fast and high accuracy multigrid solution of the three dimensional poisson equation we employ a fourth order compact finite difference scheme fos with the multigrid algorithm to solve the three dimensional poisson equation we test the influence of different orderings of the grid space and different grid transfer operators on the convergence and efficiency of our high accuracy algorithm fourier smoothing analysis is conducted to show that fos has a smaller smoothing factor than the traditional second order central difference scheme cds a new method of fourier smoothing analysis is proposed for the partially decoupled red black gauss seidel relaxation with fos numerical results are given to compare the computed accuracy and the computational efficiency of fos with multigrid against cds with multigrid

vectorized multigrid poisson solver for the cdc cyber 205 the full multigrid fmg method is applied to the two dimensional poisson equation with dirichlet boundary conditions this has been chosen as a relatively simple test case for examining the efficiency of fully vectorizing of the multigrid method data structure and programming considerations and techniques are discussed accompanied by performance details

two grid analysis of minimal residual smoothing as a multigrid acceleration technique we analyze the two level method accelerated by a minimal residual smoothing mrs technique the two grid analysis is sufficient for our purpose because our mrs acceleration scheme is only applied on the finest level of the multigrid method we prove that the mrs acceleration scheme is a semi iterative method with respect to the underlying two level iteration and that the mrs accelerated two level method is a polynomial acceleration of first order we explain why mrs may not effectively accelerate standard multigrid method for solving poisson like problems the iteration matrices for the mrs accelerated coarse grid correction operator and the mrs accelerated two level operator are obtained we give bounds for the residual reduction rates of the accelerated two level method numerical experiments are employed to support the analytical results

estimating equations for removal data analysis we consider the problem of estimating a population size from successive catches taken during a removal experiment and propose two estimating functions approaches the traditional quasi likelihood tql approach for dependent observations and the conditional quasi likelihood cql approach using the conditional mean and conditional variance of the catch given previous catches asymptotic covariance of the estimates and the relationship between the two methods are derived simulation results and application to the catch data from smallmouth bass show that the proposed estimating functions perform better than other existing methods especially in the presence of overdispersion

correlation matching by finite length sequences the problem of matching a finite length data sequence to a set of not necessarily uniformly spaced correlation lags is considered a characterization of the set of correlations that can be derived from sequences of a given length is presented using this characterization an algorithm called the expanding hull algorithm is presented for determining the minimum sequence length and a sequence of this length which matches a given set of correlation values is obtained this sequence has a z transform which is the lowest order correlation matching moving average model the sequence also generates the minimum length correlation extension the expanding hull algorithm also provides a method for extendibility testing of missing lag and multidimensional correlation sequences numerical examples are provided

on red black sor smoothing in multigrid optimal relaxation parameters are obtained for red black gauss seidel relaxation in multigrid solvers of a family of elliptic equations the resulting relaxation schemes are found to retain very high efficiency over an appreciable range of coefficients of the elliptic differential operator yielding simple inexpensive and fully parallelizable smoothers in many situations where less cost effective block and alternating direction schemes are commonly used

minimal residual smoothing in multi level iterative method a minimal residual smoothing mrs technique is employed to accelerate the convergence of the multi level iterative method by smoothing the residuals of the original iterative sequence the sequence with smoothed residuals is re introduced into the multi level iterative process the new sequence generated by this acceleration procedure converges much faster than both the sequence generated by the original multi level method and the sequence generated by mrs technique the cost of this acceleration scheme is independent of the original operator and in many cases is negligible the emphasis of this paper is on the practical implementation of mrs acceleration techniques in the multi level method the discussions are focused on the two level method because the acceleration scheme is only applied on the finest level of the multi level method numerical experiments using the proposed mrs acceleration scheme to accelerate both the two level and multi level methods are conducted to show the efficiency and the cost effectiveness of this acceleration scheme

estimating equations with nonignorably missing response data troxel lipsitz and brennan 1997 biometrics 53 857 869 considered parameter estimation from survey data with nonignorable nonresponse and proposed weighted estimating equations to remove the biases in the complete case analysis that ignores missing observations this paper suggests two alternative modifications for unbiased estimation of regression parameters when a binary outcome is potentially observed at successive time points the weighting approach of robins rotnitzky and zhao 1995 journal of the american statistical association 90 106 121 is also modified to obtain unbiased estimating functions the suggested estimating functions are unbiased only when the missingness probability is correctly specified and misspecification of the missingness model will result in biases in the estimates simulation studies are carried out to assess the performance of different methods when the covariate is binary or normal for the simulation models used the relative efficiency of the two new methods to the weighting methods is about 3 0 for the slope parameter and about 2 0 for the intercept parameter when the covariate is continuous and the missingness probability is correctly specified all methods produce substantial biases in the estimates when the missingness model is misspecified or underspecified analysis of data from a medical survey illustrates the use and possible differences of these estimating functions

on the autocorrelation of finite length sequences lt is well known that all autocorrelation sequences are positive semi definite the correlation sequence is further constrained when as is often the case it is obtained from correlating a finite length data sequence with itself in this paper we study the effect of the data sequence length on the attainable correlations we derive a formula for the sequence length needed to match an autocorrelation value at a given lag the formula indicates that inordinately long sequences are needed to obtain a normalized autocorrelation coefficient near unity we conclude that all zero moving average models may be ill suited for highly correlated signals the formula for the single lag case is then used to obtain a bound on the sequence length needed to match a set of autocorrelations simultaneously

stability of linear predictors and numerical range of a linear operator corresp the zeros of a predictor polynomial are shown to belong to the numerical range of a linear operator associated with the particular prediction problem considered application of this result to the autocorrelation and postwindowed cases shows that the predictor polynomials enjoy a well defined stability margin which depends in particular on the length of the data sequence the generalization of these results to the multichannel case is also discussed

steplength optimization and linear multigrid methods recently steplength parameters have been used in linear multigrid methods in this paper we give a theoretical analysis of the effects of steplength optimization in a rather general framework which covers two different implementations of steplength optimization in standard multigrid methods

locations of zeros of predictor polynomials this work shows that the zeros of the predictor polynomial determined by a finite data least squares linear prediction problem lie inside an irregular polygon contained in the unit circle of the complex plane the polygon is independent of the data only depending on the length of the data and the order of the predictor the results are an analytic statement of the resolution limitations of spectral estimates based on finite data least squares linear predictors

modelling ordered categorical data recent advances and future challenges this article summarizes recent advances in the modelling of ordered categorical ordinal response variables we begin by reviewing some models for ordinal data introduced in the literature in the past 25 years we then survey recent extensions of these models and related methodology for special types of applications such as for repeated measurement and other forms of clustering we also survey other aspects of ordinal modelling such as small sample analyses power and sample size considerations and availability of software throughout we suggest problem areas for future research and we highlight challenges for statisticians who deal with ordinal data

on matching correlation sequences by parametric spectral models the correlation matching property of all pole models does not extend readily to pole zero models it is proved that if the number of poles in a pole zero model is strictly less than the number of given correlations a pole zero model that matches the given correlations may not exist irrespective of the number of zeros in the model it is also shown that if one adds a set of cepstral constraints a maximum entropy pole zero model that matches the correlation and cepstral constraints may not exist it is concluded that pole zero modeling minimizing some error criterion might be preferable to exact constraint matching

a quasi likelihood approach for ordered categorical data with overdispersion quasi likelihood ql methods are often used to account for overdispersion in categorical data this paper proposes a new way of constructing a ql function that stems from the conditional mean variance relationship unlike traditional ql approaches to categorical data this ql function is in general not a scaled version of the ordinary log likelihood function a simulation study is carried out to examine the performance of the proposed ql method fish mortality data from quantal response experiments are used for illustration

an extended quasi likelihood function summary wedderburn s original definition of quasi likelihood for generalized linear models is extended to allow the comparison of variance functions as well as those of linear predictors and link functions the relationship between generalized linear models and the use of transformations of the response variable is explored and the ideas are illustrated by three examples generalized linear models by allowing the full distributional assumption about the random component in the model to be replaced by a much weaker assumption in which only the first and second moments were defined in making this extension wedderburn widened the scope of generalized linear models in a way very similar to that of gauss when he replaced the assumption of normality in classical linear models by that of equal variance for generalized linear models with distributions in the exponential family likelihood ratio and score tests are used for testing hypotheses concerning nested subsets of covariates in the linear predictor and for assessing hypothesized link functions these methods are also applicable with wedderburn s form of quasi likelihood however neither of these methods is suitable for the comparison of different variance functions in this paper we introduce an extended quasi likelihood function which allows for the comparison of various forms of all the components of a generalized linear model i e the linear predictor the link function and the variance function we then apply the ideas to the analysis of several sets of data

acceleration of five point red black gauss seidel in multigrid for poisson equation abstract a new relaxation analysis and two acceleration schemes are proposed for the five point red black gauss seidel smoothing in multigrid for solving a two dimensional poisson equation for a multigrid v cycle we discovered that underrelaxation is applicable to restriction half cycle and overrelaxation is applicable to interpolation half cycle numerical experiments using modified multigrid v cycle algorithms show that our simple acceleration schemes accelerate the convergence rate by as much as 34 with negligible cost this result is contrary to the existing belief that sor is not suitable for using as a smoother in multigrid for poisson equation because the gain in computational savings would not pay for the cost of implementing it more important is the idea of employing different parameters to accelerate the reduction of low and high frequency errors separately our discovery offers a new way for sor smoothing in multigrid

a growth curve model approach to multivariate quantal bioassay the typical quantal bioassay is used to estimate that concentration which will affect a given response in loop of the subjects of interest when p 50 we speak of ec50 for median effective concentration or lc50 for a lethal effect more explicitly such a controlled laboratory experiment usually involves subjecting nj experimental units to a concentration cj and recording the number zj of experimental units affected by this concentration level for an increasing sequence of concentrations cj j 1 2 d with different corresponding sets of nj subjects a sequence zj of the number of subjects responding is obtained when a lethal effect occurs the ratios zj nj pj say become mortality rates and the problem is to estimate the concentration lc50 which corresponds to p 50 various univariate methods can be used to solve this problem the most popular being probit analysis see for example finney 1971 or hubert 1984 if the responses are recorded after a time of t hours the parameter to be estimated is referred to as the t hour lc50 when the experiment is monitored over a sequence of time points the corresponding response variables become functions of concentration and time however the application of univariate probit analysis at each time point neglects the dependency in time and also the possible interaction of time and concentration on the response the analysis of indirect quantitative assays with correlated data has been described by box and hay 1953 their analysis assumes a repeated measures model that is the correlations between observations on the same experimental unit are assumed to be equal elashoff 1981 studied the effect of heterogeneity of variances and correlations on this analysis v0lund 1980 treated the case in which these correlations are arbitrary thereby extending the work of rao 1954 multivariate indirect quantal assays have been studied by kolakowski and bock 1981 and by kooijman 1981 kolakowski and bock 1981 assumed a latent structural model with independence among the responses over time kooijrman 1981 assumed that the mortality counts follow a multinomial distribution with the probability of death being a function of time and concentration such a model does not differentiate between sampling units and experimental units for example if concentrations of copper are administered to tanks of 20 fish the model requires the assumption that there is no error variation among tanks of fish some work on the inclusion of this extra variation in the model for experiments involving litters of subjects has been done by segreti and munson 1981

on stability and root location of linear prediction models simple algebraic proofs are given for some results pertaining to the location of the roots of linear prediction models also called all pole or autoregressive models

a cost effective multigrid projection operator a novel heuristic residual analysis is proposed to derive a computationally cost effective residual projection operator in multigrid with the five point red black gauss seidel relaxation for solving the two dimensional poisson equation this optimal residual injection operator is as cheap as the trivial injection operator but is more efficient than the costly full weighting operator and achieves near optimal convergence rate

high accuracy multigrid solution of the 3d convection diffusion equation we present an explicit fourth order compact finite difference scheme for approximating the three dimensional 3d convection diffusion equation with variable coefficients this 19 point formula is defined on a uniform cubic grid fourier smoothing analysis is performed to show that the smoothing factor of certain relaxation techniques used with the scheme is smaller than 1 we design a parallelization oriented multigrid method for fast solution of the resulting linear system using a four color gauss seidel relaxation technique for robustness and efficiency and a scaled residual injection operator to reduce the cost of multigrid inter grid transfer operator numerical experiments on a 16 processor vector computer are used to test the high accuracy of the discretization scheme as well as the fast convergence and the parallelization or vectorization efficiency of the solution method several test problems are solved and highly accurate solutions of the 3d convection diffusion equations are obtained for small to medium values of the grid reynolds number effects of using different residual projection operators are compared on both vector and serial computers

simple bounds on the extreme eigenvalues of toeplitz matrices simple bounds are presented on the extreme eigenvalues of n n dimensional hermitian toeplitz matrices such a matrix say t sub n is determined by its first row the proposed bounds have low complexity o n furthermore examples are presented for which the proposed bounds are tighter than the slepian landau bounds at their best i e when the extreme eigenvalues of the submatrix obtained by deleting the first row and first column of t sub n are known exactly the bounds are first presented on the extreme eigenvalues of hermitian toeplitz matrices the corresponding bounds for real symmetric toeplitz matrices follow as a special case then these bounds are extended to hermitian toeplitz interval matrices

zero location of lacunary prediction polynomials this correspondence discusses the stability of some lacunary prediction polynomials and rederives in passing known upper bounds for the normalized autocorrelation coefficients

residual scaling techniques in multigrid ii practical applications this paper focuses on the practical applications of the multigrid residual scaling techniques and is the continuation of a companion paper residual scaling techniques in multigrid i equivalence proof appl math comput 86 283 303 1997 we discuss the computational issues of some residual scaling techniques which have been proven mathematically equivalent a heuristic residual analysis technique based on the geometry of the grid points and the relaxation pattern is introduced to estimate the optimal residual scaling factor for a high order multigrid method we compare the performance of a typical pre optimization pre acceleration technique with a typical post optimization post acceleration technique and show that the pre optimization is preferable in both convergence and efficiency our numerical results support the theoretical conclusions made in the companion paper and demonstrate the full advantage of the pre optimization technique over the post optimization technique

residual scaling techniques in multigrid i equivalence proof this paper addresses the residual scaling techniques coarse grid correction optimization techniques in multigrid methods we surveyed recent developments in this area and prove the equivalence of the overweighted residual technique and the overcorrection technique this leads to the proof of mathematical equivalence of the prescaling and postscaling acceleration techniques two theorems have been proved to unify the concept of the residual scaling techniques these theoretical results clear the way for developing efficient prescaling acceleration techniques for practical applications those practical prescaling acceleration techniques are discussed in a companion paper residual scaling techniques ii practical applications

multigrid smoothing factors for red black gauss seidel relaxation applied to a class of elliptic operators analytic formulae are obtained for the smoothing factors yielded by gauss seidel relaxation in two color ordering for a class of scalar elliptic operators block and point relaxations in conjunction with full or partial coarsening are encompassed for operators with general constant positive coefficients in general dimensions and for an arbitrary number of relaxation sweeps it is found that there is no direct dependence of the smoothing factors on the dimension and that the effect of the number of relaxation sweeps on the smoothing factor is usually independent of the operator coefficients and of the relaxation scheme the results are compared with computed results of two level analyses smoothing strategies implied by the formulae are discussed