An asymptotically distribution-free test of symmetry

A procedure, based on sample spacings, is proposed for testing whether a univariate distribution is symmetric about some unknown value. The proposed test is a modification of a sign test suggested by Antille and Kersting [1977. Tests for symmetry. Z. Wahrscheinlichkeitstheorie verw. Gebiete 39, 235–255], but unlike Antille and Kersting's test, our modified test is asymptotically distribution-free and is usable in practice. A simulation study indicates that the proposed test maintains the nominal level of significance, α$\alpha$ fairly accurately even for samples of size as small as 20, and a comparison with the classical test based on sample coefficient of skewness, shows that our test has good power for detecting different asymmetric distributions.     

A class of count models and a new consistent test for the Poisson distribution

We define a class of count distributions which includes the Poisson as well as many alternative count models. Then the empirical probability generating function is utilized to construct a test for the Poisson distribution, which is consistent against this class of alternatives. The limit distribution of the test statistic is derived in case of a general underlying distribution, and efficiency considerations are addressed. A simulation study indicates that the new test is comparable in performance to more complicated omnibus tests.     

Goodness-of-fit tests for a heavy tailed distribution

We study the Kolmogorov–Smirnov test, Berk–Jones test, score test and their integrated versions in the context of testing the goodness-of-fit of a heavy tailed distribution function. A comparison of these tests is conducted via Bahadur efficiency and simulations.     

On the asymptotic behaviour of location-scale invariant Bickel–Rosenblatt tests

Location-scale invariant Bickel–Rosenblatt goodness-of-fit tests (IBR tests) are considered in this paper to test the hypothesis that f, the common density function of the observed independent d-dimensional random vectors, belongs to a null location-scale family of density functions. The asymptotic behaviour of the test procedures for fixed and non-fixed bandwidths is studied by using an unifying approach. We establish the limiting null distribution of the test statistics, the consistency of the associated tests and we derive its asymptotic power against sequences of local alternatives. These results show the asymptotic superiority, for fixed and local alternatives, of IBR tests with fixed bandwidth over IBR tests with non-fixed bandwidth.     

A NONPARAMETRIC TEST OF SYMMETRY VERSUS ASYMMETRY FOR RANKED-SET SAMPLES

This paper introduces a nonparametric test of symmetry for ranked-set samples to test the asymmetry of the underlying distribution. The test statistic is constructed from the Cramér-von Mises distance function which measures the distance between two probability models. The null distribution of the test statistic is established by constructing symmetric bootstrap samples from a given ranked-set sample. It is shown that the type I error probabilities are stable across all practical symmetric distributions and the test has high power for asymmetric distributions.     

Regression model checking with Berkson measurement errors

This paper discusses asymptotically distribution free tests for the lack-of-fit of a parametric regression model in the Berkson measurement error model. These tests are based on a martingale transform of a certain marked empirical process of calibrated residuals. A simulation study is included to assess the effect of measurement error on the proposed test. It is observed that empirical level is more stable across the chosen measurement error variances when fitting a linear model compared to when fitting a nonlinear model, while, in both cases, the empirical power decreases as this error variance increases, against all chosen alternatives.     

Combined multiple testing by censored empirical likelihood

We propose a new procedure for combining multiple tests in samples of right-censored observations. The new method is based on multiple constrained censored empirical likelihood where the constraints are formulated as linear functionals of the cumulative hazard functions. We prove a version of Wilks’ theorem for the multiple constrained censored empirical likelihood ratio, which provides a simple reference distribution for the test statistic of our proposed method. A useful application of the proposed method is, for example, examining the survival experience of different populations by combining different weighted log-rank tests. Real data examples are given using the log-rank and Gehan-Wilcoxon tests. In a simulation study of two sample survival data, we compare the proposed method of combining tests to previously developed procedures. The results demonstrate that, in addition to its computational simplicity, the combined test performs comparably to, and in some situations more reliably than previously developed procedures. Statistical software is available in the R package ‘emplik’.     

Semiparametric estimation for count data through weighted distributions

This paper is concerned with semiparametric discrete kernel estimators when the unknown count distribution can be considered to have a general weighted Poisson form. The estimator is constructed by multiplying the Poisson estimate with a nonparametric discrete kernel-type estimate of the Poisson weight function. Comparisons are then carried out with the ordinary discrete kernel probability mass function estimators. The Poisson weight function is thus a local multiplicative correction factor, and is considered as the uniform measure to detect departures from the equidispersed Poisson distribution. In this way, the effects of dispersion and zero-proportion with respect to the standard Poisson distribution are also minimized. This method of estimation is also applied to the weighted binomial form for the count distribution having a finite support. The proposed estimators, in addition to being simple, easy-to-implement and effective, also outperform the competing nonparametric and parametric estimators in finite-sample situations. Two examples illustrate this new semiparametric estimation.     

Robust permutation tests for two samples

We consider robust permutation tests for a location shift in the two sample case based on estimating equations, comparing the test statistics based on a score function and an M-estimate. First we obtain a form for both tests so that the exact tests may be carried out using the same algorithms as used for permutation tests based on the mean. Then we obtain the Bahadur slopes of the tests in these two statistics, giving numerical results for two cases equivalent to a test based on Huber scores and a particular case of this related to a median test. We show that they have different Bahadur slopes with neither exceeding the other over the whole range. Finally, we give some numerical results illustrating the robustness properties of the tests and confirming the theoretical results on Bahadur slopes.     

Estimating the distribution function using k-tuple ranked set samples

The basic assumption underlying the concept of ranked set sampling is that actual measurement of units is expensive, whereas ranking is cheap. This may not be true in reality in certain cases where ranking may be moderately expensive. In such situations, based on total cost considerations, k-tuple ranked set sampling is known to be a viable alternative, where one selects k units (instead of one) from each ranked set. In this article, we consider estimation of the distribution function based on k-tuple ranked set samples when the cost of selecting and ranking units is not ignorable. We investigate estimation both in the balanced and unbalanced data case. Properties of the estimation procedure in the presence of ranking error are also investigated. Results of simulation studies as well as an application to a real data set are presented to illustrate some of the theoretical findings.     

Some nonparametric precedence-type tests based on progressively censored samples and evaluation of power

In this paper, we consider the problem of testing the equality of two distributions when both samples are progressively Type-II censored. We discuss the following two statistics: one based on the Wilcoxon-type rank-sum precedence test, and the second based on the Kaplan–Meier estimator of the cumulative distribution function. The exact null distributions of these test statistics are derived and are then used to generate critical values and the corresponding exact levels of significance for different combinations of sample sizes and progressive censoring schemes. We also discuss their non-null distributions under Lehmann alternatives. A power study of the proposed tests is carried out under Lehmann alternatives as well as under location-shift alternatives through Monte Carlo simulations. Through this power study, it is shown that the Wilcoxon-type rank-sum precedence test performs the best.     

Bayesian mixture modeling for spatial Poisson process intensities,with applications to extreme value analysis

We propose a method for the analysis of a spatial point pattern, which is assumed to arise as a set of observations from a spatial nonhomogeneous Poisson process. The spatial point pattern is observed in a bounded region, which, for most applications, is taken to be a rectangle in the space where the process is defined. The method is based on modeling a density function, defined on this bounded region, that is directly related with the intensity function of the Poisson process. We develop a flexible nonparametric mixture model for this density using a bivariate Beta distribution for the mixture kernel and a Dirichlet process prior for the mixing distribution. Using posterior simulation methods, we obtain full inference for the intensity function and any other functional of the process that might be of interest. We discuss applications to problems where inference for clustering in the spatial point pattern is of interest. Moreover, we consider applications of the methodology to extreme value analysis problems. We illustrate the modeling approach with three previously published data sets. Two of the data sets are from forestry and consist of locations of trees. The third data set consists of extremes from the Dow Jones index over a period of 1303 days.     

On tests for normality of experimental error in ridge regression

Considered are tests for normality of the errors in ridge regression. If an intercept is included in the model, it is shown that test statistics based on the empirical distribution function of the ridge residuals have the same limiting distribution as in the one-sample test for normality with estimated mean and variance. The result holds with weak assumptions on the behavior of the independent variables; asymptotic normality of the ridge estimator is not required.     

Asymptotic properties of the MAMSE adaptive likelihood weights

The weighted likelihood is a generalization of the likelihood designed to borrow strength from similar populations while making minimal assumptions. If the weights are properly chosen, the maximum weighted likelihood estimate may perform better than the maximum likelihood estimate (MLE). In a previous article, the minimum averaged mean squared error (MAMSE) weights are proposed and simulations show that they allow to outperform the MLE in many cases. In this paper, we study the asymptotic properties of the MAMSE weights. In particular, we prove that the MAMSE-weighted mixture of empirical distribution functions converges uniformly to the target distribution and that the maximum weighted likelihood estimate is strongly consistent. A short simulation illustrates the use of bootstrap in this context.     

Assessing additivity in nonparametric models —A kernel‐based method

In this article, we address the testing problem for additivity in nonparametric regression models. We develop a kernel‐based consistent test of a hypothesis of additivity in nonparametric regression, and establish its asymptotic distribution under a sequence of local alternatives. Compared to other existing kernel‐based tests, the proposed test is shown to effectively ameliorate the influence from estimation bias of the additive component of the nonparametric regression, and hence increase its efficiency. Most importantly, it avoids the tuning difficulties by using estimation‐based optimal criteria, while there is no direct tuning strategy for other existing kernel‐based testing methods. We discuss the usage of the new test and give numerical examples to demonstrate the practical performance of the test. The Canadian Journal of Statistics 39: 632–655; 2011. © 2011 Statistical Society of Canada     

A bias correction and acceleration approach for the problem of regions

For testing the problem of regions in the space of distribution functions, this paper considers approaches to modify the bootstrap probability to be a second-order accurate p$p$-value based on the familiar bias correction and acceleration method. It is shown that Shimodaira's [2004a. Approximately unbiased tests of regions using multistep-multiscale bootstrap resampling. Ann. Statist. 32, 2616–2641] twostep-multiscale bootstrap method works even in the problem of regions in functional space. In this paper the bias correction quantity is estimated by his onestep-multiscale bootstrap method. Instead of using the twostep-multiscale bootstrap method, the acceleration constant is estimated by a newly proposed jackknife method which requires first-level bootstrap resamplings only. Some numerical examples are illustrated, in which an application to testing significance in model selection is included.     

A bibliography of statistical tolerance regions,II

This bibiliography (completing the previous one, JILEK 1981a) contains about 130 references dealing with statistical tolerance regions. References are classified according to the type of probability distribution concerned and according to some other criteria as well     

An L-statistic approach to a test of exponentiality against IFR alternatives

In this note we consider the problem of testing exponentiality against IFR alternatives. A measure of deviation from exponentiality is developed and a test statistic constructed on the basis of this measure. It is shown that the test statistic is an L-statistic. The asymptotic as well as the exact distribution of the test statistic is obtained and the test is shown to be consistent.     

The Bernstein–von Mises theorem in semiparametric competing risks models

Semiparametric Bayesian models are nowadays a popular tool in event history analysis. An important area of research concerns the investigation of frequentist properties of posterior inference. In this paper, we propose novel semiparametric Bayesian models for the analysis of competing risks data and investigate the Bernstein–von Mises theorem for differentiable functionals of model parameters. The model is specified by expressing the cause-specific hazard as the product of the conditional probability of a failure type and the overall hazard rate. We take the conditional probability as a smooth function of time and leave the cumulative overall hazard unspecified. A prior distribution is defined on the joint parameter space, which includes a beta process prior for the cumulative overall hazard. We first develop the large-sample properties of maximum likelihood estimators by giving simple sufficient conditions for them to hold. Then, we show that, under the chosen priors, the posterior distribution for any differentiable functional of interest is asymptotically equivalent to the sampling distribution derived from maximum likelihood estimation. A simulation study is provided to illustrate the coverage properties of credible intervals on cumulative incidence functions.     

Nonparametric methods in multivariate factorial designs for large number of factor levels

We propose different multivariate nonparametric tests for factorial designs and derive their asymptotic distribution for the situation where the number of replications is limited, whereas the number of treatments goes to infinity (large a, small n case). The tests are based on separate rankings for the different variables, and they are therefore invariant under separate monotone transformations of the individual variables.