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The article presents a mathematical analysis of the power function distribution using a positive and bounded random variable with absolutely continuous distribution function F(x). The result of a random sample of size n from F(x) using an order statistics is extended with an independent statistics X(r)/X(s) and X(t). This independent statistics is also used as further proof of the condition for F(x) as the density function of the power function distribution.

This article reviews some of the history of robust estimation, emphasizing the period 1885-1920 and the work of Newcomb, Edgeworth, Sheppard, and Daniell. Particular attention is paid to lines of development which have excited recent interest, including linear functions of order statistics and mixtures of normal densities as models for heavy-tailed populations. [ABSTRACT FROM AUTHOR]

Presents letters addressed to the editor regarding educational statistics. Reliability of the coefficient of consistence; Usability of compartment model as an introduction to the techniques of stochastic processes; Discussion on the simulation of order statistics.

We construct confidence intervals for the ratio of two exponential scale parameters and present a test of hypothesis concerning these parameters. The procedures are based on the selection of subsets of two independent ordered samples from exponential populations. The discussion centers around the distribution of the ratio of the BLUE's (the ABLE's in large samples) of these parameters, and a simple approximation to that distribution. Optimal selection of the data is discussed. In an example, the power function of the proposed test is compared with the power function of the same test based on complete samples. [ABSTRACT FROM AUTHOR]

A queuing model is presented in which delay stems from a disparity between the order at which service is provided to customers and the order at which they may leave the system. Consider, for example, wives waiting in cars-single file-for the arrival of their husbands by a common train. A car is ready to leave when its passenger takes his seat but may have to wait for the departure of all cars preceding it in the lane. Initially, the probability function of waiting is determined for a single, finite queue. Next, the consequences of splitting the single queue into several parallel queues with identical discipline are explored. The results of the analysis are applied to a parking lot design problem. [ABSTRACT FROM AUTHOR]

Soient x [ABSTRACT FROM AUTHOR]

Considérons un échantillon aléatoire de taille n provenant d'une population distribuée selon la fonction de densité f(x) et appelons X [ABSTRACT FROM AUTHOR]

Given a random sample from a Pareto type distribution with the cdf F(x) = I - [1 + (x - alpha)/beta][sup - gamma] for x is greater than or equal to alpha, the problems of estimating (1) the scale parameter beta when a and 3' are known, (2) the location parameter alpha when beta and gamma are known, (3) alpha and beta when gamma is known, are considered. Particular attention is given to the best linear unbiased estimates based on the complete sample and (for (1) and (3)) the asymptotically best linear unbiased estimates based on a few selected order statistics, but some alternative estimates are also considered. The problem of optimum spacing of the order statistics is solved analytically and numerically. [ABSTRACT FROM AUTHOR]

Coefficients, variances and efficiencies are provided for the best linear unbiased estimates (BLUE's) of the mean and standard deviation of the logistic distribution, with both unknown, based on four order statistics with ranks giving the maximum joint efficiency among all BLUE's based on four order statistics. Asymptotic joint efficiencies are compared for asymptotically best linear estimates (ABLE's) based on order statistics with ranks determined by various spacings. With one parameter known, the BLUE's and ABLE's, using order statistics with ranks determined by the optimum spacings of the ABLE's, have efficiencies close to those of BLUE's based on optimum ranks. [ABSTRACT FROM AUTHOR]

Let X[sub (1)], X[sub (2)], ..., X[sub (n)] be the order statistics of a sample of size n from the density [Multiple line equation(s) cannot be represented in ASCII text] . The statistic [Multiple line equation(s) cannot be represented in ASCII text] which is the best invariant estimate of the scale parameter a, is shown to be inadmissible under squared error loss. It is shown to be strictly dominated by an estimate that is not translation invariant. Similarity between this estimation problem and the normal scale estimation problem treated by Stein [6] is discussed. [ABSTRACT FROM AUTHOR]

Originally, Tukey justified the use of the trimmed mean as a robust estimator because of its high efficiency in large samples from contaminated normal data. The means, variances and covariances of the order statistics in small samples (of size is less than or equal to 20) from a standard normal distribution contaminated by various fractions (.01, .05 and .10) of a normal distribution with mean 0 and scale parameter 3 are derived. The small sample behavior of various robust linear estimators is then discussed. At sample size 20, the variances of the usual robust linear estimators are seen to be well approximated by asymptotic theory. If one assumes that the observations come from one of the following distributions: Cauchy, double-exponential, normal, contaminated normal or logistic, then the asymptotically maximin efficient estimator, the 27 1/2 percent trimmed mean, is the maximin efficient estimator in our family at sample size 16. However, its minimum relative efficiency is less at sample size 16 than asymptotic theory suggests. With the Cauchy distribution deleted from the above family, the maximin efficient linear estimator is the 20 percent trimmed mean for all sample sizes studied. [ABSTRACT FROM AUTHOR]

Let X[sub n] and X[sub 1] be the largest and smallest order statistics, respectively, of a random sample of size n. Quite generally, X[sub n] and X[sub 1] are approximately independent for n sufficiently large. Minimum n for attaining at least specified levels of independence are developed. Level of independence is measured by the maximum difference between the true values of P(X[sub 1] is less than or equal to x[sub 1], X[sub n] is less than or equal to x[sub n]) and the corresponding values assuming independence of X[sub n] and X[sub 1]. The results apply to all possible distributions for the population sampled. The value of minimum n is the smallest allowable n for the continuous case but can be too large otherwise. Minimum n is finite for all nonzero differences. [ABSTRACT FROM AUTHOR]

The expectations of order statistics of a logarithmic-normal distribution are expressed as functions which are linear in a location and a scale parameter but nonlinear in a shape parameter of the distribution. The regression of the observed small-sample order statistics on approximations to their expectations is considered and an iterative procedure is proposed which leads to approximate weighted least squares estimates of all three parameters. The performance of the iterative procedure as well as the properties of the estimators are studied by means of Monte Carlo simulation experiments involving samples of size 10, 15 and 20 from standardized distributions (zero median, unit geometric mean) with shape parameter (natural logarithmic variance) in the range zero to unity. It is shown that the iterative procedure always converges and that almost unbiased estimates may be obtained. The variances of the location and scale parameter estimates compare favorably with those of the well-known order-statistic estimates of a standard normal distribution. The parameterization of the lognormal distribution which is adopted leads naturally to use of the latter estimates in the limiting case of samples which exhibit symmetry or even negative skewness. While only complete samples are considered explicitly, the simple extension required to take care of censored samples is indicated.' [ABSTRACT FROM AUTHOR]

A progressive-censoring model arises from a life test of a sample of items in which one or more of the survivors may be removed from the test at the time of any failure. Such a model is often more realistic for actual failure data which must be analyzed by a statistician than one in which all survivors are assumed to be removed from test simultaneously. This paper deals with the situation in which the underlying failure-time distribution for the population sampled is the two-parameter Weibull distribution. The reliable life for the population is defined to be the 100(1-R) percent point of the failure-time distribution, where R is a specified population survival proportion, or reliability. An exact confidence bound on reliable life based on three observed ordered failure times is derived for this progressive-censoring model. The criterion used for selecting the order numbers of the three failure times upon which the bound is based depends upon computed values of the power function of the test associated with the bound. A table from which lower bounds can be obtained is given for R equal to .95, confidence level .90, sample size equal to 2, 3, . . . , 6, and all possible censorings. [ABSTRACT FROM AUTHOR]

The asymptotic joint distribution of an arbitrary number of linear systematic statistics (that is, linear combinations of order statistics), when observations are made on a random vector, is shown to be normal under fairly general conditions. The linear systematic statistics may correspond to the same or to different components of the vector. Formulas for evaluating the parameters of the asymptotic normal distribution are derived. As an illustration, these are applied to the case of trimmed means when the distribution sampled is bivariate normal. [ABSTRACT FROM AUTHOR]

The aim of this paper is two-fold: (1) To give a unified treatment of the theory of order statistics when the parent distribution is not necessarily continuous. (2) To assess the effects of grouping on the distribution of order statistics and to indicate the convenience, under suitable conditions, of using order statistics for the estimation of parameters from grouped data with or without censoring. [ABSTRACT FROM AUTHOR]

Some optimum nonparametric procedures for estimating and testing contrasts in two-way layouts are proposed and studied. These procedures are based on the Chernoff-Savage [4] type of rank order statistics which include the Wilcoxon and normal scores statistics among others. In the first three sections, the properties of the proposed point estimators of contrasts, such as symmetry, invariance and asymptotic normality, are studied and their asymptotic relative efficiencies with respect to the corresponding least-squares estimators are obtained. In particular, it is shown that the procedures based on the normal scores statistics are asymptotically at least as efficient as the corresponding procedures based on the method of least squares, whatever be parent cumulative distribution functions. In sections 4 and 5, the corresponding problems of testing and confidence intervals are discussed, and generalized to two-way layouts with several observations per cell. [ABSTRACT FROM AUTHOR]

Formulae for the first two moments of the order statistics from a geometric distribution are presented in closed form. Brief illustrative tables of the first two moments of the order statistics from a geometric distribution are included and two related dice games are discussed. The distribution of the range of a set of independent observations from a geometric distribution is presented in closed form. A system of redundant components which performs a given task repetitively at discrete times and its statistical equivalent, an inverse sampling scheme, are studied. The distribution of the number of stages until the scheme terminates or the system fails is shown to be identical to the distribution of a specific order statistic from a geometric distribution. A Markov chain formulation is presented which facilitates certain system reliability and cost analyses. [ABSTRACT FROM AUTHOR]

The problem of estimating a location parameter from a random sample when the form of distribution is unknown or there is contamination of the target distribution is attacked by deriving estimators which are efficient over a class of two or more forms ("pencils") of continuous symmetric unimodal distributions. The pencils considered are the normal, double exponential, Cauchy, parabolic, triangular, and rectangular (a limiting case). The estimators considered are special symmetrical linear combinations of order statistics: trimmed means, Winsorized means, "linearly weighted" means, and a combination of the median and two other order statistics. These are also compared asymptotically with a Hodges-Lehmann estimator. The theory required for deriving asymptotic variances is outlined. Efficiences are tabulated for sample sizes of 4 or 5, 8 or 9, 16 or 17, and Infinity. Asymptotic efficiences of at least 0.82 relative to the best estimator for any single pencil are achieved by using the best trimmed mean or linearly weighted mean over a range of pencils of distributions from the normal to the Cauchy. However, the combination of the median and two other order statistics is almost as efficient (0.80) over the same range. [ABSTRACT FROM AUTHOR]

In a recent paper in this Journal, Rothenberg, Fisher and Tilanus [1] discuss a class of estimators of tile location parameter of the Cauchy distribution, taking the form of the arithmetic average of a central subset, of the sample order statistics. They show that the average of roughly the middle quarter of the ordered sample has minimum asymptotic variance within this class, and that asymptotically it eliminates about. 36 per cent of the efficiency loss of the median (the most commonly used estimate,r) in comparison to the maximum likelihood estimator (m.l.e.). Of course both the m.l.e, and the best linear unbiased estimator based on the order statistics (BLUE) achieve full asymptotic efficiency in the Cramer-Rao sense and there can be no dispute about the relative merits of the three estimators asymptotically, or about the inferiority of the median (with asymptotic efficiency 8/pi[sup 2] * This character cannot be converted in ASCII text) 0.8 compared with about 0.88 for the estimator of Rothenberg et al.). In any practical situation however, we will be concerned with estimation from samples of finite size and asymptotic properties will not necessarily give any guidance here. We are essentially concerned with two points in assessing the relative merits of estimators in small samples, their case of application and "small-sample efficiency" which is conveniently measured as the ratio of the Cramer-Rao lower bound to the variance of the estimator. In this paper various estimators of the location of the Cauchy distribution arc compared in these two respects for samples of up to 20 observations. The small-sample properties of the m.l.e. have been extensively discussed elsewhere (Barnett [2]) and relevant results are summarized where necessary. The main purpose of the paper is to discuss general linear estimators based on the order statistics, and to assess their utility in the present context. Since this paper was prepared a further interesting 'quick estimator', b [ABSTRACT FROM AUTHOR]

Abstract The order statistics of an arbitrary size random sample of variates which originate from a logistic distribution with an arbitrary degree of truncation, both one-sided and two-sided, are considered. A general expression for the product moments of the order statistics in the one-sided truncation case is derived by application of a formula obtained from the Euler Transformation. A generalization of this formula is used to obtain an expression for the product moments of the order statistics with an arbitrary amount of truncation at either tail of the distribution. All above expressions for the product moments as well as for the first and second ordinary moments are given in terms of finite sums, the terms of which involve only the logarithm and dilogarithm of P, i.e., the constant of truncation. [ABSTRACT FROM AUTHOR]

One-sided tolerance limits are developed for the three-parameter generalized gamma distribution of Stacy, one parameter at a time being assumed unknown. This distribution includes the exponential, gamma, and Weibull distributions as special cases. Confidence limits for the unknown parameter are also given in each case. Limits are developed based on the statistic SIGMA[sup n, sub I=1] x[sup f, sub I] and also on the r-th order statistic. The limits based on a single order statistic take into account the distributional form of the variable and are not the familiar nonparametric tolerance limits of Wilks which do not depend on the functional form of the variable. Some comparisons of the limits are considered in special cases and also some of the limits are shown to satisfy the "most stable" criterion of Goodman and Madansky. [ABSTRACT FROM AUTHOR]

This paper presents the results of a limited investigation which brings into focus the difficulties encountered in developing exact distribution free methods for stratified simple random sampling. It is shown that exact distribution-free confidence intervals for a population median can be obtained from a stratified sample under very special circumstances. The principal result concerns the order statistics of the combined separate strata samples, where proportional allocation is employed. It is proved that a pair of symmetric order statistics provides a confidence interval for the population median whose confidence coefficient is not less than the confidence coefficient obtained from the corresponding order statistics of a random sample of the same total size drawn from the entire population. Furthermore, it is shown by means of a numerical example that this result necessarily holds only when proportional allocation is employed. [ABSTRACT FROM AUTHOR]

The present paper contrasts the relative efficacies of the traditional mean square or variance, and a time-series statistic, the mean square successive difference, as descriptors of cyclic heart rate variability. Estimates of fluctuation in the R-R interval during the prestimulus and cold pressor stimulus periods were obtained for two subject samples. The first sample was composed of individuals having bilateral differences in the diameter of their pupils, a condition known as anisocoria, while the pupils of the Ss in the second sample were of equal diameter. A group sinusoidal rhythm in the mean square successive difference of R-R interval variation was observed during the cold pressor stimulus for the anisocoric sample. However, the rhythm was obscured when the traditional mean square statistic was utilized to describe the same R-R interval variation. No rhythms were identified for the non-anisocoric sample. The mean square successive difference statistic would appear to provide a method for detecting cyclic autonomic variability patterns otherwise obscured by traditional analysis. [ABSTRACT FROM AUTHOR]

The article discusses the probability distribution of order statistics from discrete populations. Order statistics have in recent years become an increasingly popular tool for applied statisticians. Many of their properties do not depend upon the distribution from which the random sample is obtained and they are particularly informative for studying properties of extreme values from an observable process. These conditions occur frequently in decision making problems of business. Some goodness of fit tests are likely to be of little power when the main body of data conforms quite well to a certain distribution but the observation in the tails do not conform. If the statistician wishes to test whether the extreme observations obey the same laws as do the more regular observations, his test will have to be based on the analysis of ordered statistics. Example computations are given.

A survey is given of recent advances, including a number of original contributions by the author, in the use of order statistics to obtain point and interval estimates of the parameters of various statistical populations from complete and from censored samples. In a few cases estimators based on order statistics are the efficient estimators, but more often they are substitute estimators that sacrifice some efficiency in the interest of computational simplicity and/or robustness in the presence of outliers. In life testing, they are often used to obtain estimates of the parameters of the life distribution before all of the items placed on test have failed. Point estimators based on order statistics may be either best linear unbiased estimators, based on all available observations from complete or censored samples or on one or more observations chosen in some optimal manner, or maximum-likelihood estimators, based on censored samples, of a single parameter or of two or more parameters jointly. Interval estimators may be based on percentage points of order statistics or of functions of order statistics, or they may be based on asymptotic variances and covariances of maximum-likelihood estimators. A summary is given of available results, together with a list of references and examples of applications to such problems as estimating the scatter of bullets aimed at a target and the reliability of an electronic device. [ABSTRACT FROM AUTHOR]

The article discusses aspects of manpower planning in industrial management. Manpower planning incorporates a specification of the types and numbers of men an organization will need to accomplish its profit, growth, and service objectives. One approach in manpower planning is the utilization of stochastic models. Experts say adequate manpower planning models cannot rely on mathematical methods alone. Another manpower planning technique discussed is the replacement table method. This approach was instituted during World War II. The flexible hierarchy model is detailed.

A linear programming problem is said to be stochastic if one or more of the coefficients in the objective function or the system of constraints or resource availabilities is known only by its probability distribution. Various approaches are available in this case, which may be classified into three broad types: 'chance constrained programming', 'two-stage programming under uncertainty' and 'stochastic linear programming'. For problems of 'stochastic linear programming' a distinction is usually made between two related approaches to stochastic programming, the passive and the active approach respectively. In the passive approach to stochastic linear programming the statistical distribution of the optimum value of the objective function is estimated either exactly or approximately by numerical methods and optimum decision rules are based on the different characteristics of the estimated distribution. In the active approach, a new set of decision variables are introduced which indicate the proportions of different resources to be allocated to the various activities. One effect of introducing this set of new decision variables in the active approach is the truncation of the statistical distribution of optimal value of the objective function of the passive approach. In this aspect the active approach is useful in suggesting criteria for changing from one optimum decision rule to another, as the probability distribution of the objective function is specified less and less incompletely for different choices of the set of new decision variables. This paper investigates some mathematical relations between the active and passive approach and derives some inequalities for the case when the elements of the coefficient matrix of the inequalities are random variables. Since an 'approximate solution' of a stochastic linear programming problem is defined usually by replacing each random element by its expected value and then solving the resulting non-stochastic program, some conditions have been derived under which the expected value of the objective function for the optimal solution will exceed the optimal value of the objective function for 'the approximate solution'. Based on the properties of the distribution of extreme values and other order statistics, some approximate bounds have also been specified for the passive approach, which can easily be extended to the active approach. [ABSTRACT FROM AUTHOR]

Point estimation for the scale and location parameters of the extreme-value (Type I) distribution by linear functions of order statistics from Type II progressively censored samples is investigated. Four types of linear estimators are considered: the best linear unbiased (BLU), an approximation to the BLU, unweighted regression, and a linearized maximum likelihood. Linear transformations of the estimators are also considered for reducing mean square errors. Exact bias, variance, and mean square error comparisons of the estimators are made for several censoring patterns. Since the natural logarithms of Weibull variates have extreme-value distributions, the investigation is applicable to estimation for Weibull distributions. [ABSTRACT FROM AUTHOR]

We obtain exact and approximate confidence intervals (tabulated for 90%, 95% and 99%) for the scale parameter, σ, of the exponential distribution in small and large samples. The exact confidence intervals are based on the distributions of the BLUE and ABLE of σ, using κ optimally selected order statistics from a random sample of size η and are tabulated for κ = 1(1)5 and various η. The approximate intervals are based on approximating chi-square distributions. We find that in large samples, the optimal quantiles for the interval estimation of σ are the same as those for the point estimation of σ, for several optimality criteria; or, in other words, these several criteria, are equivalent in large samples. [ABSTRACT FROM AUTHOR]

Procedures are developed for evaluation of various composite and simple hypotheses of exponentiality. The principles underlying these methods are similar to those used to defining and extending the W-statistic for normality (Shapiro and Wilk, 1965, 1968). The W-exponential statistic (scale and origin invariant) turns out to be the scaled ratio of the squared difference of the sample mean and smallest observation to the usual symmetric sum of squares about the mean. Procedures are developed to augment the W-exponential statistic when one or both of the location and scale parameters of the general exponential are specified. Necessary tables, some examples, and an abstract of an empirical sampling study of sensitivity are given. [ABSTRACT FROM AUTHOR]

This is an expository paper, pointing out explicitly the pseudoness of the "F-statistic" used in stepwise procedures for determining the independent variables to be used in a linear prediction equation. Unfortunately this pseudoness prevents one from obtaining any probabilistic measure, of the goodness of the final prediction equation. The use of the distribution of an order statistic is discussed as an aid to understanding the problem as well as furnishing a "better" (although still unsatisfactory) approach. [ABSTRACT FROM AUTHOR]

Discusses the moments of the log-Weibull order statistics. Definition of the moments of order statistics; Formulae for the log-Weibull distribution; Relationship between Weibull and log-Weibull variables.

This paper deals with the problem of estimating the mean and the standard deviation of the logistic distribution by using ordered observations in small samples. Linear unbiased estimators with minimum variance based on ordered observations are constructed both for the complete and the censored samples for sample sizes n = 2, 5, 10, 15, 20 and 25. The censored sample case consists of the n - r[sub 1] - r[sub 2] observations, where r[sub 1] observations are missing in the beginning and r[sub 2] observations are missing at the end. The efficiencies of these estimators relative to the moment estimator and the Cramèr-Rao bound are evaluated and graphed. A table of variances and covariances of all order statistics for n = 10(5)25 is also given. [ABSTRACT FROM AUTHOR]

Gupta [2] derived expressions for the moments of order statistics, moments of pairs of order statistics and the modal values of order statistics from a gamma distribution when the shape-parameter is a positive integer. But, there are several situations (e.g., see [3]) when the data can arise from a gamma population with any positive shape-parameter. To meet these situations, the results of Gupta [2] are extended in this note for a gamma population with any positive shape-parameter. [ABSTRACT FROM AUTHOR]

This paper presents the small sample optimum choice of the k(≤ τ[sub 2]) order statistics for the best linear unbiased estimate (BLUE) of the parameters μ and σ or σ alone (μ known) when the sample is Type II censored on the right. For n = 2(1)10, k = 1(1)τ[sub 2] and τ[sub 2] = {[.50n]+1} (1)n, the optimum ranks, the coefficients of the BLUEs have been presented in Table I. [ABSTRACT FROM AUTHOR]

In this paper a c-sample slippage analogue of the Wilcoxon [11] test is considered. Given a sample of size n for each of c populations, the test rejects the hypothesis that the c populations are identical when max[sub 1≤i≤c ∑[sub h] τ[sub ik] > λ, where τ[sub i1], …, τ[sub in] are the ranks of the observations from the i-th population in the combined sample of size cn. The small and large sample distributions of the test statistic are derived. Tables of the exact distribution are given for c = 2(1)5, n = 2(1)5. Tables of critical values are given for c = 2(1)6, n = 2(1)8 for values of α = 0.001, 0.005, 0.01, 0.025, 0.05, 0.10, and 0.20. [ABSTRACT FROM AUTHOR]

Suggests some alternative estimates for the parameters of the extreme value distribution. Moments of order statistics; Efficiencies of linear unbiased estimates for the extreme value distribution; Method of distinguishing two Weibull distributions; Use of two-standard error rule.

A derivation is given of the maximum likelihood estimator &θcirc;, based on the first m out of n ordered observations, of the scale parameter θ of a Weibull population with known shape parameter K. It is shown that 2m &θcirc;[sup K]/θ[sup K] has a chi-square distribution with 2m degrees of freedom (independent of n). Use is made of this fact to set upper confidence bounds with confidence level 1 - P (lower confidence bounds with confidence level P) on the scale parameter θ. Formulas are given for the mean squared deviations of the upper and lower confidence bounds from the true value of the parameter. From these one can obtain expressions for the efficiency of confidence bounds and confidence intervals. The expected value of &θcirc; is also determined and from it the unbiasing factor &θtilde;/&θcirc; by which &θcirc; must be multiplied to obtain an unbiased estimator &θtilde;. An expression for the variance of the unbiased estimator &θtilde; is found. Values of the unbiasing factor and of the variance of the unbiased estimator, both of which are independent of n, are tabled for m = 1(1)100 and K = 0.5(0.5)4.0(1.0)8.0. A section on use of the table and a numerical example are included. [ABSTRACT FROM AUTHOR]

Asymptotically best linear unbiased estimates (ABLUE) of the location and scale parameters (when both are unknown) of the Cauchy distribution, based on k order statistics selected from a large sample, are considered. It is shown that the joint asymptotic relative efficiency (JARE) of the two estimates is maximized when the sample quantiles are equally spaced, and expressions are derived for the coefficients used in these estimates. In estimating the scale parameter using a reasonably large number of quantiles, it is found that not much weight is attached to the extreme observations or those close to the median. [ABSTRACT FROM AUTHOR]

Let X[sub 1], ..., X[sub n], n is greater than or equal to 2 be n i.i.d. r.v.'s, each having a geometric distribution, and let Y[sub 1] less than or equal to ... less than or equal to Y[sub n], be the corresponding order statistics. Write Z = SIGMA[sup eta, sub 1-2] (Y[sub i] -- Y[sub 1]). Then it is well known that Y[sub 1] and Z are independent. In this article, we show that a very weak form of this property is a characterizing property of the geometric distribution. Other characterizations of the geometric distribution are also obtained by similar properties. [ABSTRACT FROM AUTHOR]

This paper represents a first detailed study of the application of a generalized probability distribution, the Burr distribution, to control charts for the maximum and minimum value in a sample of size n. The Burr distribution is a two parameter probability distribution with cumulative distribution function represented by Multiple line equation(s) cannot be represented in ASCII text This paper illustrates the application of the Burr distribution to control charts for the maximum and minimum value in sampling from a normal distribution and compares results with those previously derived using the standard normal distribution. [ABSTRACT FROM AUTHOR]

Considérons n variables aléatoires indépendantes {X [ABSTRACT FROM AUTHOR]

It is well known that we tend to have to wait longer than we expect. Special cases of this paradox have been discussed by Feller and by Zelen. Here the paradox is further explored, and generalizations are considered to describe the effect of the observer on the expected length of an interval between successive events in a life test. [ABSTRACT FROM AUTHOR]

Comments on an article by M.A.H. Taha (1964) and A.M. Mood (1954) on the use of rank test for detecting scale parameter changes for asymmetrical one-sided distributions. Basis of the squared rank tests used; Derivation of the exact reduction due to tied observations.

This article focuses on the range of order statistics. It presents several mathematical formulas regarding the statistics. It further says that P.A. Samuelson has shown that one of the given theorems yields an inequality sharper than that of Chebyshev. It is of interest to note that there is a discrepancy between Samuelson's second theorem and the corresponding theorem of this article.

This article focuses on the recurrence relation for distribution functions from bivariate distributions. In the univariate case, a number of recurrence relations between the densities and the moments of the order statistics are available ([1], [2], [3], [4], [5]). In this paper, a similar result is derived for bivariate distributions. Let F(x, y) be a bivariate distribution function with continuous margins F1(x) and F2(y).

Investigates the machine-generation of order statistics for Monte Carlo computations. Steps in solving distributional problems involving order statistics; Use of exponential distribution; Specification of sample sizes.

Examines the existence of moments of order statistics of a random sample from a Cauchy distribution. Representation of the order statistics; Techniques for achieving the arithmetic mean; Presentation of derived formulas.