The inverse probability weighting approach isdoubly robust and highly efficient and also allows for theincorporation of auxiliary covariates. Using martingale theory and semiparametric theory for missing data problems, the asymptotic properties of these estimators are developed and the semiparametric efficiency of relevant estimators is proved. Simulation studies are carried out to assess the performance of these estimators in finite samples. The approaches are also illustrated using the data from a clinical trial in elderly women with stage II breast cancer. The inverse probability weighted doubly robust semiparametric estimator is recommended for itssimplicity, flexibility, robustness and high efficiency.<P>

In some neurological postmortem brain tissue studies, repeated measures are observed. These observations are taken on the same experimental subject and are therefore correlated within the subject. Furthermore, each observation can be viewed as coming from one of a pre-specified number of populations where each population corresponds to a possible type of neurons. In this dissertation, we propose several mixture models with two components to model such repeated data. In the first model, we include subject-specific random effects in the component distributions to account for the within-subject correlation present in the data. The mixture components are generalized linear models with random effects, while the mixing proportions are governed by a logistic regression. In the second proposed model, the mixture components are generalized linear models, while the component-indicator variables are modeled by a multivariate Bernoulli distribution that depends on covariates. The within-subject observations are taken to be correlated through the latent component indicator random variables. As a special case of the second model, we focus on multivariate Bernoulli mixtures of normals, where the component-indicator variables are modeled by logistic regressions with random effects, and the mixture components are linear regressions. The third proposed model combines the first and second models, so that the within-subject correlation is built into the model not only through the component distributions, but also through the latent component indicator variables. The focus again is on a special case of the third model, where the mixture components are linear regressions with random effects while the mixing proportions are logistic regressions with another group of random effects. For each model, model fitting procedures, based on MCMC methods for sampling from the posterior distribution of the parameters, are developed. The second and third model are used to compare schizophrenic and control subjects with regard to the somal volumes of deep layer 3 pyramidal cells in the auditory association cortex. As a preliminary analysis, we start by employing classic mixture models and mixtures-of-experts to analyze such data neglecting the within-subject correlation. We also provide a discussion of the statistical and computational issues concerning estimation of classic Poisson mixtures.

The main concern of financial time series analysis is how to forecast future values of financial variables, based on all available information. One of the special features of financial variables, such as stock prices and exchange rates, is that they show changes in volatility, or variance, over time. Several statistical models have been suggested to explain volatility in data, and among them Stochastic Volatility models or SV models have been commonly and successfully used. Another feature of financial variables I want to consider is the existence of several missing data. For example, there is no stock price data available for regular holidays, such as Christmas, Thanksgiving, and so on. Furthermore, even though the chance is small, stretches of data may not available for many reasons. I believe that if this feature is brought into the model, it will produce more precise results. The goal of my research is to develop a new technique for estimating parameters of SV models when some parts of data are missing. By estimating parameters, the dynamics of the process can be fully specified, and future values can be estimated from them. SV models have become increasingly popular in recent years, and their popularity has resulted in several different approaches proposed regarding the problem of estimating the parameters of the SV models. However, as of yet there is no consensus on this problem. In addition there has been no serious consideration of the missing data problem. A new statistical approach based on the EM algorithm and particle filters is presented. Moreover, I expand the scope of application of SV models by introducing a slight modification of the models.