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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 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]

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

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.