Lack-of-Fit Tests for Generalized Linear Models via Splines

Cubic B-splines are used to estimate the nonparametric component of a semiparametric generalized linear model. A penalized log-likelihood ratio test statistic is constructed for the null hypothesis of the linearity of the nonparametric function. When the number of knots is fixed, its limiting null distribution is the distribution of a linear combination of independent chi-squared random variables, each with one df. The smoothing parameter is determined by giving a specified value for its asymptotically expected value under the null hypothesis. A simulation study is conducted to evaluate its power performance; a real-life dataset is used to illustrate its practical use.     

Using -splines to test the linearity of partially linear models

The nonparametric component in a partially linear model is estimated by a linear combination of fixed-knot cubic B-splines with a second-order difference penalty on the adjacent B-spline coefficients. The resulting penalized least-squares estimator is used to construct two Wald-type spline-based test statistics for the null hypothesis of the linearity of the nonparametric function. When the number of knots is fixed, the first test statistic asymptotically has the distribution of a linear combination of independent chi-squared random variables, each with one degree of freedom, under the null hypothesis. The smoothing parameter is determined by specifying a value for the asymptotically expected value of the test statistic under the null hypothesis. When the number of knots is fixed and under the null hypothesis, the second test statistic asymptotically has a chi-squared distribution with K=q+2 degrees of freedom, where q is the number of knots used for estimation. The power performances of the two proposed tests are investigated via simulation experiments, and the practicality of the proposed methodology is illustrated using a real-life data set.     

A Partially Linear Model with Its Applications

The nonparametric component in a partially linear model is approximated via cubic B-splines with a second-order difference penalty on the adjacent B-spline coefficients to avoid undersmoothing. A Wald-type spline-based test statistic is constructed for the null hypothesis of no effect of a continuous covariate. When the number of knots is fixed, the limiting null distribution of the test statistic is the distribution of a linear combination of independent chi-squared random variables, each with one degree of freedom. A real-life dataset is provided to illustrate the practical use of the test statistic.     

Statistical Inference on the Parametric Component in Partially Linear Spatial Autoregressive Models

A statistical test procedure is proposed to check whether the parameters in the parametric component of the partially linear spatial autoregressive models satisfy certain linear constraint conditions, in which a residual-based bootstrap procedure is suggested to derive the p-value of the test. Some simulations are conducted to assess the performance of the test and the results show that the bootstrap approximation to the null distribution of the test statistic is valid and the test is of satisfactory power. Furthermore, a real-world example is given to demonstrate the application of the proposed test.     

A nonparametric hypothesis test for heteroscedasticity

In this paper, a hypothesis test for heteroscedasticity is proposed in a nonparametric regression model. The test statistic, which uses the residuals from a nonparametric fit of the mean function, is based on an adaptation of the well-known Levene's test. Using the recent theory for analysis of variance when the number of factor levels goes to infinity, the asymptotic distribution of the test statistic is established under the null hypothesis of homocedasticity and under local alternatives. Simulations suggest that the proposed test performs well in several situations, especially when the variance is a nonlinear function of the predictor.     

Testing for a Change of the Innovation Distribution in Nonparametric Autoregression: The Sequential Empirical Process Approach

We consider a nonparametric autoregression model under conditional heteroscedasticity with the aim to test whether the innovation distribution changes in time. To this end, we develop an asymptotic expansion for the sequential empirical process of nonparametrically estimated innovations (residuals). We suggest a Kolmogorov–Smirnov statistic based on the difference of the estimated innovation distributions built from the first ?ns?and the last n ? ?ns? residuals, respectively (0 ≤ s ≤ 1). Weak convergence of the underlying stochastic process to a Gaussian process is proved under the null hypothesis of no change point. The result implies that the test is asymptotically distribution‐free. Consistency against fixed alternatives is shown. The small sample performance of the proposed test is investigated in a simulation study and the test is applied to a data example.     

Testing for Linear Regression Relationships in Randomly Right-Censored Varying-Coefficient Models

A test statistic is constructed to test linear relationships in randomly right-censored varying-coefficient models. A residual-based bootstrap procedure is employed to derive the p-value of the test. The performance of the test is examined by extensive simulations. The simulation results show that the bootstrap estimate of the null distribution of the test statistic is approximately valid and the test method with the residual-based bootstrap works satisfactorily for at least moderate censoring rates of the response. Furthermore, the proposed test is applied to the Stanford heart transplant data for exploring a linear regression relationship between the logrithm of the survival time and the age of the patients.     

Comparison of Nonparametric Components in Two Partially Linear Models

In this article, we are concerned with whether the nonparametric functions are parallel from two partial linear models, and propose a test statistic to check the difference of the two functions. The unknown constant α is estimated by using moment method under null models. Nonparametric functions under both null and full models are estimated by using local linear method. The asymptotic properties of parametric and nonparametric components are derived. The test statistic under the null hypothesis is calculated and shown to be asymptotically normal.     

Adaptive tests of regression functions via multiscale generalized likelihood ratios

Many applications of nonparametric tests based on curve estimation involve selecting a smoothing parameter. The author proposes an adaptive test that combines several generalized likelihood ratio tests in order to get power performance nearly equal to whichever of the component tests is best. She derives the asymptotic joint distribution of the component tests and that of the proposed test under the null hypothesis. She also develops a simple method of selecting the smoothing parameters for the proposed test and presents two approximate methods for obtaining its P‐value. Finally, she evaluates the proposed test through simulations and illustrates its application to a set of real data.     

Profile Likelihood Inferences on the Partially Linear Model with a Diverging Number of Parameters

In this article, we study the profile likelihood estimation and inference on the partially linear model with a diverging number of parameters. Polynomial splines are applied to estimate the nonparametric component and we focus on constructing profile likelihood ratio statistic to examine the testing problem for the parametric component in the partially linear model. Under some regularity conditions, the asymptotic distribution of profile likelihood ratio statistic is proposed when the number of parameters grows with the sample size. Numerical studies confirm our theory.     

Estimation and inference for varying coefficient partially nonlinear errors-in-variables models

In this article, we study the varying coefficient partially nonlinear model with measurement errors in the nonparametric part. A local corrected profile nonlinear least-square estimation procedure is proposed and the asymptotic properties of the resulting estimators are established. Further, a generalized likelihood ratio (GLR) statistic is proposed to test whether the varying coefficients are constant. The asymptotic null distribution of the statistic is obtained and a residual-based bootstrap procedure is employed to compute the p-value of the statistic. Some simulations are conducted to evaluate the performance of the proposed methods. The results show that the estimating and testing procedures work well in finite samples.     

Asymptotocally nonparametric tests with censored paired data

A class of asymptotically nonparametric test with contains a test proposed by Wei(1980), is considered for testing the equality of two continuous distribution funcitons when paired observations are subject to arbitrary right censorship. It is shown that under the null hypothesis each test statistic converges in distribution to the standard normal random variable. Furthermore. the Monte Carlo simulation results indicate that some tests in this class are more powerful than Wei's test. A generalization to incomplete censored paired data is also included.     

Testing the linearity of negative binomial regression models

The negative binomial (NB) is frequently used to model overdispersed Poisson count data. To study the effect of a continuous covariate of interest in an NB model, a flexible procedure is used to model the covariate effect by fixed-knot cubic basis-splines or B-splines with a second-order difference penalty on the adjacent B-spline coefficients to avoid undersmoothing. A penalized likelihood is used to estimate parameters of the model. A penalized likelihood ratio test statistic is constructed for the null hypothesis of the linearity of the continuous covariate effect. When the number of knots is fixed, its limiting null distribution is the distribution of a linear combination of independent chi-squared random variables, each with one degree of freedom. The smoothing parameter value is determined by setting a specified value equal to the asymptotic expectation of the test statistic under the null hypothesis. The power performance of the proposed test is studied with simulation experiments.     

Statistical inference for restricted partially linear varying coefficient errors-in-variables models

As a useful extension of partially linear models and varying coefficient models, the partially linear varying coefficient model is useful in statistical modelling. This paper considers statistical inference for the semiparametric model when the covariates in the linear part are measured with additive error and some additional linear restrictions on the parametric component are available. We propose a restricted modified profile least-squares estimator for the parametric component, and prove the asymptotic normality of the proposed estimator. To test hypotheses on the parametric component, we propose a test statistic based on the difference between the corrected residual sums of squares under the null and alterative hypotheses, and show that its limiting distribution is a weighted sum of independent chi-square distributions. We also develop an adjusted test statistic, which has an asymptotically standard chi-squared distribution. Some simulation studies are conducted to illustrate our approaches.     

A nonparametric assessment of model adequacy based on Kullback-Leibler divergence

A discrepancy measure to assess model fitness against a nonparametric alternative is proposed. First, a Polya tree prior is constructed so that the centering distribution is the null. Second, the prior is updated in the light of data to obtain the posterior centering distribution as the alternative. Third, a Kullback-Leibler divergence type of test statistic is derived to assess the discrepancy between the two centering distributions. The properties of the test statistic are derived, and a power comparison with several well-known test statistics is conducted. The use of the test statistic is illustrated using network traffic data.     

A NONPARAMETRIC TEST OF CHANGING CONDITIONAL VARIANCES IN AUTOREGRESSIVE TIME SERIES

A nonparametric test for detecting changing conditional variances in stationary AR(p) time series is proposed in this paper. For AR(1) models, the test statistic is a Kolmogorov-Smirnov type statistic and the asymptotic theory is developed under both the null and the alternative hypotheses. For AR(p) models (p ≥ 2), an approximate test procedure is proposed. The empirical upper percentage points for our test are tabulated for both p = 1 and p = 2 cases and a bootstrap procedure is suggested for the p ≥ 3 case. Monte Carlo simulations demonstrate that the test has very good powers for finite samples under both normal and non-normal errors.     

A CONSISTENT MODEL SPECIFICATION TEST BASED ON THE KERNEL SUM OF SQUARES OF RESIDUALS

This paper constructs a consistent model specification test based on the difference between the nonparametric kernel sum of squares of residuals and the sum of squares of residuals from a parametric null model. We establish the asymptotic normality of the proposed test statistic under the null hypothesis of correct parametric specification and show that the wild bootstrap method can be used to approximate the null distribution of the test statistic. Results from a small simulation study are reported to examine the finite sample performance of the proposed tests.     

Tests for Symmetric Error Distribution in Linear and Nonparametric Regression Models

In linear and nonparametric regression models, the problem of testing for symmetry of the distribution of errors is considered. We propose a test statistic which utilizes the empirical characteristic function of the corresponding residuals. The asymptotic null distribution of the test statistic as well as its behavior under alternatives is investigated. A simulation study compares bootstrap versions of the proposed test to other more standard procedures.     

Testing variance components in balanced linear growth curve models

It is well known that the testing of zero variance components is a non-standard problem since the null hypothesis is on the boundary of the parameter space. The usual asymptotic chi-square distribution of the likelihood ratio and score statistics under the null does not necessarily hold because of this null hypothesis. To circumvent this difficulty in balanced linear growth curve models, we introduce an appropriate test statistic and suggest a permutation procedure to approximate its finite-sample distribution. The proposed test alleviates the necessity of any distributional assumptions for the random effects and errors and can easily be applied for testing multiple variance components. Our simulation studies show that the proposed test has Type I error rate close to the nominal level. The power of the proposed test is also compared with the likelihood ratio test in the simulations. An application on data from an orthodontic study is presented and discussed.     

Statistical Inference in Partially Linear Varying-Coefficient Models with Missing Responses at Random

This article considers statistical inference for partially linear varying-coefficient models when the responses are missing at random. We propose a profile least-squares estimator for the parametric component with complete-case data and show that the resulting estimator is asymptotically normal. To avoid to estimate the asymptotic covariance in establishing confidence region of the parametric component with the normal-approximation method, we define an empirical likelihood based statistic and show that its limiting distribution is chi-squared distribution. Then, the confidence regions of the parametric component with asymptotically correct coverage probabilities can be constructed by the result. To check the validity of the linear constraints on the parametric component, we construct a modified generalized likelihood ratio test statistic and demonstrate that it follows asymptotically chi-squared distribution under the null hypothesis. Then, we extend the generalized likelihood ratio technique to the context of missing data. Finally, some simulations are conducted to illustrate the proposed methods.