Showing papers in "Technometrics in 1973"
TL;DR: In this article, the generalized inverse of matrices and its applications are discussed and discussed in terms of generalized inverse of matrix and its application in the context of generalization of matrix matrices.
Abstract: (1973). Generalized Inverse of Matrices and Its Applications. Technometrics: Vol. 15, No. 1, pp. 197-197.
2,402 citations
TL;DR: In this paper, a generalization of the Poisson distribution with two parameters λ1 and λ2 is obtained as a limiting form of the generalized negative binomial distribution, where the variance of the distribution is greater than, equal to or smaller than the mean according as λ 2 is positive, zero or negative.
Abstract: A new generalization of the Poisson distribution, with two parameters λ1 and λ2, is obtained as a limiting form of the generalized negative binomial distribution. The variance of the distribution is greater than, equal to or smaller than the mean according as λ2 is positive, zero or negative. The distribution gives a very close fit to supposedly binomial, Poisson and negative-binomial data and provides with a model suitable to most unimodel or reverse J-shaped distributions. Diagrams showing the variations in the form of the distribution for different values of λ1 and λ2 are given.
518 citations
TL;DR: The estimation of parameters in nonlinear algebraic models is considered for a general class of problems where both the independent and dependent variables are subject to error and problems where the model is algebraically implicit.
Abstract: The estimation of parameters in nonlinear algebraic models is considered for a general class of problems. Included are problems where both the independent and dependent variables are subject to error and problems where the model is algebraically implicit. The maximum likelihood principle leads to an equality constrained minimization problem. An algorithm to achieve this minimization is obtained through the use of Lagrange multipliers.
431 citations
TL;DR: The Theory of Optimal Stopping as mentioned in this paper is a theory of optimal stopping that is based on the Great Expectations theory of stopping, and it can be seen as an alternative to optimal stopping.
Abstract: (1973). Great Expectations: Theory of Optimal Stopping. Technometrics: Vol. 15, No. 3, pp. 652-653.
382 citations
TL;DR: In this paper, the authors present the theory and development of an algorithm associated with the exploration of a dual response surface system, where a user can generate simple two dimensional plots to determine the conditions of constrained maximum primary response regardless of the number of independent variables in the system.
Abstract: The purpose of this paper is to present the theory and develop an algorithm associated with the exploration of a dual response surface system. The approach is to find conditions on a set of independent or “design” variables which maximize (or minimize) a “primary response” function subject to the condition that a “constraint response” function takes on some specified or desirable value. A method is outlined whereby a user can generate simple two dimensional plots to determine the conditions of constrained maximum primary response regardless of the number of independent variables in the system. He thus is able to reduce to simple plotting the complex task of exploring the dual response system. The procedure that is used to generate the plots depends on the nature of the individual univariate response functions. In certain situations it becomes necessary to apply the additional constraint that the located operating conditions are a certain “distance” from the origin of the independent variables (or the cent...
287 citations
TL;DR: In this article, the ratio of maximized likelihoods provides a good test for selecting one of these distributions, and a table of the necesqary critical values is given, which may also be used for discriminnting between the normal and the type 1 extreme value distributions.
Abstract: The log-normal and the Weibull are often considered for situations in which a skewed distribution for a non-negative random variable is needed. The ratio of maximized likelihoods provides a good test for selecting one of these. A table of the necesqary critical values is given. The table may also be used for discriminnting between the normal and the type 1 extreme value distributions.
211 citations
TL;DR: In this article, the instability of ordinary least squares estimates of linear regression coefficients is demonstrated for mortality rates regressed around various socioeconomic, weather and pollution variables, and two methods of variable elimination are compared.
Abstract: The instability of ordinary least squares estimates of linear regression coefficients is demonstrated for mortality rates regressed around various socioeconomic, weather and pollution variables. A ridge regression technique presented by Hoer1 and Kennard (Technometrics 12 (1970) 69–82) is employed to arrive at “stable” regression coefficients which, in some instances, differ considerably from the ordinary least squares estimates. In addition, two methods of variable elimination are compared—one based on total squared error and the other on a ridge trace analysis.
TL;DR: In this paper, the ridge estimators are viewed as a subclass of the class of linear transforms of the least squares estimator and an alternative class of estimators, labeled shrunken estimators is considered, which satisfy the admissibility condition proposed by Hoer1 and Kennard.
Abstract: Hoer1 and Kennard introduced a class of biased estimators (ridge estimators) for the parameters in an ill-conditioned linear model. In this paper the ridge estimators are viewed as a subclass of the class of linear transforms of the least squares estimator. An alternative class of estimators, labeled shrunken estimators is considered. It is shown that these estimators satisfy the admissibility condition proposed by Hoer1 and Kennard. In addition, both the ridge estimators and shrunken estimators are derived as minimum norm estimators in the class of linear transforms of the least squares estimators. The former minimizes the Euclidean norm and the latter minimizes the design dependent norm. The class of estimators which are minimum variance linear transforms of the least squares estimator is obtained and the members of this class are shown to be stochastically shrunken estimators. An example is computed to show the behavior of the different estimators.
TL;DR: In this paper some problems associated with fitting multiresponse models are identified and discussed, and a method is described for detecting and handling dependencies in the data.
Abstract: Experience has shown that unless special care is exercised in analyzing multiresponse data serious mistakes can be made. In this paper some problems associated with fitting multiresponse models are identified and discussed. In particular, three kinds of dependencies are considered: dependence among the errors, linear dependencies among the expected values of the responses, and linear dependencies in the data. Since ignoring such dependencies can lead to difficulties, a method is described for detecting and handling them. The concepts involved are illustrated with a chemical example.
TL;DR: The W transformation as discussed by the authors is a matrix transformation for the mixed analysis of variance model to compute maximum likelihood estimates of the variance components and fixed parameters, which does not depend upon the number of observations in any way.
Abstract: The W transformation, a matrix transformation, is developed and applied for the mixed analysis of variance model to compute maximum likelihood estimates of the variance components and fixed parameters. This transformation not only eliminates the need for the explicit computation of the n × n inverse matrix H−l but permits handling the iterative calculations such that they do not depend upon n (the number of observations) in any way. Although not wedded to a particular numerical method, the W transformation is implemented in conjunction with a modified Newton-Raphson method in which variance components are restricted to being non-negative.
TL;DR: In this article, the minimum variance unbiased estimator of Pr {Y < X} under the assumption that X and Y are independently normally distributed is derived, and approximate approximations to this estimator are suggested as alternatives to the asymptotically equivalent estimator used by Church and Harris.
Abstract: The minimum variance unbiased estimator of Pr {Y < X} under the assumption that X and Y are independently normally distributed is derived. Approximations to this “best” estimator are obtained, and these approximations are suggested as alternatives to the asymptotically equivalent estimator used by Church and Harris (1970) to obtain confidence intervals for that probability.
TL;DR: In this paper, a discussion is given of some of the pertinent literature for estimating variances in errors of measurement, or the "imprecisions" of measurement when two or three instruments are used to take the same observations on a series of items or characteristics.
Abstract: A very important and yet widely misunderstood concept or problem in science and technology is that of precision and accuracy of measurement. It is therefore necessary to define the terms precision and accuracy (or imprecision and inaccuracy) clearly and analytically if possible. Also, we need to establish and develop appropriate statistical tests of significance for these measures, since generally a relatively small number of measurements will be made or taken in most investigations. In this paper a discussion is given of some of the pertinent literature for estimating variances in errors of measurement, or the “imprecisions” of measurement, when two or three instruments are used to take the same observations on a series of items or characteristics. Also, present techniques for comparing the imprecision of measurement of one instrument with that of a second instrument through the use of statistical tests of significance are reviewed, as well as procedures for detecting the significance of the difference i...
TL;DR: In this paper, modifications of the two-sided one-sample Kolmogorov-Smirnov "goodness-of-fit" test, for use with censored and truncated samples, are suggested.
Abstract: Modifications of the two-sided one-sample Kolmogorov-Smirnov “goodness-of-fit” test, for use with censored and truncated samples, are suggested. Tables of the distributions of the modified statistics are given. Applications to life testing and reliability estimation problems are discussed.
TL;DR: In this paper, the authors deal with methods of quality control when observation-vectors are coming from a multivariate binomial or multivariate Poisson population, under certain assumptions.
Abstract: This paper deals with methods of quality control when observation-vectors are coming from a multivariate binomial or multivariate Poisson population. The case when successive observations are time-dependent has been studied under certain assumptions. Further when a sample covariance matrix is singular or near singular, a method has been proposed for translating it into a non-singular estimate under the assumption of conditional independence. Assuming approximate normality a x 2 -chart has been proposed. At the end a numerical example is given to illustrate the procedure in this paper.
TL;DR: In this article, the authors present tables for obtaining confidence bounds for the two parameters and the 90th, 95th, and 99th percentiles of the two-parameter Weibull or extreme-value distributions.
Abstract: Tables are given for obtaining confidence bounds for the two parameters and the 90th, 95th, and 99th percentiles of the two-parameter Weibull or extreme-value distribut.ions. The tables are based on best linear invariant estimlators of extreme-value location and scale parameters and apply to samples of size n, n = 3(1)16, which may be censored at the mth smallest sample observation, m = 3(1)%. More extensive tables for samples of size n, n = 3(1)25, with censoring from m = 3(1)n, may be found in [12]. Discussion is given concerning other methods of obtaining confidence and tolerance bounds for these distributions, properties of the estimators on which the bounds are based and computational procedures used.
TL;DR: In this article, the problem of selecting a model from two models with unknown location and scale parameters is considered, and it is shown that the distribution of the ratio of maximum likelihoods does not depend upon the values of the nuisance locations and scales.
Abstract: The problem of selecting a model from two models with unknown location and scale parameters is considered. It is noted that the distribution of the ratio of maximum likelihoods does not depend upon the values of the nuisance location and scale parameters. Consequently, this ratio provides a convenient (and hopefully powerful) test for discriminating between two location and scale parameter models when the parameters are unknown. Tables for discriminating between the normal and the Cauchy; the normal and the exponential; or the normal and the double exponential are presented. An empirical comparison of the power of this test with the power of the UMP invariant test for discriminating between the normal and the Cauchy is given. Some further comparisons with the chi-square goodness of fit and the Kolmogorov-Smirnov tests for normality are also given.
TL;DR: A review of the literature concerning experiments with mixtures is presented in this article, starting with the work of Scheffe in 1958, the review covers approximately twenty papers in near chronological order, the last appearing in October 1971.
Abstract: A review of the literature concerning experiments with mixtures is presented Beginning with the work of Scheffe in 1958, the review covers approximately twenty papers in near chronological order, the last appearing in October 1971 Although priority in the treatment of designs for mixtures is claimed by Quenouille [27], and while others may argue that the work of Claringbold [6] was the impetus for Scheffe's research, the choice of starting with Scheffe is made because most of the authors of the reviewed papers recognized Scheffe as the person most instrumental in pioneering the work which has appeared over the past fourteen years
TL;DR: In this paper, confidence intervals based on the maximum likelihood estimators for the location and scale parameters of the Double Exponential distribution are given by determining the distribution of the pivotal quantities ( − θ)/ and /σ.
Abstract: Confidence intervals based on the maximum likelihood estimators are given for the location and scale parameters of the Double Exponential distribution. These intervals are obtained by determining the distribution of the pivotal quantities ( – θ)/ and /σ. Exact distributions are determined for n = 3 and n = .5, and approximate distributions are provided for larger n. The asymptotic distributions are also given and the accluacy of these approximations are investigated. The powers of the associated tests of hypotheses are given and tolerance limits for the population are also provided. Some possible applications are indicated.
TL;DR: In this paper, a simple unbiased estimator, based on a censored sample, has been proposed for the scale parameter of the Extreme-value distribution, which is shown to have high efficiency and to be approximately distributed as a chi-square variable if substantial censoring occurs.
Abstract: A simple, unbiased estimator, based on a censored sample, has been proposed by Rain [1] for the scale parameter of the Extreme-value distribution. This estimator was shown to have high efficiency and to be approximately distributed as a chi-square variable if substantial censoring occurs. Further small sample and asymptotic properties of this estimator are considered in this paper. The estimator is modified so that it is more applicable to the complete sample case and a close chi-square approximation is established for all cases. The estimator is also shown to be related to the maximum likelihood estimator.
TL;DR: In this article, a procedure of studentizing or standardizing the residuals by dividing them by their estimated standard deviations is proposed for testing for outliers in simple linear regression, and it is shown that these values are extremely close to the values in Grubbs' (1950) Table IA and independent of the values taken for the independent variable.
Abstract: A procedure of studentizing or standardizing the residuals by dividing them by their estimated standard deviations is proposed for testing for outliers in simple linear regression. A simulation study yielded the critical values of the test statistic, and it is shown that these values are extremely close to the values in Grubbs' (1950) Table IA and independent of the values taken for the independent variable.
TL;DR: In this paper, it is shown that mixture models often can be simplified by observing near equalities among the estimated coefficients and that expressing the components in terms of pseudocomponents will sometimes improve the fit of Becker's models and improve the numerical analysis.
Abstract: Techniques for the analysis of mixture data are discussed. It is shown that mixture models often can be simplified by observing near equalities among the estimated coefficients. Expressing the components in terms of pseudocomponents will sometimes improve the fit of Becker's models [1] and improve the numerical analysis. Scatter plots are useful in detecting dominant components in a mixture system. The ratio model is shown to be a useful alternative to the polynomial and Becker's models. Examples are presented to illustrate these methods of analysis.
TL;DR: Several zone tests for quality-control charts are considered, and a simple means is given for finding the average-run-length of procedures that combine a zone test with the standard 3σ-limit.
Abstract: Several zone tests for quality-control charts are considered. One test calls for initiating a search for an assignable cause whenever n out of N consecutive points fall in some predetermined warning zone. An algorithm is given for obtaining the average-run-length of this procedure; and several alternative procedures, for which the average-run-lengths are explicitly given, are suggested. Also, a simple means is given for finding the average-run-length of procedures that combine a zone test with the standard 3σ-limit.
TL;DR: In this article, a modification of the "V" mask control scheme for cusum charts is proposed, where a parabolic section is included in the mask to provide better performance when the process undergoes a large change in the mean from goal conditions, it is shown that the modified mask can be implemented either in conventional graphic form or in an algorithmic form suitable for a digital computer.
Abstract: A modification of the “V”' mask control scheme for cusum charts is proposed. In this modified scheme, a parabolic section is included in the mask to provide better performance when the process undergoes a large change in the mean from goal conditions, It is shown that the modified “V” mask can be implemented either in conventional graphic form, or in an algorithmic form suitable for a digital computer. Average run lengths are given for a typical range of circumstances.
TL;DR: In this paper, the problem of estimating Y (1), the smallest of a future sample of k observations from the Weibull distribution, based on an observed sample from the same distribution, is considered.
Abstract: The problem of estimating Y (1), the smallest of a future sample of k observations from the Weibull distribution, based on an observed sample from the same distribution, is considered. A conditional confidence interval solution is proposed for the estimation of Y (1). The results have direct application in reliability theory, where the time until the first failure in a group of k items in service provides a measure of assurance regarding the operation of the items.
TL;DR: In this article, the authors used the C p search technique to find a new form of the equation for manufacturing and marketing data in which either new form is found or a new insight is gained into the limitations and strengths of the fitted equation.
Abstract: Componentrplus-residual plots, plots of the estimated effect of each independent variable at each observed data point plus corresponding residuals, are used as an aid (1) to choose an appropriate form of the equation, (2) to observe the distribution of the observations over the range of each independent variable and (3) to estimate the influence of each observation on each component of the equation With the help of indicator variables and the C p search technique, far-out observations can be tested to see if their responses are compatible with those of the remainder of the data Examples are given from studies of manufacturing and marketing data in which either a new form of the equation is found or a new insight is gained into the limitations and strengths of the fitted equation
TL;DR: The trivariate theory and some suggested forms of tabulating the plans are given and two fields of potential application are proposed, one concerning hazardous substances in foods and the other amounts of active ingredient in dosage forms of drugs.
Abstract: The 3-Class Attributes Plan is defined by recognizing three categories of quality, say good, marginal and bad. This device makes it possible to avoid calling an item simply acceptable or unacceptable as is done in the usual a-Class Attributes Plan. The trivariate theory and some suggested forms of tabulating the plans are given. Two fields of potential application are proposed, one concerning hazardous substances in foods and the other amounts of active ingredient in dosage forms of drugs.