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Consider the usual balanced one-way fixed effects analysis of variance (ANOVA) model [X.sub.ij] = [[mu].sub.i] + [[epsilon].sub.ij] (1 [less than or equal to] j [less than or equal to] n; 1 [less than or equal to] i [less than or equal to] k), where the [[epsilon].sub.ij] are independent N(0, [[sigma].sup.2]) random variables and [[mu].sub.i] is the mean of the ith treatment (1 [less than or equal to] i [less than or equal to] k). The [[mu].sub.i] and [[sigma].sup.2] are unknown parameters. Let [[bar.X].sub.i] be the sample mean of the ith treatment based on n observations (1 [less than or equal to] i [less than or equal to] k), and let [S.sup.2] be an unbiased estimate of [[sigma].sup.2], which is distributed independently of the [[bar.X].sub.i], as a [[sigma].sup.2][[chi].sub.v.sup.2]/v random variable. Usually, the ANOVA mean squared error with v = k(n - 1) df is used as the estimate [S.sup.2]. A problem that has received considerable attention is that of testing the null hypothesis [H.sub.0]: [[mu].sub.1] = *** = [[mu]s.ub.k] against the simple ordered alternative hypothesis [H.sub.A]: [[mu].sub.1] [less than or equal to] *** [less than or equal to] [[mu].sub.k] with at least one strict inequality [see, e.g., Barlow, Bartholomew, Bremner, and Brunk (1972) and Robertson, Wright, and Dykstra (1988)]. Various test procedures have been proposed for this problem, such as the likelihood ratio test (LRT) of Bartholomew (1961), which are particularly sensitive to configurations of the treatment means satisfying the order restriction. Many of the test procedures, however, suffer from the disadvantage of not having a convenient inversion to a set of simultaneous confidence intervals for useful contrasts of the treatment means. In this article the following simple multiple contrast test procedure is considered, which may be thought of as a 'one-sided' studentized range test (OSRT). The OSRT rejects the null hypothesis [H.sub.0] iff the statistic [max[[less than or equal to]i KEY WORDS: Analysis of variance; Critical point calculation; Least favorable configuration; Order restricted inference; Simultaneous confidence intervals; Unbalanced models., 1. SIMULTANEOUS ONE-SIDED CONFIDENCE INTERVALS The pioneering work in the development of simultaneous confidence intervals for restricted settings was carried out by Bohrer (1967) and Bohrer and Francis (1972). In [...]

Suppose that [X.sub.i] are independent random variables, and that [X.sub.i] has cdf [F.sub.i](x), 1 [less than or equal to] i [less than or equal to] k. Many statistical problems involve the probability Pr{[X.sub.1] [less than] [X.sub.2] [less than] . . . [less than] [X.sub.k]}. In this note a numerical method is proposed for computing this probability. KEY WORDS: Multivariate probability; Statistical computing., Suppose that [X.sub.i] are independent random variables (discrete or continuous), and that [X.sub.i] has cdf [Fi.sub.](x), 1 [less than or equal to] i [less than or equal to] k. We [...]

The comparison of two treatments with normally distributed data is considered. Inferences are considered based upon the difference between single potential future observations from each of the two treatments, which provides a useful and easily interpretable assessment of the difference between the two treatments. These methodologies combine information from a standard confidence interval analysis of the difference between the two treatment means, with information available from standard prediction intervals of future observations. Win-probabilities, which are the probabilities that a future observation from one treatment will be superior to a future observation from the other treatment, are a special case of these methodologies. The theoretical derivation of these methodologies is based upon inferences about the non-centrality parameter of a non-centralt-distribution. Equal and unequal variance situations are addressed, and extensions to groups of future observations from the two treatments are also considered. Some examples and discussions of the methodologies are presented. [ABSTRACT FROM AUTHOR]

This article considers the problem of making exact inferences about a standard two parameter Weibull model. Exact inferences are obtained from the construction of a confidence set for the two parameters that has an exact specified confidence level. This exact confidence set is obtained from Kolmogorov's bounds on the empirical cumulative distribution function of the data set. Simple and compact equations are derived that define the confidence set in terms of a set of linear inequalities. The confidence set can be used, for example, to generate confidence bands for the reliability function, confidence intervals for the moments and quantiles of the distribution, and confidence intervals for the parameters themselves. The methodology can be extended to different models such as other two-parameter and three-parameter Weibull models. [ABSTRACT FROM AUTHOR]

This article addresses the problem of confidence band construction for a standard multiple linear regression model. An “independence point” method of construction is developed which generalizes the method of Gafarian (1964) for a simple linear regression model to a multiple linear regression model. Wynn (1984) pioneered the approach of basing confidence bands for a polynomial regression on a set of nodes where the function estimates are independent, and this approach is exploited in this article. This method requires only critical points from t-distributions so that the confidence bands are easy to construct. Both one-sided and two-sided confidence bands can be constructed using this method. An illustration of the new method is provided, and comparisons are made with other procedures. [ABSTRACT FROM AUTHOR]

The evaluation of the cumulative distribution function of a multivariate normal distribution is considered. The multivariate normal distribution can have any positive definite correlation matrix and any mean vector. The approach taken has two stages. In the first stage, it is shown how non–centred orthoscheme probabilities can be evaluated by using a recursive integration method. In the second stage, some ideas of Schläfli and Abrahamson are extended to show that any non–centred orthant probability can be expressed as differences between at most (m - 1)! non–centred orthoscheme probabilities. This approach allows an accurate evaluation of many multivariate normal probabilities which have important applications in statistical practice. [ABSTRACT FROM AUTHOR]

A frequentist simultaneous confidence interval procedure requires the predetermination of the comparisons and their corresponding forms of confidence intervals before viewing the data in order that the error probability is controlled at a preassigned level. This often renders it less sensitive to detecting actual true differences and may result in it including many noninformative inferences. On the other hand, by taking a Bayesian approach, we can select the comparisons of interest and construct corresponding joint credible intervals after having viewed the data. This enables us to focus on those significant differences of interest and consequently to be able to make sharper inferences. The joint posterior probability of the credible intervals play a similar role as the joint coverage probability of the simultaneous confidence intervals, that is, to guarantee, with at least that probability, all the inferences made using the intervals are correct at the same time. In this article, we consider some standard problems in simultaneous inference and discuss how a Bayesian approach may be implemented. The methodologies are illustrated with examples. [ABSTRACT FROM AUTHOR]