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Showing papers in "Biometrika in 1975"


Journal ArticleDOI

462 citations


Journal ArticleDOI
TL;DR: In this article, the test statistic X2(1bbl)+X2(b2) is considered in normal sampling, where X(1bl) and X(bbl) are standardized normal equivalents to the sample skewness, Jbl, and kurtosis, b2, statistics.
Abstract: SUMMARY The test statistic X2(1bbl)+X2(b2), where X(1bl) and X(b2) are standardized normal equivalents to the sample skewness, Jbl, and kurtosis, b2, statistics, is considered in normal sampling. Using the Johnson system, Su and SB, as approximate normalizing distributions, contours in the (1bl, b2) plane of the test statistic are set up for sample sizes ranging from 20 to 1000.

386 citations


Journal ArticleDOI

384 citations


Journal ArticleDOI
TL;DR: In this article, two graphical methods are proposed for identifying bivariate observations that may unduly influence the sample correlation coefficient and robust estimators of correlation are developed and a Monte Carlo comparative study is made of these and other wellknown estimators.
Abstract: Two graphical methods are proposed for identifying bivariate observations that may unduly influence the sample correlation coefficient. Secondly, robust estimators of correlation are developed and a Monte Carlo comparative study is made of these and other wellknown estimators. Also considered are methods for developing positive-definite estimates of correlation matrices and extensions of robustness to other problems such as regression are mentioned.

355 citations


Journal ArticleDOI
TL;DR: In this paper, a Bayesian approach is considered to the problem of making inferences about the point in a sequence of random variables at which the underlying distribution changes, based on the posterior probabilities of the possible change-points.
Abstract: SUMMARY A Bayesian approach is considered to the problem of making inferences about the point in a sequence of random variables at which the underlying distribution changes Inferences are based on the posterior probabilities of the possible change-points Detailed analyses are given for cases in which the distributions are binomial and normal, and numerical illustrations are provided An informal sequential procedure is also noted

334 citations


Journal ArticleDOI
TL;DR: In this paper, the D-optimum design theory has been extended to the problem of discriminating between any number of models, where the design points xi are known and the random variables Cik are independently normally distributed with zero mean and constant variance 0y2.
Abstract: where the design points xi are known and the random variables Cik are independently normally distributed with zero mean and constant variance 0y2. In the theoretical development, but not in the numerical examples, we shall, without loss of generality, take 0-2 to be unity. The function st(x) is one of two known functions 81(x, 01) and 82(x, 02), where 01 and 02 are sets of unknown parameters of dimension m1 and in2. The purpose of the experiment is to determine which of the two models is true. The design of experiments for discriminating between any number of models has been investigated by several authors. References to this work and some comments on general aspects of the problem are given in the recent paper of Atkinson & Cox (1974) and in the ensuing discussion. Designs specific for discriminating between only two models have been derived by Fedorov & Malyutov (1972) and Fedorov & Uspensky (1975). In the present paper we collect, exemplify and generalize these results on designs for two models and describe the relationship with the designs of Atkinson & Cox, which are based on an extension of D-optimum design theory. In ? 2 we describe nonsequential designs which are the limits to which the sequential designs of ? 3 converge as the number of trials increases. In ? 4 we discuss the relationship between the two approaches when both models are linear. The least squares estimates of the parameters in the two models, which in general need not be linear, are the solutions of the equations

302 citations


Journal ArticleDOI
TL;DR: In this paper, two widely used methods, one replacing the unknown parameter by an efficient estimate and so termed estimative and the other using a mixture of the possible density functions and commonly termed predictive, are compared.
Abstract: SUMMARY Fitting a parametric model or estimating a parametric density function plays an important role in a number of statistical applications. Two widely-used methods, one replacing the unknown parameter by an efficient estimate and so termed estimative and the other using a mixture of the possible density functions and commonly termed predictive, are compared. On a general criterion of closeness of fit based on a discriminating information measure the predictive method is shown to be preferable. Explicit measures of the relative closeness of predictive and estimative fits are obtained for gamma and multinormal models.

259 citations


Journal ArticleDOI
TL;DR: In this article, the properties of simple estimates based on equitailed order statistics are derived, such as transformation of exponential and gamma random variables, and errors in previous work are discovered and partially corrected.
Abstract: SUMMARY Transformations to symmetry, or approximate symmetry, are considered. In particular, properties of simple estimates based on equitailed order statistics are derived. Examples include transformation of exponential and gamma random variables. Errors in previous work are discovered and partially corrected.

252 citations


Journal ArticleDOI
TL;DR: In this paper, the authors consider experimental designs for discriminating between three or more rival regression models and show that the results are similar to those of the earlier one, except in some simple cases, a straightforward generalization of those when there are only two models.
Abstract: In this paper we consider experimental designs for discriminating between three or more rival regression models. In an earlier paper (Atkinson & Fedorov, 1975) we described optimal designs for discriminating between two models. Although the approach and nomenclature of the present paper are similar to those of the earlier one, the results are not, except in some simple cases, a straightforward generalization of those when there are only two models. We now suppose that there are v rival regression models

243 citations


Journal ArticleDOI
TL;DR: In this paper, the modulus of the difference between the empirical and theoretical characteristic functions is weighted and integrated over the real line; this function of the sample values and the parameters is then minimized with respect to the parameters via a gradient search routine nested within a sequential search.
Abstract: SUMMARY Estimation of the parameters of the stable laws is considered for samples of size at least 50. The modulus of the difference between the empirical and theoretical characteristic functions is weighted and integrated over the real line; this function of the sample values and the parameters is then minimized with respect to the parameters via a gradient search routine nested within a sequential search. Extensive simulation experiments validate the effectiveness of the estimation procedure over the entire parameter space. Application of the procedure to stock market price behaviour is made.

201 citations


Journal ArticleDOI
TL;DR: In this paper, the authors developed tests for alternatives representing decreasing mean residual life and the property "new better than used in expectation" and obtained consistent and asymptotic relative efficiency results for the tests based on V* and K*.
Abstract: SUMMARY In this paper we develop tests for alternatives representing decreasing mean residual life and the property 'new better than used in expectation'. The decreasing mean residual life test statistic, V*, is new, and critical constants and a large-sample approximation are obtained to make the test readily applicable. The new better than used in expectation statistic, K*, is shown to be equivalent to the total time on test statistic; the latter is ordinarily viewed as a test statistic for alternatives of increasing failure rate. Consistency and asymptotic relative efficiency results are obtained for the tests based on V* and K*. These results lead to a reinterpretation of the total time on test statistic as a test statistic for classes of alternatives larger than the increasing failure rate class and including the 'new better than used in expectation' class.

Journal ArticleDOI
TL;DR: In this article, the shape of population is compared to the robustness of the distribution of the variables in a statistical procedure, based on extensive computer simulation, and it is shown that the shape is related to robustness.
Abstract: SUMMARY The underlying theory upon which a great number of statistical procedures are based assumes that the variable or variables sampled are normally distributed. While there has been a good deal of theoretical research on the robustness of these procedures, the results seem not to have been set out in terms which the unsophisticated user of statistical methods can easily assimilate. The present paper, based on extensive computer simulation, aims at relating diagrammatically the shape of population to the robustness of the distribution of

Journal ArticleDOI
TL;DR: In this paper, the error model is transformed and reparameterized to induce regular estimation on the boundary with one or both degrees of freedom infinite, leading to bivariate score tests for normal, extreme value and logistic special cases as well as an evaluation of these models within a more general framework.
Abstract: SUMMARY Linear models, with errors that follow the distribution of the logarithm of an F statistic, are shown to include a number of common statistical models as special cases. The error model is transformed and reparameterized to induce regular estimation on the boundary with one or both degrees of freedom infinite. This leads to bivariate score tests for normal, extreme value and logistic special cases as well as an evaluation of these models within a more general framework. In particular, the test for normality is found to reduce to the usual tests based on sample skewness and kurtosis. Sample sizes are given for pairwise discrimination among some specific models. Applications are indicated.

Journal ArticleDOI
TL;DR: In this paper, the maximum likelihood estimation for rectangular lattice autonormal schemes is discussed and the efficiency of the coding technique in testing for randomness on various types of lattice is assessed.
Abstract: SUMMARY Maximum likelihood estimation for rectangular lattice autonormal schemes is discussed. Explicit results for the first-order isotropic scheme are given and compared with those obtained using a simple coding technique. The efficiency of the coding technique in testing for randomness on various types of lattice is assessed.

Journal ArticleDOI
TL;DR: In this paper, the authors investigated theoretically the problem of normal-means distribution of the data in significance tests and found that the theoretical efficiency of the procedure is quite high and recommended the proportions into which the data should be divided.
Abstract: SUMMARY It has sometimes been suggested that to overcome difficulties arising in significance tests when the effects tested are selected in the light of the data, the data should be split randomly into two portions. The first portion is used to choose the hypothesis for test and the second portion for the evaluation of significance. After some general criticism of the idea, it is investigated theoretically on a simple problem about normal means. Recommendations are reached about the proportions into which the data should be divided and the theoretical efficiency of the procedure is assessed and found to be quite high.

Journal ArticleDOI
TL;DR: In this article, the authors considered Kolmogorov-smirnov tests of goodness of fit in the presence of unknown nuisance parameters and showed that for a wide class of cases the acceptance probability is the ratio of two densities of which the denominator density is known.
Abstract: The paper considers Kolmogorov-Smirnov tests of goodness of fit in the presence of unknown nuisance parameters. It is shown that for a wide class of cases the acceptance probability is the ratio of two densities of which the denominator density is known. Methods of calculating and inverting the Fourier transform of the numerator density are considered. The results are applied to the case of an exponential distribution which has unknown mean and perhaps also unknown lower terminal. Tables of percentage points are given for the standard Kolmogorov-Smirnov statistics {D^-_n}, {D^+_n}, {D_n}. It is shown that these tables also give the percentage points for analogous statistics derived from the sample distribution function of the spacings between uniform order statistics.

Journal ArticleDOI
TL;DR: In this article, two criteria are set up to judge the relative performance of the least squares estimator and the best linear unbiased estimator of, in the linear model y = X/, + u, where E(u) = 0, E(uu') = F.
Abstract: SUMMARY Two criteria are set up to judge the relative performance of the least squares estimator and the best linear unbiased estimator of , in the linear model y = X/, + u, where E(u) = 0, E(uu') = F. The matrices X and r are found so that the relative performance of least squares is worst. Both criteria give the same least favourable situation: when X(X) is any one of the 2k manifolds (Y1 +? Yn, ***., Yk ? Yn-k+l), where Fyi = fiyi andf1 < ... < fn are fixed, ,/(. ) denoting the subspace spanned by the columns of the relevant matrix. The case where allfi may be chosen in a preassigned interval is also discussed. The practical implications of the various results are mentioned.

Journal ArticleDOI
J. Kiefer1
TL;DR: In this paper, the advisability of comparing designs on the basis of several different criteria of goodness is discussed, including A-, D- and E-optimality, as well as one of Box & Draper.
Abstract: SUMMARY The advisability of comparing designs on the basis of several different criteria of goodness is discussed. As an illustration, designs for quadratic regression on a simplex are compared in terms of a family of criteria that includes those of A-, D- and E-optimality, as well as one of Box & Draper. Efficiency robustness under change of region, as well as of criterion, is considered. The dependence of structure of good designs on dimension is determined.

Journal ArticleDOI
TL;DR: In this article, some properties of the sample estimator of attributable risk A, defined here as the proportion of all cases of disease which may be attributed to a risk factor, are considered for the case-control study situation.
Abstract: SUMMARY Some properties of the sample estimator of attributable risk A, defined here as the proportion of all cases of disease which may be attributed to a risk factor, are considered for the case-control study situation. It is shown that log (1 -A) may be expressed in terms of the prevalences of the factor in cases and healthy controls, that the bias of the estimator is minimized when 2 is added to the cell frequencies corresponding to nonexposed persons in the usual 2 x 2 contingency table, and that the distribution of I - A is asymptotically log normal. Examples of the calculations, and a discussion of the results, are given for a number of risk factors for childhood leukemia.

Journal ArticleDOI
TL;DR: In this article, the duality relationship between D-type optimal designs and problems of covering a subset of Euclidean space by central ellipsoids and ellipseidal cylinders is discussed and consequences that aid the solution of either problem are discussed.
Abstract: SUMMARY The duality relationships between D-type optimal designs and problems of covering a subset of Euclidean space by central ellipsoids and ellipsoidal cylinders are described and consequences that aid the solution of either problem are discussed. It is shown that there are also design duals for the problems of finding minimal covering spheres and ellipsoids or cylinders whose centres or axes are chosen optimally.

Journal ArticleDOI
TL;DR: In this article, the powers of several two-sample tests are compared by simulation for small samples from exponential and Weibull distributions with and without censoring, and the results for this case are essentially the same as in the case of exponential distributions.
Abstract: SUMMARY The powers of several two-sample tests are compared by simulation for small samples from exponential and Weibull distributions with and without censoring. The tests considered include the F test, a modification for samples that are from Weibull distributions, Cox's test, Peto & Peto's log rank test, their generalized Wilcoxon test, a modified log rank test, and a generalized Wilcoxon test of Gehan. When samples are from exponential distributions, with or without censoring, the F test is the most powerful followed by two general groupings of tests: first the three non-Wilcoxon tests and then the two Wilcoxon tests. There is little difference in the power characteristics of the tests within each grouping. Estimates of the asymptotic relative efficiencies of the various tests relative to F are obtained from the normal probability plots of the power curves. When the samples are taken from Weibull distributions with constant hazard ratio, the F test is not robust and a modification is used. The results for this case are essentially the same as in the case of exponential distributions, with the modified F test as the most powerful. However, when the hazard ratio is nonconstant, the two generalizations of the Wilcoxon test have more power than the other tests.

Journal ArticleDOI
TL;DR: In this article, the authors apply the theory developed by Lindley & Smith to the estimation problem for growth curves, and also consider the problem of prediction, given a sample from this model.
Abstract: SUMMARY Recent work on the analysis of growth curves has concentrated on the generalized growth model put forward by Potthoff & Roy (1964), the model being studied from a Bayesian viewpoint by Geisser (1970). The analysis of such data however falls naturally within the scope of the general Bayesian linear model proposed and analyzed by Lindley & Smith (1972). Here we apply the theory developed by Lindley & Smith to the estimation problem for growth curves, and also consider the problem of prediction, given a sample from this model.

Journal ArticleDOI
TL;DR: In this article, a certain class of nested halving procedures is shown to be highly efficient and the saving over one-at-a-time procedures is even greater for the estimation problem than for the previously treated group-testing problems of classifying all the units in a given finite set.
Abstract: SUMMARY The ideas of group testing, i.e. of testing units in batches instead of individually, where each test only indicates whether the tested batch contains only good units or whether it contains at least one defective, is applied to the problem of estimating the probabilityp of an arbitrary unit being defective. A certain class of nested halving procedures is shown to be highly efficient and the saving over the one-at-a-time procedures is even greater for the estimation problem than for the previously treated group-testing problems of classifying all the units in a given finite set. The main criterion used for evaluating efficiency in this paper is the asymptotic cost per unit information when a large number of tests are carried out. This paper deals only with the large-sample aspects of the problem. Tables are given that explicitly describe how to carry out the procedure and other tables indicate how close the results are to those of optimal procedures for certain input parameters.

Journal ArticleDOI
TL;DR: In this paper, the bias introduced into the least squares estimators by these errors, first when they were regarded as fixed and second when they are regarded as random, is investigated.
Abstract: SUMMARY Suppose that the independent variables in a linear regression are subject to error. This paper is concerned with the bias introduced into the least squares estimators by these errors, first when they are regarded as fixed and second when they are regarded as random. Simple criteria are introduced for deciding whether the bias is likely to be serious. The paper also considers the effect of the occasional large error in either the dependent or independent variables.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the time between the first infection and the last removal in the closed stochastic epidemic and proved limit theorems for the distribution of this time, as the population size becomes large, and then used the limits to provide an approximate distribution.
Abstract: SUMMARY The paper discusses the time between the first infection and the last removal in the closed stochastic epidemic. The method is to prove limit theorems for the distribution of this time, as the population size becomes large, and then to use the limits to provide an approximate distribution: the epidemic is allowed to start either with a large number of immigrant infectives, or with a single case. The accuracy of the approximation in finite populations is illustrated by some examples, and the method of proof also gives a theoretical estimate of the rate of convergence to the limit. The problem is typical of a much wider class of boundary problems, and the method used can be adapted to them without difficulty.

Journal ArticleDOI
TL;DR: In this paper, the robustness of distance estimators of density is evaluated against two stochastic models, from extreme regularity, through randomness, to extreme aggregation, and their robustness is assessed analytically.
Abstract: SUMMARY Distance estimators of density may exhibit serious bias unless the population under consideration forms a completely random spatial pattern, i.e. the estimators are not robust. In this paper some new estimators are proposed, and their robustness is assessed analytically against two stochastic models, which together embrace a continuous range of spatial pattern, from extreme regularity, through randomness, to extreme aggregation.

Journal ArticleDOI
TL;DR: In this article, the asymptotic expansions of the distributions of the likelihood ratio criterion and Wald's statistic are derived for a composite hypothesis under a sequence of local alternative hypotheses converging to the null hypothesis when the sample size tends to infinity.
Abstract: SUMMARY The asymptotic expansions of the distributions of the likelihood ratio criterion and Wald's statistic are derived for a composite hypothesis under a sequence of local alternative hypotheses converging to the null hypothesis when the sample size tends to infinity. Comparisons between the two statistics are made.


Journal ArticleDOI
TL;DR: In this paper, a model is constructed in which its properties are represented in terms of a Bayesian prior distribution, and the model is analyzed to give parameter estimates and predictions of further observations.
Abstract: A method is presented which, in many cases, appears to be an improvement over the standard approach to the polynomial regression problem. This improvement is achieved by focusing attention on the deviation of the polynomial representation from the true underlying function. By fully utilizing the nature of this deviation, a model is constructed in which its properties are represented in terms of a Bayesian prior distribution. The model is analyzed to give parameter estimates and predictions of further observations. Comparisons are made with standard least squares procedures when the true underlying model is (a) quadratic and (b) linear and quadratic with a superimposed sine wave.

Journal ArticleDOI
TL;DR: Bloomfield and Watson as discussed by the authors gave a proof for the minimum efficiency of least squares inequality, which up to the present has remained a conjecture, and Bloomfield & Watson have independently obtained a proof of the same inequality.
Abstract: Watson (1955) gave an inequality for the minimum efficiency of least squares as measured by the ratio of the generalized variances of the efficient and the inefficient estimators of the vector of regression coefficients. The inequality had been conjectured by J. Durbin. The proof given by Watson was faulty and was acknowledged as such by Watson (1967). This paper gives a proof for the inequality, which up to the present has remained a conjecture. The paper owes much to conversations over several months with J. Durbin and K. G. Binmore. Bloomfield & Watson (1975) have independently obtained a proof of the same inequality. The inequality asserts that