On a further generalization of the savage test

A statistic based on the frequencies within the k+1 intervals specified by k arbitrary quantiles is proposed for a LMP test against Lehmann alternatives generalizing the Savage test for the two-sample problem. The maximum efficiency relative to the Savage test for optimally chosen k quantiles is also provided for k=l(2)l5. The asymptotic normality of the statistic follows from the asymptotic multinomial distribution of the frequencies in the classes determined by the k quantiles.     

Goodness‐of‐Fit Test for Monotone Functions

Abstract. In this article, we develop a test for the null hypothesis that a real‐valued function belongs to a given parametric set against the non‐parametric alternative that it is monotone, say decreasing. The method is described in a general model that covers the monotone density model, the monotone regression and the right‐censoring model with monotone hazard rate. The criterion for testing is an ‐distance between a Grenander‐type non‐parametric estimator and a parametric estimator computed under the null hypothesis. A normalized version of this distance is shown to have an asymptotic normal distribution under the null, whence a test can be developed. Moreover, a bootstrap procedure is shown to be consistent to calibrate the test.     

Confidence limits for stress–strength reliability involving Weibull models

The problem of interval estimation of the stress–strength reliability involving two independent Weibull distributions is considered. An interval estimation procedure based on the generalized variable (GV) approach is given when the shape parameters are unknown and arbitrary. The coverage probabilities of the GV approach are evaluated by Monte Carlo simulation. Simulation studies show that the proposed generalized variable approach is very satisfactory even for small samples. For the case of equal shape parameter, it is shown that the generalized confidence limits are exact. Some available asymptotic methods for the case of equal shape parameter are described and their coverage probabilities are evaluated using Monte Carlo simulation. Simulation studies indicate that no asymptotic approach based on the likelihood method is satisfactory even for large samples. Applicability of the GV approach for censored samples is also discussed. The results are illustrated using an example.     

Ordered Alternatives and the 1 1/2-Tail Test

Under proper conditions, two independent tests of the null hypothesis of homogeneity of means are provided by a set of sample averages. One test, with tail probability P ₁, relates to the variation between the sample averages, while the other, with tail probability P ₂, relates to the concordance of the rankings of the sample averages with the anticipated rankings under an alternative hypothesis. The quantity G = P ₁ P ₂ is considered as the combined test statistic and, except for the discreteness in the null distribution of P ₂, would correspond to the Fisher statistic for combining probabilities. Illustration is made, for the case of four means, on how to get critical values of G or critical values of P ₁ for each possible value of P ₂, taking discreteness into account. Alternative measures of concordance considered are Spearman's ρ and Kendall's τ. The concept results, in the case of two averages, in assigning two-thirds of the test size to the concordant tail, one-third to the discordant tail.     

Nonparametric control charts based on order statistics: Some advances

ABSTRACT

In this article, we introduce new nonparametric Shewhart-type control charts that take into account the location of two order statistics of the test sample as well as the number of observations in that sample that lie between the control limits. Exact formulae for the alarm rate, the run length distribution and the average run length (ARL) are all derived. A key advantage of the new charts is that, due to its nonparametric nature, the false alarm rate (FAR) and in-control run length distribution is the same for all continuous process distributions. Tables are provided for the implementation of the proposed charts for some typical FAR and ARL values. Furthermore, a numerical study carried out reveals that the new charts are quite flexible and efficient in detecting shifts to Lehmann-type out-of-control situations, while they seem preferable from a robustness point of view in comparison with the distribution-free control chart of Balakrishnan et al. (2009).     

Tests of Multivariate Independence for Ordinal Data

Population and sample versions of Kendall and Spearman measures of association suitable for multivariate ordinal data are defined. The latter generalize the indices of dependence of Ruymgaart and van Zuijlen (1978 Ruymgaart , F. H. , van Zuijlen , M. C. A. ( 1978 ). Asymptotic normality of multivariate linear rank statistics in the non-i.i.d. case . Ann. Statist. 6 : 588 – 602 .[Crossref], [Web of Science ®] , [Google Scholar]), Joe (1990 Joe , H. ( 1990 ). Multivariate concordance . J. Multivariate Anal. 35 : 12 – 30 .[Crossref], [Web of Science ®] , [Google Scholar]), and Schmid and Schmidt (2007 Schmid , F. , Schmidt , R. ( 2007 ). Multivariate extensions of Spearman's rho and related statistics . Statist. Probab. Lett. 77 : 407 – 416 .[Crossref], [Web of Science ®] , [Google Scholar]) by allowing atoms in the underlying distribution. The representation of the proposed empirical measures as U-statistics enables to establish their asymptotic normality under general distributions. A special attention is given to tests of independence for multivariate ordinal data, where the power of the new methodologies are investigated under fixed and contiguous alternatives.     

Adaptively combining independent jonckheere statistics in a randomized block design with unequal scales

Jonckheere (1954) proposed a test statistic which is commonly used in testing for ordered alternatives in block designs.- We consider the application of Jonckheere's test statistic in block designs which have unequal scale parameters for the blocks. Estimates of the unknown scale parameters ar-fcrmed and are used to construct a modification of Jonckheere's test statistic using adaptive ideas. A Monte Carlo study shows that the modified Jonckheere is significantly more powerful than the original Jonckheere in many unequal scale situations.     

A james-stein type detour of U-statistics

For a vector of estimable parameters, a modified version of the James-Stein rule (incorporating the idea of preliminary test estimators) is utilized in formulating some estimators based on U-statistics and their jackknifed estimator of dispersion matrix. Asymptotic admissibility properties of the classical U-statistics, their preliminary test version and the proposed estimators are studied.     

Distribution-free tests in randomized blocks

Classes of distribution-free tests are proposed for testing homogeneity against order restricted as well as unrestricted alternatives in randomized block designs with multiple observations per cell. Allowing for different interblock scoring schemes, these tests are constructed based on the method of within block rankings. Asymptotic distributions (cell sizes tending to infinity) of these tests are derived under the assumption of homogeneity. The Pitman asymptotic relative efficiencies relative to the least squares statistics are studied. It is shown that when blocks are governed by different distributions, adaptive choice of scores within each block results in asymptotically more efficient tests as compared with methods that ignore such information. Monte Carlo simulations of selected designs indicate that the method of within block rankings is more power robust with respect to differing block distributions.