Nonparametric Estimation of Additive Models with Shape Constraints

Monotone additive models are useful in estimating productivity curves or analyzing disease risk where the predictors are known to have monotonic effects on the response. Existing literature mainly focuses on univariate monotone smoothing. Available methods for the estimation of monotone additive models are either difficult to interpret or have no asymptotic guarantees. In the first part of this dissertation, we propose a one-step backfitted constrained polynomial spline method for the estimation of monotone additive models. In our proposed method, we obtain monotone estimators by imposing a set of linear constraints on the spline coefficients for each additive component. In the second part of the dissertation, we extend the constrained polynomial spline method to estimate the production frontier that is used to quantify the maximum production output in econometrics. The estimation of frontier functions is more challenging since it is the boundary of the support rather than the mean output function to be estimated. Here, we develop a two-step shape constrained polynomial spline method for the frontier estimation. The first step is to capture the shape of frontier while the second step is to estimate the location of frontier. Both proposed methods in this dissertation give smooth estimators with the desired shape constraints (monotonicity or/and concavity). They are easily implementable and computationally efficient by taking advantage of linear programming. Most importantly, our methods are applicable for multi-dimensions where some existing methods fail to work. For the assessment of properties of the proposed estimators, asymptotic theory is also developed. In addition, the simulation studies and application of our methods to analyze Norwegian Farm data in both parts suggest that our proposed methods have better numerical performance than the existing methods, especially when the data has outliers

Model estimation, identification and inference for next-generation functional data and spatial data

This dissertation is composed of three research projects focused on model estimation, identification, and inference for next-generation functional data and spatial data. The first project deals with data that are collected on a count or binary response with spatial covariate information. In this project, we introduce a new class of generalized geoadditive models (GGAMs) for spatial data distributed over complex domains. Through a link function, the proposed GGAM assumes that the mean of the discrete response variable depends on additive univariate functions of explanatory variables and a bivariate function to adjust for the spatial effect. We propose a two-stage approach for estimating and making inferences of the components in the GGAM. In the first stage, the univariate components and the geographical component in the model are approximated via univariate polynomial splines and bivariate penalized splines over triangulation, respectively. In the second stage, local polynomial smoothing is applied to the cleaned univariate data to average out the variation of the first-stage estimators. We investigate the consistency of the proposed estimators and the asymptotic normality of the univariate components. We also establish the simultaneous confidence band for each of the univariate components. The performance of the proposed method is evaluated by two simulation studies and the crash counts data in the Tampa-St. Petersburg urbanized area in Florida. In the second project, motivated by recent work of analyzing data in the biomedical imaging studies, we consider a class of image-on-scalar regression models for imaging responses and scalar predictors. We propose to use flexible multivariate splines over triangulations to handle the irregular domain of the objects of interest on the images and other characteristics of images. The proposed estimators of the coefficient functions are proved to be root-$n$ consistent and asymptotically normal under some regularity conditions. We also provide a consistent and computationally efficient estimator of the covariance function. Asymptotic pointwise confidence intervals (PCIs) and data-driven simultaneous confidence corridors (SCCs) for the coefficient functions are constructed. A highly efficient and scalable estimation algorithm is developed. Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed method. The proposed method is applied to the spatially normalized Positron Emission Tomography (PET) data of Alzheimer\u27s Disease Neuroimaging Initiative (ADNI). In the third project, we propose a heterogeneous functional linear model to simultaneously estimate multiple coefficient functions and identify groups, such that coefficient functions are identical within groups and distinct across groups. By borrowing information from relevant subgroups, our method enhances estimation efficiency while preserving heterogeneity. We use an adaptive fused lasso penalty to shrink subgroup coefficients to shared common values within each group. We also establish the theoretical properties of our adaptive fused lasso estimators. To enhance the computation efficiency and incorporate neighborhood information, we propose to use a graph-constrained adaptive lasso. A highly efficient and scalable estimation algorithm is developed. Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed method. The proposed method is applied to a dataset of hybrid maize grain yields from the Genomes to Fields consortium

ISBIS 2016: Meeting on Statistics in Business and Industry

This Book includes the abstracts of the talks presented at the 2016 International Symposium on Business and Industrial Statistics, held at Barcelona, June 8-10, 2016, hosted at the Universitat Politècnica de Catalunya - Barcelona TECH, by the Department of Statistics and Operations Research. The location of the meeting was at ETSEIB Building (Escola Tecnica Superior d'Enginyeria Industrial) at Avda Diagonal 647. The meeting organizers celebrated the continued success of ISBIS and ENBIS society, and the meeting draw together the international community of statisticians, both academics and industry professionals, who share the goal of making statistics the foundation for decision making in business and related applications. The Scientific Program Committee was constituted by: David Banks, Duke University Amílcar Oliveira, DCeT - Universidade Aberta and CEAUL Teresa A. Oliveira, DCeT - Universidade Aberta and CEAUL Nalini Ravishankar, University of Connecticut Xavier Tort Martorell, Universitat Politécnica de Catalunya, Barcelona TECH Martina Vandebroek, KU Leuven Vincenzo Esposito Vinzi, ESSEC Business Schoo