Search

We study a nonparametric test procedure based on order statistics for testing the null hypothesis of equality of two continuous distributions. The exact null distribution of the proposed test statistic is obtained using an enumeration method and a novel combinatorial argument. A recurrence relation for the probability generating function and a sequential approach for computing the mean and variance of the distribution are given. Critical values and characteristics of the distribution for selected small sample sizes are presented. For the Lehmann alternative family, the exact power function of the new test is derived, and its power performance is examined. We also study the power performance of the proposed test under the location-shift and scale-shift alternatives using Monte Carlo simulations and observe its superior performance when compared to commonly used nonparametric tests under various scenarios. A generalization of the proposed procedure for unequal sample sizes is discussed. An illustrative example and some concluding remarks are provided. [ABSTRACT FROM AUTHOR]

In this note we propose a new semi-parametric bootstrap procedure for hypothesis tests about a statistical function and termed bootstrap warping. This procedure was motivated by empirical likelihood and bootstrap tilting techniques. The procedure is computationally efficient and has a fixed number of parameters. We show that the warping procedure has good type I error control and has monotone power as a function of sample size and shift alternatives. [ABSTRACT FROM AUTHOR]

Decadal prediction results are analyzed for the predictability and skill of annual mean temperature. Forecast skill is assessed in terms of correlation, mean square error (MSE) and mean square skill score. The predictability of the forecast system is assessed by calculating the corresponding "potential" skill measures based on the behaviour of the forecast ensemble. The expectation is that potential skill, where the model predicts its own evolution, will be greater than the actual skill, where the model predicts the evolution of the real system, and that the difference is an indication of the potential for forecast improvement. This will depend, however, on the agreement of the second order climate statistics of the forecasts with those of the climate system. In this study the forecast variance differs from the variance of the verifying observations over non-trivial parts of the globe. Observation-based values of variance from different sources also differ non-trivially. This is an area of difficulty independent of the forecasting system and also affects the comparison of actual and potential mean square error. It is possible to scale the forecast variance estimate to match that of the verifying data so as to avoid this consequence but a variance mismatch, whatever its source, remains a difficulty when considering forecast system improvements. Maps of actual and potential correlation indicate that over most of the globe potential correlation is greater than actual correlation, as expected, with the difference suggesting, but not demonstrating, that it might be possible to improve skill. There are exceptions, mainly over some land areas in the Northern Hemisphere and at later forecast ranges, where actual correlation can exceed potential correlation, and this behaviour is ascribed to excessive noise variance in the forecasts, at least as compared to the verifying data. Sampling error can also play a role, but significance testing suggests it is not sufficient to explain the results. Similar results are obtained for MSE but only after scaling the forecasts to match the variance of the verifying observations. It is immediately clear that the forecast system is deficient, independent of other considerations, if the actual correlation is greater than the potential correlation and/or the actual MSE is less than the potential MSE and this gives some indication of the nature of the deficiency in the forecasts in these regions. The predictable and noise components of an ensemble of forecasts can be estimated but this is not the case for the actual system. The degree to which the difference between actual and potential skill indicates the potential for improvement of the forecasting can only be judged indirectly. At a minimum the variances of the forecasts and of the verifying data should be in reasonable accord. If the potential skill is greater than the actual skill for a forecasting system based on a well behaved model it suggests, as a working hypothesis, that forecast skill can be improved so as to more closely approach potential skill. [ABSTRACT FROM AUTHOR]

We propose a consistent test for discrimination between a simple hypothesis and a left-sided alternative about the tails of Weibull-type distributions. The test is constructed using just k higher-order statistics of the sample. [ABSTRACT FROM AUTHOR]

In this article, having observed the generalized order statistics in a sample, we construct a test for the hypothesis that the underlying distribution is the Pareto I distribution. The Shannon entropy of generalized order statistics is used to test the null hypothesis. [ABSTRACT FROM AUTHOR]

X1,X2, …,Xkarek(k⩾ 2) uniform populations which eachXifollowsU(0, θi). This note shows the test statistic for the null hypothesisH0: θ1= θ2= ⋅⋅⋅ = θkby using the order statistics. [ABSTRACT FROM AUTHOR]

In this article, we investigate some statistical procedures for testing homogeneity in a multinomial population. First we focus on the simpler problem of testing homogeneity in the entire vector of multinomial proportions. When the null hypothesis of uniformity is violated, some of the cell frequencies will tend to be small and some will tend to be large. We propose to investigate the performance of tests based onX(k)= max {X1, ....,Xk} orX(1)= min {X1, ....,Xk}, which are functions of the minimal sufficient statistics for this problem. The performance of the likelihood ratio test implemented by the asymptotic chi-square distribution and the simulated exact distribution will be compared to a test which is based on the maximum, the minimum, the pair maximum and minimum, the range, and the ratio statistic:X(k)/X(1). We also investigate the more difficult problem of testing homogeneity of a subset of the p-vector. For this problem, we focus only on likelihood ratio tests and investigate two implementations. Simulation studies are presented to evaluate the accuracy of achieving the targeted significance level and to compare the powers of the test statistics mentioned above for a range of alternatives. Based on the numerical results, recommendations are given for employing these testing procedures. [ABSTRACT FROM PUBLISHER]

Recursive formulas are provided for computing probabilities of a multinomial distribution. Firstly, a recursive formula is provided for computing rectangular probabilities which include the cumulative distribution function as a special case. These rectangular probabilities can be used to provide goodness-of-fit tests for the cell probabilities. The probability that a certain cell count is the maximum of all cell counts is also considered, which can be used to assess the probability that the maximum cell count corresponds to the cell with the maximum probability. Finally, a recursive formula is provided for computing the probability that the cell counts satisfy a certain ordering, which can be used to assess the probability that the ordering of the cell counts corresponds to the ordering of the cell probabilities. The computational intensity of these recursive formulas is linear in the number of cells, and they provide opportunities for calculating probabilities that would otherwise be computationally challenging. Some examples of the applications of these formulas are provided. [ABSTRACT FROM AUTHOR]

For classification problems where the test data are labeled sequentially, the point at which all true positives are first identified is often of critical importance. This article develops hypothesis tests to assess whether all true positives have been labeled in the test data. The tests use a partial receiver operating characteristic (ROC) that is generated from a labeled subset of the test data. These methods are developed in the context of unexploded ordnance (UXO) classification, but are applicable to any binary classification problem. First, the likelihood of the observed ROC given binormal model parameters is derived using order statistics, leading to a nonlinear parameter estimation problem. I then derive the approximate distribution of the point on the ROC at which all true instances are found. Using estimated binormal parameters, this distribution can be integrated up to a desired confidence level to define a critical false alarm rate (FAR). If the selected operating point is before this critical point, then additional labels out to the critical point are required. A second test uses the uncertainty in binormal parameters to determine the critical FAR. These tests are demonstrated with UXO classification examples and both approaches are recommended for testing operating points. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider the problem of testing exponentiality against IFR alternatives. A measure of deviation from exponentiality is developed and a class of test statistics are constructed on the basis of this measure. It is shown that the test statistic is an L-statistic. The asymptotic as well as the exact distributions of the test statistics are obtained and the test statistics are proved to be consistent. The Pitman efficiency has also been studied. [Copyright &y& Elsevier]

ABSTRACT We show that the distribution of luminosities of brightest cluster galaxies (BCGs) in a Sloan Digital Sky Survey (SDSS)-based group catalogue suggests that BCG luminosities are just the statistical extremes of the group galaxy luminosity function. The latter happens to be very well approximated by the all-galaxy luminosity function (restricted to M r < −19.9), provided one uses a parametrization of this function that is accurate at the bright end. A similar analysis of the luminosity distribution of the brightest satellite galaxies (BSGs) suggests that they are best thought of as being the second brightest pick from the same luminosity distribution of which BCGs are the brightest. That is, BSGs are not the brightest of some universal satellite luminosity functions, in contrast to what halo model analyses of the luminosity dependence of clustering suggest. However, we then use mark correlations to provide a novel test of these order statistics, showing that the hypothesis of a universal luminosity function (i.e. no halo mass dependence) from which the BCGs and BSGs are drawn is incompatible with the data, despite the fact that there was no hint of this in the BCG and BSG luminosity distributions themselves. We also discuss why, since extreme value statistics are explicitly a function of the number of draws, the consistency of BCG luminosities with extreme value statistics is most clearly seen if one is careful to perform the test at fixed group richness N. Tests at e.g. fixed total group luminosity Ltot will generally be biased and may lead to erroneous conclusions. [ABSTRACT FROM AUTHOR]

Consider a sensor network made of remote nodes connected to a common fusion center. In a recent work, Blum and Sadler proposed the idea of ordered transmissions—sensors with more informative measurements deliver their messages first—and they proved that optimal detection performance can be achieved using only a subset of the measurements available to the system. Taking to one extreme this approach, we show that using only one transmission the detection error can be made as small as desired, provided that the network size n is large enough. Indeed, we design a distributed detection scheme and prove its asymptotic consistency with respect to n, when the decision is made using just one—but the best—out of n collected samples. [ABSTRACT FROM PUBLISHER]

We discuss the asymptotic properties of the deviation between order statistics and p-quantile of an independently and identically distributed sample of size n. The large and moderate deviation principle and Bahadur's asymptotic efficiency of the deviation X(k) - ξp are obtained under some weak conditions. [ABSTRACT FROM AUTHOR]

There are a number of situations in which an observation is retained only if it is a record value, which include studies in industrial quality control experiments, destructive stress testing, meteorology, hydrology, seismology, athletic events and mining. When the number of records is fixed in advance, the data are referred to as inversely sampled record-breaking data. In this paper, we study the problems of constructing the nonparametric confidence intervals for quantiles and quantile intervals of the parent distribution based on record data. For a single record-breaking sample, the confidence coefficients of the confidence intervals for the pth quantile cannot exceed p and 1-p, on the basis of upper and lower records, respectively; hence, replication is required. So, we develop the procedure based on k independent record-breaking samples. Various cases have been studied and in each case, the optimal k and the exact nonparametric confidence intervals are obtained, and exact expressions for the confidence coefficients of these confidence intervals are derived. Finally, the results are illustrated by numerical computations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of determining non-parametric confidence intervals for quantiles when available data are in the form of k-records. Distribution-free confidence intervals as well as lower and upper confidence limits are derived for fixed quantiles of an arbitrary unknown distribution based on k-records of an independent and identically distributed sequence from that distribution. The construction of tolerance intervals and limits based on k-records is also discussed. An exact expression for the confidence coefficient of these intervals are derived. Some tables are also provided to assist in choosing the appropriate k-records for the construction of these confidence intervals and tolerance intervals. Some simulation results are presented to point out some of the features and properties of these intervals. Finally, the data, representing the records of the amount of annual rainfall in inches recorded at Los Angeles Civic Center, are used to illustrate all the results developed in this paper and also to demonstrate the improvements that they provide on those based on either the usual records or the current records. [ABSTRACT FROM AUTHOR]

Abstract: In the situation of a multi-sample experiment consisting of differently equipped sequential k-out-of-n systems, scale parameters of underlying distributions from a general location-scale family of distributions are estimated under an order restriction. In each sample, the case of missing the smallest observations is included. Moreover, based on a profile likelihood a homogeneity test against an ordered alternative is proposed and analyzed. This work extends an approach of Bhattacharya [2007. Testing for ordered failure rates under general progressive censoring. J. Statist. Plann. Inference 137, 1775–1786] in the progressive Type-II censoring framework. [Copyright &y& Elsevier]

Abstract: A harmonic new better than used in expectation (HNBUE) variable is a random variable which is dominated by an exponential distribution in the convex stochastic order. We use a recently obtained condition on stochastic equality under convex domination to derive characterizations of the exponential distribution and bounds for HNBUE variables based on the mean values of the order statistics of the variable. We apply the results to generate discrepancy measures to test if a random variable is exponential against the alternative that is HNBUE, but not exponential. [Copyright &y& Elsevier]

In this paper, we consider the application of order statistics to establish the optimality in stochastic and heuristic optimization algorithms. A method for estimating the minimum value with an associated confidence interval is developed using the formalism of the theory of order statistics for i.i.d. variables; we examine it by computer simulation. We build a method for the estimation of confidence intervals of the minimum value using order statistics, implemented for optimality testing and stopping in Markov type random search algorithms. The efficiency of this approach is discussed, using the results of application to stochastic approximation and simulated annealing. [ABSTRACT FROM AUTHOR]

The Kulback-Leibler information has been considered for establishing goodness-of-fit test statistics, which have been shown to perform very well (Arizono & Ohta, 1989; Ebrahimi et al., 1992, etc). In this paper, we propose censored Kullback-Leibler information to generalize the discussion of the Kullback-Leibler information to the censored case. Then we establish a goodness-of-fit test statistic based on the censored Kullback-Leibler information with the type 2 censored data, and compare the test statistics with some existing test statistics for the exponential and normal distributions. [ABSTRACT FROM AUTHOR]

We express the joint entropy of progressively censored order statistics in terms of an incomplete integral of the hazard function, and provide a simple estimate of the joint entropy of progressively Type-II censored data. We then construct a goodness-of-fit test statistic based on Kullback-Leibler information with progressively Type-II censored data. Finally, by using Monte Carlo simulations, the power of the test is estimated, and compared against several alternatives under different progressive censoring schemes. [ABSTRACT FROM AUTHOR]

The ranked-set sampling literature includes both inference procedures that rely on the assumption of perfect rankings and inference procedures that are robust to violations of this assumption. Procedures that assume perfect rankings tend to be more efficient when rankings are in fact perfect, but they may be invalid when perfect rankings fail. As a result, users of ranked-set sampling must decide between efficiency and robustness, and there is at present little to guide their decision. In this article we introduce three rank-based goodness-of-fit tests that may be consulted in making these decisions. Our strategy in producing these tests is to think of the judgment order statistic classes as separate samples, compute the ranks of the units from each sample within the combined sample, and use these ranks to test whether the judgment rankings are perfect. Consideration of both power and ease of use leads us to recommend use of a test that rejects when the concordance between the vector of mean ranks and its null expectation is small. Tables of critical values and appropriate asymptotic theory for applying this test are provided, and we illustrate the use of the tests by applying them to a biological dataset. [ABSTRACT FROM AUTHOR]

In a recent paper, approximate and simulated-based generalized confidence limits for the reliability function of a two-parameter exponential model under failure censoring have been proposed by Roy and Mathew [Roy, A. and Mathew, T., 2005. A generalized confidence limit for the reliability function of a two-parameter exponential distribution. Journal of Statistical Planning and Inference, 128, 509-517]. Nevertheless, in some cases, those limits may be greater than 1. In order to overcome this difficulty, the generalized pivot used by Roy and Matthew must be conveniently truncated. In this article, the distribution and density functions of the new pivotal quantity are presented in closed forms. Confidence limits and intervals of shortest length with exact nominal coverages are found immediately by merely solving the appropriate nonlinear equations. Reliability testing is easily carried out using the generalized pivot. The results derived can also be applied to related distributions, as the Pareto model. A numerical example is included for illustrative purposes. [ABSTRACT FROM AUTHOR]

The problem of testing if a given probability distribution fits to a set of independent and identically distributed observations is usually treated by categorizing the data range. Discretization can be done by means of relative frequencies or by using sample quantiles. In this article, quantile-based test statistics are proposed to test the hypothesis of uniformity in the interval (0, 1). Exact critical values of the family of Rukhin's statistics are estimated. A Monte Carlo simulation experiment is carried out to calculate powers of these tests in different alternatives. Results obtained from each kind of categorization are compared to give several recommendations about the use of Rukhin's statistics and type of categorization. [ABSTRACT FROM AUTHOR]

Focuses on goodness-of-fit tests using dual versions of characterizations via moments of order statistics. Use of dual versions of characterizations to construct good-of-fit tests; Discusses of related theorems, corollaries and equation.

We study the problem of testing whether a hazard function is monotonic or not. The proposed test statistics, a global test and four localized tests, are all based on normalized spacings. The global test is in fact just the test statistic [Proschan, F. and Pyke, R. (1967). Tests for monotone failure rate. Fifth Berkeley Symposium , 3 , 293-313], introduced for testing a constant hazard function versus a nondecreasing nonconstant hazard function. This global test is powerful for detecting global departures of the null hypothesis, but lacks power when there are local departures from the null hypothesis. By localizing the global test, we obtain tests that respond to this drawback. We also show how the testing procedures can be used when dealing with Type II censored data. We evaluate the performance of the test statistics via simulation studies and illustrate them on some data sets. [ABSTRACT FROM AUTHOR]

One of the most common types of acceleration of tests, along with high usage rate, overstress, degradation and specimen design (Nelson, W. (1990). Accelerated Testing. New York: Wiley & Sons) is right censoring, which occurs when tests are terminated before all specimens placed on test run to failure. In this article, a test procedure in sequential setting based on the Wald-Wolfowitz (Wald, A., Wolfowitz, J. (1940). On a test whether two samples are from the same population. The Annals of Mathematical Statistics 11:147-162) run test with possibly right censored data is discussed. Simulation results are also included. [ABSTRACT FROM AUTHOR]

The Simes method for testing intersection of more than two hypotheses is known to control the probability of type I error only when the underlying test statistics are independent. Although this method is more powerful than the classical Bonferroni method, it is not known whether it is conservative when the test statistics are dependent. This article proves that for multivariate distributions exhibiting a type of positive dependence that arise in many multiple-hypothesis testing situations, the Simes method indeed controls the probability of type I error. This extends some results established very recently in the special case of two hypotheses. [ABSTRACT FROM AUTHOR]

In a nonrandom sample or a nonrandomized comparison of treated and control groups, methods are developed for displaying the sensitivity of confidence intervals for population quantiles to the magnitude of the departure from random selection. Quantiles of a single distribution and for a shift in two distributions are both discussed. These confidence intervals are based on a new inequality for the distribution of the number of successes in a nonrandom sample or nonrandomized comparison. The inequality has additional applications. In particular, it greatly accelerates the computations in a sensitivity analysis for the Mantel-Haenszel statistic in a 2 × 2 × S contingency table. Examples include a nonrandom collection of blood samples from a town with comparatively high levels of radon gas, an observational study of chromosome damage from mercury, and a study of the drug allopurinol as a cause of skin rash. The proof of the inequality makes use of Savage's lattice for ranks and Holley's inequality, which gives sufficient conditions for stochastic ordering in a lattice. [ABSTRACT FROM AUTHOR]

A study is made of some unusual location estimates first proposed in 1977 by J S Maritz, M Wu, and R G Staudte, Jr., who established some strong robustness properties of these estimates, such as redescending influence functions and, in some cases, full efficiency at the Cauchy model It is further shown here that the estimates, called M-W-S estimates, are derivable from a rank-based scheme founded on convex functions and hence are regular, are unique, and have associated tests and confidence intervals The tests belong to a family of signed-rank-type tests, and their distributions have nice combinatorial properties It is not the case, therefore, that a redescending influence function need imply all the computational irregularities possessed by redescending M estimates. In addition, M-W-S estimates are shown to have breakdown point of 5, a very strong robustness property. [ABSTRACT FROM AUTHOR]

Ranked-set samples have been shown to lead to improved methods of estimation in parametric settings under specific distributional forms when actual measurement of the sample observations is difficult but ranking them is relatively easy. The earliest work with ranked-set data concentrated on estimating a population mean or variance. More recently, a ranked-set sample estimator of a cumulative distribution function was developed and used to obtain a simultaneous confidence interval for the function. In this article, we take the next logical step and use this ranked-set empirical distribution function to construct distribution-free competitors to the standard Mann-Whitney-Wilcoxon estimation and testing procedures. The appropriate null distribution tables for the associated test are presented for the case of perfect ranking. Asymptotic relative efficiency comparisons between the simple random sample Mann-Whitney-Wilcoxon procedures and their ranked-set analogues are discussed, and the results of a small-sample Monte Carlo simulation study of the same arc presented. [ABSTRACT FROM AUTHOR]

A quantile function based analog to Neyman's smooth tests for goodness-of-fit hypotheses is developed. These procedures possess asymptotic and small-sample power properties similar to those for smooth tests. They are, however, computationally simple, applicable to both complete and Type H censored data, and easily focused in the direction of any specific set of alternatives. These results are used to develop a new test for exponentiality. Only location/scale null hypotheses will be considered. The ideas are easily adapted to other types of nulls, however. Neyman first proposed smooth tests for the goodness-of-fit hypothesis. As stated by Kopecky and Pierce and by Rayner and Best, the basic idea behind these tests is to embed the null density in a larger exponential family. The procedure is then based upon the likelihood-ratio test statistic for testing that the additional parameters equal zero. Smooth type tests may be viewed as a compromise between omnibus test procedures, with generally low power in all directions, and procedures whose power is focused in the direction of a specific alternative. It is easily shown that any of the smooth type tests are inconsistent for some alternatives. Further, as stated by Koziol, one of the main criticisms of these tests has been that the larger exponential families employed seem to be selected more for mathematical convenience than as statistically reasonable alternative distributions. Results of Monte Carlo studies by Miller and Quesenberry and others, how- ever, indicate that smooth tests perform well for a wide range of alternatives. The proposed tests are developed by embedding the quantile function associated with the null hypothesis into a larger linear model and then using typical regression techniques to test that the coefficients of the added terms in this model vanish. These statistics are interpreted and asymptotically have many of the properties of the corresponding regression tests. Further, as with Neyman's smooth tests, if the alternative model is properly selected, the resulting procedure provides a good omnibus test. [ABSTRACT FROM AUTHOR]

Let pi[sub N] be a finite population of size N whose elements have distinct X values X[sub (1)] < ... < X[sub (N)]. Let chi ([sub 1]) < ... < x(sub (n)] denote the order statistics of a simple random sample of size n taken from pi [sub N] without replacement. An outer confidence interval is formed for the quantile interval [X ([sub t]), X [sub (u)] (1

The techniques of exploratory data analysis include a resistant rule for identifying possible outliers in univariate data. Using the lower and upper fourths, F[sub L] and F[sub U] (approximate quartiles), it labels as "outside" any observations below F[sub L] - 1.5(F[sub U] - F[sub L]) or above F[sub U] + 1.5( F[sub U] - F[sub L]). For example, in the ordered sample -5, -2, 0, l, 8, F[sub L] = -2 and F[sub U] = 1, so any observation below -6.5 or above 5.5 is outside. Thus the rule labels 8 as outside. Some related rules also use cutoffs of the form F[sub L] -k(F[sub U] - F[sub L]) and F[sub U] + k(F[sub U] - F[sub L]). This approach avoids the need to specify the number of possible outliers in advance; as long as they are not too numerous, any outliers do not affect the location of the cutoffs. To describe the performance of these rules, we define the some-outside rate per sample as the probability that a sample will contain one or more outside observations. Its complement is the all-inside rate per sample. We also define the outside rate per observation as the average fraction of outside observations. For Gaussian data the population all-inside rate per sample (0) and the population outside rate per observation (.7%) substantially understate the corresponding small-sample values. Simulation studies using Gaussian samples with n between 5 and 300 yield detailed information on the resistant rules. The main resistant rule (k = 1.5) has an all-inside rate per sample between 67% and 86% for 5 less than or equal to n less than or equal to 20, and corresponding estimates of its outside rate per observation range from 8.6% to 1.7%. Both characteristics vary with n in ways that lead to good empirical approximations. Because of the way in which the fourths are defined, the sample sizes separate into four classes, according to whether dividing n by 4 leaves a remainder of 0, 1, 2, or 3. Within these... [ABSTRACT FROM AUTHOR]

Suppose that y[sub 1] ..... y[sub k] are normal random variables with ordered means u[sub 1] less than or equal to u[sub 2] less than or equal to ... less than or equal to u[sub k].. A confidence bound is found for each u[sub i] that takes advantage of the ordering. The upper and lower bounds are the maximum and minimum values of x such that the hypotheses that x < u[sub i] and u, < x are accepted by their respective likelihood ratio tests. These bounds are often more precise than the usual bounds and can be used to determine the safety of a toxin and to bound the virtual safe dose in a low-dose extrapolation. Simultaneous intervals for u[sub 1],..., u[sub k] are also derived. [ABSTRACT FROM AUTHOR]

In multiple comparison procedures often the ordering of populations is of more interest than the testing of hypotheses. In such a situation it is important to control the probability of a Type III error (introduced by Hatter 1957 to indicate that one population is concluded to be better than another when actually it is worse). The studentized differences between means may be used to give ordering conclusions on all pairs of populations under consideration. Although this may be done by setting up confidence intervals using Tukey's honest significant difference, a more efficient method (when the confidence intervals themselves are not of interest) is suggested here, and tables are provided for its implementation. [ABSTRACT FROM AUTHOR]

Let pi[sub I], pi[sub 2] ..... pi[sub k] be k is greater than or equal to 2 sources of observations (treatments, populations) and suppose the "goodness" of treatment pi[sub I] is characterized by the size of an unknown real-valued parameter theta[sub I]. Let theta[sub k] = max[sub 1 less than or equal to I less than or equal to k] theta[sub I] If pi[sub I] is preferred to pi[sub j] when theta[sub I] > theta[sub j], the parameters delta[sub l] = theta[sub [k]] - theta[sub I], I = 1, 2 ..... k reflect in an inverse sense the "goodness" of each treatment relative to the "best" treatment. A general technique for obtaining simultaneous confidence intervals on the delta[sub I] is demonstrated with several examples. This technique can be applied in any setting where comparison-with-control intervals can be computed regarding any pi[sub j] as the control. These results have special importance in ranking and selection problems in that the process of generating upper bounds on the delta[sub I] generates traditional confidence statements of both the indifference zone and the subset selection schools, simultaneously, as established by Hsu (1981). [ABSTRACT FROM AUTHOR]

This article explores the use of t statistics based on two- or four-order statistics. The functions of the order statistics, which are used to define the t statistics, are the pivots and bipivots. The pivots are exact order statistics, while the bipivots are the means of two adjacent order statistics. Two t statistics, based on the pivots and bipivots, are examined and compared to other t statistics, including Student's t, under various sampling situations. We conclude that the proposed t statistics are not only simple to use but are robust and perform quite well in terms of the lengths of their confidence intervals. We recommend that these t statistics be used for sample sizes between 4 and 20, because they perform well and are easy to use. Tables of percentage points are provided to facilitate the use of the t statistics. [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]

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]

Wilks [6] introduced distribution-free confidence intervals based on order statistics for quantile intervals. For the rth and sth order statistics in a sample of size n (1 less than or equal to r < s less than or equal to n), upper and lower bounds and an exact expression for the confidence coefficient are derived. Numerical examples comparing the bounds and the exact solution are provided. A connection with distribution-free tolerance intervals is indicated. [ABSTRACT FROM AUTHOR]

Order statistics from a strongly unimodal distribution are strongly unimodal. In this note, authors focus on the strong unimodality of order statistics. Alam points out the relevance of the unimodality of order statistics for the construction of shortest confidence intervals for the largest of several location parameters. Under some regularity conditions, distribution of an order statistic is known to converge to a unimodal distribution. It is thus not surprising that some order statistics are unimodal even though the population may be far from unimodal. For instance, the bimodal density of an example, where all the order statistics, beginning with samples of size 2, are unimodal. On the other hand, order statistics from unimodal populations need not be unimodal. The main result of this note points out that order statistics from strongly unimodal distributions are necessarily strongly unimodal. As a consequence, normal order statistics are not only unimodal, they are strongly unimodal.