Search

Applications in operations research often employ models which contain linear functions. These linear functions may have some components (coefficients and variables) which are random. (For instance, linear functions in mathematical programming often represent models of processes which exhibit randomness in resource availability, consumption rates, and activity levels.) Even when the linearity assumptions of these models is unquestioned, the effects of the randomness in the functions is of concern. Methods to accommodate, or at least estimate for a linear function the implications of randomness in its components typically make several simplifying assumptions. Unfortunately, when components are known to be random in a general, multivariate dependent fashion, concise specification of the randomness exhibited by the linear function is, at best, extremely complicated, usually requiring severe, unrealistic restrictions on the density functions of the random components. Frequent stipulations include assertion of normality, or of independence-yet, observed data, accepted collateral theory and common sense may dictate that a symmetric distribution with infinite domain limits is inappropriate, or that a dependent structure is definitely present. (For example, random resource levels may be highly correlated due to economic conditions, and nonnegative for physical reasons.) Often, an investigation is performed by discretizing the random components at point quantile levels, or by replacing the random components by their means-methods which give a deterministic "equivalent" model with constant terms, but possibly very misleading results. Outright simulation can be used, but requires considerable time investment for setup and debugging (especially for generation of dependent sequences of pseudorandom variates) and gives results with high parametric specificity and computation cost. This paper shows how to use elementary methods w estimate the mean and variance of a linear... [ABSTRACT FROM AUTHOR]

A recurring statistical problem is the complete ordering of a set of k distributions (or populations) with respect to some parameter. A related problem is the selection of the largest or smallest parameter of k distributions. In this paper we examine the characteristics of an ad hoc procedure for populations which admit a x² statistic, as a function of an estimator for the population parameter, and use these statistics to order the populations. Lower bounds for the probability of correct ranking, of selecting the smallest, and of selecting the largest parameter are provided as a function of degrees of freedom and the number of populations to be ordered, k, for k = 3(1)8. [ABSTRACT FROM AUTHOR]

This paper compares two point estimators of fraction defective of a normal distribution when both population parameters are unknown; the minimum variance unbiased estimator, &Fmacr;(χ), and the maximum likelihood estimator, &Fcirc;(χ). Using minimum mean squared error as a criterion, it is shown that the choice of estimator depends upon the true value of F(χ), and the sample size. In the domain .0005 ≤ F(χ) ≤ .50, the maximum likelihood estimator is generally superior even for small sample sizes, except for F(χ) less than about 0.01, or greater than 0.25. Furthermore, the bias in the m.l.e. is slight over much of the domain where this estimator has smaller mean squared error. As a practical solution to the estimation problem, it is suggested that the m.v.u.e. be calculated, and if this estimate is between 0.01 and 0.25, it should be replaced with the m.l.e. This combined estimator is shown to be nearly as efficient as the better of the m.v.u.e. and m.l.e. throughout the domain of F(χ). [ABSTRACT FROM AUTHOR]

The occurrence of early failures in a fixed-sample acceptance test, where the sample observations are obtained sequentially, presents an interesting decision problem. It may be desirable to abandon the test at an early stage if the conditional probability of passing is small and the testing cost is high. This paper presents a stopping rule based on the maximum-likelihood estimate of total costs involved in the decision to continue beyond an early failure. A Bernoulli model, an exponential model, and a Weibull model are examined. [ABSTRACT FROM AUTHOR]

A method is suggested for obtaining a multiple linear regression equation which permits the variance, as well as the mean, of normally distributed random variables, Y, to be a function of known constants X[sub 1] ... X[sub p]. The method is applicable to large samples, and yields an approximation of maximum likelihood estimates of regression coefficients, which may be used to construct confidence intervals for the parameters of the normal distribution and tolerance intervals for individual Y. A likelihood ratio test may be used to test this model against the usual homoscedastic least squares model. Two examples of this type of analysis on data from the literature are presented. [ABSTRACT FROM AUTHOR]

The maximum likelihood estimator and minimum variance unbiased estimator of the probability of zero failures in m binomial trials were compared on the basis of expected squared estimation error. The choice of estimator proved to be a function of the region of parameter space. However, the m. l. e. was preferred throughout the regions likely to be encountered in practice (where the probability of zero failures exceeded 0.5). [ABSTRACT FROM AUTHOR]

Minimum variance unbiased estimators of reliability are developed for a series, parallel, and standby redundant system where the components are from a single exponentially distributed population. It is assumed that sampling information has been censored with respect to number of failures or test time. [ABSTRACT FROM AUTHOR]

An article by D.S. Tull described the results of a regression analysis relating forecast and actual sales and profits for sixty-three new products. While it was recognized that the data were heteroscedastic, a traditional least-squares regression model was used including the assumption of homoscedasticity. H C. Rutemiller and D. A. Bowers have since developed a least-squares regression model which allows for heteroscedasticity represented through an equation. This note reports the results of a regression analysis of the same data using the Rutemiller-Bowers model and compares them with the results obtained in the original study. The technique utilized by Tull was to fit a linear regression line-- with the forecasts as a dependent variable and the actual sales or profits as an independent variable--by the method of least squares. The regression coefficients were tested jointly for significance, using as the null hypothesis an intercept of zero and a slope of unity (perfect forecasting).