asymptotic expansion of the quasiconfluent hypergeometric function the asymptotic expansion of the hypergeometric function 2f1 a b c z b in the case of quasiconfluence i e for b is revised a very simple expansion in terms of a semiasymptotic sequence of polynomials is presented some properties of those polynomials are discussed

a generalization of n matrices abstract ky fan defines an n matrix to be a matrix of the form a ti b b 0 t b where b is the spectral radius of b and is the maximum of the spectral radii of all principal submatrices of b of order n 1 in this paper we define the closure n 0 matrices of n matrices by letting t it is shown that if a z and a 1 a n certain inequalities of n matrices are shown to hold for n 0 matrices and a method for constructing an n matrix from an m matrix is given

confluence expansions of the generalized hypergeometric function by confluencing a subset of upper and lower parameters in the generalized hypergeometric function f p q a 1 a p c 1 c q z with the variable z one obtains a lower order hypergeometric

on classes of inverse z matrices abstract recently a classification of matrices of class z was introduced by fiedler and markham this classification contains the classes of m matrices and the classes of n 0 and f 0 matrices studied by fan g johnson and smith the problem of determining which nonsingular matrices have inverses which are z matrices is called the inverse z matrix problem for special classes of z matrices such as the m and n 0 matrices there exist at least partial results i e special classes of matrices have been introduced for which the inverse of such a matrix is an m matrix or an n 0 matrix here we define a system of classes of matrices for which the inverse of each matrix of each class belongs to one class of the classification of z matrices defined by fiedler and markham moreover certain properties of the matrices of each class are established e g inequalities for the sum of the entries of the inverse and the structure of certain schur complements we also give a necessary and sufficient condition for regularity the class of inverse n 0 matrices given here generalizes the class of inverse n 0 matrices discussed by johnson all results established here can be applied to a class of distance matrices which corresponds to a nonarchimedean metric this metric arises in p adic number theory and in taxonomy

likelihood based methods for missing covariates in the cox proportional hazards model problems associated with missing covariate data are well known but often ignored we present a method for estimating the parameters in the cox proportional hazards model when the missing data are missing at random mar and censoring is noninformative due to the computational burden of this method we introduce an approximation that allows us to use a weighted expectation maximization em algorithm to estimate the parameters more easily when the missing covariates are continuous rather than categorical we implement a monte carlo version of the em algorithm along with the gibbs sampler to obtain parameter estimates we also give the asymptotic distribution of these estimates the primary advantage of this method over complete case analysis is that it produces more efficient parameter estimates and corrects for bias in the mar setting to motivate the methodology we present an analysis of a phase iii melanoma clinical trial conducted by the eastern cooperative oncology group

posterior propriety and computation for the cox regression model with applications to missing covariates full likelihood bayesian approaches in which a gamma process prior is specified for the cumulative baseline hazard we examine characterizations of posterior propriety under completely observed data settings as well as for missing covariates latent variables are introduced to facilitate a straightforward gibbs sampling scheme in the bayesian computation a real dataset is presented to illustrate the proposed methodology

further results for systems with repeated scalar nonlinearities a class of nonlinear systems described by a discrete time state equation containing a repeated scalar nonlinearity is considered this paper sharpens several previous results on the performance synthesis and model reduction for such systems extensions to the case that the nonlinearity is not an odd function are also discussed

nonparametric bayes estimators based on beta processes in models for life history data the article studies the problem of finding bayes estimators for cumulative hazard rates and related quantities w r t prior distributions that correspond to cumulative hazard rate processes with nonnegative independent increments a particular class of prior processes termed beta processes is introduced and is shown to constitute a conjugate class an important bonus of the approach based on cumulative hazards is that more complicated models for life history data than the simple life table situation can be treated for example time inhomogeneous markov chains we find posterior distributions and derive bayes estimators in such models and present a semiparametric bayesian analysis of the cox regression model

an algorithm for determining if the inverse of a strictly diagonally dominant stieltjes matrix is strictly ultrametric it was recently shown that the inverse of a strictly ultrametric matrix is a strictly diagonally dominant stieltjes matrix on the other hand as it is well known that the inverse of a strictly diagonally dominant stieltjes matrix is a real symmetric matrix with nonnegative entries it is natural to ask conversely if every strictly diagonally dominant stieltjes matrix has a strictly ultrametric inverse examples show however that the converse is not true in general i e there are strictly diagonally dominant stieltjes matrices in n n for everyn 3 whose inverses are not strictly ultrametric matrices then the question naturally arises if one can determine which strictly diagonally dominant stieltjes matrices in n n n 3 have inverses which are strictly ultrametric here we develop an algorithm based on graph theory which determines if a given strictly diagonally dominant stieltjes matrixa has a strictly ultrametric inverse where the algorithm is applied toa and requires no computation of inverse moreover if this given strictly diagonally dominant stieltjes matrix has a strictly ultrametric inverse our algorithm uniquely determines this inverse as a special sum of rank one matrices

a bayesian justification of cox s partial likelihood summary in this paper we establish both naive and formal bayesian justifications of cox s 1975 partial likelihood and its various modifications we extend the original work of kalbfieisch 1978 who showed that the partial likelihood is a limiting marginal posterior under noninformative priors for baseline hazards we extend the result to scenarios with timedependent covariates and time varying regression parameters we establish results for continuous time as well as grouped survival data in addition we present a bayesian justification of a modified partial likelihood for handling ties we also present tools for simplification of the gibbs sampling algorithm for implementing partial likelihood based bayesian inference in various practical applications

an expansion of the hypergeometric function in bessel functions an expansion of the hypergeometric function 2f1 a b c 1 z2 4ab in bessel functions of argument z is derived this expansion can be used to obtain an asymptotic expansion of the hypergeometric function for large absolute values of a and b

generalized ultrametric matrices a class of inverse m matrices abstract recently marti nez michon and san marti n introduced the new class of symmetric strictly ultrametric matrices they proved that the inverse of a strictly ultrametric matrix is a strictly row and strictly column diagonally dominant stieltjes matrix here we generalize their result by introducing a class of nonsymmetric matrices calledgeneralized ultrametric matrices we give a necessary and sufficient condition for the regularity of these matrices and prove that the inverse of a nonsingular generalized ultrametric matrix is a row and column diagonally dominant m matrix we establish that a nonnegative matrix is a generalized ultrametric matrix if and only if the matrix is a certain sum of at most rank two matrices moreover we give a characterization of generalized ultrametric matrices based on weighted trees the entries of generalized ultrametric matrices then arise as certain distances between the leaves and the root of the tree

an asymptotic expansion for the hypergeometric function 2f1 a b c x watson s treatise expresses the ordinary bessel function as a limit of a hypergeometric function where the first two parameters go to infinity while simultaneously the argument goes to zero we have extended watson s method of proof to derive an asymptotic expression for the hypergeometric function to second order in the inverse of the first two parameters with the ordinary bessel function as its leading order term we show as an example that the use of this new result is pivotal in showing the correspondence between quantal and classical results of the differential cross section in coulomb excitation

estimating equations with incomplete categorical covariates in the cox model incomplete covariate data is a common occurrence in many studies in which the outcome is survival time when a full likelihood is specified a useful technique for obtaining parameter estimates is the em algorithm we propose a set of estimating equations to estimate the parameters of cox s proportional hazards model when some covariate values are missing these estimating equations can be solved by an algorithm similar to the em algorithm because of the computational burden of finding a solution to these estimating equations we propose obtaining parameter estimates via monte carlo methods asymptotic variances of the parameter estimates are also derived we present a clinical trials example with three covariates two of which have some missing values

computational methods for multiplicative intensity models using weighted gamma processes proportional hazards marked point processes and panel count data we develop computational procedures for a class of bayesian nonparametric and semiparametric multiplicative intensity models incorporating kernel mixtures of spatial weighted gamma measures a key feature of our approach is that explicit expressions for posterior distributions of these models share many common structural features with the posterior distributions of bayesian hierarchical models using the dirichlet process using this fact along with an approximation for the weighted gamma process we show that with some care one can adapt efficient algorithms used for the dirichlet process to this setting we discuss blocked gibbs sampling procedures and polya urn gibbs samplers we illustrate our methods with applications to proportional hazard models poisson spatial regression models recurrent events and panel count data

semiparametric methods for response selective and missing data problems in regression summary suppose that data are generated according to the model f ylx 0 g x where y is a response and x are covariates we derive and compare semiparametric likelihood and pseudolikelihood methods for estimating 0 for situations in which units generated are not fully observed and in which it is impossible or undesirable to model the covariate distribution the probability that a unit is fully observed may depend on y and there may be a subset of covariates which is observed only for a subsample of individuals our key assumptions are that the probability that a unit has missing data depends only on which of a finite number of strata that y x belongs to and that the stratum membership is observed for every unit applications include case control studies in epidemiology field reliability studies and broad classes of missing data and measurement error problems our results make fully efficient estimation of 0 feasible and they generalize and provide insight into a variety of methods that have been proposed for specific problems

the collapsed gibbs sampler in bayesian computations with applications to a gene regulation problem abstract this article describes a method of grouping and collapsing in using the gibbs sampler and proves from an operator theory viewpoint that the method is in general beneficial the norms of the forward operators associated with the corresponding nonreversible markov chains are used to discriminate among different simulation schemes when applied to bayesian missing data problems the idea of collapsing suggests skipping the steps of sampling parameter s values in standard data augmentation by doing this we obtain a predictive update version of the gibbs sampler a procedure of calculating the posterior odds ratio via the collapsed gibbs sampler when incomplete observations are involved is presented as an illustration of possible applications three examples along with a bayesian treatment for identifying common protein binding sites in unaligned dna sequences are provided

a general framework for random effects survival analysis in the cox proportional hazards setting summary the use of random effects modeling in statistics has increased greatly in recent years the introduction of such modeling into event time analysis has proceeded more slowly however previously random effects models for survival data have either required assumptions regarding the form of the baseline hazard function or restrictions on the classes of models that can be fit in this paper we develop a method of random effect analysis of survival data the hierarchical cox model that is an extension of cox s original formulation in that the baseline hazard function remains unspecified this method also allows an arbitrary distribution for the random effects we accomplish this using markov chain monte carlo methods in a bayesian setting the method is illustrated with three models for a dataset with times to multiple occurrences of mammory tumors for 48 rats treated with a carcinogen and then randomized to either treatment or control this analysis is more satisfying than standard approaches such as studying the first event for each subject which does not fully use the data or assuming independence which in this case would overestimate the precision

inverse m matrix inequalities and generalized ultrametric matrices abstract we use weighted directed graphs to introduce a class of nonnegative matrices which under a simple condition are inverse m matrices we call our class the generalized ultrametic matrices since it contains the class of symmetric ultrametric matrices and some unsymmetric matrices we show that a generalized ultrametric matrix is the inverse of a row and column diagonally dominant m matrix if and only if it contains no zero row and no two of its rows are identical this theorem generalizes the known result that a symmetric strictly ultrametric matrix is the inverse of a strictly diagonally dominant m matrix we also present inequalities and conditions for equality among the entries of the inverse of a row diagonally dominant m matrix some of these inequalities and conditions for equality generalize results of stieltjes on inverses of symmetric diagonally dominant m matrices

on using the cox proportional hazards model with missing covariates summary we propose two methods for handling missing covariates in using the cox proportional hazards model the maximum partial likelihood estimator based only on study subjects having complete covariates does not utilise all available information also it is biased when the probability of missingness depends on the failure or censoring time our suggestion is to impute the conditional expectation of the statistic involving missing covariates given the available information the proposed method provides a consistent regression parameter estimator when the probability of missingness depends on the failure or censoring time as well as on the observed covariates also the proposed estimator is more efficient than the estimator suggested previously by lin ying 1993 when data are missing completely at random

on the zeros of a class of generalized hypergeometric polynomials we obtain the asymptotic behavior of the zeros of a class of generalized hypergeometric polynomials for this purpose we make use of a mehler heine type formula for these polynomials we illustrate these results with numerical experiments and some figures

b spline collocation method for the singular perturbation problem using artificial viscosity in this paper we develop a b spline collocation method using artificial viscosity for solving singularly perturbed equations given by eu x a x u x b x u x f x a x a 0 b x b 0 0 1 u 0 a u 1 b we use the artificial viscosity to capture the exponential features of the exact solution on a uniform mesh and use b spline collocation method which leads to a tridiagonal linear system the convergence analysis is given and the method is shown to have uniform convergence of second order the design of artificial viscosity parameter is confirmed to be a crucial ingredient for simulating the solution of the problem known test problems have been studied to demonstrate the accuracy of the method numerical results show the behaviour of the method with emphasis on treatment of boundary conditions results shown by the method are found to be in good agreement with the exact solution

inverse m matrices abstract this is an attempt at a comprehensive expository study of those nonnegative matrices which happen to be inverses of m matrices and is aimed at an audience conversant with basic ideas of matrix theory a theme is the parallels anddifferences between the class of m matrices and the class of inverse m matrices among the primary tools used are diagonal multiplications the neumann expansion and the form of the inverse of a partitioned matrix some of the results seem not to have appeared before and several unsolved problems are mentioned

log linear analysis of censored survival data with partially observed covariates abstract log linear models provide a flexible means of extending life table techniques for the analysis of censored survival data with categorical covariates as discussed by holford 1980 and laird and olivier 1981 we extend this methodology to incorporate cases in which one or more of the categorical covariates are sometimes missing maximum likelihood estimates of the parameters are calculated using data from all cases this can result in large gains in efficiency over standard methods that require the exclusion of cases with incomplete data with this approach we assume that the hazard function conditional on the covariates is a stepwise function over disjoint intervals of time the model has two parts a log linear model describing the hazard parameters and a multinomial model describing the probabilities in the contingency table defined by the covariates the main interest is in the model for the hazard parameters we show how to calculate maximum likelihood estimates of parameters of the mo

bounds on the spectrum of nonnegative matrices and certain z matrices spectral bounds are obtained for a type of z matrix using perron frobenius theory on the inverse these lead to spectral bounds for nonnegative matrices and m matrices these results are then extended using the result of rothblum and tan concerning the coefficient of ergodicity of a nonnegative matrix

asymptotic expansion of 2f1 a b c x in terms of confluent hypergeometric functions we have developed a new asymptotic formula for 2f1 a b c x in which a and c are finite b tends to infinity and the argument x tends to zero we make use of this result to obtain asymptotic expansions of the electric dipole e1 quantal differential excitation function for the adiabaticity parameter f i and the sommerfeld parameter f tending to infinity simultaneously while the parameter i remains finite

auxiliary covariate data in failure time regression summary we consider the problem of missing covariate data in the context of censored failure time relative risk regression auxiliary covariate data which are considered informative about the missing data but which are not explicitly part of the relative risk regression model may be available full covariate information is available for a validation set an estimated partial likelihood method is proposed for estimating relative risk parameters this method is an extension of the estimated likelihood regression analysis method for uncensored data pepe 1992 pepe fleming 1991 a key feature of the method is that it is nonparametric with respect to the association between the missing and observed including auxiliary covariate components asymptotic distribution theory is derived for the proposed estimated partial likelihood estimator in the case where the auxiliary or mismeasured covariates are categorical asymptotic efficiencies are calculated for exponential failure times using an exponential relative risk model the estimated partial likelihood estimator compares favourably with a fully parametric maximum likelihood analysis comparisons are also made with a standard partial likelihood analysis which ignores the incomplete observations important efficiency gains can be made with the estimated partial likelihood method small sample properties are investigated through simulation studies

a conditional model for incomplete covariates in parametric regression models summary incomplete covariate data arise in many data sets when the missing covariates are categorical a useful technique for obtaining parameter estimates is the em algorithm by the method of weights proposed in ibrahim 1990 this method requires the estimation of many nuisance parameters for the distribution of the covariates unfortunately in data sets when the percentage of missing data is high and the missing covariate patterns are highly non monotone the estimates of the nuisance parameters can lead to highly unstable estimates of the parameters of interest we propose a conditional model for the covariate distribution that has several modelling advantages for the e step and provides a reduction in the number of nuisance parameters thus providing more stable estimates in finite samples we present a clinical trials example with six covariates five of which have some missing values

maximum likelihood methods for cure rate models with missing covariates we propose maximum likelihood methods for parameter estimation for a novel class of semiparametric survival models with a cure fraction in which the covariates are allowed to be missing we allow the covariates to be either categorical or continuous and specify a parametric distribution for the covariates that is written as a sequence of one dimensional conditional distributions we propose a novel em algorithm for maximum likelihood estimation and derive standard errors by using louis s formula louis 1982 journal of the royal statistical society series b 44 226 233 computational techniques using the monte carlo em algorithm are discussed and implemented a real data set involving a melanoma cancer clinical trial is examined in detail to demonstrate the methodology

cox regression with incomplete covariate measurements abstract this article provides a general solution to the problem of missing covariate data under the cox regression model the estimating function for the vector of regression parameters is an approximation to the partial likelihood score function with full covariate measurements and reduces to the pseudolikelihood score function of self and prentice in the special setting of case cohort designs the resulting parameter estimator is consistent and asymptotically normal with a covariance matrix for which a simple and consistent estimator is provided extensive simulation studies show that the large sample approximations are adequate for practical use the proposed approach tends to be more efficient than the complete case analysis especially for large cohorts with infrequent failures for case cohort designs the new methodology offers a variance covariance estimator that is much easier to calculate than the existing ones and allows multiple subcohort augmentations to improve efficiency real data taken f

a mean score method for missing and auxiliary covariate data in regression models summary we consider regression analysis when incomplete or auxiliary covariate data are available for all study subjects and in addition for a subset called the validation sample true covariate data of interest have been ascertained the term auxiliary data refers to data not in the regression model but thought to be informative about the true missing covariate data of interest we discuss a method which is nonparametric with respect to the association between available and missing data allows missingness to depend on available response and covariate values and is applicable to both cohort and case control study designs the method previously proposed by flanders greenland 1991 and by zhao lipsitz 1992 is generalised and asymptotic theory is derived our expression for the asymptotic variance of the estimator provides intuition regarding performance of the method optimal sampling strategies for the validation set are also suggested by the asymptotic results