A goodness-of-fit test for the stratified proportional hazards model for survival data

Abstract

Goodness-of-fit testing is addressed in the stratified proportional hazards model for survival data. A test statistic based on within-strata cumulative sums of martingale residuals over covariates is proposed and its asymptotic distribution is derived under the null hypothesis of model adequacy. A Monte Carlo procedure is proposed to approximate the critical value of the test. Simulation studies are conducted to examine finite-sample performance of the proposed statistic.     

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.     

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.     

On Testing Changes in Autoregressive Parameters of a VAR Model

The article deals with the problem of testing a change in autoregressive matrices of the p-th order vector autoregressive process, VAR(p). The proposed test statistics are based on the likelihood ratio concept and are studied under the null hypothesis of no change in parameters. Their asymptotic behavior is derived under minimal moment assumptions in both cases where the time point of possible change is known a priori and is undefined. The Gumbel-type approximation of the test statistic is also developed, which previous papers on VAR(p) models do not cover.     

Testing transition probability matrix of a multi-state model with censored data

In this paper, we develop procedures to test hypotheses concerning transition probability matrices arising from certain nonhomogeneous Markov processes. It is assumed that the data consist of sample paths, some of which are observed until a certain terminal state, and the other paths are censored. Problems of this type arise in the context of multi-state models relevant to Health Related Quality of Life (HRQoL) and Competing Risks. The test statistic is based on the estimator for the associated intensity matrix. We show that the asymptotic null distribution of the proposed statistic is Gaussian, and demonstrate how the procedure can be adopted for HRQoL studies and competing risks model using real data sets. Finally, we establish that the test statistic for the HRQoL has greatest local asymptotic power against a sequence of proportional hazards alternatives converging to the null hypothesis.     

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.     

Nonlinear regression models with general distortion measurement errors

This paper considers nonlinear regression models when neither the response variable nor the covariates can be directly observed, but are measured with both multiplicative and additive distortion measurement errors. We propose conditional variance and conditional mean calibration estimation methods for the unobserved variables, then a nonlinear least squares estimator is proposed. For the hypothesis testing of parameter, a restricted estimator under the null hypothesis and a test statistic are proposed. The asymptotic properties for the estimator and test statistic are established. Lastly, a residual-based empirical process test statistic marked by proper functions of the regressors is proposed for the model checking problem. We further suggest a bootstrap procedure to calculate critical values. Simulation studies demonstrate the performance of the proposed procedure and a real example is analysed to illustrate its practical usage.     

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.     

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.     

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.     

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.     

A rank test for bivariate location and scale problem for elliptically symmetric populations ∗

A nonparametric rank test for the two-sample location and scale bivariate problem is proposed. The asymptotic distribution of the statistic of the test is derived under the null hypothesis and under a class of contiguous alternatives. The asymptotic relative efficiency is given and a simulation study gives the performance of the test and some competitors.     

A Non‐parametric ANOVA‐type Test for Regression Curves Based on Characteristic Functions

This article studies a new procedure to test for the equality of k regression curves in a fully non‐parametric context. The test is based on the comparison of empirical estimators of the characteristic functions of the regression residuals in each population. The asymptotic behaviour of the test statistic is studied in detail. It is shown that under the null hypothesis, the distribution of the test statistic converges to a finite combination of independent chi‐squared random variables with one degree of freedom. The coefficients in this linear combination can be consistently estimated. The proposed test is able to detect contiguous alternatives converging to the null at the rate n ^{? 1 ∕ 2}. The practical performance of the test based on the asymptotic null distribution is investigated by means of simulations.     

Empirical phi-divergence test statistics for testing simple and composite null hypotheses

The main purpose of this paper is to introduce first a new family of empirical test statistics for testing a simple null hypothesis when the vector of parameters of interest is defined through a specific set of unbiased estimating functions. This family of test statistics is based on a distance between two probability vectors, with the first probability vector obtained by maximizing the empirical likelihood (EL) on the vector of parameters, and the second vector defined from the fixed vector of parameters under the simple null hypothesis. The distance considered for this purpose is the phi-divergence measure. The asymptotic distribution is then derived for this family of test statistics. The proposed methodology is illustrated through the well-known data of Newcomb's measurements on the passage time for light. A simulation study is carried out to compare its performance with that of the EL ratio test when confidence intervals are constructed based on the respective statistics for small sample sizes. The results suggest that the ‘empirical modified likelihood ratio test statistic’ provides a competitive alternative to the EL ratio test statistic, and is also more robust than the EL ratio test statistic in the presence of contamination in the data. Finally, we propose empirical phi-divergence test statistics for testing a composite null hypothesis and present some asymptotic as well as simulation results for evaluating the performance of these test procedures.     

A test for bivariate symmetry based on the empirical distribution function

Hollander (1970) proposed a conditionally distribution-free test of bivariate symmetry based on the empirical distribution function. In this paper Hollander’s test statistic is examined In greater detail: in particular; its conditional asymptotic distribution is derived under the null hypothesis as well as under a sequence of local alternatives. Percentage points of the asymptotic distribution are presented; a power comparison between Hollander’s statistic and the likelihood ratio criterion in testing a variant of the sphericity hypothesis in multivariate analysis is made.     

A Model Specification Test For GARCH(1,1) Processes

We provide a consistent specification test for generalized autoregressive conditional heteroscedastic (GARCH (1,1)) models based on a test statistic of Cramér‐von Mises type. Because the limit distribution of the test statistic under the null hypothesis depends on unknown quantities in a complicated manner, we propose a model‐based (semiparametric) bootstrap method to approximate critical values of the test and to verify its asymptotic validity. Finally, we illuminate the finite sample behaviour of the test by some simulations.     

Book Review: Introducing Monte Carlo Methods with R

In this article we propose a nonparametric test for poolability in large dimensional semiparametric panel data models with cross-section dependence based on the sieve estimation technique. To construct the test statistic, we only need to estimate the model under the alternative. We establish the asymptotic normal distributions of our test statistic under the null hypothesis of poolability and a sequence of local alternatives, and prove the consistency of our test. We also suggest a bootstrap method as an alternative way to obtain the critical values. A small set of Monte Carlo simulations indicate the test performs reasonably well in finite samples.     

Testing for linearity in Markov switching models: a bootstrap approach

Testing for linearity in the context of Markov switching models is complicated because standard regularity conditions for likelihood based inference are violated. In particular, under the null hypothesis of linearity, some parameters are not identified and scores are identically zero. Thus, the asymptotic distribution of the relevant test statistic does not possess the standard χ ²-distribution. A bootstrap resampling scheme to approximate the distribution of the relevant test statistic under the null of linearity is proposed. The procedure is relatively easy to program and computation requirements are reasonable. The performance of the bootstrap-based test is investigated by means of Monte Carlo simulations. Results show that this test works well and outperforms the Hansen test and the Carrasco et al. test.     

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.     

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.