There are many situations in which researchers want to consider a set of response variables simultaneously rather than just one response variable. For instance, a possible example is when a researcher wishes to determine the effects of an exercise and diet program on both the cholesterol levels and the weights of obese subjects. Dietz (1989) proposed two multivariate generalizations of the Jonckheere test for ordered alternatives. In this study, we propose k-sample tests for nondecreasing ordered alternatives for bivariate normal data and compare their powers with Dietz's sum statistic. The proposed k-sample tests are based on transformations of bivariate data to univariate data. The transformations considered are the sum, maximum and minimum functions. The ideas for these transformations come from the Leconte, Moreau, and Lellouch (1994). After the underlying bivariate normal data are reduced to univariate data, the Jonckheere-Terpstra (JT) test (Terpstra, 1952 and Jonckheere, 1954) and the Modified Jonckheere-Terpstra (MJT) test (Tryon and Hettmansperger, 1973) are applied to the univariate data. A simulation study is conducted to compare the proposed tests with Dietz's test for k bivariate normal populations (k=3, 4, 5). A variety of sample sizes and various location shifts are considered in this study. Two different correlations are used for the bivariate normal distributions. The simulation results show that generally the Dietz test performs the best for the situations considered with the underlying bivariate normal distribution. The estimated powers of MJT sum and JT sum are often close with the MJT sum generally having a little higher power. The sum transformation was the best of the three transformations to use for bivariate normal data.

<p> Statistical analysis of panel count data is an important topic to a number of applied fields including biology, engineering, econometrics, medicine, and public health. Panel count data include observations on subjects over multiple time points where the response variable is a count or recurrent event process when only the numbers of events occurring between observation time points are available. The choice of method for analyzing panel count data usually depends on the relationship between the observation times and the response variable and questions of interest. Most of the previous research was done when the observation times are fixed. If the observation times are random, the data structure becomes more challenging since the observation times for individual subjects vary in addition to the incompleteness of observations. The model-based approach was used to deal with such data. However, this method relies on extra assumptions on the observation scheme and thus is restrictive in practice. In this dissertation, we discuss the problem of multi-sample nonparametric comparison of counting processes with panel count data, which arise naturally when recurrent events are considered. For the problem considered, we develop some new nonparametric tests.</p> <p> First, we construct a class of nonparametric test statistics based on the integrated weighted differences between the estimated mean functions of the count processes, where the isotonic regression estimate is used for the mean functions. The asymptotic distributions of the proposed statistics are derived and their finite-sample properties are examined through Monte Carlo simulations. A panel count data from a cancer study is analyzed and presented as an illustrative example.</p> <p>As shown through Monte Carlo simulations, the nonparametric maximum likelihood estimator (NPMLE) of the mean function is more efficient than the nonparametric maximum pseudo-likelihood estimator (NPMPLE). However, no nonparametric tests have been discussed in the literature for panel count data based on the NPMLE since the NPMLE is more complicated both theoretically and computationally. It is, therefore, particularly important to develop nonparametric tests based on the NPMLE for panel count data.</p> <p> In the second part of the dissertation, we focus on the situation when treatment indicators can be regarded as independent and identically distributed random variables and propose a nonparametric test in this case using the maximum likelihood estimator. The asymptotic property of the test statistic is derived. Simulation studies are carried out which suggest that the proposed method works well for practical situations, and is more powerful than the existing tests based on the NPMPLEs of the mean functions.</p> <p>In the third part of the dissertation, we consider more general situations. We construct a class of nonparametric tests based on the accumulated weighted differences between the rates of increase of the estimated mean functions of the counting processes over observation times, where the nonparametric maximum likelihood approach is used to estimate the mean functions instead of the nonparametric maximum pseudolikelihood. The asymptotic distributions of the proposed statistics are derived and their finite-sample properties are evaluated by means of Monte Carlo simulations. The simulation results show that the proposed methods work quite well and the tests based on NPMLE are more powerful than those based on NPMPLE. Two real data sets are analyzed and presented as illustrative examples.</p> <p>The last part of the dissertation discusses a special type of panel count data, namely, current status or case 1 interval-censored data. Such data often occur in tumorigenicity experiments. For nonparametric two-sample comparison based on censored or interval-censored data, most of the existing methods have focused on testing the hypothesis that specifies the two population distributions to be identical under the assumption that observation or censoring times have the same distribution. We consider the nonparametric Behrens-Fisher hypothesis (NBFH) under this settings. For this purpose, we study the asymptotic property of the nonparametric maximum likelihood estimator of the probability that an observation from the first distribution exceeds an observation from the second distribution. A nonparametric test for the NBFH is proposed and the asymptotic normality of the proposed test is established. The method is evaluated using simulation studies and illustrated by a set of real data from a tumorigenicity experiment.</p> / Thesis / Doctor of Philosophy (PhD)

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Nonparametric Bayesian models have become increasingly popular in speech recognition tasks such as language and acoustic modeling due to their ability to discover underlying structure in an iterative manner. These methods do not require a priori assumptions about the structure of the data, such as the number of mixture components, and can learn this structure directly. Dirichlet process mixtures (DPMs) are a widely used nonparametric Bayesian method which can be used as priors to determine an optimal number of mixture components and their respective weights in a Gaussian mixture model (GMM). Because DPMs potentially require an infinite number of parameters, inference algorithms are needed to make posterior calculations tractable. The focus of this work is an evaluation of three of these Bayesian variational inference algorithms which have only recently become computationally viable: Accelerated Variational Dirichlet Process Mixtures (AVDPM), Collapsed Variational Stick Breaking (CVSB), and Collapsed Dirichlet Priors (CDP). To eliminate other effects on performance such as language models, a phoneme classification task is chosen to more clearly assess the viability of these algorithms for acoustic modeling. Evaluations were conducted on the CALLHOME English and Mandarin corpora, consisting of two languages that, from a human perspective, are phonologically very different. It is shown in this work that these inference algorithms yield error rates comparable to a baseline Gaussian mixture model (GMM) but with a factor of up to 20 fewer mixture components. AVDPM is shown to be the most attractive choice because it delivers the most compact models and is computationally efficient, enabling its application to big data problems. / Electrical and Computer Engineering

The primary aim of this research is to shed more light on the issue of model uncertainty in applied econometrics in general and cross-country growth as well as happiness and well-being regressions in particular. Model uncertainty consists of three main types: theory uncertainty, focusing on which principal determinants of economic growth or happiness should be included in a model; heterogeneity uncertainty, relating to whether or not the parameters that describe growth or happiness are identical across countries; and functional form uncertainty, relating to which growth and well-being regressors enter the model linearly and which ones enter nonlinearly. Model averaging methods including Bayesian model averaging and Frequentist model averaging are the main statistical tools that incorporate theory uncertainty into the estimation process. To address functional form uncertainty, a variety of techniques have been proposed in the literature. One suggestion, for example, involves adding regressors that are nonlinear functions of the initial set of theory-based regressors or adding regressors whose values are zero below some threshold and non-zero above that threshold. In recent years, however, there has been a rising interest in using nonparametric framework to address nonlinearities in growth and happiness regressions. The goal of this research is twofold. First, while Bayesian approaches are dominant methods used in economic empirics to average over the model space, I take a fresh look into Frequentist model averaging techniques and propose statistical routines that computationally ease the implementation of these methods. I provide empirical examples showing that Frequentist estimators can compete with their Bayesian peers. The second objective is to use recently-developed nonparametric techniques to overcome the issue of functional form uncertainty while analyzing the variance of distribution of per capita income. Nonparametric paradigm allows for addressing nonlinearities in growth and well-being regressions by relaxing both the functional form assumptions and traditional assumptions on the structure of error terms. / Ph. D.

One form of model robust regression (MRR) predicts mean response as a convex combination of a parametric and a nonparametric prediction. MRR is a semiparametric method by which an incompletely or an incorrectly specified parametric model can be improved through adding an appropriate amount of a nonparametric fit. The combined predictor can have less bias than the parametric model estimate alone and less variance than the nonparametric estimate alone. Additionally, as shown in previous work for uncorrelated data with linear mean function, MRR can converge faster than the nonparametric predictor alone. We extend the MRR technique to the problem of predicting mean response for clustered non-normal data. We combine a nonparametric method based on local estimation with a global, parametric generalized estimating equations (GEE) estimate through a mixing parameter on both the mean scale and the linear predictor scale. As a special case, when data are uncorrelated, this amounts to mixing a local likelihood estimate with predictions from a global generalized linear model. Cross-validation bandwidth and optimal mixing parameter selectors are developed. The global fits and the optimal and data-driven local and mixed fits are studied under no/some/substantial model misspecification via simulation. The methods are then illustrated through application to data from a longitudinal study. / Ph. D.

In typical normal theory regression, the assumption of homogeneity of variances is often not appropriate. Instead of treating the variances as a nuisance and transforming away the heterogeneity, the structure of the variances may be of interest and it is desirable to model the variances. Aitkin (1987) proposes a parametric dual model in which a log linear dependence of the variances on a set of explanatory variables is assumed. Aitkin's parametric approach is an iterative one providing estimates for the parameters in the mean and variance models through joint maximum likelihood. Estimation of the mean and variance parameters are interrelatedas the responses in the variance model are the squared residuals from the fit to the means model. When one or both of the models (the mean or variance model) are misspecified, parametric dual modeling can lead to faulty inferences. An alternative to parametric dual modeling is to let the data completely determine the form of the true underlying mean and variance functions (nonparametric dual modeling). However, nonparametric techniques often result in estimates which are characterized by high variability and they ignore important knowledge that the user may have regarding the process. Mays and Birch (1996) have demonstrated an effective semiparametric method in the one regressor, single-model regression setting which is a "hybrid" of parametric and nonparametric fits. Using their techniques, we develop a dual modeling approach which is robust to misspecification in either or both of the two models. Examples will be presented to illustrate the new technique, termed here as Dual Model Robust Regression. / Ph. D.

The problem considered in this dissertation is the following: let π₁ and π₂ be two bivariate populations having unknown cumulative distribution functions F₁(x₁, x₂) and F₂(x₁, x₂), respectively. Assume that F₁ and F₂ are continuous and identical except possibly in location parameters. It is desired to test the null hypothesis H₀: F₁(x₁, x₂) ≡ F₂(x₁, x₂) against the alternative H₀: F₁(x₁, x₂) ≠ F₂(x₁, x₂) It cannot be assumed that the variables x₁ and x₂ are statistically independent. Suppose there are n₁pairs of observations (x₁₁, x₂₁),..., (x_1n₁, x_2n₁) from the population π₁ and n₂ pairs of observations (x_ln+1. X_2n₁+1),..., (x_1N, x_2N) from population π₂, where N = n₁ + n₂. The x₁ᵢ (i = 1,2,..., N) are ranked according to magnitude, the largest being assigned rank 1 and the smallest assigned rank N. In a similar manner, ranks are assigned to the observations x₂ᵢ (i = 1, 2, …, N). It is assumed that there are no ties in ranks. Let u₁ᵢ and u₂ᵢ denote the ranks assigned to x₁ᵢ and x₂ᵢ if these observations belong to population π₁, and let u’₁ᵢ and u’₂ᵢ denote the ranks of the same observations if they belong to population π₂. Since the sum of the first N integers is (N(N+1))/2, it follows that Σ_k=1^n₁ u_ik + Σ_{k=n₁ + 1}^N u_ik’ = (N(N+1))/2 If the N pairs of ranks are plotted on a plane, it is likely that the n₁ points from population π₁ and the n₂ points from population π₂ will be interspersed forming a circular or elliptical pattern under the assumption that F₁(x₁, x₂) and F₂(x₁, x₂) are identical. Under the alternative hypothesis, it is likely that there will be a segregation of the points into two groups. A test statistic, S₁² is constructed to measure the extent of this segregation . The S₁²-statistic proposed here, is based on the Euclidean distance between the centroids of the ranks belonging to π₁ and π₂, in particular S₁²= (ū₁-ū₁')² + (ū₂-ū₂')² where ūᵢ = n₁⁻¹ Σ_k=1^n₁ u_ik , uᵢ’ = n₂⁻¹ Σ_{k=n₁ + 1}^N u_ik’ The first two moments of S₁² are derived under the following conditional randomization procedures keeping the ranks paired as given in the sample, n₁ pairs are selected at random (with equal probabilities) from among the N = n₁ + n₂ pairs and assigned to population π₁; the remaining n₂ pairs are assigned to population π₂. It is shown that E(S₁²) = (N²(N+1))/6n₁n₂ and σ²_S₁² = a₀₀ + a₁₁A₁₁+ a₁₂A₁₂ + a₂₁A₂₁ + a₂₂A₂₂ + a₁₁,₁₁A²₁₁ Where A_rs = Σ_k=1^N₁ u_1k^ru_2k^s are parameters depending on the sample, and the coefficients a₀₀, a₁₁, a₁₂, a₂₁, a₂₂ and a₁₁,₁₁ have been tabulated for values of n₁ and n₂ up to 20. The exact sampling distribution of S₁² is unknown However, it is sho•Nn that the distribution of (kE(S₁²))/ σ²_S₁² is approximately χ² with (2[E(S₁²)]²/ σ²_S₁² degrees of freedom. A rank analogue of Wald’s modification of Hotelling's T² is given and the first two moments obtained. Also, a multivariate extension is considered and a statistic, S₁²(k,2), constructed. The expectation and variance of S₁²(k,2) are derived. A multi-populatiun extension for the case of bivariate populations is given and the expectation is derived for a statistic, S₁²(2,p). A statistic, S₁²(k,p) is constructed for the most general case and its expectation is given. An alternative approach to the problem, also investigated, is by means of discriminant analysis. In this case simplified formulas are given for the calculation of the components of a vector which provides optimum discrimination. It is shown that this method is not a fruitful one for the construction of tests of significance pertaining to the original null hypothesis. / Doctor of Philosophy