scispace - formally typeset
Search or ask a question

Showing papers in "Technometrics in 1970"


Journal ArticleDOI
TL;DR: In this paper, the use of ridge regression methods is discussed and recommendations are made for obtaining a better regression equation than that given by ordinary least squares estimation. But the authors focus on the RIDGE TRACE which is a two-dimensional graphical procedure for portraying the complex relationships in multifactor data.
Abstract: This paper is an exposition of the use of ridge regression methods. Two examples from the literature are used as a base. Attention is focused on the RIDGE TRACE which is a two-dimensional graphical procedure for portraying the complex relationships in multifactor data. Recommendations are made for obtaining a better regression equation than that given by ordinary least squares estimation.

2,345 citations


Journal ArticleDOI
TL;DR: Tests are grouped together primarily according to general type of mathematical derivation or type of statistical "information" used in'conducting the test, and mathematical interrelationships among the tests are indicated.
Abstract: As a result of an extensive survey of the literature, a large number of distribution-free statistical tests are examined. Tests are grouped together primarily according to general type of mathematical derivation or type of statistical \"information\" used in'conducting the test. Each of the more important tests is treated under the headings: Rationale, Null Hypothesis, Assumptions, Treatment of Ties, Efficiency, Application, Discussion, Tables, and Sources. Derivations are given and mathematical interrelationships among the tests are indicated. Strengths and weaknesses of individual tests, and of distribution-free tests as a class compared to parametric tests, are discussed.

2,104 citations


Journal ArticleDOI
Donald W. Marquaridt1
TL;DR: In this article, the authors discuss a class of biased linear estimators employing generalized inverses and establish a unifying perspective on nonlinear estimation from nonorthogonal data.
Abstract: A principal objective of this paper is to discuss a class of biased linear estimators employing generalized inverses. A second objective is to establish a unifying perspective. The paper exhibits theoretical properties shared by generalized inverse estimators, ridge estimators, and corresponding nonlinear estimation procedures. From this perspective it becomes clear why all these methods work so well in practical estimation from nonorthogonal data.

1,828 citations


Journal ArticleDOI
TL;DR: Weak Convergence in Metric Spaces as discussed by the authors is one of the most common modes of convergence in metric spaces, and it can be seen as a form of weak convergence in metric space.
Abstract: Weak Convergence in Metric Spaces. The Space C. The Space D. Dependent Variables. Other Modes of Convergence. Appendix. Some Notes on the Problems. Bibliographical Notes. Bibliography. Index.

1,564 citations


Journal ArticleDOI
TL;DR: In this paper, the authors present an introduction to matrices with applications in statistics, and present a set of matrices that can be used in statistics applications in the field of computer vision.
Abstract: (1970). Introduction to Matrices with Applications in Statistics. Technometrics: Vol. 12, No. 4, pp. 929-931.

518 citations


Journal ArticleDOI
TL;DR: In this article, the authors show that the advantages of the variable selection scheme in which independent variables are successively discarded one at a time from the original full set are not known to workers in this field.
Abstract: Recent reviews have dealt with the subject of which variables to select and which to discard in multiple regression problems. Lindley (1968) emphasized that the method to be employed in any analysis should be related to the use intended for the finally fitted regression. In the report by Beale et al. (1967), the emphasis is on selecting the best subset for any specified number of retained independent variables. Here we will be concerned with pointing out the advantages of the variable selection scheme in which independent variables are successively discarded one at a time from the original full set. While these advantages are not unknown to workers in this field, they are however not appreciated by the statistical community in general. For the purposes of this demonstration it is assumed that we are in the nonsingular case so that the number of observations exceeds the number of regressor variables. Let us begin by considering economy of effort. Suppose that we were using a step-up regression procedure, ignoring for the while its theoretical deficiencies (to be discussed later). We should then first fit k simple regressions, one for each of the k regressor variables considered, selecting the single most significant individual regressor variable. Having made this selection we would proceed with k - 1 additional fits to determine which of the remaining variables in conjunction with the first selected yielded the greatest reduction in residual variation. This process is continued on so as to provide a successive selection and ordering of variables. We may even require the ordering of all k variables, leaving for later decision what critical juncture is to be employed in determining which of the k variables to retain, which to reject-if we do so we shall have made a total of k(k + 1)/2 fits, albeit they may have differed greatly in their degree of complexity. A complete stepdown regression procedure however requires but k fits, as will now be indicated. Suppose we have done a multiple regression on all k variables and wish to consider the k possible multiple regressions on all sets of k - 1 variables, that is where 1 variable has been deleted. The results for these k possible multiple regressions are implicit in the initial k-variable regression, provided we have secured the inverse matrix, or at least its diagonal, necessary for testing the significance of the fitted partial regression coefficients. The case

256 citations


Journal ArticleDOI
TL;DR: In this paper, the confidence intervals for P (Y < X) are obtained under the assumption that X and Y are independently normally distributed and t.he distribution of Y is known.
Abstract: Confidence intervals for P (Y < X) are obtained under the assumptions that X and Y are independently normally distributed and t.he distribution of Y is known. The procedures of this paper are compared with a procedure suggested by Z. Govindarajulu.

213 citations


Journal ArticleDOI
TL;DR: In this paper, the authors present the results of a study of the maximum likelihood estimator, (t), of the reliability, R(t), when the two-parameter Weibull distribution is assumed.
Abstract: This paper presents the results of a study of the maximum likelihood estimator, (t), of the reliability, R(t), when the two-parameter Weibull distribution is assumed. It is shown that the distribution of (t) depends only upon R(t) and n. It is observed that (t) is very nearly unbiased and has a variance that is practically equal to the Cramer-Rao lower bound for the variance of an unbiased estimator. Tables of lower confidence limits for the reliability are also provided. For an observed value of (t), the lower confidence limit can be read directly from the table for confidence levels of .75, 30, 35, .90, .95, and .98. The large sample normal approxmation for (t) is also investigated. Tolerance limits based on the maximum likelihood estimators of the Weibull parameters are developed. It is found that the tables needed for obtaining confidence interval for R(t) also enable one to obtain lower tolerance intervals. An example is given to help clarify the procedure.

110 citations


Journal ArticleDOI
TL;DR: In this paper, the authors present a procedure for answering the question of whether a mixture of two normal distributions, with five known parameters, is unimodal or bimodal.
Abstract: This paper presents a procedure for answering the question of whether a mixture of two normal distributions, with five known parameters μ1, μ2, σ1, σ2, p, is unimodal or not. The approach of the study is based on a simple transformation, which might be useful for further investigation of the properties of such mixtures. It is shown, by a geometric interpretation, that a mixture of two normal distributions is either unimodal or bimodal. For finding the modes, a simple iterative procedure is given, which always converges. A brief table of the modes is provided for some values of the parameters. Some conditions under which we have a unique mode are studied. It is shown that, for this purpose, a general sufficient condition is |μ1 – μ2| ≤ 2 min (σ1, σ2). For the special case in which μ1 = μ2 = μ, a necessary and sufficient condition is presented, and from that the simple sufficient condition |μ1 – μ2| ≤ 2σ is derived.

109 citations


Journal ArticleDOI
TL;DR: In this article, the problem of selecting the best subsets of independent variables in linear regression analysis is reduced to identifying subsets for which the number of variables is small and applying these criteria to all possible subsets is not feasible.
Abstract: A number of criteria have been proposed for selecting the best subset or subsets of independent variables in linear regression analysis. Applying these criteria to all possible subsets is, in general, not feasible if the number of variables is large. Many of the criteria are monotone functions of the residual sum of squares hence the problem is reduced to identifying subsets for which this quantity is small. In an earlier paper (Selection of the Best Subset in Regression Analysis by R. R. Hocking and R. N. Leslie, 1967) a method was described for identifying such subsets without considering all possible subsets. However, the amount of computation required if more than fifteen independent variables were considered was excessive. The present paper extends the basic ideas in that paper so that moderately large problems can how be treated with what appears to be a minimum of computation.

97 citations


Journal ArticleDOI
TL;DR: In this paper, the authors compared the performance of Krutchkoff's method with the classical method from other points of view, such as consistency, consistency, and mean square error of the relevant asymptotic distributions.
Abstract: A procedure suggested by Krutchkoff (1967) f or inverse estimation in linear regression is compared with the classical procedure from other points of view than that taken by Krutchkoff, i.e. comparative mean square error. In particular, comparisons are made on the basis of “closeness” in estimation (Pitman, 1937), consistency (in a setting where this concept is relevant), and mean square error of the relevant asymptotic distributions. It is found that, for large samples, Krutchkoff's estimate is superior in the sense of “closeness” if values of the independent variable are restricted to a certain closed interval around the mean of the independent variates in the experiment and inferior elsewhere. However, the width of this interval varies inversely as the product of the absolute value of the standardized slope (i.e. scaled by the error standard deviation) and the standard deviation of the independent variables used in the experiment. As a practical matter the parameter tends to be large so that the interv...

Journal ArticleDOI
TL;DR: In this article, the authors describe a method of process improvement which supplements the more orthodox studies and is run in the normal course of production by plant personnel themselves, and the basic philosophy is introduced that industrial processes should be run so as to generate not only product, but also information on how the product can be improved.
Abstract: The rate at which industrial processes are improved is limited by the present shortage of technical personnel. Dr Box describes a method of process improvement which supplements the more orthodox studies and is run in the normal course of production by plant personnel themselves. The basic philosophy is introduced that industrial processes should be run so as to generate not only product, but also information on how the product can be improved.

Journal ArticleDOI
TL;DR: These results are valid also for centrosymmetric matrices, which include Toeplitz matrices as a special case, and if a kernel is an even function of its vector argument (x, t), then it can be discretely approximated by a centroSymmetric matrix.
Abstract: (1970). The Inverse of a Centrosymmetric Matrix. Technometrics: Vol. 12, No. 4, pp. 925-928.

Journal ArticleDOI
TL;DR: In this paper, the authors compare the results of the approximate results with the exact ones in order to determine the quality of the approximations, and conclude that the latter is better than the former.
Abstract: It may be of some interest in certain statistical problems to have available percentage points of the sample coefficient of variation (S.C.V.). If the parent population is assumed to be normal, the obtaining of such percentage points from tables of the non-central t distribution might be thought to be simple. However, one does encounter difficulty in using these tables to calculate the percentage points of the S.C.V. One can, as an alternative, use the exact table of Iglewicz (1967). A number of approximations for the percentage points of the S.C.V. exists in the literature. These approximations may be useful as alternatives to either of the tables mentioned above. The applicability of several of these approximations is further enhanced by their simplicity. In this paper we compare these approximate results with the exact ones in order to determine the quality of the approximations.

Journal ArticleDOI
TL;DR: In this article, the optimal experimental designs for estimating the parameters in certain nonlinear models in various situations are given, and it is noted that these designs usually consist of suitable replicates of a small set of experiments.
Abstract: The optimal experimental designs for estimating the parameters in certain nonlinear models in various situations are given, and it is noted that these designs usually consist of suitable replicates of a small set of experiments, the number of which is equal to the number of unknown parameters. In view of the computational difficulty in obtaining the optimal experimental designs, it is suggested that the experience gained with these examples be regarded as typical, and on this basis computational “short cuts” to the optimal (or at least near optimal) designs are available. Also discussed is a grave disadvantage of replicated optimal designs, namely that such experimentation is of little use in highlighting inadequacies of the assumed model, if any exist.

Journal ArticleDOI
TL;DR: In this article, a method for estimation of the parameters of a compound Weibull distribution with two shape parameters, two scale parameters and a proportionality factor is presented. But the model is not suitable for the analysis of atmospheric data, as the distributions encountered are often a result of combining two or more component distributions.
Abstract: The two-parameter Weibull distribution has been recognized as a useful model for survival populations associated with reliability studies and life testing experiments. In the analysis of atmospheric data, the distributions encountered are often a result of combining two or more component distributions. These compound distributions are consequently of interest to aerospace scientists. Presented is a method for estimation of the parameters of a compound Weibull distribution with two shape parameters, two scale parameters and a proportionality factor.

Journal ArticleDOI
TL;DR: In this paper, it was shown that it is possible to estimate the parameters using a general computer program in which the derivatives of the residuals with respect to the independent variables, which are needed to compute the effective variance covariance matrices, are estimated numerically.
Abstract: and secondly to show that in these situations it is possible to estimate the parameters using a general computer program in which the derivatives of the residuals with respect to the independent variables, which are needed to compute the effective variance-covariance matrices, are estimated numerically. This eliminates much tedious algebraic manipulation, and provides the user with a statistically advanced technique which is no more difficult to use than "least squares" estimation.

Journal ArticleDOI
TL;DR: In this paper, the problem of identifying the population of origin of each observation in a sample thought to be the result of mixing a random sample of size N 1, from a gamma distribution with scale parameter σ 1 and an independent random sample from another gamma distribution having scale parameters σ 2 was studied.
Abstract: The gamma distribution, known also as the Erlangian distribution, and its special case the exponential distribution arise in many technological applications of statistics. The present note is on the problem of identifying the population of origin of each observation in a sample thought to be the result of mixing a random sample of size N 1, from a gamma distribution with scale parameter σ1 and an independent random sample of size N 2 from another gamma distribution with scale parameter σ2. We shall also be interested in the estimation of σ1 and σ2. The method of moments and the maximum likelihood method are applied to the solution of these problems.

Journal ArticleDOI
TL;DR: The authors suggests that Mantel exaggerates the advantages of the backward elimination or %tepdown procedure, and suggests that the reverse elimination procedure is more efficient than the % tepdown method.
Abstract: Mantel (1970) has pointed out that many procedures are now available for selecting variables in multiple regression analyses. This note reviews the more important ones briefly, and suggests that Mantel exaggerates the advantages of the backward elimination or %tepdown” procedure.

Journal ArticleDOI
TL;DR: Sample size tables for tolerance limits on a normal distribution are given in this article, where the criterion used for determining sample size is as follows: for a tolerance limit such that Pr (coverage ≥ P) = γ, choose P′ > P and δ (small) and require Pr (cover ≥ P′) ≤ δ.
Abstract: Sample size tables are given for tolerance limits on a normal distribution. Wald-Wolfowitz two-sided limits and one-sided limits are considered. The criterion used for determining sample size is as follows: For a tolerance limit such that Pr (coverage ≥ P) = γ, choose P′ > P and δ (small) and require Pr (coverage ≥ P′) ≤ δ. Five levels of P, three levels of γ, three levels of P′, and three levels of δ are used in the tables. The tables are given for the common case where the degrees of freedom for the x2 is one less than the sample size, but it is shown how to use the tables for other cases which occur in simple linear regression and some experimental designs. Examples are given to illustrate the use of the tables.

Journal ArticleDOI
TL;DR: In this paper, the authors discuss some ranking and selection procedures for multivariate normal populations in terms of measures of dispersion using the generalized variance as the measure of the dispersion, a procedure is defined to select a subset of the populations which would include the population with the smallest generalized variance with a certain probability.
Abstract: The present paper discusses some ranking and selection procedures for multivariate normal populations in terms of measures of dispersion. Using the generalized variance as the measure of dispersion, a procedure is defined to select a subset of the populations which would include the population with the smallest generalized variance with a certain probability. Two approximations to the distribution of the sample generalized variance are discussed. A discussion of the properties of the above procedure is also included.

Journal ArticleDOI
TL;DR: In this paper, the possibility of practical use of the concept of D-optimality in response surface design was reviewed and a number of different second order response surface designs, previously built, have been compared.
Abstract: The purpose of this article is to review the possibility of practical use of the concept of D-optimality in response surface design. Second order quasi-D-optimum designs on a cube have been built and their high efficiency proved. Using the computer, a number of different second order response surface designs, previously built, have been compared. It has been proved that the D-optimum concept, being more general than the rotatable one, can be used as the theoretical basis for building and comparing response surface designs in use.

Journal ArticleDOI
TL;DR: A new algorithm based upon implicit enumeration is presented and comparative computational experience shows that the new algorithm requires significantly less computer time than existing algorithms.
Abstract: The problem of enumerating all proper cuts of a linear graph arises in several reliability applications and is usually solved by algebraic algorithms. We present a new algorithm based upon implicit enumeration. In addition, we present comparative computational experience which shows that the new algorithm requires significantly less computer time than existing algorithms.

Journal ArticleDOI
Nancy R. Mann1
TL;DR: In this paper, the problem of obtaining exact confidence bounds for the shape parameter and for reliable life is considered in detail, and a two-parameter Weibull model is assumed.
Abstract: A two-parameter Weibull model is assumed, and the problem of obtaining exact confidence bounds for the shape parameter and for reliable life is considered in detail. Analytically derived bounds for both of these parameters based on only a few ordered observations are shown to be highly efficient with respect to those derivable by Monte Carlo procedures using all the ordered observations. Tables are given for obtaining bounds on and estimates for the shape parameter and reliable life for two values of a specified survival proportion.

Journal ArticleDOI
TL;DR: In this article, a new lower bound procedure based on the modular decomposition of a coherent structure is proposed, which is shown to provide a sharper lower bound estimate of the system reliability than the Esary Proschan procedure and is computationally more efficient.
Abstract: Since the computation of system reliability is often a difficult task, approximation procedures are needed. Esary and Proschan have developed a procedure for finding a lower bound estimate of the system reliability of a coherent structure. In this paper, a new lower bound procedure based on the modular decomposition of a coherent structure is proposed. It is shown that this procedure provides a sharper lower bound estimate of the system reliability of a coherent structure than the Esary Proschan procedure and is computationally more efficient.

Journal ArticleDOI
TL;DR: In this article, the problem of determining the probability that an assignable cause occurs for the first time following the kth sample is solved by inverting a matrix, and only one element of the inverse is required.
Abstract: A zone test in quality control calls for a search for an assignable cause whenever n out of N consecutive points fall in some predetermined zone, say, above μ + cσ. Determining the probability that this event occurs for the first time following the kth sample is a classic problem. The generating function of this probability (and hence the average run length) can be obtained by inverting a matrix. Since only one element of the inverse is required, the calculations can be greatly simplified.

Journal ArticleDOI
TL;DR: Based on a completely random three-stage nested model, five fundamental sampling structures are defined and 61 designs were enumerated such that each design contained no more than three fundamental structures and a multiple of twelve third-stage samples such that the designs would permit the ANOVA estimation of all three variance components.
Abstract: Based on a completely random three-stage nested model, five fundamental sampling structures are defined. From the five fundamental structures, 61 designs were enumerated such that each design contained no more than three fundamental structures and a multiple of twelve third-stage samples such that the designs would permit the ANOVA estimation of all three variance components. The table of designs contains in addition to the design structure, a design code for classifying the designs as well as a set of four coefficients that simpliiy the computation of the expected mean squares for the design. General formulas for the variances and covariances of the ANOVA estimators are presented in an appendix. Three criteria derived from the covariance matrix of the variance component estimators are proposed and then used to compare the designs as well as determine the optimum design for 49 diierent variance component configurations and 10 diierent sample sizes.

Journal ArticleDOI
TL;DR: In this article, a Monte Carlo design is presented for estimating the variance and cumulative distribution function of translation and scale invariant statistics based on independent Student random variables, and the method itself amounts to suppressing some of the variability in the sampled objects by integrating these objects over appropriate regions of the underlying probability space.
Abstract: A Monte Carlo design is presented for estimating the variance and cumulative distribution function of translation and scale invariant statistics based on independent Student random variables. One obvious application is studying estimates of the location parameter from a symmetric, possibly long-tailed distribution. The method itself amounts to suppressing some of the variability in the sampled objects by integrating these objects over appropriate regions of the underlying probability space. Indications are that, in cases of interest, the variability is thereby considerably reduced, as is illustrated in an application concerning trimmed and Winsorieed means.

Journal ArticleDOI
TL;DR: In this article, some rough tests for bivariate normality are employed in an attempt to quantify the intuitive notion that coordinate transformations to normality produce distributions which are "more bivariate normal" than the original variables.
Abstract: Some rough tests for bivariate normality are employed in an attempt to quantify the intuitive notion that coordinate transformations to normality produce distributions which are “more bivariate normal” than the original variables. These tests are not rigorous procedures but are intuitively satisfying, based on natural statistics, and provide numerical measures of the “distance” of a bivariate distribution from the normal model. It is shown that, for a wide class of non-normal (X, Y) distributions, coordinate transformations to normality decrease this distance as measured by these tests. It is indicated how one may estimate the coordinate transformations and applications to correlation theory are explored.

Journal ArticleDOI
TL;DR: In this paper, a comparison of two measuring devices whose readings are subject to random variations involves comparing their means to determine if they differ significantly and comparing their variances to determine the difference significantly.
Abstract: The comparison of two measuring devices whose readings are subject to random variations involves 1) comparing their means to determine if they differ significantly and 2) comparing their variances to determine if they differ significantly. For the problem considered, the available data consist of one reading with one device and two additional independent readings with a second device on each of a number of units. An estimate of the difference in means can be readily obtained from the difference in the average readings for the two devices, and an estimate of the reading error variance for the second device is obtained from the differences in the two readings with that device. A method is also developed for estimating the reading error variance for the measuring device with only a single reading per unit. Procedures are given for testing hypotheses concerning the average difference in readings between the two devices, for testing the hypothesis that the reading error standard deviations are identical, for o...