A class of distribution-free tests for the two-sample slippage problem

A class of distribution-free tests for the two-sample slippage problem, when the random variables take only nonnegative values, is proposed. These tests are consistent and unbiased against the general slippage alternative. Recurrence relations for generating small sample significance points are given. The tests have been compared with the Savage test, the Wilcoxon test and the appropriate locally most powerful rank test by considering Pitman asymptotic relative efficiencies for several alternative hypotheses. Some of these tests exhibit considerable robustness in terms of efficiency for the various alternative hypotheses which are considered.     

A parametric bootstrap solution to the MANOVA under heteroscedasticity

In this article, we consider the problem of comparing several multivariate normal mean vectors when the covariance matrices are unknown and arbitrary positive definite matrices. We propose a parametric bootstrap (PB) approach and develop an approximation to the distribution of the PB pivotal quantity for comparing two mean vectors. This approximate test is shown to be the same as the invariant test given in [Krishnamoorthy and Yu, Modified Nel and Van der Merwe test for the multivariate Behrens–Fisher problem, Stat. Probab. Lett. 66 (2004), pp. 161–169] for the multivariate Behrens–Fisher problem. Furthermore, we compare the PB test with two existing invariant tests via Monte Carlo simulation. Our simulation studies show that the PB test controls Type I error rates very satisfactorily, whereas other tests are liberal especially when the number of means to be compared is moderate and/or sample sizes are small. The tests are illustrated using an example.     

Discordancy Tests Based on the Likelihood Ratio for the Bipolar Watson Distribution on the Hypersphere

As the Watson distribution is frequently used for modeling axial data, it is important to investigate the existence of possible outliers in samples from this distribution. Then, we develop for the bipolar Watson distribution defined on the hypersphere, some tests of discordancy of an outlier or several outliers en bloc based on the likelihood ratio, supposing an alternative model of contamination of slippage type. We evaluate the performance of these tests of discordancy of an outlier and we also compare some tests of discordancy of an outlier available for this distribution.     

Analysis of means using ranks for the randomized complete block design

A nonparametric test procedure is proposed for the analysis of randomized complete block designs. Such a procedure may be carried out graphically in the form of a Shewhart control chart. Exact and asymptotic critical values are given for the implementation of the proposed procedure. A Monte Carlo study is made to compare the powers of the proposed procedure to those of analysis of variance, the analysis of means, and the Friedman procedures. Results of the study indicate that the proposed procedure has superior power performance when testing against slippage alternative hypotheses under heavy-tailed distributions such as the Cauchy distribution. However, when testing against symmetric alternatives under light-tailed distributions, the proposed procedure does not perform well     

Data-driven smooth test for a location-scale family

A combination of a smooth test statistic and (an approximate) Schwarz's selection rule has been proposed by Inglot, T., Kallenberg, W. C. M. and Ledwina, T. ((1997). Data-driven smooth tests for composite hypotheses. Ann. Statist. 25, 1222–1250) as a solution of a standard goodness-of-fit problem when nuisance parameters are present. In the present paper we modify the above solution in the sense that we propose another analogue of Schwarz's rule and rederive properties of it and the resulting test statistic. To avoid technicalities we restrict our attention to location-scale family and method of moments estimators of its parameters. In a parallel paper [Janic-Wróblewska, A. (2004). Data-driven smooth tests for the extreme value distribution. Statistics, in press] we illustrate an application of our solution and advantages of modification when testing of fit to extreme value distribution.     

Tests for the skewness parameter of two-piece double exponential distribution in the presence of nuisance parameters

In this paper, tests for the skewness parameter of the two-piece double exponential distribution are derived when the location parameter is unknown. Classical tests like Neyman structure test and likelihood ratio test (LRT), that are generally used to test hypotheses in the presence of nuisance parameters, are not feasible for this distribution since the exact distributions of the test statistics become very complicated. As an alternative, we identify a set of statistics that are ancillary for the location parameter. When the scale parameter is known, Neyman–Pearson's lemma is used, and when the scale parameter is unknown, the LRT is applied to the joint density function of ancillary statistics, in order to obtain a test for the skewness parameter of the distribution. Test for symmetry of the distribution can be deduced as a special case. It is found that power of the proposed tests for symmetry is only marginally less than the power of corresponding classical optimum tests when the location parameter is known, especially for moderate and large sample sizes.     

Data-driven smooth tests for the extreme value distribution

In this paper we present data-driven smooth tests for the extreme value distribution. These tests are based on a general idea of construction of data-driven smooth tests for composite hypotheses introduced by Inglot, T., Kallenberg, W. C. M. and Ledwina, T. [(1997). Data-driven smooth tests for composite hypotheses. Ann. Statist., 25, 1222–1250] and its modification for location-scale family proposed in Janic-Wróblewska, A. [(2004). Data-driven smooth test for a location-scale family. Statistics, in press]. Results of power simulations show that the newly introduced test performs very well for a wide range of alternatives and is competitive with other commonly used tests for the extreme value distribution.     

Bootstrap method in ranking and slippage problems

In this paper, the bootstrap method of Efron (1979) is given for a ranking and a slippage problem, where the ranking (or slippage) is with respect to the mean of the distributions. The method is also applied to obtain a confidence interval for the largest mean.     

Inference for the Birnbaum–Saunders Lifetime Regression Model with Applications

The Birnbaum–Saunders distribution is a positively skewed distribution that is frequently used for analyzing lifetime data. Regression analysis is widely used in this context when some covariates are involved in the life-test. In this article, we discuss the maximum likelihood estimation of the model parameters and associated inference. We discuss the likelihood-ratio tests for some hypotheses of interest as well as some interval estimation methods. A Monte Carlo simulation study is then carried out to examine the performance of the proposed estimators and the interval estimation methods. Finally, some numerical data analyses are done for illustrating all the inferential methods developed here.     

Estimation and testing under exchangeable exponential model with a single outlier

Several estimators of mean of an exponential distribution, when an unidentified single outlier in a sample of size n is present* are discussed. It is assumed that n?1 of these observations have a mean σ, While one could have a mean σ/α. The estimation of σ has been considered in detail with some reference to the estimation of α. Finally, tests of hypotheses about σare briefly mentioned.     

An analysis of pulsation periods of long period variable stars

Summary.  We report the results of a period change analysis of time series observations for 378 pulsating variable stars. The null hypothesis of no trend in expected periods is tested for each of the stars. The tests are non-parametric in that potential trends are estimated by local linear smoothers. Our testing methodology has some novel features. First, the null distribution of a test statistic is defined to be the distribution that results in repeated sampling from a population of stars. This distribution is estimated by means of a bootstrap algorithm that resamples from the collection of 378 stars. Bootstrapping in this way obviates the problem that the conditional sampling distribution of a statistic, given a particular star, may depend on unknown parameters of that star. Another novel feature of our test statistics is that one-sided cross-validation is used to choose the smoothing parameters of the local linear estimators on which they are based. It is shown that doing so results in tests that are tremendously more powerful than analogous tests that are based on the usual version of cross-validation. The positive false discovery rate method of Storey is used to account for the fact that we simultaneously test 378 hypotheses. We ultimately find that 56 of the 378 stars have changes in mean pulsation period that are significant when controlling the positive false discovery rate at the 5% level.     

Detection of a changepoint,a mean-shift accompanied with a trend change,in short time-series with autocorrelation

In this study, a changepoint model, which can detect either a mean shift or a trend change when accounting for autocorrelation in short time-series, was investigated with simulations and a new method is proposed. The changepoint hypotheses were tested using a likelihood ratio test. The test statistic does not follow a known distribution and depends on the length of the time-series and the autocorrelation. The results imply that it is not possible to detect autocorrelation and that the estimate of the autocorrelation parameter is biased. It is therefore recommended to use critical values from the empirical distribution for a fixed autocorrelation.     

A note on homogeneity tests when combining 2×2 tables

This paper examines the use of homogeneity tests prior to tests of overall association among g 2 x 2 tables. When placed in the context of a one-way analysis of variance, hypotheses of overall association and homogeneity can be viewed as hypotheses regarding mean and treatment effects, respectively. In this context, the need for homogeneity tests is presented. What constitutes a relevant test of homogeneity is also examined. The conclusion is that some of the difficulties raised in the literature regarding tests of homogeneity stem from differences in the hypothesis of association being examined.     

Bayesian testing of an exponential point null hypothesis

The problem of testing a point null hypothesis involving an exponential mean is The problem of testing a point null hypothesis involving an exponential mean is usual interpretation of P-values as evidence against precise hypotheses is faulty. As in Berger and Delampady (1986) and Berger and Sellke (1987), lower bounds on Bayesian measures of evidence over wide classes of priors are found emphasizing the conflict between posterior probabilities and P-values. A hierarchical Bayes approach is also considered as an alternative to computing lower bounds and “automatic” Bayesian significance tests which further illustrates the point that P-values are highly misleading measures of evidence for tests of point null hypotheses.     

Likelihood ratio test for discordancy with slippage alternatives

A likelihood ratio test for discordancy in the sample is considered with slippage alternatives. It is shown for a wide class of univariate distributions that only the extreme observations in the sample need to be tested for discordancy. This result provides a firmer support to many commonly used discordancy tests that take only extreme observations as candidates. The problem of testing multiple discordant observations is also discussed.     

Testing for a separation between two normal distributions

Three tests are proposed for testing for a specified degree of overlap between two normal distributions, The hypotheses considered are an extension of the Behrens-Fisher problem, A simulation study of the performance of the tests is presented.     

Aligned rank statistics for repeated measurement models with orthonormal design, employing a Chernoff–Savage approach

In this paper, aligned rank statistics are considered for testing hypotheses regarding the location in repeated measurement designs, where the design matrix for each set of measurements is orthonormal. Such a design may, for instance, be used when testing for linearity. It turns out that the centered design matrix is not of full rank, and therefore it does not quite satisfy the usual conditions in the literature. The number of degrees of freedom of the limiting chi-square distribution of the test statistic under the null hypothesis, however, is not affected, unless rather special hypotheses are tested. An independent derivation of this limiting distribution is given, using the Chernoff–Savage approach. In passing, it is observed that independence of the choice of aligner, which in the location problem is well-known to be due to cancellation, may in scale problems occur as a result of the type of score function suitable for scale tests. A possible extension to multivariate data is briefly indicated.     

Robustness of tests for directional mean

In this paper we study the robustness of the likelihood ratio, circular mean and circular trimmed mean test functionals in the context of tests of hypotheses regarding the mean direction of circular normal and wrapped normal distributions. We compute the level and power breakdown properties of the three test functionals and compare them. We find that the circular trimmed mean test functional has the best robustness properties for both the above-mentioned distributions. The level and power properties of the test statistics corresponding to these functionals are also studied. Two examples with real data are given for illustration. We also consider the problem of testing the mean direction of the von-Mises–Fisher distribution on the unit sphere and explore the robustness properties of the spherical mean direction and likelihood ratio test functionals.     

Percentage points of the statistics for testing hypotheses on mean vectors of multivariate normal distributions with missing observations

A problem of testing of hypotheses on the mean vector of a multivariate normal distribution with unknown and positive definite covariance matrix is considered when a sample with a special, though not unusual, pattern of missing observations from that population is available. The approximate percentage points of the test statistic are obtained and their accuracy has been checked by comparing them with some exact percentage points which are calculated for complete samples and some special incomplete samples. The approximate percentage points are in good agreement with exact percentage points. The above work is extended to the problem of testing the hypothesis of equality of two mean vectors of two multivariate normal distributions with the same, unknown covariance matrix     

On the Use of Bayes Factor in Frequentist Testing of a Precise Hypothesis

This article deals with Bayes factors as useful Bayesian tools in frequentist testing of a precise hypothesis. A result and several examples are included to justify the definition of Bayes factor for point null hypotheses, without merging the initial distribution with a degenerate distribution on the null hypothesis. Of special interest is the problem of testing a proportion (joint with a natural criterion to compare different tests), the possible presence of nuisance parameters, or the influence of Bayesian sufficiency on this problem. The problem of testing a precise hypothesis under a Bayesian perspective is also considered and two alternative methods to deal with are given.