A cubic smoothing spline based lack of fit test for nonlinear regression models

The use of a statistic based on cubic spline smoothing is considered for testing nonlinear regression models for lack of fit. The statistic is defined to be the Euclidean squared norm of the smoothed residual vector obtained from fitting the nonlinear model, The asymptotic distribution of the statistic is derived under suitable smooth local alternatives and a numerical example is presented.     

Asymptotic properties of the score test for varying dispersion in exponential family nonlinear models

Wei et al. [B.C. Wei, J.Q. Shi, W.K. Fung, and Y.Q. Hu, Testing for varying dispersion in exponential family nonlinear models, Ann. Inst. Statist. Math. 50 (1998), pp. 277–294.] developed the score diagnostics for varying dispersion in exponential family nonlinear models, such as the normal, inverse Gaussian, and gamma models, and investigated the powers of these tests through Monte Carlo simulations. In this paper, the asymptotic behaviours, including asymptotic chi-square and approximate powers under local alternatives of the score tests, are studied and examined by Monte Carlo simulations. The methods to estimate local powers of the score tests are illustrated with Grass yield data [P. McCullagh, and J.A. Nelder, Generalized Linear Models, Chapman and Hall, London (1989).].     

On choosing between two nonlinear models estimated robustly. Some Monte Carlo evidence

The article considers the problem of choosing between two (possibly) nonlinear models that have been fitted to the same data using M-estimation methods. An asymptotically normally distributed test statistic that takes into account the fact that the models are fitted robustly is given. The new procedure is compared with other test statistics using a Monte Carlo study. We found that the presence of a competitive model either in the null or the alternative hipothesis affects the distributional properties of the tests, and that in the case that the data contains outlying observations the new procedure had a significantly higher power than the rest of the tests.     

Robust test for independence in high dimensions

This article develops a new test based on Spearman’s rank correlation coefficients for total independence in high dimensions. The test is robust to the non normality and heavy tails of the data, which is a merit that is not shared by the existing tests in literature. Simulation results suggest that the new test performs well under several typical null and alternative hypotheses. Besides, we employ a real data set to illustrate the use of the new test.     

Testing parametric models in linear‐directional regression

This paper presents a goodness‐of‐fit test for parametric regression models with scalar response and directional predictor, that is, a vector on a sphere of arbitrary dimension. The testing procedure is based on the weighted squared distance between a smooth and a parametric regression estimator, where the smooth regression estimator is obtained by a projected local approach. Asymptotic behaviour of the test statistic under the null hypothesis and local alternatives is provided, jointly with a consistent bootstrap algorithm for application in practice. A simulation study illustrates the performance of the test in finite samples. The procedure is applied to test a linear model in text mining.     

Implications of uncorrelatedness and independence in linear regression models

The assumption of normally distributed disturbances in the linear regression model implies that the disturbances are both uncorrelated and independent. Recently however, attention has focussed on possibly nonnonnally distributed disturbances, and in this case a distinction needs to be made between only uncorrelated disturbances and independently distributed disturbances. In this paper, general specification errors associated with misspecifying uncorrelatedness and independence for student - t distributed disturbances is examined. This class of distributions is a reasonable way of modelling tails that are fatter than those of the normal distribution which has applications to the modelling of series such as prices in financial and commodity markets, growth -curve models and astronomical data. Specification tests which test for only uncorrelatedness versus independence are also discussed.     

Contributions to-asymptotic theory in regression models with linear covariance structure 1

Asymptotic results are .presented- for estimating the parameters in variance-components models both with linear and with nonlinear regression functions.     

On a test of independence for contingency tables

Using the concept of distributional distance, a test statistic is proposed FOR the hypothesis of independence in multidimensional contingency tables. A Monte Carlo Study is done to empirically compare the power of the proposed test to the Pearson x² and the likelihood ratio test- Further, the nonnull distribution under various spike alternatives is tabulated     

Optimal design for nonlinear regression models

Two results for D _θ-optimal designs for nonlinear regression models are shown to follow directly from approximate design theory. The first result considered is one concerning the replication of exact designs with minimum support, first established by Atkinson and Hunter and by M.J. Box in 1968, while the second pertains to a heteroscedastic model introduced by Velilla and Llosa in 1992. An illustrative example is provided.     

A residual-based test for autocorrelation in quantile regression models

Quantile regression (QR) models have been increasingly employed in many applied areas in economics. At the early stage, applications in the QR literature have usually used cross-sectional data, but the recent development has seen an increase in the use of QR in both time-series and panel data sets. However, testing for possible autocorrelation, especially in the context of time-series models, has received little attention. As a rule of thumb, one might attempt to apply the usual Breusch–Godfrey LM test to the residuals of a baseline QR. In this paper, we demonstrate analytically and by Monte Carlo simulations that such an application of the LM test can result in potentially large size distortions, especially in either low or high quantiles. We then propose a correct test (named the QF test) for autocorrelation in QR models, which does not suffer from size distortion. Monte Carlo simulations demonstrate that the proposed test performs fairly well in finite samples, across either different quantiles or different underlying error distributions.     

A parametric bootstrap approach for two-way error component regression models

In this article, the two-way error component regression model is considered. For the nonhomogenous linear hypothesis testing of regression coefficients, a parametric bootstrap (PB) approach is proposed. Simulation results indicate that the PB test, regardless of the sample sizes, maintains the Type I error rates very well and outperforms the existing generalized variable test, which may far exceed the intended significance level when the sample sizes are small or moderate. Real data examples illustrate the proposed approach work quite satisfactorily.     

A comparison of two exact confidence regions for partially nonlinear regression models

Exact confidence regions for all the parameters in nonlinear regression models can be obtained by comparing the lengths of projections of the error vector into orthogonal subspaces of the sample space. In certain partially nonlinear models an alternative exact region is obtained by replacing the linear parameters by their conditional estimates in the projection matrices. An ellipsoidal approximation to the alternative region is obtained in terms of the tangent-plane coordinates, similar to one previously obtained for the more usual region. This ellipsoid can be converted to an approximate region for the original parameters and can be used to compare the two types of exact confidence regions.     

Influence analysis of additive mixed-effects nonlinear regression models via EM algorithm

This paper presents a unified method for influence analysis to deal with random effects appeared in additive nonlinear regression models for repeated measurement data. The basic idea is to apply the Q-function, the conditional expectation of the complete-data log-likelihood function obtained from EM algorithm, instead of the observed-data log-likelihood function as used in standard influence analysis. Diagnostic measures are derived based on the case-deletion approach and the local influence approach. Two real examples and a simulation study are examined to illustrate our methodology.     

Some blum-kiefer-rosenblatt type tests for the joint independence of variables

Testing the joint independence of variables has long been an interesting issue in statistical inferences. Blum, Kiefer and Rosenblatt (1961) suggested a test based on a sample distribution function. To overcome the sparseness of data points in high-dimensional space and deal with general cases, we in this paper suggest several extended versions of B-K-R tests via projection pursuit. Bootstrap method is applied to determine the critical values and for computational reason, an approximation derived by Number-theoretic method, for the bootstrap statistics is suggested. Several simulation experiments are performed and a real-life example is investigated.     

Simultaneous confidence bands for nonlinear regression models

The maximization and minimization procedure for constructing confidence bands about general regression models is explained. Then, using an existing confidence region about the parameters of a nonlinear regression model and the maximization and minimization procedure, a generally conservative simultaneous confidence band is constructed about the model. Two examples are given, and some problems with the procedure are discussed     

A nonparametric specification test for the volatility functions of diffusion processes

This paper develops a new test for the parametric volatility function of a diffusion model based on nonparametric estimation techniques. The proposed test imposes no restriction on the functional form of the drift function and has an asymptotically standard normal distribution under the null hypothesis of correct specification. It is consistent against any fixed alternatives and has nontrivial asymptotic power against a class of local alternatives with proper rates. Monte Carlo simulations show that the test performs well in finite samples and generally has better power performance than the nonparametric test of Li (2007 Li, F. (2007). Testing the parametric specification of the diffusion function in a diffusion process. Econometric Theory 23(2):221–250.[Crossref], [Web of Science ®] , [Google Scholar]) and the stochastic process-based tests of Dette and Podolskij (2008 Dette, H., Podolskij, M. (2008). Testing the parametric form of the volatility in continuous time diffusion models–a stochastic process approach. Journal of Econometrics 143(1):56–73.[Crossref], [Web of Science ®] , [Google Scholar]). When applying the test to high frequency data of EUR/USD exchange rate, the empirical results show that the commonly used volatility functions fit more poorly when the data frequency becomes higher, and the general volatility functions fit relatively better than the constant volatility function.     

A consistent test for heteroscedasticity in semi-parametric regression with nonparametric variance function based on the kernel method

It is important to detect the variance heterogeneity in regression models. Heteroscedasticity tests have been well studied in parametric and nonparametric regression models. This paper presents a consistent test for heteroscedasticity for nonlinear semi-parametric regression models with nonparametric variance function based on the kernel method. The properties of the test are investigated through Monte Carlo simulations. The test methods are illustrated with a real example.     

A robust class of homoscedastic nonlinear regression models

In this paper, we examine a nonlinear regression (NLR) model with homoscedastic errors which follows a flexible class of two-piece distributions based on the scale mixtures of normal (TP-SMN) family. The objective of using this family is to develop a robust NLR model. The TP-SMN is a rich class of distributions that covers symmetric/asymmetric and lightly/heavy-tailed distributions and is an alternative family to the well-known scale mixtures of skew-normal (SMSN) family studied by Branco and Dey [35]. A key feature of this study is using a new suitable hierarchical representation of the family to obtain maximum-likelihood estimates of model parameters via an EM-type algorithm. The performances of the proposed robust model are demonstrated using simulated and some natural real datasets and also compared to other well-known NLR models.     

A modified chi-square test of independence against a class of ordered alternatives in an r x c contingency table

A modified chi-square test statistic is constructed for testing the hypothesis of independence in a two-way contingency table against a class of ordered alternatives defined in terms of pooled cross-product ratios. The test procedure can also be used to test for positive quadrant dependence in a two-way contingency table. The asymptotic distribution of the test statistic under the null hypothesis is obtained. Some power comparisons with known test procedures are given. A numerical example is given to illustrate the use of this test.     

A fiducial p-value approach for comparing heteroscedastic regression models

To study the equality of regression coefficients in several heteroscedastic regression models, we propose a fiducial-based test, and theoretically examine the frequency property of the proposed test. We numerically compare the performance of the proposed approach with the parametric bootstrap (PB) approach. Simulation results indicate that the fiducial approach controls the Type I error rates satisfactorily regardless of the number of regression models and sample sizes, whereas the PB approach tends to be a little of liberal in some scenarios. Finally, the proposed approach is applied to an analysis of a real dataset for illustration.