Jiahua Chen

The parametric likelihood method has been a popular platform for inference. When the model is regular, the maximum likelihood estimator possesses many nice properties. However, when the model is not accurate or regular, these properties may no longer hold. Because of this, researchers turn to non-parametric modelling, which imposes weaker model assumptions.Empirical likelihood is a popular non-parametric platform for inference. The resulting profile empirical likelihood function has many similar properties to its parametric counterpart. The resulting empirical likelihood confidence regions are data driven, range respecting and transformation invariant. The empirical likelihood confidence regions tend to have coverage probability below the nominal level. To address this issue, researchers have proposed various methods to improve the coverage precision. However, these methods need to estimate a theoretical tuning parameter, which is difficult to estimate robustly. In my thesis, we develop a computer experiment data-driven approach to improve the coverage precision of empirical likelihood confidence regions.The maximum empirical likelihood estimator, just like its parametric counterpart, shares many nice properties. However, the optimal properties cannot be utilized unless we know that the local maximum at hand is close to the unknown true parameter value. To overcome this obstacle, we first identify a set of conditions under which the global maximum is consistent. We then develop a `global maximum test’ to ascertain if the local maximum at hand is, in fact, the global maximum. Furthermore, we invent a `global consistency remedy’ to ensure global consistency by expanding the set of estimating functions under empirical likelihood.For non-regular models such as finite normal mixture models, the MLE no longer enjoys these nice properties. In fact, the MLE is not well-defined because of the unboundedness of likelihood. To address this issue, researchers have proposed penalized likelihood and constrained MLE methods for consistently estimating the mixing distribution. However, the consistency of these methods is established under the assumption that the subpopulations are non-degenerate. We relax this restriction to demonstrate that the penalized MLE remains consistent even when some subpopulations degenerate. We also develop a hypothesis testing procedure for detecting degenerate subpopulations in finite Gaussian mixture models.

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Finite mixture models are widely used to model data that exhibit heterogeneity. In machine learning, they are often used as probabilistic models for clustering analysis. In the application of finite mixtures to real datasets, the most fundamental task is to learn model parameters. In modern applications, datasets are often too large to be stored in a single facility and are distributed across data centres. To learn models on these distributed datasets, split-and-conquer (SC) approaches are often used. SC approaches consist of two steps: they first locally learn one model per data centre and then send these local results to a central machine to be aggregated. Since the parameter spaces of mixtures are non-Euclidean, existing aggregation methods are not appropriate. We develop a novel computationally efficient aggregation approach for SC learning of finite mixtures. We show that the resulting estimator is root-n-consistent under some general conditions. Experiments show the proposed approach has comparable statistical performance with the global estimator based on the full dataset, if the latter is feasible. It also has better statistical and computational performance than some existing methods for learning mixtures on large datasets.Finite mixtures are also widely used to approximate density functions of various shapes. When mixtures are used in graphical models to approximate density functions, the order of the mixture increases exponentially due to recursive procedures and the inference becomes intractable. One way to make the inference tractable is to use mixture reduction, that is, to approximate the mixture by one with a lower order. We propose a novel reduction approach by minimizing the Composite Transportation Divergence (CTD) between two mixtures. The optimization problem can be solved by a majorization minimization algorithm. We show that many existing algorithms for reduction are special cases of our approach. This finding allows us to provide theoretical support for these existing algorithms. Our approach also allows flexible choices for cost functions in CTD. This flexibility allows our approach to have better performance than existing approaches. We also discuss other related learning issues under finite mixtures.

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In many applications, we collect independent samples from interconnected populations. These population distributions share some latent structures, so it is advantageous to jointly analyze the samples. Recently, many researchers have advocated the use of the semiparametric density ratio model (DRM) to account for the latent structures these distributions share and have developed more efficient data analysis procedures based on pooled data. The advantages and several asymptotic properties of the DRM-based inferences have been demonstrated in many fields and studies, and they show that the DRM helps to improve statistical efficiency. In this thesis, we investigate several inference problems related to the DRM. The first research problem we study is on the efficiency of the inference under a two-sample DRM. We consider a scenario where we have two samples whose sizes grow to infinity at different rates. The DRM-based inferences for the smaller-sized sample are studied. We find that some DRM-based estimators achieve the same asymptotic efficiency as the parametric estimators under some parametric model assumptions. Our simulation studies support our theoretical results. Our second work studies hypothesis test problems on population quantiles when we have multiple samples whose population distributions are connected via a DRM. We explore the use of the empirical likelihood ratio test for these hypotheses, which fills a gap in the literature in this context. Our major contribution is the derivation of the limiting chi-square distribution of the test statistic. Simulation experiments and a real-data example illustrate the efficacy of the proposed method. Finally, we solve an important open problem in the literature of DRM. The DRM postulates that the log density ratios are linear combinations of prespecified basis functions. The benefit of DRM relies on correctly specifying the basis functions. However, in applications, we do not have complete knowledge to enable a perfect choice of the basis functions. A data-adaptive choice can alleviate the risk of severe model misspecification. We propose a data-adaptive approach to the choice of basis functions based on functional principal component analysis. Our simulations and real-data analyses demonstrate that our proposed method leads to an efficiency gain.

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This thesis studies two types of research problems under finite mixture models. The first type is mixing distribution estimation. It is well-known that the maximum likelihood estimator (MLE) fails under some finite mixture models because their likelihood function is unbounded. This unboundedness occurs, for instance, under the finite normal mixture model, the finite gamma mixture model, and the finite location-scale mixture model. In the literature, different estimation methods have been developed by modifying the likelihood functions to restore consistency. The penalized MLE is one of the popular remedies. Though the consistency of penalized MLE has been studied extensively by many researchers, the results were often acquired under finite mixture models with some specific forms of kernel distribution. We provide a route to establish the consistency of penalized MLE under a unified framework that covers many finite mixture models. This route summarizes and helps to improve the existing results. We also study a novel way to modify the likelihood function based on data augmentation. This modified likelihood function produces a consistent estimator, the augmented MLE, under those finite mixture models with unbounded likelihood functions. In some circumstances, the augmented MLE is more efficient than its competitors in the literature.The second type of research problem is hypothesis testing for homogeneity un- der finite mixture models. We develop two tests for this purpose. First, our work migrates the Expectation-Maximization (EM) test to a finite vector-parameter mixture model with structural parameters. The EM test is a numeric and analytically tractable likelihood-based test, which has been applied to many finite mixture models. The second test is Neyman’s C(α) test, a score test variant. We generalize this test to finite vector-parameter mixture models following the principles of Neyman. Our generalization aligns with some existing results in the literature, though they are motivated differently. Yet we find this generalized test can be asymptotically biased under the finite gamma mixture model. To overcome this deficiency, we develop another C(α) test that we conjecture to be asymptotically unbiased.

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Dose-response experiments and subsequent data analyses are often carried out according to optimal designs for the purpose of accurately determining a specific effective dose (ED) level. If the interest is the dose-response relationship over a range of ED levels, many existing optimal designs are not accurate. In this dissertation, we propose a new design procedure, called two-stage sequential ED-design, which directly and simultaneously targets several ED levels. We use a small number of trials to provide a tentative estimation of the model parameters. The doses of the subsequent trials are then selected sequentially, based on the latest model information, to maximize the efficiency of the ED estimation over several ED levels. Although the commonly used logistic and probit models are convenient summaries of the dose-response relationship, they can be too restrictive. We introduce and study a more flexible albeit slightly more complex three-parameter logistic dose-response model. We explore the effectiveness of the sequential ED-design and the D-optimal design under this model, and develop an effective model fitting strategy. We develop a two-step iterative algorithm to compute the maximum likelihood estimate of the model parameters. We prove that the algorithm iteration increases the likelihood value, and therefore will lead to at least a local maximum of the likelihood function. We also study the numerical solution to the D-optimal design for the three-parameter logistic model. Interestingly, all our numerical solutions to the D-optimal design are three-point-support distributions.We also discuss the use of the ED-design when experimental subjects become available in groups. We introduce the group sequential ED-design, and demonstrate how to construct this design. The ED-design has a natural extension to more complex model and can satisfy a broad range of the demands that may arise in applications.

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Maintaining a high quality of lumber products is of great social and economic importance. This thesis develops theories as part of a research program aimed at developing a long term program for monitoring change in the strength of lumber. These theories are motivated by two important tasks of the monitoring program, testing for change in strength populations of lumber produced over the years and making statistical inference on strength populations based on Type I censored lumber samples. Statistical methods for these inference tasks should ideally be efficient and nonparametric. These desiderata lead us to adopt a semiparametric density ratio model to pool the information across multiple samples and use the nonparametric empirical likelihood as the tool for statistical inference.We develop a dual empirical likelihood ratio test for composite hypotheses about the parameter of the density ratio model based on independent samples from different populations. This test encompasses testing differences in population distributions as a special case. We find the proposed test statistic to have a classical chi-square null limiting distribution. We also derive the power function of the test under a class of local alternatives. It reveals that the local power is often increased when strength is borrowed from additional samples even when their underlying distributions are unrelated to the hypothesis of interest. Simulation studies show that this test has better power properties than all potential competitors adopted to the multiple sample problem under the investigation, and is robust to model misspecification. The proposed test is then applied to assess strength properties of lumber with intuitively reasonable implications for the forest industry.We also establish a powerful inference framework for performing empirical likelihood inference under the density ratio model when Type I censored samples are present. This inference framework centers on the maximization of a concave dual partial empirical likelihood function, and features an easy computation. We study the properties of this dual partial empirical likelihood, and find its corresponding likelihood ratio test to have a simple chi-square limiting distribution under the null model and a non-central chi-square limiting distribution under local alternatives.

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Feature selection plays a pivotal role in knowledge discovery and contemporary scientific research. Traditional best subset selection or stepwise regression can be computationallyexpensive or unstable in the selection process, and so various penalized likelihood methods (PLMs) have received much attention in recent decades. In this dissertation, we develop approaches based on PLMs to deal with the issues of feature selection arising from several application fields.Motivated by genomic association studies, we first address feature selection in ultra-high-dimensional situations, where the number of candidate features can be huge. Reducing the dimension of the data is essential in such situations. We propose a novel screening approach via the sparsity-restricted maximum likelihood estimator that removes most of the irrelevant features before the formal selection. The model after screening serves as an excellent starting point for the use of PLMs. We establish the screening and selection consistency of the proposed method and develop efficient algorithms for its implementation.We next turn our attention to the analysis of complex survey data, where the identification of influential factors for certain behavioral, social, and economic indices forms a variable selection problem. When data are collected through survey sampling from a finite population, they have an intrinsic dependence structure and may provide a biased representation of the target population. To avoid distorted conclusions, survey weights are usually adopted in these analyses. We use a pseudo-likelihood to account for the survey weights and propose a penalized pseudo-likelihood method for the variable selection of survey data. The consistency of the proposed approach is established for the joint randomization framework.Lastly, we address order selection for finite mixture models, which provides a flexible tool for modeling data from a heterogeneous population. PLMs are attractive for such problems. However, this application requires maximizations over nonsmooth and nonconcave objective functions, which are computationally challenging. We transform the original multivariate objective function into a sum of univariate functions and design an iterative thresholding-based algorithm to efficiently solve the sparse maximization without ad hoc steps. We establish the convergence of the new algorithm and illustrate its efficiency through both simulations and real-data examples.

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