Search

In a paper by R. G. Krutchkoff, “Classical and Inverse Regression Methods of Calibration,” two methods of Calibration are compared by means of the Monte Carlo technique. He comes to the conclusion that the inverse approach to the Calibration problem has a uniformly smaller mean square error than the classical approach. Our theoretical treatment of the problem shows that the conclusion is true only for small samples. It is shown that when more than one observation is made on y the advantage of the inverse approach is reduced. A large sample formula for the ratio of the two mean square errors is derived which mainly conforms to the result by Krutchkoff.

In a paper in this Journal, Watson (1961) has discussed the use of group screening methods for detecting effective factors. The methods are analogous to those suggested by Dorfman (1943) for the medical problem of detecting a rare defect among members of a large population by using pooled blood samples. One step in Watson's derivation of the probability that an effective factor is declared significant is invalid. The purpose of this note is to derive the correct expression for this probability and show how the correction affects the subsequent theory. None of Watson's qualitative statements about the efficiency of group screening methods nor his numerical example are seriously affected.

A number of useful tests for contingency tables and finite stationary Markov chains are presented in this paper based on the use of the notions of information theory. A consistent and simple approach is used in developing the various test procedures and the results are given in the form of analysis-of-information tables. Beginning with tests of hypotheses for a one-way table, tests of hypotheses of specified probabilities, independence, conditional independence, homogeneity of classifications, and symmetry are developed for contingency tables of two, three, four, and higher order classihcations. For the Markov chains, the tests include the hypotheses of a specified matrix of transition probabilities, Markovity, and homogeneity of several realizations of Markov chains. Worked examples are given throughout the paper.

McLean and Anderson [1] described extreme vertices designs and an algorithm for determining the extreme vertices. In three component mixtures, conventional triangular graph paper includes the entire factor space and any design can be easily visualized by plotting the vertices and centroids of the design. In four component systems, the entire factor space is included within a pyramid, the base of which is equivalent to triangular graph paper. The following describes the construction, on transparent sheets, of triangles on various horizontal planes within the pyramid, on which the vertices and centroids of any design can be plotted. With proper positioning of these sheets on a frame, any four component design can be visualized. Let the levels of the factors X1, X2 and X3 vary within the horizontal planes of the pyramid and let vertical distances within the pyramid represent X4 . For simplicity of explanation, the procedure will be described starting with the case X4 = 0, (Figure 1), even though this might not be one of the defined

This paper deals with a problem that often arises in visual scanning experiments in particle physics, viz. that of estimating the number of undetected particles from the scanning record. This problem is formulated here as one in estimating the size of a multinomial population from an incomplete observation of the cell totals under constraints on the cell probabilities. These constraints differ according to the assumptions made about the scanners and the particles, thus giving rise to different probability models. Several models are considered here-existing ones as well as a new generalized model (Sections 2, 4, 5) Estimation procedures corresponding to these models are discussed (Sections 6–10). A discussion of the applicability of the techniques presented here to other areas is also included (Section 1). A table of the notation used in this paper is provided in Section 3.

This paper gives tables of the exact operating characteristic (O. C.) function, of the exact average sampler number (A. S. N.) function, and of the probability of termination at any step for certain one-sided truncated sequential tests of the mean of a normal distribution when the variance is known. By looking at the smallness of the probability that a sequential test terminates after the equivalent fixed sample test it is observed that truncated sequential tests are more economical over a wider range of possible values of the mean than previously recognized. The O. C. and A. S. N. tables presented here may be used for approximate two-sided truncated sequential tests in the same manner as in [7]. Also, the scope of thii paper is contrasted with the work of other authors. Aroian's direct numerical methods are used [2]. These methods emphasize the use of digital computers to calculate the distributions of those test statistics required for computation of the operating characteristic (O. C.) function and the...

In this paper an estimator of the location parameter of the Weibull distribution is proposed which is independent of its scale and shape parameters. Several properties of this estimator are established which suggest a proper choice of three ordered sample observations insuring a permissible estimate of the location parameter. This result is valid for every distribution which has the location parameter acting as the origin or threshold parameter. Asymptotic properties of such an estimator of the location parameter of the Weibull distribution is discussed. Finally, the paper contains a brief discussion on a percentile estimator of the location parameter of the Weibull distribution and includes some numerical illustration.

This paper presents a method for determining the shapes of two overlapping functions f 1(x) and f 2(x) from an observed set of additive mixtures, [α i f 1(x) + β i f 2(x); i = 1, …, n), of the two functions. This type of problem arises in the fields of spectrophotometry, chromatography, kinetic model building, and many others. The methods described by this paper are based on the use of principal component techniques, and produce two bands of functions, each of which contains one of the unknown, underlying functions. Under certain mild restrictions on the fj (x), each band reduces to a single curve, and the fi (x) are completely determined by the analysis.

Procedures are given for determining statistically whether the highest observation, the lowest observation, the highest and lowest observations, the two highest observations, the two lowest observations, or more of the observations in the sample are statistical outliers. Both the statistical formulae and the application of the procedures to examples are given, thus representing a rather complete treatment of tests for outliers in single samples. This paper has been prepared primarily as an expository and tutorial article on the problem of detecting outlying observations in much experimental work. We cover only tests of significance in thii paper.

A number of recent publications have dealt with problems of analyzing the performance of multi-component systems and evaluating their reliability. For example, a comprehensive theory of two-terminal networks was presented in [I] by Moore and Shannon who, among other results, have developed methods for obtaining highly reliable systems using components of low reliability; some of their procedures are credited to earlier work by von Neumann [2]. Several of the concepts and results of the present paper are generalizations of the corresponding concepts and results of the Moore-Shannon paper. A discussion of complex systems interpreted as Boolean functions may be found in the paper [3] by Mine. The present study deals with general classes of systems which contain two-terminal networks and most other kinds of systems considered previously as special cases, and investigates their combinatorial properties and their reliability. These classes consist, with several variants, of systems such that the more components...

This paper is a sequel to two previous papers on the subject of the Folded Normal Distribution appearing in Technometrics, S, pp. 543–550 and 551–562, (1961). In these earlier papers methods for estimating the parameters of the Folded Normal Distribution were proposed. Thii paper gives approximations for the standard errors for the maximum liklihood estimates of these parameters.

The purpose of this paper is to revive interest in the periodogram approach to time series analysis which, at present, is only of historical interest and is seldom used. During the late 1940's, when it was realized that the smoothed periodogram could be used to estimate the spectral density of a stationary time series, the method was impractical because of the amount of computations. This is no longer the case, but not realized by many applied workers. In this paper the smoothed periodogram is considered from the point of view of its spectral window. The results are compared with two standard spectral windows by a method which avoids defining a bandwidth. The spectral windows are normalized so that they have the same variance, and plotted. The user can then choose the window which best suits his needs. Rejection filtering, trigonometric regression and cross-spectra analysis are discussed. An example is given in which the spectra and cross-spectrum of a bivariate time series are estimated.

A two-stage procedure using shrinkage techniques is used for estimation in the simple linear regression model. A direct estimation of the predicted response as well as estimation of the parameters in the model are considered in this paper. Throughout the paper, ‘a priori’ values of the parameters are assumed to be available in the form of prior estimates or realistic guessed values based on the experimenter's knowledge and past experience. Some real problem examples are given to illustrate the use of this procedure. This technique may have some application to response surface analysis.

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.

This paper extends some of the results obtained in a recent paper by Kale and Sinha [3] for the exponential distribution. The problem of selecting an efficient estimator of the expected value in the presence of an outlying observation with higher expected value is discussed. An iterative procedure for the estimation of the mean is provided and the method is illustrated by considering an example.

The present paper is based on an earlier study of the author [11], and its primary objective is to present the methodology of a Bayesian design of stratified sampling. In order to avoid abstract mathematics, and make the exposition more concrete, we focus attention on a problem of estimating a linear function of the proportion defectives in k finite lots of a certain product. In particular, consider the following sampling inspection problem, k lots (k > 2) of a certain product are subject to sampling inspection. The sizes of the lots, N1, N2, * * * , Nk say, are known. Let Mi (i = 1, -.. , k) be the number of defective items in the i-th lot. The objective is to estimate the total number of defective items o =l.Mk, in the k lots. A total sample of a fixed size, n, is specified. The problem is how to allocate the sample over the k given lots. We study this allocation problem in a Bayesian framework with a squared-error loss. Thus, whatever sampling plan is adopted, the estimator of 0 to be used after observing the sample is the Bayes estimator for the squared-error loss. The question is, however, what sampling plan to adopt. The principle one following is that of determining a sampling plan which minimizes the associated prior risk. In a recent paper [12] the author discussed the problem of the optimal Bayes selection of a sample from a finite population. It has been shown there that the optimal Bayes sampling selection is nonrandomized, without replacement, and should often be performed sequentially. However, to obtain such an optimal Bayes sampling plan, it is required to specify a prior joint distribution to all the N variates (Xi, ... , XN) of the population. This requirement is very often impractical. It is rarely the case that the sampling designer can specify such a joint prior distribution. Bayesians tend often to specify a simple joint prior distribution, which can be conveniently manipulated. Under such simplified prior assumptions one is liable to attain trivial designs. A further elaboration of this point can be found in Solomon and Zacks [10]. We feel that most practitioners, especially, in the field of quality control, where they have to inspect

In a recent paper, Tiao and Guttman (1967) discussed the use of adjusted residuals to analyze the behaviour in moderate to large samples of the premium and protection of Anscombe's rule (herein designated as the A(k)-rule) when sampling is from the N(μ, σ) distribution, μ is to be estimated;and where k outlying observations are suspected of being spurious (k = 1 and 2). A discussion of two other rules, Semi-Winsorization (S (1)-rule) and Winsorization (W (1)-rule), is given in Guttman and Smith (1969, 1971). This paper investigates the behaviour, for moderate to large n, of the A (k)), S (k)) and W (k))- rules, k = 1 and 2. To do this, we define rules based on adjusted residuals, which we shall denote as the A k , S k and W k rules. Expressions for the premium and protection of the S k and W k rules are derived, and contrasted with these characteristics of the A k rule, obtained by Tiao and Guttman (1967). Some discussion of the case when σ2 is unknown is also included. Here we assume that there is an ind...

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.

In a previous paper in Technometrics, Vol. 1, 1959, the author derived the maximum liklihood estimates of the mean and variance for simply truncated or simply censored samples drawn from a Normal distribution. This paper extends considerably the tables originally published, and contains a further worked example.

The main results of this paper, apart from the limit theorems, were given in two lectures at the University of London, and in a seminar at Imperial College of Science and Technology, London, January 1959. The complete paper was presented for discussion in a meeting at Imperial College in February 1960. The paper reviews present sampling inspection plans for attributes placing particular emphasis on their underlying assumptions. A model is then proposed based upon prior distributions and costs, and optimum sampling plans are derived which minimize the average costs for any prior distribution. Tables and examples are provided.

At first glance the subject-matter of this paper may appear to be rather trivial. No questions of offense or defense strategies are involved; one is interested solely in calculating the probability that a point target is destroyed by one or more weapons in a salvo. If the target, has an extended area, the probability of destruction is replaced by the expected fraction of the target destroyed. One might reasonably conclude that a few simple mathematical arguments involving independent random events are all that is required. However, appearances are deceptive. Since the second world war a large number of authors have dealt with problems of this type and the results of their researches are widely scattered through the mathematical literature under the general name of coverage problems. A few answers can be obtained in closed form, but the majority run into diffkulties which can be overcome only by numerical integration or simulation. This paper attempts to classify these researches into a more-or-less logica...

This paper is concerned with the study of time series models which are linear combinations of normal variables with zero means and variances σ2 t(t = 0, ±1, ±2, …). If xt (t = 1, 2, …, N) is a realization of a time series, the periodogram of X 2 t is employed to discover the dominant frequencies in σ2 t, where σ2 t is expressed in a finite Fourier series. In Part I of the paper it is proved that this periodogram is, in some sense discussed in the paper, a consistent estimator of a product of the absolute value of the Fourier coefficients and a function of the linear model coefficients; then an approximate method for estimating the Fourier coefficients separately is given. In Part II a large sample testing procedure, depending on the Kolmogorov-Smirnov Statistic, is presented for the homogeneity of the variances in the models studied in Part I. Artificial examples are given for the support of the results of the paper.

Detailed examination of the lifetimes of components and assemblies by means of the usual sequential probability ratio tests is often not feasible because of the prohibitive time or cost involved in such an examination. Situations commonly arise, however, where extraneous information is available (perhaps in the form of experience of similar situations or concerning the reputation of the supplier of the components or assemblies) which reflects on the current situation. Such information might be incorporated in a Bayesian analysis, but little work seems to have been done in this area. This paper presents a possible sequential life test incorporating prior information, applied to the simplest situation of components with exponentially distributed lifetimes, tested individually. The results for conventional sequential lifetests are used as a rough yardstick against which to measure the properties of the Bayesian lifetest discussed in the paper.

Almost all the statistical procedures in current use for evaluating the reliability of components or equipment rest on the assumption that the failure times follow the exponential distribution. However, in practical situations one rarely has enough data to determine whether failure times are actually exponential. This paper studies the behavior of several statistical life testing procedures based on the exponential failure law if the true failure law is the Weibull distribution. It is found that these statistical techniques, which are widely used, are very sensitive to departures from initial assumptions. Applying these techniques to lie test data when the exponential failure law is not satisfied may result in substantially increasing the probability of accepting components or equipments having poor mean-timsto-failure. This paper also develops convenient analytic techniques for approximating (i) the distribution of sums of independent random variables, and (ii) the characteristics of sequential procedure...

In a paper in this Journal, McMillan (1971) compared three procedures for the detection of outliers in samples from a Normal population. One of these was a sequential application of a maximum residual test. One step in McMillan's derivation of the probability of detecting two outliers using this test is incorrect. A correct derivation is presented in this note and some numerical results in McMillan's paper are appropriately amended. McMillan's qualitative statements on the performance of the test are essentially unaffected.

Two papers “Rejection of Outliers” by F. J. Anscombe and “Locating Outliers in Factorial Experiments” by C. Daniel were presented at the Christmas, 1959 meetings of the American Statistical Association. These papers, plus a discussion by William Kruskal, Thomas S. Ferguson, John W. Tukey and E. J. Gumbel were published in the hlay 1960 issue of Technometrics. The following paper by Dr. Bross continues the exchange of ideas on the subject of “Outliers”.

In a 1959 paper, A. E. Hoerl discussed a method for examining a second order response surface. Thii paper provides a mathematically simpler derivation of the technique and proofs of some stated properties.

In many analyses of variance it is appropriate to consider the effects as randomly chosen from a population; thus providing a model II analysis in the terminology of Eisenhart. Interest then centers on the variances of the effects and on the components of variance. The usefulness of this technique is limited by the frequent occurrence of negative estimates of the variance components which are by definition non-negative quantities. This paper presents a method of data analysis, applicable to many of the more popular models, which always yields non-negative estimates. Recently a simple algorithm, the "pool-the-minimum-violator" algorithm, has been developed [11] as a result of maximizing the likelihood function of the mean squares subject to a set of constraints. This paper has been prepared as a less mathematical companion article to [11] in order to treat the more applied aspects of the problem. 2. AN EXAMPLE

This paper compares four experimental procedures for estimating the probability p a defective item fails in a field test. Let there be S defective items available for test and k pieces of test equipment. Assume that failure of an item during the test causes the loss of the testing equipment. Then it becomes necessary to limit the test to as few pieces of equipment as possible while obtaining an estimate of the portion of defective failures, p, with the desired precision. The paper compares two estimators of the probability of a defective failure, namely, the maximum likelihood estimator and an unbiased estimator. The estimator to be used depends on the probability, p. The experimental procedure to be used is the one which uses as many pieces of test equipment as is economically possible and gives the desired precision of the estimate.

In the decoding of received messages to transmitted messages, one of the usual assumptions is that the transmitted messages are sent with equal relative frequencies. In many practical situations this assumption is not valid. This paper is concerned with the investigation of the effects of removing the assumption of uniform input, on the optimum assignment of the received messages to a specific set of transmitted messages. The results show that if the deviation from uniformity is not too extreme, the optimum assignment will be one of the possible optimum assignments under the uniform input assumption. A measure of deviation from uniformity and a criterion for extreme deviation is included in the paper.

Part I of this paper was published in the previous issue of Technometrics, (Vol. 2, No. 1, February 1960). In Part I Dr. Epstein describes several graphical and analytical procedures for testing the assumption that the underlying distribution of life is exponential. This portion of the paper contains numerical examples illustrative of these procedures.

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.

It is fairly commonplace in reliability analyses to encounter data which is incompatible with the exponential, Weibull, and other familiar probability models. Such data motivates research to enlarge the group of probability distributions which are useful to the reliability analyst. In this paper, we examine a three-parameter generalization of the gamma distribution and derive parameter estimation techniques for that distribution. Those techniques, in the general case, depend upon method of moments considerations which lead to simultaneous equations for which closed form solutions are not available. Graphic solution is proposed and aids to the computations are provided. Major concepts in the paper are summarized by means of a numerical example. Details are given for the special case in which only the scale parameter is unknown. Three unbiased estimators for that parameter are derived along with their variance formulas. Minimum variance considerations are discussed by application of the Cramer-Rao Theorem.

This paper unifies and extends previously published characterizations of moving average, geometric moving average, and cumulative sum control chart procedures. It presents comparable characterizations of two procedures based on tests described but not evaluated in earlier papers. One of these procedures is based on a test devised by Girshick and Rubin that is optimal under a particular set of idealized conditions. The other procedure is based on run sum tests, which are generalizations of the type of run test that counts the number of consecutive points that exceed a limit, the generalization taking into account the extent that points in such a run exceed the limit.

In this paper we consider the method of Johns and Lieberman (1966) for estimating the shape and the log of the scale parameters of the two parameter Weibull distribution. The estimates are linear, asymptotically jointly normal and efficient. The main thrust of the paper is that unlike most linear estimating procedures of the Weibull parameters, that require different sets (tables) of weights for the observations for each size and often for each different number of observations from the sample that are available, the present technique requires only the knowledge of the proportion of observations from the sample available for the estimates. Given this proportion and four numbers obtained from a table look-up, the weights of the observations are then expressible in simple closed forms. Further we include a simple scheme to update the estimates when more observations become available. Finally, small sample computations of mean squared errors indicate that these estimates compare very Pdvorably even here with ...

Sampling plans for truncated life tests from the normal and lognormal distributions are obtained. The tables of this paper give the minimum sample size necessary to assure a certain mean or median life when the experiment time is tied in advance. The modification necessary to assure any other quantile (percentile) of the distributions is obtained. Thus, sampling plans for establishing other percentiles of the distributions are shown to be obtainable from the tables of this paper. The operating characteristic functions of these plans are obtained and for a wide range of values of practical interest these functions are graphed in order to facilitate selection of an appropriate plan in a given situation. Producer's risk is discussed and a table is given for the ratio of the true median life to the specified median life (or the difference between the true mean lie and the specified mean life) to insure that the producer's risk does not exceed α = .l0, .05. An approximation is given for the minimum sample size...

A recent paper by Feldman and Fox (1968) discussed the estimation of the parameter n in the binomial distribution when the other parameter p is known. This paper gives a Bayesian treatment of the problem both when p is known and when p is unknown. The Bayesian approach is a very practical one and is especially easy to apply in small problems because less computing is involved. It also does not require any appeal to asymptotic distributional results. (Author)

This paper reviews three basic methods of non-linear least squares estimation and the best known modifications to the popular Gauss-Newton procedure. With regard to the Gauss-Newton procedure the paper identifies the problem of poor parameterization and gives simple examples to show how it may arise. A simple example is also given to show that one popular method of handling constrained boundaries can lead to erroneous results. A modification, based on stepwise linear regression techniques, to handle both of these problems is presented and discussed and an example is given to illustrate its effectiveness. The required fundamentals of stepwise linear regression are reviewed.

Estimation of the Poisson parameter from various types of truncated and censored samples was studied in an earlier paper by the writer [1]. More recently, considerable attention [2], [3], [4], [5], [7], [8], [9], [11] has been directed to the special case in which zero values of the random variable are missing. In this paper we are concerned with samples that are truncated on the right at a terminus which we designate as d. In order that any given member of a population might be included in a sample of the type under consideration, it is necessary that x d are restricted from observation. Manufacturing processes in which x, the number of defects per item, is a Poisson distribution random variable often give rise to samples of this type. It is not uncommon to find an inspection being performed which results in rejecting all items for which x > d, where here d is the maximum number of defects permissible per item. Samples selected from the screened (accepted) production are accordingly truncated on the right at d.

This paper deals with large sample estimation of the location parameter (α1 and the scale parameter α2 in the gamma distribution with known shape parameter. Best linear unbiased estimates based on k sample quantiles are used. For a given k, the optimum spacings of the sample quantiles can be replaced by simpler “nearly optimum” spacings at virtually no loss of asymptotic efficiency. The theory behind the nearly optimum spacings is briefly reviewed. The major part of the paper concerns estimation of α2 when α2 is known. Nearly optimum spacings together with the coefficients to be used in computing the estimates are presented in a number of tables for k = 1(1) 10, and various values of the shape parameter. The paper also contains brief discussions of estimation of α1, when α2 is known, and simultaneous estimation of α1 and α2.

A modification of the square root method of matrix inversion has been recently advocated. The present paper shows the conditions under which the original method, or the proposed modification, is preferred by comparing the round off error of the two methods when calculations are to be made to a fixed number of decimals with a desk calculator. The paper recommends the discontinuance of the use of incomplete numbers in reporting results where calculations on approximate numbers are made to a fixed decimal position.

This paper investigates the errors of prediction of multiple regression equations, in which the regressor variables are regarded as drawn at random from a multivariate normal population. In most theoretical treatments of multiple regression it is assumed that one or more "dependent" variables are to be predicted, given the observed values of a number of "independent" variables. The independent or regressor variables are treated as constants. However, in many applications, it is more reasonable to regard them as random variables, drawn, for example, from a multivariate normal population. In this case the term "independent" variables is confusing, and it seems better to call them regressor variables. It has been suggested by Ehrenberg (1963) that regression or stochastic regressor variables is useless. This paper shows that although such regression has it limitations, especially if the number of regressor variables is large, it may be useful if the limitations are understood. The particular case to be discussed is that in which the regressor variables are drawn from a multivariate normal population. The k-dimensional vectors X1 , X2, ... X *... X, are given as a random

This paper represents a first detailed study of the application of a generalized probabi1ity distribution, the Burr distribution, to control charts for the maximum and minimum value in a sample of size n. The Burr distribrltion is a two parameter probability distribution with cumulative distribution function represented by This paper illustrates the application of the Burr distribution to control charts for the maximum and minimum value in sampling from a normal distribution and compares results with those previously derived using the standard normal distribution.

In the statistical literature, Booth et al. (Ref. 1) and Hartley (Ref. 2) developed modifications of the well-known Gauss-Newton method of iterative solution and applied it to estimate a set of parameters for nonlinear regression problems by least squares. To ensure convergence of his method, Hartley restricts its applicability to problems that satisfy his assumptions. Herein, we describe a method that removes two of Hartley's three assumptions, thereby making it applicable to those problems for which his method fails to converge. The method we propose is based upon an algorithm due to Davidon (Ref. 3). It is an iterative descent method whose rate of convergence is quadratic in the limit. It was later modified by Fletcher and Powell (Ref. 4), and has been used for locating an unconstrained local minimum of a function of several variables. Fletcher and Powell's account of Davidon's method has been found to be very useful when first derivatives of the function are available. However, in realistic situations it frequently is practically impossible to calculate first derivatives. Therefore, the method we describe is a modification of the basic Fletcher and Powell method due to McGill (Ref. 5) and Taylor (Ref. 6) and includes the case for which the gradient is not given analytically. Our concern is with the application and introduction of the modified Davidon method (MDM) to nonlinear regression problems of the type discussed by Hartley. We have omitted all proofs concerning stability and rate of convergence of the method and refer the reader to the excellent paper of Fletcher and Powell for such proofs. Included is a description of the MDM, a description of ways of terminating the iterations, and an illustration of the method with the numerical example Hartley used in his paper. We show the results for both the case of analytic and approximate gradient. Also included are results obtained in applying both methods to a second example, which involves the estimation of six parameters for an exponential regression function. For this example, it is shown that the method described in this note converges when Hartley's method fails to converge.

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.