Showing papers in "Communications in Statistics-theory and Methods in 1973"
TL;DR: In this paper, a general system of distributions and the reciprocal transformation is considered and the parameters c,k determine μ,σ and the standardized moments α3,α4. The purpose of this paper is to provide c, k values which cover a wide grid of α 3,α 4.
Abstract: This paper is concerned with a general system of distributions , and the reciprocal transformation . Parameters c,k determine μ,σ and the standardized moments α3,α4. The purpose of this paper is to provide c,k values which cover a wide grid of α3,α4. values, for each c,k matching an α3,α4. point, μ and σ are also provided, The grid of α3,α4 values is sufficiently detailed that interpolation in moat cases appears unnecessary.
102 citations
TL;DR: In this paper, the Monte Carlo method was used to estimate the parameters of a mixture of two univariate normal distributions with, and sample sizes less than 300, where the components are not well separated and the sample size is small.
Abstract: There are few results In the literature on the properties of the maximum likelihood estimates of the parameters of a mixture of two normal distributions when the components are not well separated and the sample size is small. In the present investigation mixtures of two univariate normal distributions with , and sample sizes less than 300 are studied by the Monte Carlo method. For the cases considered, empirical evidtnca is given that the method of maximum likelihood should be used with extreme caution or not at all.
68 citations
TL;DR: In this paper, a class of density estimates of the histogram type based on differences of the form Yj+k−Yj,k≧l,j=l,n−k are proposed and studied.
Abstract: Let Y1,Y2,…,Yn,(Y1≦ Y2≦ … ≦Yn) be the order statistics of a random sample from a distribution F with density f on the real line. A class of density estimates of the histogram type based on differences of the form Yj+k−Yj,k≧l,j=l,…n−k are proposed and studied. The estimates are shown to be both weakly and strongly consistent at ail points x=C(f), the continuity set of f, under suitable conditions.
66 citations
TL;DR: In this article, a generalization of the Durbin-Watson statistic, which tests for higher order dependence among residuals is proposed, and the corresponding significance tables based on Jacobi orthogonal polynomials and the beta density are given.
Abstract: The Durbin-Watson statistic is used for testing the first order serial correlation among residuals in a linear model. It is based on the residuals from a corresponding regression analysis. In this paper a generalization of the statistic which tests for higher order dependence among residuals is proposed. The paper gives a brief review of the Durbin-Watson theory and the construction of the corresponding significance tables based on Jacobi orthogonal polynomials and the beta density. The distribution of the Durbin-Watson statistic based on Tmhof's distribution of Quadratic forms is indicated as an alternative method of directly computing the distribution function of the Durbin-Watson statistic, as well as its generalization, Significance values for 5 percent level of significance for lags 2 to 4 are given in Table II.
66 citations
TL;DR: In this article, the authors derived the derivative of the α th largest sharactsristic root of a symmetric matrix S = (srs) with respect to srs (r ≦ s) at S = A = diag(λ 1,…, λp) wher λ 1 ≧… ≧ λ p and λα is assumed to be simple.
Abstract: Derivatives of the α th largest sharactsristic root of a symmetric matrix S = (srs) with respect to srs (r ≦ s) at S = A = diag(λ1,…, λp) are given in this paper, whers λ1 ≧… ≧ λp and λα is assumed to be simple.The first application lies in deriving the partial differential equation for zonal polynomials given by James [13] and further new partial differential aquation of fourth degree for zonal polynomials.The second application lies inegiving the asymptotic expansions of the distribution of the a, th largest root of a Nishart matrix having Wp (n, ∑), when a th root of ∑ is simple.It is given by normal distribution function and its derivatives, If the a th root is not simple, non-normal limitiing distribution is obtained when p = 2, The similar results for the derivatives of a Heraitianmatrix and for a root or a complex Wishart matrix are also given.
65 citations
TL;DR: In this paper, it was shown that the transformation x = y g(x) occurs naturally in the distribution of the number of customers served during a busy period which implies that at least one particular family of these Lagraagian distributions must play a basic role in queueing theory.
Abstract: Several discrete Lagrangian probability distributions have been generated by Consul and Shanton (1972) by using the Lagrange expansion in y of a probability generating function f(x) under the transformation x = y g(x) where g(x) is another pgf. By using probabilistic arguments the authors show that the transformation x = y g(x) occurs naturally in the distribution of the number of customers served during a busy period which implies that at least one particular family of these Lagraagian distributions must play a basic role in queueing theory. It has also been proved that under one set of conditions all discrete Lagrangian distributions approach to the normal density function while under another set of conditions they approach the inverse Gaussiam density function.
51 citations
TL;DR: In this paper, an hyperbolic regression model is proposed for two-phase piecawise linear regression with a smooth transition between regines and compared with an altarnative model proposed by Bacon and Watts.
Abstract: An hyperbolic regression model is proposed for two-phase piecawise linear regression with a smooth transition between regines and compared with an altarnative model proposed by Bacon and Watts . The modal may be generalised to a situation in which there are several possible points of transition between the linear regines, each representing some 'critical level' for a particular data set. In particular this way apply when the secon linear regine corresponds to an asymptotic level of the dependent variable
50 citations
TL;DR: In this article, the population genetic hypothesis that mating is random with respect to genotype at a specific two allele locus is equivalent to the statistical hypothesis that the genotypic frequency distribution is given by a binomial expansion (p+q)2.
Abstract: The population genetic hypothesis that mating is random with respect to genotype at a specific two allele locus is equivalent to the statistical hypothesis that the genotypic frequency distribution is given by a binomial expansion (p+q)2. An exact,small sample, conditional test of this hypothesis is derived by tabulating critical values of the number of hetero-zygotes in a sample of size n containing given frequencies of each allele. for amy fixed value of relative gene frequency in the sample this conditional distribution rapidly approaches normality, and the conventional chi-square test becomes valid.
48 citations
TL;DR: The problem of clustering individuals is considered within the context of a mixture of distributions, where a parametric family of distributions is considered, a set of parameter values being associated with each population.
Abstract: The problem of clustering individuals is considered within the context of a mixture of distributions. A modification of the usual approach to population mixtures is employed. As usual, a parametric family of distributions is considered, a set of parameter values being associated with each population. In addition, with each observation is associated an identification parameter, Indicating from which population the observation arose. Theresulting likelihood function is interpreted in terms of the conditional probability density of a sample from a mixture of populations, given the identification parameter of each observation. Clustering algorithms are obtained by applying a method of iterated maximum likelihood to this like-lihood function.
43 citations
TL;DR: In this paper, a monte Carlo study was used to select an estimator as good as or better than the ordinary least squares estimator in terms of generalized mean square error with high frequency.
Abstract: When a researcher is confronted with multicollinearity in the standard linear model, he should consider restrictions his estimates by the linear restriction implied by the dilation of that set of principal components of the independent variables which maximizes R2. This paper shows via the use of a monte carlo study that this procedure will select an estimator as good as or better than the ordinary least-squares estimator in terms of generalized mean square error with a relatively high frequency.
42 citations
TL;DR: In this paper, five alternative methods for evaluating the relative discriminatory power of individual variables, six methods for selecting variable subsets, and a method for comparing the significance of different size subsets are examined in the context of an empirical problem.
Abstract: Five alternative methods for evaluating the relative discriminatory power of individual variables, six methods for selecting variable subsets, and a method for comparing the significance of different size subsets are examined in the context of an empirical problem. Results imply that the assessments of the relative importance of individual variables and subset composition vary with both the method and performance criterion employed.
TL;DR: In this paper, a linear model with special reference to its association with the experimental context is proposed for unbalanced cross classifications, a situation in which the definitional ambiguities of the usual linear models pose serious problems.
Abstract: The association of linear models with the analysis of complex sets of data dates back to Gauss (about 1800). But linear models assumed a major role in statistics only after Fisher's colleagues introduced then in explaining the analysis of variance. Sirica then it has become a common practice to describe experimental situations by associated linear models* The emergence of the concept of estiimability and its associated pedagogical difficulties accompanied this practice. This paper reconsiders the definition of a linear model with special reference to its association with the experimental context. The parameters of the resulting linear model submit to simple estimation without definitional ambiguity, These ideas are illustrated by considering the analysis of unbalanced cross classifications, a situation in which the definitional ambiguities of the usual linear models pose serious problems. Finally, the proposed model is compared to the usual less than full rank model.
TL;DR: In this paper, a class of fractional factorial designs of the 27 series, which are of resolutionV, are presented, allowing the estimation of the general mean, the main effects and the two factors interactions (29 parameters in all for the 27 factorial) assuming that the higher order effects are negligible.
Abstract: In this paper, we present a class of fractional factorial designs of the 27 series, which are of resolutionV. Such designs allow the estimation of the general mean, the main effects and the two factors interactions (29 parameters in all for the 27 factorial) assuming that the higher order effects are negligible. For every value ofN (the number of runs) such that 29≦N≦42, we give a resolutionV design that is optimal (with respect to the trace criterion) within the subclass of balanced designs. Also, for convenience of analysis, we present for each design, the covariance matrix of the estimates of the various parameters. As a by product, we establish many interesting combinatorial theorems concerning balanced arrays of strength four (which are generalizations of orthogonal arrays of strength four, and also of balanced incomplete block designs with block sizes not necessarily equal).
TL;DR: In this article, it was shown that a sufficient condition for an autoregressive scheme to be stationary is that the roots of the indicia1 polynomial lie within the unit circle.
Abstract: We show that a sufficient condition for an autoregrassive scheme to be stationary is that the roots of the indicia1 polynomial lie within the unit circle The importance of this theorem is that its proof yields some interesting results The main ones are: (i) when fitting an autoregressive scheme via the Yule-Walker equations, the fitted scheme is guaranteed to be stationary, and (ii) an easily applicable test to check whether a given scheme is stationary or not is found
TL;DR: For the univariate case, no more than the first few terms in any of the expansions have been published, with the exception of the Cornish-Fisher expansions.
Abstract: An investigation into previous work dealing with various series expansions about the normal distribution revealed little in the way of exact formulas for the terms in these expansions. In fact, for the univariate case, no more than the first few terms in any of the expansions have been published, with the exception of the Cornish-Fisher expansions, for which the first six terms are given in the Cornish-Fisher (1960) paper. In order to use the expansions in other problems, we first needed the exact formulas for some higher order terms. This paper is devoted to the presentation of these new terms and to the techniques used to obtain them. After this work was begun, a paper by Hill and Davis (1968) dealt with the theory of the same problem and some unpublished tables were referenced. The present paper, based one equivalent theory, but done independently using different computing algorithms, provides exact formulas for, at least, the first eight terms in four important expansions. The unpublished results of H...
TL;DR: This article derived simple expressions for the approximate relative errors in the use of either of two binomial approximations to the hypergeometric, and also of the Poisson to either the hyper-geometric or the binomial.
Abstract: The approximations herein derived are for individual terms of the hypergeometric, binomial and Poisson distributions. They give,simple expressions for the approximate relative errors in the use of either of two binomial approximations to the hypergeometric, and also of the Poisson to either the hypergeometric or the binomial. They may be used to estimate errors involved or as corrections to provide closer approximations to desired probabilities.
TL;DR: In this paper, the Darmois-Skitovic Theorem on characterization of normal lew through independence of linear functions of random variables is extended to independence of more general functions satisfying an addition theorem.
Abstract: The Darmois-Skitovic Theorem on characterization of normal lew through independence of linear functions of random variables is extended to independence of more general functions satisfying an addition theorem.
TL;DR: In this article, the exact distribution of the test criterion used in testing that the probabilities in a one way table are equal was computed and a set of tables was given to facilitate picking out the critical region corresponding to 13, 33, and 10th type 1 error.
Abstract: The author has computed the exact distribution of the test criterion used in testing that the probabilities in a one way table are equal. A set of tables is given to facilitate picking out the critical region corresponding to 13, 33, and 10th type 1 error. With the aid of graphs, it is shown that the continuity corrections that reduce the value of the computed test criterion are actually making the error in the chi-square approxitation verse.
TL;DR: A power study of the D test (D' Agostino, 1971) for sample size n=100 is presented in this paper, where power values for the contaminated normals of Chen (1971b) for n=20, 50 and 100.
Abstract: Due to the extension of the Shapiro and Wilk W test by Shapiro and Francia (1972) a new power study of the D test (D' Agostino, 1971) is presented for sample size n=100. For most alternatives the D testcompares favorably with the.W' test of Shapiro and Francia and the traditional tests. However, itspowers for some skewed alternatives with less than normal kurtosis (β2<3) are low. Also presented are power values for the- contaminated normals of Chen (1971b) for n=20, 50 and 100. Forlocation contaminated normals the W and W' tests are either as goodorbetter than D. While for scale contaminated normals D is as good orbetter than W and W'.
TL;DR: In this article, the joint distribution of a certain set of Kn + 1 of the variables Y1(n),… Yn(n) is examined and it is shown that for all asymptotic probability calculations, this joint distribution can be assumed to be normal.
Abstract: For each n, X1(n),…Xn(n) are independent and identically distributed random variables.Y1(n) < … < Yn(n) are the ordered values of X1(n),… Xn(n). Kn is a positive integer, with . The joint distribution of a certain set of Kn +1 of the variables Y1(n),… Yn(n) is examined.It is shown that for all asymptotic probability calculations, this joint distribution can be assumed to be normal. Applications to asymptotic statistical inference are given.
TL;DR: In this paper, the relationship between strong unimodality and canonical exponential families was investigated and some considerations pertaining to the duality viewpoint as applied to the sampling and the likelihood aspects of statistical models.
Abstract: Relationships between uninodality, strong unimodality and canonical exponential families axe studied. Furthermore, some considerations axe given pertaining to the duality viewpoint as applied to the sampling and the likelihood aspects of statistical models.
TL;DR: In this paper, the authors consider a situation where the random Tariables are such that (n-1) of them are independent and identically distributed with a probability density function (p.d.), and with probability, the remaining one is distributed with p.d.
Abstract: Me consider a situation where the random Tariables are such that (n-1) of them are independent and identically distributed with a probability density function (p.d.f.) , and with probability , the remaining one is distributed with p.d.f. . We will study the behaviour of the Bayesian estimators of the nuisance parameter, mean life and the reliability function under different prior distributions of a and a, The corresponding solutions when the p.d.f. of X's are , will also be outlined.
TL;DR: In this paper, a sequential sampling procedure was developed to estimate the scale parameter θ of the Pareto distribution when the shape parameter is unknown using the cost function, where A is a positive constant.
Abstract: A sequential sampling procedure is developed to estimate the scale parameter θ of the Pareto distribution when the shape parameterθ is unknown using the cost function , where A is a positive constant. The probability distribution of the stopping time for this procedure is tabulated and the expected stopping time and cost under the sequential sampling procedure are computed and compared with the optimum sample size and minimum cost under the fixed sample size procedure (for the case when the shape parameter is known).
TL;DR: In this paper, the effects of aurocorrelated errors on the parameter estimates of a linear regression model with varying degrees of multicollinearity among the regressors and another linear model with a distributed lag are evaluated.
Abstract: Sampling experiments are conduced to determine the effects of aurocorrelated errors on the parameter estimates of a linear regression model with varying degrees of multicollinearity among the regressors and another linear model with a distributed lag. Various estimators are employed. When there is a high degree of autocorrelation among the errors, the better performing estimators are the ones proposed by Cochrane-Orcutt, Durbin, and a nonlinear estimator.
TL;DR: In this paper, it was shown that the distribution of the maximum likelihood estimator of γ is independent of the location and scale parameters, α and β, and some examples of problems in which this result may have useful application are discussed.
Abstract: Consider a three-parameter density function of the form , where g is a known function and -∞ , < α <, ∞ , β < 0.. It is shown that the distribution of the maximum likelihood estimator of γ is independent of the location and scale parameters, α and β. Some examples of problems in which this result may have useful application are discussed.
TL;DR: In this paper, the authors proved the infinite divisibility of a linear combination of correlated noncentral quadratic forms is normal variables and derived a generalization of the noncentral multivariate t-distribution and also obtained as exact expression for the density of the difference of two correlated chi-square random variables with same degrees of freedom.
Abstract: In this paper, the authors proved the infinite divisibility of a linear combination of correlated noncentral quadratic forms is normal variables Further, using the property of the infinite divisibility, tvo representations are given for the distribution of the above linear combination Finally, the authors derived a generalization of the noncentral multivariate t‐distribution and also obtained as exact expression for the density of the difference of two correlated chi‐square random variables with same degrees of freedom
TL;DR: In this paper, the information matrix for the variance components of a linear model involve, under normality, the inverse of the variance-covariance matrix of the vector of observations, and attempts at finding this inverse for the 2-way crossed classification random model, unbalanced data, are described.
Abstract: Elements of the information matrix for the variance components of a linear model involve, under normality, the inverse of the variance-covariance matrix of the vector of observations. Attempts at finding this inverse for the 2-way crossed classification random model, unbalanced data, are described.
TL;DR: In this paper, McNemar's 1947 test for significant change in a 2×2 table is discussed and related to the parameter spaces associated with Stuart's 1955 test for homogeneity of marginal distributions and Bowkar's 1948 test for symetry.
Abstract: In this paper tets for change in exc contingency tables are considered. These tests are extansions of McNemar's 1947 test for significant change in a 2×2 table. The null parameter spaces associated with these tests are discussed and related to the parameter spaces associated with Stuart's 1955 test for homogeneity of marginal distributions and Bowkar's 1948 test for symetry.
TL;DR: In this article, tests for uniform clustering and randomness based on runs observed in lines or core samples are presented for a dichotomous classification of particles, but extensions can be made.
Abstract: In statistical problems associated with particulate matter,geographic areas and stersology one is often interested in clustering patterns of certain phasesParticularly, one may be interested in whether or not the cluster sizes are relatively unifrom throughout the mediam and whether or not particules and to occur in random arrangementThis paper presents tests for uniform clustering and randomness based on runs observed in lines or core samples The results are presented for a dichotomous classification of particles,but extensions can be made
TL;DR: In this paper, the power function of the two-sided test proposed by Roy to test the sphericity hypothesis against a specific alternative for the covariance matrix of a bivariate normal distribution is computed.
Abstract: Tables are computed for the power function of the two sided test proposed by Roy to test the sphericity hypothesis against a specific alternative for the covariance matrix of a bivariate normal distribution