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The determination of plotting positions on probability graph paper so that the associated weighted least squares estimators of the scale parameter and the percentiles of a continuous distribution have certain properties is discussed. Necessary and sufficient conditions are given for an invariant optimal plot for percentile estimation. Also discussed is the derivation of ordered plotting positions. [ABSTRACT FROM AUTHOR]

The year 2008 marks the 100th anniversary of the publication of The Probable Error of a Mean by William Sealy Gosset, nom de plume "Student." Gosset's work and his relationships with the leading statisticians of his day have been considered by several authorities. Despite the extensive documentation, and the seminal nature of the work, modern-day statistics textbooks give him, and this 1908 article, short shrift. Thus, few of today's students-or their teachers-are aware of the "z" statistic whose sampling distribution he actually derived, the mathematical derivation, his simulations to check his work, the material used in the simulations, the table he produced, the "one-line" missing proof supplied by the 22-year-old Fisher (still a student himself) or the subsequent switch, in collaboration with Fisher, from the z to the t statistic. We remind readers of these aspects, and rework his calculations using 21st century computing power. We hope that the next generation of statisticians come to know more about the man and his work than simply that "he worked for the Guinness brewery," and appreciate that not all statistical distributions are derived in a single pass. Research students would do well to use his 1908 article as a model when writing their first statistical article. [ABSTRACT FROM AUTHOR]

This paper deals with simulation-based estimation of the probability distribution for completion time in stochastic activity networks. These distribution functions may be valuable in many applications. A simulation method, using importance-sampling techniques, is presented for estimation of the probability distribution function. Separating the state space into two sets, one which must be sampled and another which need not be, is suggested. The sampling plan of the simulation can then be decided after the probabilities of the two sets are adjusted. A formula for the adjustment of the probabilities is presented. It is demonstrated that the estimator is unbiased and the upper bound of variance minimized. Adaptive sampling, utilizing the importance sampling techniques, is discussed to solve problems where there is no information or more than one way to separate the state space. Examples are used to illustrate the sampling plan. [ABSTRACT FROM AUTHOR]

This paper proposes the use of approximate posterior distributions resulting from operational prior distributions chosen with regard to the realized likelihood function. L.J. Savage's "precise measurement" is generalized for approximation in terms of an arbitrary operational prior density, including mixed-type prior distributions with positive probabilities on singular subsets. A new approximation is also given relating such distributions to absolutely continuous distributions with high local concentrations of density, Mixed-type distributions constructed from the natural conjugate prior distributions are proposed and illustrated in the normal-sampling case for unified Bayesian inference in testing and estimation contexts. [ABSTRACT FROM AUTHOR]

The article presents information on a nomogram for the "Student"-fisher t test. A nomogram is given for estimating the probability (P) for a given value of the "Student"-Fisher t test. W.S. Gosset, an employee of the Guiness brewing company in Dublin, published papers in 1908 in which he correctly solved three problems: the probable error of a mean, the distribution of the mean divided by its estimated standard deviation and the distribution of the estimated correlation coefficient between independent variates. Later "Student" and economist R.A. Fisher calculated tables of the relevant t distribution and Fisher gives a table of t and probabilities, corresponding to various degrees of freedom. Fisher and F. Yates, scholar provide in addition a column for P. It seemed that presentation of the P, degrees of freedom, t relationship in the form of a nomogram would be advantageous. It makes possible a fairly exact estimate of probabilities less than 0.0001 and makes it possible to get an estimate of P for any value of t from 1 to 65, instead merely of selected values.

This paper deals with certain aspects of variance component estimation for the unbalanced one-way random classification where the number (N[sub I]) of observations in the ith class is treated as a random variable not necessarily independent of the class effect (A[sub iota]). It is assumed that in general P(N[sub I] = 0) > 0. The conditional expectations (given the number of observations in each class) of all estimators of the between variance component (sigma[sup 2, sub alpha]) belonging to a certain class of estimators are derived. A general expression is found for the expected value of that estimator of sigma[sup 2, sub alpha] yielded by analysis of variance of class means. The limit of this expression (as the number of classes arrow right Infinity) is given; and it is shown that, if the bivariate distribution function of A[sub I], N[sub I] belongs to a certain class of distribution functions, then this limit is less than sigma[sup 2, sub a]. Numerical approximations to the expected values of two estimators of sigma[sup 2, sub a] are presented for one subclass of such distribution functions. [ABSTRACT FROM AUTHOR]

The experiments to be discussed in this article was designed to investigate empirically the distribution of the sample version of the measure of association (G), &b.gamma;. Professors, William H. Kruskal and Leo A. Goodman have published an article giving interpretive motivation for the coefficient, and have developed the large sample theory for G. In the present research, sampling experiments were performed to test the adequacy of the large sample theory for 5X5 population cross classifications and samples of size ten, twenty-five and fifty. The numerical experiment described in this paper was based on a number of 25 cells multinomial populations, each considered as represented by the probabilities in a 5X5 population cross classification. Gamma is defined by Goodman and Kruskal in terms of probabilities. The 100 population cross classifications were chosen to be representative of practical situations in psychology. The method use to obtain them was to begin with 5X5 cross classification having complete association along the main diagonal and with a variety of marginal distributions.

Ever since the introduction of the correlation coefficient in 1888, there has been some confusion between zero correlation and statistical independence. We examine this, with emphasis on Student's famous 1908 paper leading to the t-test, and indicate some subsequent developments. [ABSTRACT FROM AUTHOR]

The detection and treatment of outliers in data has represented an important area of research. The general approach that has been adopted involves observing that outliers contribute to the "fatness" in the tails of the error distribution and using an appropriate model of the error term lo lake this into account. This article introduces a general class of distributions to capture nonnormalities in the data based on the generalized exponential family. This family of distributions provides great flexibility in modeling not only symmetric Tat-tailed distributions, but also skewed and possibly even multimaxial distributions. The approach taken is to use subordinate distributions within the generalized exponential family to model the empirical distribution. This family represents a generalization of the unimodal exponential family, which contains as subordinates the normal, gamma, beta, Student, and so forth. The parameters of the generalized exponential distribution can be estimated using maximum likelihood, weighted least squares, or minimum chi-squared methods, depending on the nature of the data. Special attention is given to analyzing the distributional properties of two subordinates of this family; namely, the generalized Student (and generalized lognormal distributions. The flexibility of these distributions is illustrated by applying them to two empirical problems and comparing the results with previous methods. [ABSTRACT FROM AUTHOR]