Search

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]

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.