Bootstrap-assisted tests of symmetry for dependent data*

The paper considers the problem of testing for symmetry (about an unknown centre) of the marginal distribution of a strictly stationary and weakly dependent stochastic process. The possibility of using the autoregressive sieve bootstrap and stationary bootstrap procedures to obtain critical values and P-values for symmetry tests is explored. Bootstrap-assisted tests for symmetry are straightforward to implement and require no prior estimation of asymptotic variances. The small-sample properties of a wide variety of tests are investigated using Monte Carlo experiments. A bootstrap-assisted version of the triples test is found to have the best overall performance.     

The analysis of multicentre clinical trials when there is heterogeneity between centres

Two-treatment multicentre clinical trials are very common in practice. In cases where a non-parametric analysis is appropriate, a rank-sum test for grouped data called the van Elteren test can be applied. As an alternative approach, one may apply a combination test such as Fisher's combination test or the inverse normal combination test (also called Liptak's method) in order to combine centre-specific P-values. If there are no ties and no differences between centres with regard to the groups’ sample sizes, the inverse normal combination test using centre-specific Wilcoxon rank-sum tests is equivalent to the van Elteren test. In this paper, the van Elteren test is compared with Fisher's combination test based on Wilcoxon rank-sum tests. Data from two multicentre trials as well as simulated data indicate that Fisher's combination of P-values is more powerful than the van Elteren test in realistic scenarios, i.e. when there are large differences between the centres’ P-values, some quantitative interaction between treatment and centre, and/or heterogeneity in variability. The combination approach opens the possibility of using statistics other than the rank sum, and it is also a suitable method for more complicated designs, e.g. when covariates such as age or gender are included in the analysis.     

Empirical Bayes testing for guarantee lifetime: Non identical components case

This article considers an empirical Bayes testing problem for the guarantee lifetime in the two-parameter exponential distributions with non identical components. We study a method of constructing empirical Bayes tests under a class of unknown prior distributions for the sequence of the component testing problems. The asymptotic optimality of the sequence of empirical Bayes tests is studied. Under certain regularity conditions on the prior distributions, it is shown that the sequence of the constructed empirical Bayes tests is asymptotically optimal, and the associated sequence of regrets converges to zero at a rate O(n^{? 1 + 1/[2(r + α) + 1]}) for some integer r ? 0 and 0 ? α ? 1 depending on the unknown prior distributions, where n is the number of past data available when the (n + 1)st component testing problem is considered.     

BAHADUR SLOPES FOR COMPARISONS OF TESTS BASED ON RESAMPLING

Let X₁,…, X_n be random variables symmetric about θ from a common unknown distribution F_θ(x) =F(x–θ). To test the null hypothesis H₀:θ= 0 against the alternative H₁:θ > 0, permutation tests can be used at the cost of computational difficulties. This paper investigates alternative tests that are computationally simpler, notably some bootstrap tests which are compared with permutation tests. Of these the symmetrical bootstrap-f test competes very favourably with the permutation test in terms of Bahadur asymptotic efficiency, so it is a very attractive alternative.     

Some estimators and tests for accelerated hazards model using weighted cumulative hazard difference

For a censored two-sample problem, Chen and Wang [Y.Q. Chen and M.-C. Wang, Analysis of accelerated hazards models, J. Am. Statist. Assoc. 95 (2000), pp. 608–618] introduced the accelerated hazards model. The scale-change parameter in this model characterizes the association of two groups. However, its estimator involves the unknown density in the asymptotic variance. Thus, to make an inference on the parameter, numerically intensive methods are needed. The goal of this article is to propose a simple estimation method in which estimators are asymptotically normal with a density-free asymptotic variance. Some lack-of-fit tests are also obtained from this. These tests are related to Gill–Schumacher type tests [R.D. Gill and M. Schumacher, A simple test of the proportional hazards assumption, Biometrika 74 (1987), pp. 289–300] in which the estimating functions are evaluated at two different weight functions yielding two estimators that are close to each other. Numerical studies show that for some weight functions, the estimators and tests perform well. The proposed procedures are illustrated in two applications.     

An adaptive two-sample location-scale test of lepage type for symmetric distributions

For the two-sample location and scale problem we propose an adaptive test which is based on so called Lepage type tests. The well known test of Lepage (1971) is a combination of the Wilcoxon test for location alternatives and the Ansari-Bradley test for scale alternatives and it behaves well for symmetric and medium-tailed distributions. For the cae of short-, medium- and long-tailed distributions we replace the Wilcoxon test and the .Ansari-Bradley test by suitable other two-sample tests for location and scale, respectively, in oder to get higher power than the classical Lepage test for such distribotions. These tests here are called Lepage type tests. in practice, however, we generally have no clear idea about the distribution having generated our data. Thus, an adaptive test should be applied which takes the the given data set inio consideration. The proposed adaptive test is based on the concept of Hogg (1974), i.e., first, to classify the unknown symmetric distribution function with respect to a measure for tailweight and second, to apply an appropriate Lepage type test for this classified type of distribution. We compare the adaptive test with the three Lepage type tests in the adaptive scheme and with the classical Lepage test as well as with other parametric and nonparametric tests. The power comparison is carried out via Monte Carlo simulation. It is shown that the adaptive test is the best one for the broad class of distributions considered.     

Nonparametric Phase-II monitoring for detecting monotone trend based on inverse sampling

Recently, Mukherjee and Bandyopadhyay (J Stat Plan Inference, 2011, doi:10.1016/j.jspi.2011.02.017) introduced some partially sequential tests for detecting liner trend among the incoming series of observations when a training sample is available a-priori. Their work is very useful in econometric or environmental monitoring under certain situations. The present work is intended for generalization of their tests for any monotone trend. We develop two nonparametric tests for the identity of some unknown univariate continuous distribution functions against monotone or unidirectional trend in location. One of these two tests is based on usual ranks and the other is based on sequential ranks. These are typical nonparametric tests for monitoring structural changes. Performance of the two tests are compared using asymptotic studies as well as through some numerical results based on Monte-Carlo simulations. An illustration is offered using a real data on monthly production of certain beverage.     

Goodness‐of‐fit tests for the hyperbolic distribution

The authors give tests of fit for the hyperbolic distribution, based on the Cramér‐von Mises statistic W². They consider the general case with four parameters unknown, and some specific cases where one or two parameters are fixed. They give two examples using stock price data.     

The Use of Posterior Predictive P-Values in Testing Goodness-of-Fit

In this article, we introduce two goodness-of-fit tests for testing normality through the concept of the posterior predictive p-value. The discrepancy variables selected are the Kolmogorov-Smirnov (KS) and Berk-Jones (BJ) statistics and the prior chosen is Jeffreys’ prior. The constructed posterior predictive p-values are shown to be distributed independently of the unknown parameters under the null hypothesis, thus they can be taken as the test statistics. It emerges from the simulation that the new tests are more powerful than the corresponding classical tests against most of the alternatives concerned.     

Testing for spherical symmetry via the empirical characteristic function

Kolmogorov–Smirnov-type and Cramér–von Mises-type goodness-of-fit tests are proposed for the null hypothesis that the distribution of a random vector X is spherically symmetric. The test statistics utilize the fact that X has a spherical symmetric distribution if, and only if, the characteristic function of X is constant over surfaces of spheres centred at the origin. Both tests come in convenient forms that are straightforwardly applicable with the computer. The asymptotic null distribution of the test statistics as well as the consistency of the tests is investigated under general conditions. Since both the finite sample and the asymptotic null distribution depend on the unknown distribution of the Euclidean norm of X, a conditional Monte Carlo procedure is used to actually carry out the tests. Results on the behaviour of the test in finite-samples are included along with a real-data example.     

Bootstrap methods for single structural change tests: power versus corrected size and empirical illustration

This paper assesses the performance of tests for a single structural change at unknown date when regressors are stationary, trending and when they have a break in mean. Size and power of the test procedures are compared in a simulation setup particularly aimed at autoregressive models using their limiting distribution and some bootstrap approximations. The comparisons are performed using graphical methods, namely P value discrepancy plots and size–power curves. The simulation study gives some interesting insights to the test procedures. Indeed, it documents that tests based on the conventional asymptotic distribution are oversized in small samples. The size correction is achieved by some bootstrap methods which appear to possess reasonable size properties. For the power study, the proposed bootstrap method improves on the asymptotic approximations of some tests for heteroskedastic regression errors especially when there is a mean-shift in the regressors. This result has not been found for the case of i.i.d. errors where the bootstrap tests have the same power properties as the tests based on the asymptotic approximations. We finally study the relationship between two monthly US interest rates. The results show that such relationship has been altered by a regime-shift located in May 1981.     

Adaptive Test for Testing the Difference in Survival Distributions

An adaptive test is proposed for the problem of testing the difference in survival distributions when the shape of the hazard ratio is unknown, hence the efficient test is unknown. The proposed adaptive test selects a test statistic from a finite set of the weighted logrank statistics T on the basis of the estimates of the efficiencies of the tests in T for given data. The efficiency estimator uses the length of the test based nonparametric confidence interval for the shift in a time transformed shift model. The suggested adaptive test is shown to be asymptotically efficient among the tests in T under the time transformed shift model and conditions commonly used in survival analysis. Simulations demonstrate that the adaptive test enjoys good small sample properties and in most situations is more powerful than the test using the maximum of the tests in T.     

A new test for the mean vector in large dimension and small samples

In this article, we consider the problem of testing the mean vector in the multivariate normal distribution, where the dimension p is greater than the sample size N. We propose a new test T_Block and obtain its asymptotic distribution. We also compare the proposed test with other two tests. The simulation results suggest that the performance of the new test is comparable to the existing two tests, and under some circumstances it may have higher power. Therefore, the new statistic can be employed in practice as an alternative choice.     

Testing multiple slippages

A class of invariant Bayes rules is derived for testing homogeneity of k (≥2) different populations against (^k_t) slippage alternatives that some (unknown) subset of size t of the given populations has parameter larger than the remaining k-t, where t is a given integer between 1 and k-1. For a similar problem in nonparametric situations, locally best tests based on ranks are derived.     

Bootstrapping analogs of the one way MANOVA test

Abstract

Analogs of the classical one way MANOVA model have recently been suggested that do not assume that population covariance matrices are equal or that the error vector distribution is known. These tests are based on the sample mean and sample covariance matrix corresponding to each of the p populations. We show how to extend these tests using other measures of location such as the trimmed mean or coordinatewise median. These new bootstrap tests can have some outlier resistance, and can perform better than the tests based on the sample mean if the error vector distribution is heavy tailed.     

Powerful Parametric Tests Based on Sum‐Functions of Spacings

Assume that we have a sequence of n independent and identically distributed random variables with a continuous distribution function F, which is specified up to a few unknown parameters. In this paper, tests based on sum‐functions of sample spacings are proposed, and large sample theory of the tests are presented under simple null hypotheses as well as under close alternatives. Tests, which are optimal within this class, are constructed, and it is noted that these tests have properties that closely parallel those of the likelihood ratio test in regular parametric models. Some examples are given, which show that the proposed tests work also in situations where the likelihood ratio test breaks down. Extensions to more general hypotheses are discussed.     

Testing inference in heteroskedastic linear regressions: a comparison of two alternative approaches

We consider the issue of performing testing inferences on the parameters that index the linear regression model under heteroskedasticity of unknown form. Quasi-t test statistics use asymptotically correct standard errors obtained from heteroskedasticity-consistent covariance matrix estimators. An alternative approach involves making an assumption about the functional form of the response variances and jointly modelling mean and dispersion effects. In this paper we compare the accuracy of testing inferences made using the two approaches. We consider several different quasi-t tests and also z tests performed after estimated generalized least squares estimation which was carried out using three different estimation strategies. The numerical evidence shows that some quasi-t tests are typically considerably less size distorted in small samples than the tests carried out after the jointly modelling of mean and dispersion effects. Finally, we present and discuss two empirical applications.     

Rank tests in the two-sample scale problem with unequal and unknown locations

The two-sample scale problem is studied in the case of unequal and unknown location parameters. The method proposed is based on the idea of Moses (1963) and it is distribution-free. The two samples are separated into random subgroups of the same sizek. It is proposed to choosek=4 and to apply the Wilconxon test or the Savage test to the ranges or sample variances of the subgroups. The asymptotic power functions of the tests are compared. For small and moderate sample sizes simulations are carried out. Relations to some other procedures, especially to the method of Compagnone and Denker (1996) are briefly discussed.     

Statistical inference for the size-biased Weibull distribution

In this paper, statistical inferences for the size-biased Weibull distribution in two different cases are drawn. In the first case where the size r of the bias is considered known, it is proven that the maximum-likelihood estimators (MLEs) always exist. In the second case where the size r is considered as an unknown parameter, the estimating equations for the MLEs are presented and the Fisher information matrix is found. The estimation with the method of moments can be utilized in the case the MLEs do not exist. The advantage of treating r as an unknown parameter is that it allows us to perform tests concerning the existence of size-bias in the sample. Finally a program in Mathematica is written which provides all the statistical results from the procedures developed in this paper.