Repository logo

Browsing by Author "Meyer, Mary, advisor"

Now showing 1 - 2 of 2
  • Thumbnail Image
    ItemOpen Access
    Applications of least squares penalized spline density estimator
    (Colorado State University. Libraries, 2024) Jing, Hanxiao, author; Meyer, Mary, advisor; Cooley, Daniel, committee member; Kokoszka, Piotr, committee member; Berger, Joshua, committee member
    The spline-based method stands as one of the most common nonparametric approaches. The work in this dissertation explores three applications of the least squares penalized spline density estimator. Firstly, we present a novel hypothesis test against the unimodality of density functions, based on unimodal and bimodal estimates of the density function, using penalized splines. The test statistic is the difference in the least-squares criterion, between these fits. The distribution of the test statistics under the null hypothesis is estimated via simulated data sets from the unimodal fit. Large sample theory is derived and simulation studies are conducted to compare its performance with other common methods across various scenarios, alongside a real-world application involving neuro-transmission data from guinea pig brains. Secondly, we tackle the deconvolution density estimation problem, introducing the penalized splines deconvolution estimator. Building upon the results gained from piecewise constant splines, we achieve a cube-root convergence rate for piecewise quadratic splines and uniform errors. Moreover, we derive large sample theories for the penalized spline estimator and the constrained spline estimator. Simulation studies illustrate the competitive performance of our estimators compared to the kernel estimators across diverse scenarios. Lastly, drawing inspiration from the preceding applications, we develop a hypothesis test to discern whether the underlying density is unimodal or multimodal, given data with measurement error. Under the assumption of uniform errors, we introduce the test and derive the test statistic. Simulations are conducted to show the performance of the proposed test under different conditions.
  • Thumbnail Image
    ItemOpen Access
    Bayesian shape-restricted regression splines
    (Colorado State University. Libraries, 2011) Hackstadt, Amber J., author; Hoeting, Jennifer, advisor; Meyer, Mary, advisor; Opsomer, Jean, committee member; Huyvaert, Kate, committee member
    Semi-parametric and non-parametric function estimation are useful tools to model the relationship between design variables and response variables as well as to make predictions without requiring the assumption of a parametric form for the regression function. Additionally, Bayesian methods have become increasingly popular in statistical analysis since they provide a flexible framework for the construction of complex models and produce a joint posterior distribution for the coefficients that allows for inference through various sampling methods. We use non-parametric function estimation and a Bayesian framework to estimate regression functions with shape restrictions. Shape-restricted functions include functions that are monotonically increasing, monotonically decreasing, convex, concave, and combinations of these restrictions such as increasing and convex. Shape restrictions allow researchers to incorporate knowledge about the relationship between variables into the estimation process. We propose Bayesian semi-parametric models for regression analysis under shape restrictions that use a linear combination of shape-restricted regression splines such as I-splines or C-splines. We find function estimates using Markov chain Monte Carlo (MCMC) algorithms. The Bayesian framework along with MCMC allows us to perform model selection and produce uncertainty estimates much more easily than in the frequentist paradigm. Indeed, some of the work proposed in this dissertation has not been developed in parallel in the frequentist paradigm. We begin by proposing a semi-parametric generalized linear model for regression analysis under shape-restrictions. We provide Bayesian shape-restricted regression spline (Bayes SRRS) models and MCMC estimation algorithms for the normal errors, Bernoulli, and Poisson models. We propose several types of inference that can be performed for the normal errors model as well as examine the asymptotic behavior of the estimates for the normal errors model under the monotone shape-restriction. We also examine the small sample behavior of the proposed Bayes SRRS model estimates via simulation studies. We then extend the semi-parametric Bayesian shape-restricted regression splines to generalized linear mixed models. We provide a MCMC algorithm to estimate functions for the random intercept model with normal errors under the monotone shape restriction. We then further extend the semi-parametric Bayesian shape-restricted regression splines to allow the number and location of the knot points for the regression splines to be random and propose a reversible jump Markov chain Monte Carlo (RJMCMC) algorithm for regression function estimation under the monotone shape restriction. Lastly, we propose a Bayesian shape-restricted regression spline change-point model where the regression function is shape-restricted except at the change-points. We provide RJMCMC algorithms to estimate functions with change-points where the number and location of interior knot points for the regression splines are random. We provide a RJMCMC algorithm to estimate the location of an unknown change-point as well as a RJMCMC algorithm to decide between a model with no change-points and model with a change-point.