On Order Statistics from Sine Distribution

We derive closed form expressions for the first two moments of order statistics from the sine distribution. For the higher moments, a recurrence relation is given. We also give a recurrence relation for the product moments. These relations will be useful for moment computations based on ordered data.     

On Conditionally Independent Random Variables,Copula and Order Statistics

The copula representations for conditionally independent random variables and the distribution properties of order statistics of these random variables are studied.     

A Note on the Regression of Order Statistics

This paper examines the relationships between the mean residual life functions of parallel and k-out-of-n systems with the regression of order statistics. Using these relationships, the results and properties about the mean residual life function of those systems can be used for the regression of order statistics and vice versa. Finally, the paper proposes a definition for the mean residual life function of a k-out-of-n system when the number of failed components of the system is known.     

Approximations of Variances and Covariances for Order Statistics from the Standard Extreme Value Distribution

We consider simple approximations of variances and covariances for order statistics from the standard extreme value distribution. Exact values and simulation results of the variances and covariances for certain sample sizes are used to determine the validity of the suggested approximations.     

Some Properties of Order Statistics When the Sample Size Is Random

ABSTRACT

The study of r-out-of-n systems is of utmost importance in reliability theory. In this note, we study closure of different partial orders under the formation of r-out-of-N and (N ? s)-out-of-N systems when the number of components N, forming the system, is a random variable having support {k, k + 1,…}, where k is a fixed positive integer, r ∈ {1,…, k} and s ∈ {0, 1,…, k ? 1}. This generalizes quite a few results already known in the literature. We also study the closure of different partial orders when two systems are formed out of different random number of components.     

Bounds for Moments of the Maximum of Concomitants of Selected Order Statistics With Application

We present sharp bounds for moments of the maximum of concomitants of selected order statistics. The dependence between pair components is modeled by copulas. We use the bounds to compare some insurance premiums.     

On the Characterization of a Mixture of Generalized Power Function Distributions by Conditional Expectation of Order Statistics

In the present article, we give some theorems to characterize the mixture of two generalized power function distributions based on conditional expectation of order statistics.     

Pitman Closeness of Order Statistics to Population Quantiles

In this article, Pitman closeness of sample order statistics to population quantiles of a location-scale family of distributions is discussed. Explicit expressions are derived for some specific families such as uniform, exponential, and power function. Numerical results are then presented for these families for sample sizes n = 10,15, and for the choices of p = 0.10, 0.25, 0.75, 0.90. The Pitman-closest order statistic is also determined in these cases and presented.     

Estimation of the Parameters of Type-I Generalized Logistic Distribution Using Order Statistics

In this work, we propose a technique of estimating the location parameter μ and scale parameter σ of Type-I generalized logistic distribution by U-statistics constructed by using best linear functions of order statistics as kernels. The efficiency comparison of the proposed estimators with respect to maximum likelihood estimators is also made.     

Baker- Lin-Huang Type Bivariate Distributions Based on Order Statistics

Baker (2008 Baker, R. (2008). An order-statistics-based method for constructing multivariate distributions with fixed marginals. J. Multivariate Anal. 99: 23122327.[Crossref], [Web of Science ®] , [Google Scholar]) introduced a new class of bivariate distributions based on distributions of order statistics from two independent samples of size n. Lin and Huang (2010 Lin, G.D., Huang, J.S. (2010). A note on the maximum correlation for Baker’s bivariate distributions with fixed marginals. J. Multivariate Anal. 101: 22272233.[Crossref], [Web of Science ®] , [Google Scholar]) discovered an important property of Baker’s distribution and showed that the Pearson’s correlation coefficient for this distribution converges to maximum attainable value, i.e., the correlation coefficient of the Fréchet upper bound, as n increases to infinity. Bairamov and Bayramoglu (2013 Bairamov, I., Bayramoglu, K. (2013). From Huang-Kotz distribution to Baker’s distribution. J. Multivariate Anal. 113: 106115.[Crossref], [Web of Science ®] , [Google Scholar]) investigated a new class of bivariate distributions constructed by using Baker’s model and distributions of order statistics from dependent random variables, allowing higher correlation than that of Baker’s distribution. In this article, a new class of Baker’s type bivariate distributions with high correlation are constructed based on distributions of order statistics by using an arbitrary continuous copula instead of the product copula.     

Sharp Bounds for Generalized Order Statistics via Logarithmic Moments

ABSTRACT

We present sharp bounds for expectations of generalized order statistics with random indices. The bounds are expressed in terms of logarithmic moments E X ^a (log max {1, X}) ^b of the underlying observation X. They are attainable and provide characterizations of some non trivial distributions. No restrictions are imposed on the parameters of the generalized order statistics model.     

Some Applications of Order Statistics to Multivariate Analysis II

ABSTRACT

Harter (1979) summarized applications of order statistics to multivariate analysis up through 1949. The present paper covers the period 1950–1959. References in the two papers were selected from the first and second volumes, respectively, of the author's chronological annotated bibliography on order statistics [Harter (1978, 1983)]. Tintner (1950a) established formal relations between four special types of multivariate analysis: (1) canonical correlation, (2) principal components, (3) weighted regression, and (4) discriminant analysis, all of which depend on ordered roots of determinantal equations. During the decade 1950–1959, numerous authors contributed to distribution theory and/or computational methods for ordered roots and their applications to multivariate analysis. Test criteria for (i) multivariate analysis of variance, (ii) comparison of variance–covariance matrices, and (iii) multiple independence of groups of variates when the parent population is multivariate normal were usually derived from the likelihood ratio principle until S. N. Roy (1953) formulated the union–intersection principles on which Roy & Bose (1953) based their simultaneous test and confidence procedure. Roy & Bargmann (1958) used an alternative procedure, called the step–down procedure, in deriving a test for problem (iii), and J. Roy (1958) applied the step–down procedure to problem (i) and (ii), Various authors developed and applied distribution theory for several multivariate distributions. Advances were also made on multivariate tolerance regions [Fraser & Wormleighton (1951), Fraser (1951, 1953), Fraser & Guttman (1956), Kemperman (1956), and Somerville (1958)], a criterion for rejection of multivariate outliers [Kudô (1957)], and linear estimators, from censored samples, of parameters of multivariate normal populations [Watterson (1958, 1959)]. Textbooks on multivariate analysis were published by Kendall (1957) and Anderson (1958), as well as a monograph by Roy (1957) and a book of tables by Pillai (1957).     

Nonparametric Prediction Intervals for Future Order Statistics in a Proportional Hazard Model

Consider k independent random samples with different sample sizes such that the ith sample comes from the cumulative distribution function (cdf) F _i  = 1 ? (1 ? F)^{α _i} , where α _i is a known positive constant and F is an absolutely continuous cdf. Also, suppose that we have observed the maximum and minimum of the first k samples. This article shows how one can construct the nonparametric prediction intervals for the order statistics of the future samples on the basis of these information. Three schemes are studied and in each case exact expressions for the prediction coefficients of prediction intervals are derived. Numerical computations are given for illustrating the results. Also, a comparison study is done while the complete samples are available.     

A New Estimation Method Based on Order Statistics in the Families of Symmetric Location-scale Distributions

In this study, new unbiased and nonlinear estimators based on order statistics are proposed for the family of symmetric location-scale distributions and these estimators can be computed from both uncensored and symmetric doubly Type II censored samples. In addition, other relevant unbiased estimators are proposed to estimate standard deviations of these new estimators. A simulation study has been performed to evaluate the performance of the new estimators compared to BLU estimators for small sample sizes. As a result of the simulation study, the new estimators proposed for the location-scale family in general performed nearly as good as BLU estimators. Furthermore, the computational advantage of the proposed estimators over BLU and ML estimators are worthy of notice. In addition, these new estimators have been applied to real data, and the estimation results obtained have been compatible with those of BLUE methods.     

On Distributions of Exceedances Based on Generalized Order Statistics

Distributions of exceedance statistics based on generalized order statistics are obtained for a random threshold model. The ordinary order statistics, progressively Type-II right censored order statistics and record values are considered as special cases. The results obtained in the article imply many results on exceedance statistics for the variety of models of ordered random variables.     

Analysis of Order Statistics from Distributions with Regularly Varying Tails

In this article, necessary conditions for comparing order statistics from distributions with regularly varying tails are discussed in terms of various stochastic orders. A necessary and sufficient condition for stochastically comparing tail behaviors of order statistics is derived. The main results generalize and recover some results in Kleiber (2002 Kleiber, C. (2002). Variability ordering of heavy-tailed distributions with applications to order statistics. Statist. Probab. Lett. 58:381388.[Crossref], [Web of Science ®] , [Google Scholar], 2004 Kleiber, C. (2004). Lorenz ordering of order statistics from log-logistic and related distributions. J. Statist. Plann. Infer. 120:2004.[Crossref], [Web of Science ®] , [Google Scholar]). Extensions to coherent systems are mentioned as well.     

Further Results on Order Statistics Generated by Two Simulation Methods

Uniform order statistics generated by two simulation methods are compared by means of Pitman’s measure of closeness. This measure, as a probability, is shown to be asymptotically 1/2. Some results are also established for fixed points of the cumulative distribution function (CDF) for a uniform order statistic. These fixed points are important for calculations involving the joint distribution of these order statistics.     

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]).