An adaptive model checking test for the functional linear model

Residual bootstrap tests in linear models with many regressors

Model checks for functional linear regression models based on projected empirical processes

SummaryWe consider the problem of testing functional constraints in a class of functional concurrent linear models where both the predictors and the response are functional data measured at discrete time points. We propose test procedures based on the empirical likelihood with bias-corrected estimating equations to conduct both pointwise and simultaneous inferences. The asymptotic distributions of the test statistics are derived under the null and local alternative hypotheses, where sparse and dense functional data are considered in a unified framework. We find a phase transition in the asymptotic null distributions and the orders of detectable alternatives from sparse to dense functional data. Specifically, the tests proposed can detect alternatives of √n-order when the number of repeated measurements per curve is of an order larger than nη0 with n being the number of curves. The transition points η0 for pointwise and simultaneous tests are different and both are smaller than the transition point in the estimation problem. Simulation studies and real data analyses are conducted to demonstrate the methods proposed.

In this work, the distributional properties of the goodness-of-fit term in likelihood-based information criteria are explored. These properties are then leveraged to construct a novel goodness-of-fit test for normal linear regression models that relies on a nonparametric bootstrap. Several simulation studies are performed to investigate the properties and efficacy of the developed procedure, with these studies demonstrating that the bootstrap test offers distinct advantages as compared to other methods of assessing the goodness-of-fit of a normal linear regression model. Our inferential technique can be employed using the R package, available freely via the Comprehensive R Archive Network.

Bartlett-corrected tests for heteroskedastic linear models

Linear models with stable error densities are considered, and their local asymptotic normality with respect to the regression parameter is established. We use this result, combined with Le Cam's third lemma, to obtain local powers and asymptotic relative efficiencies for various classical rank tests (the regression and analysis of variance counterparts of the Wilcoxon, van der Waerden and median tests) under α-stable densities with various values of the skewness parameter and tail index. The same results are used to construct new rank tests, based on ‘stable scores’, achieving parametric optimality at specified stable densities. A Monte Carlo study is conducted to compare their finite-sample relative performances.

In this article, we propose a new projection test for linear hypotheses on regression coefficient matrices in linear models with high-dimensional responses. We systematically study the theoretical properties of the proposed test. We first derive the optimal projection matrix for any given projection dimension to achieve the best power and provide an upper bound for the optimal dimension of projection matrix. We further provide insights into how to construct the optimal projection matrix. One- and two-sample mean problems can be formulated as special cases of linear hypotheses studied in this article. We both theoretically and empirically demonstrate that the proposed test can outperform the existing ones for one- and two-sample mean problems. We conduct Monte Carlo simulation to examine the finite sample performance and illustrate the proposed test by a real data example.

The results from an analysis of balanced data are frequently summarized in an analysis of variance (AOV) table. Each sum of squares (SS) in the AOV table is uniquely associated with testing a particular hypothesis in the linear model. These hypotheses are well known and cause no confusion among statisticians as to what is being tested. Results from an analysis of unbalanced data, however, cannot be uniquely summarized in an AOV table, and, consequently, statisticians are often confused about the hypotheses being tested. Some statisticians prefer an orthogonal partitioning of the SS (paralleling the balanced case) as the appropriate analysis; others prefer various forms of nonorthogonal analyses. The purpose of this paper is to show (and, hopefully, clarify) the hypotheses that are being tested in various unbalanced AOV tables.

Testing the constancy of regression parameters against continuous structural change

On M Test in Linear Models

Koch, K.R.: Parameter estimation and hypothesis testing in linear models

An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

NN goodness-of-fit tests for linear models

On exact and asymptotic tests of non-nested models

This dissertation consists of five chapters with three distinct but related research projects. In Chapter 1, we introduce some necessary definitions related to the research work. In Chapter 2, we develop Bayes factor based testing procedures for a general linear hypothesis of the regression coefficients in the context of the normal linear models. We propose two calibration schemes to deal with asymmetry in information of Bayes factor: (i) by controlling Type I error probability of the Bayes factor and (ii) by balancing Type I and Type II error probabilities of the Bayes factor. We evaluate the finite sample performance of the proposed Bayes factors via simulation studies and a real-data application. Experimental results have shown than the proposed Bayes factors perform well in testing the general linear hypothesis of the regression coefficient. In Chapter 3, we consider Bayesian quantile analysis for testing constrained hypotheses in linear models, in which the quantiles the parameters satisfy a simple order restriction. We develop a Bayesian hierarchical model based on the specification the asymmetric Laplace distribution for the error component. We propose a non-iterative sampling algorithm in the Expectation-Maximization (EM) structure to generate independently and identically distributed posterior samples from their posterior distributions of the parameters. Then we adopt the Savage-Dickey density ratios to conduct the multiple comparison with simply order constraints. Simulation studies were conducted to compare the finite sample performance of the proposed non-iterative sampling algorithm with the Gibbs sampling algorithm. In Chapter 4, we consider objective Bayesian analysis for the concordance correlation coefficient (CCC), which is one of the most commonly used metrics to assess agreement of different methods in many practical applications. We develop an objective Bayesian framework for estimating the CCC based a combined use of the multivariate student's t-distribution with noninformative Independence Jeffreys prior for the unknown parameters. Extensive simulation studies are conducted to compare the performance of the proposed Bayesian estimates with the ones under the subjective priors in the literature. In Chapter 5, we discuss some ongoing projects related to our research work mentioned above and some interesting problems for future work.