Characterizations of Student's t-distribution via regressions of order statistics

Utilizing regression properties of order statistics, we characterize a family of distributions introduced by Akhundov et al. [New characterizations by properties of midrange and related statistics, Commun. Stat. Theory Methods 33(12) (2004), pp. 3133–3143], which includes the t-distribution with two degrees of freedom as one of its members. Then we extend this characterization result to t-distribution with more than two degrees of freedom.     

Number of observations near the maximum in an F α-scheme

Let K _n (a) be the number of observations in the interval (M _n ,?a, M _n ), where M _n is the maximum value in a sequence of size n. We study the asymptotic properties of K _n (a) under the F ^α-scheme and discuss the influence of the associated sequence α _n on the limit behaviour of this random variable.     

Bounds on expected generalized order statistics

We establish the upper nonpositive and all the lower bounds on the expectations of generalized order statistics based on a given distribution function with the finite mean and central absolute moment of a fixed order. We also describe the distributions for which the bounds are attained. The methods of deriving the lower nonpositive (upper nonnegative) and lower nonnegative (upper nonpositive) bounds are totally different. The first one, the greatest convex minorant method is the combination of the Moriguti and well-known Hölder inequalities and the latter one is based on the maximization of some norm on the properly chosen convex set. The paper completes the results of Cramer et al. [Evaluations of expected generalized order statistics in various scale units. Appl Math. 2002;29:285–295].     

Testing parallelism of regression lines against ordered alternatives

For the problem of testing the equality of slopes of several regression lines against the alternative that the slopes are in increasing (decreasing) order of magnitude, two types of tests are proposed. These are the likelihood ratio test and a test that depends on suitable linear combination of one group statistics. Rank analogues of the two tests are also examined.     

A Survey of the Teaching of Statistics in M.B.A. Programs

A survey of business schools was conducted to obtain information about the current state of the teaching of business statistics to students enrolled in M.B.A. degree programs. The survey was undertaken for and presented at a June 1986 conference on “Making Statistics More Effective in Schools of Business,” held at the University of Chicago's Graduate School of Business. Information was elicited concerning both the required statistics sequence and elective statistics courses for M.B.A. students, as well as computer usage in these courses. This article summarizes the information obtained from the survey.     

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.     

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]).     

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.     

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).     

Joint Moment Generating Functions of Nonadjacent Dual Generalized Order Statistics from Reflected Generalized Pareto Distributions

In this article, a class of reflected generalized Pareto distributions (cf. Burkschat et al., 2003 Burkschat , M. , Cramer , E. , Kamps , U. ( 2003 ). Dual generalized order statistics . Metron LXI ( 1 ): 13 – 26 . [Google Scholar]) is considered. Recurrence relations for joint moment generating functions of higher non adjacent dual generalized order statistics based on a random sample drawn from the considered class are derived. Higher joint moments of non adjacent dual generalized order statistics (reversed ordered order statistics and lower k-records as special cases) are obtained. Recurrence relations for single and product moment generating functions and moments of higher non adjacent dual generalized order statistics are derived. Some results of higher moments of non adjacent generalized order statistics from generalized Pareto distributions (cf. Johnson et al., 1995 Johnson , N. L. , Kotz , S. , Balakrishnan , N. ( 1995 ). Continuous Univariate Distributions. , 2nd ed. Vol. 2. New York : Wiley & Sons . [Google Scholar]), are obtained by using a relation connecting higher moments of generalized order statistics and its dual.