A test for the linearity of the nonparametric part of a semiparametric logistic regression model

A semiparametric logistic regression model is proposed in which its nonparametric component is approximated with fixed-knot cubic B-splines. To assess the linearity of the nonparametric component, we construct a penalized likelihood ratio test statistic. When the number of knots is fixed, the null distribution of the test statistic is shown to be asymptotically the distribution of a linear combination of independent chi-squared random variables, each with one degree of freedom. We set the asymptotic null expectation of this test statistic equal to a value to determine the smoothing parameter value. Monte Carlo experiments are conducted to investigate the performance of the proposed test. Its practical use is illustrated with a real-life example.     

Resampling Methods for Testing a Semiparametric Random Censorship Model

This paper presents a goodness-of-fit test for a semiparametric random censorship model proposed by Dikta (1998 ). The test statistic is derived from a model-based process which is asymptotically Gaussian. In addition to test consistency, the proposed test can detect local alternatives distinct n ^-1/2 from the null hypothesis. Due to the intractability of the asymptotic null distribution of the test statistic, we turn to two resampling approximations. We first use the well-known bootstrap method to approximate critical values of the test. We then introduce a so-called random symmetrization method for carrying out the test. Both methods perform very well with a sample of moderate size. A simulation study shows that the latter possesses better empirical powers and sizes for small samples.     

Frequency Domain Tests of Semiparametric Hypotheses for Locally Stationary Processes

Abstract.  Many time series in applied sciences obey a time-varying spectral structure. In this article, we focus on locally stationary processes and develop tests of the hypothesis that the time-varying spectral density has a semiparametric structure, including the interesting case of a time-varying autoregressive moving-average (tvARMA) model. The test introduced is based on a L ₂ -distance measure of a kernel smoothed version of the local periodogram rescaled by the time-varying spectral density of the estimated semiparametric model. The asymptotic distribution of the test statistic under the null hypothesis is derived. As an interesting special case, we focus on the problem of testing for the presence of a tvAR model. A semiparametric bootstrap procedure to approximate more accurately the distribution of the test statistic under the null hypothesis is proposed. Some simulations illustrate the behaviour of our testing methodology in finite sample situations.     

Testing for a unit root in a stationary ESTAR process

This article develops a statistic for testing the null of a linear unit root process against the alternative of a stationary exponential smooth transition autoregressive model. The asymptotic distribution of the test is shown to be nonstandard but nuisance parameter-free and hence critical values are obtained by simulations. Simulations show that the proposed statistic has considerable power under various data generating scenarios. Applications to real exchange rates also illustrate the ability of our test to reject null of unit root when some of the alternative tests do not.     

On empirical likelihood for semiparametric two-sample density ratio models

We consider estimation and test problems for some semiparametric two-sample density ratio models. The profile empirical likelihood (EL) poses an irregularity problem under the null hypothesis that the laws of the two samples are equal. We show that a dual form of the profile EL is well defined even under the null hypothesis. A statistical test, based on the dual form of the EL ratio statistic (ELRS), is then proposed. We give an interpretation for the dual form of the ELRS through φ$\phi$-divergences and duality techniques. The asymptotic properties of the test statistic are presented both under the null and the alternative hypotheses, and approximation of the power function of the test is deduced.     

Change-point analysis with bathtub shape for the exponential distribution

Likelihood ratio type test statistic and Schwarz information criterion statistics are proposed for detecting possible bathtub-shaped changes in the parameter in a sequence of exponential distributions. The asymptotic distribution of likelihood ratio type statistic under the null hypothesis and the testing procedure based on Schwarz information criterion are derived. Numerical critical values and powers of two methods are tabulated for certain selected values of the parameters. The tests are applied to detect the change points for the predator data and Stanford heart transplant data.     

Semi-parametric hybrid empirical likelihood inference for two-sample comparison with censored data

Two-sample comparison problems are often encountered in practical projects and have widely been studied in literature. Owing to practical demands, the research for this topic under special settings such as a semiparametric framework have also attracted great attentions. Zhou and Liang (Biometrika 92:271–282, 2005) proposed an empirical likelihood-based semi-parametric inference for the comparison of treatment effects in a two-sample problem with censored data. However, their approach is actually a pseudo-empirical likelihood and the method may not be fully efficient. In this study, we develop a new empirical likelihood-based inference under more general framework by using the hazard formulation of censored data for two sample semi-parametric hybrid models. We demonstrate that our empirical likelihood statistic converges to a standard chi-squared distribution under the null hypothesis. We further illustrate the use of the proposed test by testing the ROC curve with censored data, among others. Numerical performance of the proposed method is also examined.     

On a life testing problem involving the two parameter exponential distribution

For a hypothesis testing problem involving the location and scale parameters of an exponential distribution, Perng (1977) proposed a test procedure based on the first r out of n observed failure times. In this paper the likelihood ratio test is determined, critical values are provided and the asymptotic null distribution is determined. An alternate test based on an F statistic is also proposed and the critical regions and power functions of the procedures are compared.     

Linear regression analysis with inequality constraints on the regression parameters via empirical likelihood

An empirical likelihood ratio test is developed for testing for or against inequality constraints on regression parameters in linear regression analysis. The proposed approach imposes no parametric model nor identically distributing assumption on the random errors. The asymptotic distribution of the proposed test statistic under null hypothesis is shown to be of chi-bar-squared type. The asymptotic power under contiguous alternatives is also briefly discussed. Moreover, an adjusted empirical likelihood method is adopted to improve the small sample size behaviour of the proposed test. Several simulation studies are carried out to assess the finite sample performance of the proposed tests. The results reveal that the proposed tests could be valuable for improving inference efficiency. A real-life example is discussed to illustrate the theoretical results.     

Testing for multiple upper and lower outliers in an exponential sample

Due to wide applicability and simplicity, the exponential distribution is the most commonly used distribution in reliability engineering and other life testing experiments. In this paper a test statistic for testing upper and lower outliers simultaneously in an exponential sample is proposed. However, the distribution of test statistic under the alternative is rather intricate, the null distribution is derived and critical values are obtained. A simulation study is also carried out to compare the performance of test and is found that the test based on this statistic is more powerful than the other two selected tests.     

Negative exponential disparity based family of goodness-of-fit tests for multinomial models

This paper investigates a new family of goodness-of-fit tests based on the negative exponential disparities. This family includes the popular Pearson's chi-square as a member and is a subclass of the general class of disparity tests (Basu and Sarkar, 1994) which also contains the family of power divergence statistics. Pitman efficiency and finite sample power comparisons between different members of this new family are made. Three asymptotic approximations of the exact null distributions of the negative exponential disparity famiiy of tests are discussed. Some numerical results on the small sample perfomance of this family of tests are presented for the symmetric null hypothesis. It is shown that the negative exponential disparity famiiy, Like the power divergence family, produces a new goodness-of-fit test statistic that can be a very attractive alternative to the Pearson's chi-square. Some numerical results suggest that, application of this test statistic, as an alternative to Pearson's chi-square, could be preferable to the I ^2/3 statistic of Cressie and Read (1984) under the use of chi-square critical values.     

Testing Serial Correlation in Partially Linear Additive Errors-in-variables Models

This article considers testing serial correlation in partially linear additive errors-in-variables model. Based on the empirical likelihood based approach, a test statistic was proposed, and it was shown to follow asymptotically a chi-square distribution under the null hypothesis of no serial correlation. Finally, some simulation studies are conducted to illustrate the performance of the proposed method.     

Testing for a finite mixture model with two components

Summary.  We consider a finite mixture model with k components and a kernel distribution from a general one-parameter family. The problem of testing the hypothesis k =2 versus k 3 is studied. There has been no general statistical testing procedure for this problem. We propose a modified likelihood ratio statistic where under the null and the alternative hypotheses the estimates of the parameters are obtained from a modified likelihood function. It is shown that estimators of the support points are consistent. The asymptotic null distribution of the modified likelihood ratio test proposed is derived and found to be relatively simple and easily applied. Simulation studies for the asymptotic modified likelihood ratio test based on finite mixture models with normal, binomial and Poisson kernels suggest that the test proposed performs well. Simulation studies are also conducted for a bootstrap method with normal kernels. An example involving foetal movement data from a medical study illustrates the testing procedure.     

Goodness-of-fit test for exponentiality based on spacings for general progressive Type-II censored data

There have been numerous tests proposed to determine whether or not the exponential model is suitable for a given data set. In this article, we propose a new test statistic based on spacings to test whether the general progressive Type-II censored samples are from exponential distribution. The null distribution of the test statistic is discussed and it could be approximated by the standard normal distribution. Meanwhile, we propose an approximate method for calculating the expectation and variance of samples under null hypothesis and corresponding power function is also given. Then, a simulation study is conducted. We calculate the approximation of the power based on normality and compare the results with those obtained by Monte Carlo simulation under different alternatives with distinct types of hazard function. Results of simulation study disclose that the power properties of this statistic by using Monte Carlo simulation are better for the alternatives with monotone increasing hazard function, and otherwise, normal approximation simulation results are relatively better. Finally, two illustrative examples are presented.     

Empirical Likelihood Ratio Test for a Change-Point in Linear Regression Model

A nonparametric method based on the empirical likelihood is proposed to detect the change-point in the coefficient of linear regression models. The empirical likelihood ratio test statistic is proved to have the same asymptotic null distribution as that with classical parametric likelihood. Under some mild conditions, the maximum empirical likelihood change-point estimator is also shown to be consistent. The simulation results show the sensitivity and robustness of the proposed approach. The method is applied to some real datasets to illustrate the effectiveness.     

On Variance Components in Semiparametric Mixed Models for Longitudinal Data

Abstract. First, to test the existence of random effects in semiparametric mixed models (SMMs) under only moment conditions on random effects and errors, we propose a very simple and easily implemented non‐parametric test based on a difference between two estimators of the error variance. One test is consistent only under the null and the other can be so under both the null and alternatives. Instead of erroneously solving the non‐standard two‐sided testing problem, as in most papers in the literature, we solve it correctly and prove that the asymptotic distribution of our test statistic is standard normal. This avoids Monte Carlo approximations to obtain p ‐values, as is needed for many existing methods, and the test can detect local alternatives approaching the null at rates up to root n. Second, as the higher moments of the error are necessarily estimated because the standardizing constant involves these quantities, we propose a general method to conveniently estimate any moments of the error. Finally, a simulation study and a real data analysis are conducted to investigate the properties of our procedures.     

Empirical likelihood ratio test for a mean change point model with a linear trend followed by an abrupt change

In this paper, a change point model with the mean being constant up to some unknown point, and increasing linearly to another unknown point, then dropping back to the original level is studied. A nonparametric method based on the empirical likelihood test is proposed to detect and estimate the locations of change points. Under some mild conditions, the asymptotic null distribution of an empirical likelihood ratio test statistic is shown to have the extreme distribution. The consistency of the test is also proved. Simulations of the powers of the test indicate that it performs well under different assumptions of the data distribution. The test is applied to the aircraft arrival time data set and the Stanford heart transplant data set.     

Testing the equality of correlation matrices

We present results that extend an existing test of equality of correlation matrices. A new test statistic is proposed and is shown to be asymptotically distributed as a linear combination of independent x ² random variables. This new formulation allows us to find the power of the existing test and our extensions by deriving the distribution under the alternative using a linear combination of independent non-central x ² random variables. We also investigate the null and the alternative distribution of two related statistics. The first one is a quadratic form in deviations from a control group with which the remaining k-1 groups are to be compared. The second test is designed for comparing adjacent groups. Several approximations for the null and the alternative distribution are considered and two illustrative examples are provided.     

ON TESTING THE GOODNESS-OF-FIT OF NONLINEAR HETEROSCEDASTIC REGRESSION MODELS

For testing the adequacy of a parametric model in regression, various test statistics can be constructed on the basis of a marked empirical process of residuals. By using a discretized version of the decomposition of the corresponding Gaussian limiting process into its principal components, we obtain a test statistic with an asymptotic chi-squared distribution under the null hypothesis. We investigate the consistency of this test statistic and of the estimators needed to compute it. Numerical experiments indicate that the distributional approximations already work for small to moderate sample sizes and reveal that the test has good power properties against a variety of alternatives. The test has a simple implementation. We present an application to a real-data example for testing the adequacy of a possible heteroscedastic exponential model.     

Using entropy distance measures for testing odds ratio parameters under logistic regression models based on case–control data

We propose two retrospective test statistics for testing the vector of odds ratio parameters under the logistic regression model based on case–control data by exploiting the density ratio structure under a two-sample semiparametric model, which is equivalent to the assumed logistic regression model. The proposed test statistics are based on Kullback–Leibler entropy distance and are particularly relevant to the case–control sampling plan. These two test statistics have identical asymptotic chi-squared distributions under the null hypothesis and identical asymptotic noncentral chi-squared distributions under local alternatives to the null hypothesis. Moreover, the proposed test statistics require computation of the maximum semiparametric likelihood estimators of the underlying parameters, but are otherwise easily computed. We present some results on simulation and on the analysis of two real data sets.