Admissibility of Test Procedures for Linear Hypotheses in General Linear Model

This article shows that an F-test procedure is admissible for testing a linear hypothesis concerning one of the split mean vectors in a general linear model and an F-test procedure is also admissible for testing a linear hypothesis concerning another of the split mean vectors in the same model. These results are proved by showing that the critical functions of the tests are unique Bayes procedures with respect to proper prior distributions set in common for the null hypotheses and for the alternative ones, respectively.     

Ranges of Measures of Association for Familial Binary Variables

Familial binary data occur in a wide range of scientific investigations. Numerous measures of association have been proposed in the literature for the study of intra-family dependence of the binary variables. These measures include correlations, odd ratios, kappa statistics, and relative risks. We study the permissible ranges of these measures of association such that a joint distribution exists for the familial binary variables. Our results are useful for developing efficient estimation methods for the measures of association.     

Are the Order Statistics Ordered? A Survey of Recent Results

The distributions of coherent systems with components with exchangeable lifetimes can be represented as mixtures of distributions of order statistics (k-out-of-n systems) from possibly dependent samples by using the concept of the signature of Samaniego (1985 Samaniego , F. ( 1985 ). On closure of the IFR class under formation of coherent systems . IEEE Trans. Reliabil. R-34 : 69 – 72 .[Crossref], [Web of Science ®] , [Google Scholar]). This representation, together with Rychlik's (1993 Rychlik , T. ( 1993 ). Bounds for expectation of L-estimates for dependent samples . Statistics 24 : 1 – 7 .[Taylor & Francis Online] , [Google Scholar]) results, can be used to obtain sharp bounds on the distribution (or the reliability) function and on the expected lifetime of the system. Also, this representation can be used to determine the asymptotic behavior of the hazard rate of the system when the order statistics are ordered in the hazard rate order. Moreover, the lifetime distributions of coherent systems (and in particular, of order statistics) can also be represented as generalized mixtures (that is, mixtures with some negative weights) of distributions of series system lifetimes by using the concept of the minimal signature defined by Navarro et al. (2007a Navarro , J. , Ruiz , J. M. , Sandoval , C. J. ( 2007a ). Properties of coherent systems with dependent components . Commun. Statist. Theory Methods 36 : 175 – 191 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). This representation can also be used to determine the final behavior of the hazard rate of the system through the behavior of the hazard rate of the series systems. In particular, it can be used to show that the order statistics are, under some conditions, asymptotically hazard rate ordered. However, in general, this result is not true, that is, the order statistics need not be hazard rate ordered.     

On the Similarity Structure of Order Statistics

In this article, we address the similarity structure between pairs of order statistics of an identically distributed independent random variables X₁,…, X_n. The overlapping coefficient (Δ) of Weitzman (1970 Weitzman , M. S. ( 1970 ). Measures of overlap of income distributions of white and Negro families in the United States . Technical Paper No. 22 , Washington , D.C. , U.S.A : Department of Commerce, Bureau of Census . [Google Scholar]), is used to assess the degree of similarity or closeness between pairs of order statistics. It appears that the degree of the similarity between any of such pairs is independent of the parent distribution. Using this notion, it is shown that for i < j, the degree of similarity between distributions of the ith and the jth order statistics decreases as i and j sunder. Some possible biometric applicability of the value of Δ are explored. In particular, the use of this measure in estimation of the number of possible strata, subgroups or natural subdivisions in a population are suggested.     

Likelihood Ratio Order of m-Spacings in Multiple-Outlier Models

This article deals with the multiple-outlier exponential model. The likelihood ratio order between m-spacings of the combined sample is developed, some results extend the conclusions on simple spacings in Wen et al. (2007 Wen , S. , Lu , Q. , Hu , T. ( 2007 ). Likelihood ratio order of spacings of heterogeneous exponential random variables . J. Multivariate Anal. 98 : 743 – 756 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Marshall–Olkin Extended Lomax Distribution and Its Application to Censored Data

The problem of making statistical inference about θ =P(X > Y) has been under great investigation in the literature using simple random sampling (SRS) data. This problem arises naturally in the area of reliability for a system with strength X and stress Y. In this study, we will consider making statistical inference about θ using ranked set sampling (RSS) data. Several estimators are proposed to estimate θ using RSS. The properties of these estimators are investigated and compared with known estimators based on simple random sample (SRS) data. The proposed estimators based on RSS dominate those based on SRS. A motivated example using real data set is given to illustrate the computation of the newly suggested estimators.     

A Note on Simultaneous Confidence Intervals for Ratio of Variances

An explicit formula for confidence intervals for ratios of variances of several populations is presented. The intervals are based on jackknife statistics and the critical point of the studentized range distribution. The asymptotic probability of coverage is not less than the nominal value provided that the distributions of the sampled populations belong to a location-scale family of probabilities with finite fourth moment.     

Mixture of Bivariate Exponential Distributions

The exponential distribution is one of the most used type of distribution because of its importance in many lifetime applications and its properties. So is its bivariate form. Simply used, there can be limitations specially for the heterogeneous type population. Its mixture form adds a lot of characters and desirable properties. We propose a mixture of bivariate exponential distribution, study properties of the associated parameters and predict the elements of the mixture. We include the presence of covariate information through a linear relationship, capturing the now famous idea by Marshall and Olkin.     

A Distribution-Free Control Chart Based on Order Statistics

In this article, we introduce a new distribution-free Shewhart-type control chart that takes into account the location of a single order statistic of the test sample (such as the median) as well as the number of observations in that test 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 chart is that, due to its nonparametric nature, the false alarm rate and in-control run length distribution are the same for all continuous process distributions, and so will be naturally robust. Tables are provided for the implementation of the chart for some typical ARL values and false alarm rates. The empirical study carried out reveals that the new chart is preferable from a robustness point of view in comparison to a classical Shewhart-type chart and also the nonparametric chart of Chakraborti et al. (2004 Chakraborti , S. , van der Laan , P. , van de Wiel , M. A. ( 2004 ). A class of distribution-free control charts . J. Roy. Statist. Soc. Ser. C-Appl. Statist. 53 ( 3 ): 443 – 462 .[Web of Science ®] , [Google Scholar]).     

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.