composite linear models for incomplete multinomial data a composite linear model clm is a matrix model for incomplete multinomial data a clm provides a unified approach for maximum likelihood inference which is applicable to a wide variety of problems involving incomplete multinomial data by formulating a model as a clm one can simplify computation of maximum likelihood estimates and asymptotic standard errors as an example we use clm to test marginal homogeneity for ordered categories subject to both ignorable and non ignorable missing data mechanisms

a nonlinear mixed finite eelement method for a degenerate parabolic equation arising in flow in porous media we study a model nonlinear degenerate advection diffusion equation having application in petroleum reservoir and groundwater aquifer simulation the main difficulty is that the true solution is typically lacking in regularity therefore we consider the problem from the point of view of optimal approximation through time integration we develop a mixed variational form that respects the known minimal regularity and then we develop and analyze two versions of a mixed finite element approximation a simpler semidiscrete time continuous version and a fully discrete version our error bounds are optimal in the sense that all but one of the bounding terms reduce to standard approximation error the exceptional term is a nonstandard approximation error term we also consider our new formulation for the nondegenerate problem showing the usual optimal l 2 error bounds moreover superconvergence is obtained under special circumstances

a finite volume scheme for nonlinear degenerate parabolic equations we propose a second order finite volume scheme for nonlinear degenerate parabolic equations which admit an entropy functional for some of these models porous media equation drift diffusion system for semiconductors it has been proved that the transient solution converges to a steady state when time goes to infinity the present scheme preserves steady states and provides a satisfying long time behavior moreover it remains valid and second order accurate in space even in the degenerate case after describing the numerical scheme we present several numerical results which confirm the high order accuracy in various regime degenerate and nondegenerate cases and underline the efficiency to preserve the large time asymptotic

distribution of distance in the spheroid the distribution of distance in the sphere is reviewed the distribution of distance in the ellipsoid is given as an integral which can be done in terms of elementary functions for the spheroid as an application maclaurin s ratio of the polar to equatorial radius of the earth due to its rotation is rederived using the distribution found here

l1 contraction and uniqueness for quasilinear elliptic parabolic equations abstract we prove the l 1 contraction principle and uniqueness of solutions for quasilinear elliptic parabolic equations of the form formula where b is monotone nondecreasing and continuous we assume only that u is a weak solution of finite energy in particular we do not suppose that the distributional derivative t b u is a bounded borel measure or a locally integrable function

horn silver distributions in d dimensional euclidean spaces a geometric interpretation is found for a class of finite discrete probability distributions that generalizes the distribution found by horn and silver to represent an unusual distribution of neutral versus charged pions in some hadronic production models connections to de finetti s representation theorem are discussed

moments of the distribution of distance it is shown that the even moments of the distribution of distance of a density are polynomials of the moments of that density it is discussed when the even moments alone determine the distribution of distance examples are computed explicitly it follows that moments of the lengths of chords and other quantities treated in geometric probability can also be computed in terms of the moments of the density these formulas are collected here the formalism is not restricted to homogeneous densities on bounded support bodies as an example the distributions of distance of the isotropic gaussian and of the cantor set are computed

multivariate logistic models for incomplete binary responses abstract in this article we describe a likelihood based regression model appropriate for analyzing incomplete multivariate binary responses we focus on marginal models that is models where the marginal mean or expectation of the binary response is related to a set of covariates the association between the binary responses is modeled in terms of conditional log odds ratios when the nonresponse mechanism is ignorable it is not necessary to specify a nonresponse model and valid inferences can be obtained provided that the likelihood for the responses has been correctly specified but when the nonresponse mechanism is nonignorable valid inferences can only be obtained by incorporating a model for nonresponse an unresolved issue with nonignorable models concerns the identifiability of the parameters so far no general and practically useful necessary and sufficient conditions for identifiability are available here we suggest some simple procedures for examining the identifiability status of nonign

regression analysis for categorical variables with outcome subject to nonignorable nonresponse abstract we develop a log linear model for categorical response subject to nonignorable nonresponse the paper differs from fay 1986 in its focus on estimation and hypothesis testing in a regression setting as opposed to imputation in a multivariate setting we present several new results concerning the existence of solutions on the boundary of the parameter space and the construction of confidence intervals for estimates we illustrate the method by estimating the proportion of voters preferring truman in a 1948 preelection poll mosteller hyman mccarthy marks and truman 1949 results may depend strongly on the model assumed for nonresponse goodness of fit tests are available for comparing alternative models

error estimates for the finite volume discretization for the porous medium equation we analyze the convergence of a numerical scheme for a class of degenerate parabolic problems modelling reactions in porous media and involving a nonlinear possibly vanishing diffusion the scheme involves the kirchhoff transformation of the regularized nonlinearity as well as an euler implicit time stepping and triangle based finite volumes we prove the convergence of the approach by giving error estimates in terms of the discretization and regularization parameter

causal models for patterns of nonresponse abstract the problem of missing data for categorical variables is examined from the perspective of modeling the mechanisms of nonresponse log linear causal models as formulated by goodman are studied for the relationship of the survey variables to response under some conditions several such models are estimable from the observed data for nested patterns of nonresponse a specific causal model represents exactly the assumption of ignorable response most causal models however imply nonignorable response mechanisms and yield alternative estimates for the distribution of the survey variables

finite element runge kutta discretizations of porous medium type equations in this paper we analyze the convergence properties of full discretizations of a class of generalized porous medium equations for the spatial and time discretizations we use continuous piecewise linear finite elements and algebraically stable runge kutta methods respectively we prove our convergence result without any assumption on the spatial regularity it is shown that under a certain stability assumption the temporal order of convergence is given by the stage order of the method whereas the spatial order is essentially one numerical experiments illustrate our stability assumption and the convergence result

identification of asymptotic decay to self similarity for one dimensional filtration equations the objective of this paper is the derivation and the analysis of a simple explicit numerical scheme for general one dimensional filtration equations it is based on an alternative formulation of the problem using the pseudoinverse of the density s repartition function in particular the numerical approximations can be proven to satisfy a contraction property for a wasserstein metric various numerical results illustrate the ability of this numerical process to capture the time asymptotic decay towards self similar solutions even for fast diffusion equations

a modified l scheme to solve nonlinear diffusion problems abstract in this work we propose a linearization technique for solving nonlinear elliptic partial differential equations that are obtained from the time discretization of a wide variety of nonlinear parabolic problems the scheme is inspired by the l scheme which gives unconditional convergence of the linear iterations here we take advantage of the fact that at a particular time step the initial guess for the iterations can be taken as the solution of the previous time step first it is shown for quasilinear equations that have linear diffusivity that the scheme always converges irrespective of the time step size the spatial discretization and the degeneracy of the associated functions moreover it is shown that the convergence is linear with convergence rate proportional to the time step size next for the general case it is shown that the scheme converges linearly if the time step size is smaller than a certain threshold which does not depend on the mesh size and the convergence rate is proportional to the square root of the time step size finally numerical results are presented that show that the scheme is at least as fast as the modified picard scheme faster than the l scheme and is more stable than the newton or the picard scheme

a multistate markov chain model for longitudinal categorical quality of life data subject to non ignorable missingness summary quality of life qol is an important outcome in clinical research particularly in cancer clinical trials typically data are collected longitudinally from patients during treatment and subsequent follow up missing data are a common problem and missingness may arise in a non ignorable fashion in particular the probability that a patient misses an assessment may depend on the patient s qol at the time of the scheduled assessment we propose a markov chain model for the analysis of categorical outcomes derived from qol measures our model assumes that transitions between qol states depend on covariates through generalized logit models or proportional odds models to account for non ignorable missingness we incorporate logistic regression models for the conditional probabilities of observing measurements given their actual values the model can accommodate time dependent covariates estimation is by maximum likelihood summing over all possible values of the missing measurements we

high order relaxation schemes for nonlinear degenerate diffusion problems several relaxation approximations to partial differential equations have been recently proposed examples include conservation laws hamilton jacobi equations convection diffusion problems and gas dynamics problems the present paper focuses on diffusive relaxation schemes for the numerical approximation of nonlinear parabolic equations these schemes are based on a suitable semilinear hyperbolic system with relaxation terms high order methods are obtained by coupling eno and weighted essentially nonoscillatory weno schemes for space discretization with implicit explicit imex schemes for time integration error estimates and a convergence analysis are developed for semidiscrete schemes with a numerical analysis for fully discrete relaxed schemes various numerical results in one and two dimensions illustrate the high accuracy and good properties of the proposed numerical schemes also in the degenerate case these schemes can be easily implemented on parallel computers and applied to more general systems of nonlinear parabolic equations in two and three dimensional cases

models for categorical data with nonignorable nonresponse abstract when categorical outcomes are subject to nonignorable nonresponse log linear models may be used to adjust for the nonresponse the models are fitted to the data in an augmented frequency table in which one index corresponds to whether or not the subject is a respondent the likelihood function is maximized over pseudo observed cell frequencies with respect to this log linear model using an em algorithm each e step of the em algorithm determines the pseudo observed cell frequencies and the m step yields the maximum likelihood estimators mle s of these pseudo observed cell frequencies this approach may produce boundary estimates for the expected cell frequencies of the nonrespondents in these cases the estimators of the log linear model parameters are not uniquely determined and may be unstable following the approach of clogg et al we propose a bayesian method that uses smoothing constants to adjust the pseudo observed cell frequencies so that the solution is not on the boundary the role

a combined finite volume nonconforming mixed hybrid finite element scheme for degenerate parabolic problems we propose and analyze a numerical scheme for nonlinear degenerate parabolic convection diffusion reaction equations in two or three space dimensions we discretize the diffusion term which generally involves an inhomogeneous and anisotropic diffusion tensor over an unstructured simplicial mesh of the space domain by means of the piecewise linear nonconforming crouzeix raviart finite element method or using the stiffness matrix of the hybridization of the lowest order raviart thomas mixed finite element method the other terms are discretized by means of a cell centered finite volume scheme on a dual mesh where the dual volumes are constructed around the sides of the original mesh checking the local peclet number we set up the exact necessary amount of upstream weighting to avoid spurious oscillations in the convection dominated case this technique also ensures the validity of the discrete maximum principle under some conditions on the mesh and the diffusion tensor we prove the convergence of the scheme only supposing the shape regularity condition for the original mesh we use a priori estimates and the kolmogorov relative compactness theorem for this purpose the proposed scheme is robust only 5 point 7 point in space dimension three locally conservative efficient and stable which is confirmed by numerical experiments

circumspheres of sets of n 1 random points in the d dimensional euclidean unit ball 1 n d in the d dimensional euclidean space any set of n 1 independent random points uniformly distributed in the interior of a unit ball of center o determines almost surely a circumsphere of center c and radius 1 n d and an n flat 1 n d 1 the orthogonal projection of o onto this flat is called o while designates the distance o c the classical problem of the distance between two random points in a unit ball corresponds to n 1 the focus is set on the family of circumspheres which are contained in this unit ball for any d 2 and 1 n d 1 the joint probability density function of the distance o c and circumradius has a simple closed form expression the marginal probability density functions of and are both products of powers and a gauss hypergeometric function stochastic representations of the latter random variables are described in terms of geometric means of two independent beta random variables for n d 1 and have a joint dirichlet distribution with parameters d d2

error analysis for implicit approximations to solutions to cauchy problems exact error analyses for two implicit approximation schemes to the cauchy problem du t dt mathcal a u t mathrel backepsilon 0 u 0 u 0 in d mathcal a are presented the operator mathcal a is required to be maximal monotone on a hilbert space mathcal h the approximations considered are u h kh i h mathcal a k u 0 and solutions u lambda to the approximating problems du lambda t dt mathcal a lambda u lambda 0 u lambda 0 u 0 where a lambda is the yosida approximation of mathcal a the special case of a being a subgradient is considered and tight estimates of the convergence of both u h and mathcal a h u h or u h and a h u h are given the new results in this case are the linear rates of convergence of u h in l infty 0 infty mathcal h and in the energy norm and the sublinear rate of convergence of a h u h in l 2 0 infty mathcal h applications to both linear and

solution of porous medium type systems by linear approximation schemes the convergence of semi discrete and discrete linear approximation schemes is analysed for nonlinear degenerate parabolic systems of porous medium type the enthalpy formulation and variational technique are used the semi discretization used reduces the original parabolic p d e to linear elliptic p d e the algebraic correction arising from nonlinearities is treated by newton like iterations in finite steps some numerical experiments are discussed and compared with the analytical solutions

inference from nonrandomly missing categorical data an example from a genetic study on turner s syndrome abstract the process leading to partial classification with categorical data is sometimes nonrandom a particular model accounting for incomplete data which allows the probability of uncertain classification to depend on category identity is utilized for an analysis of data obtained from a genetic study on turner s syndrome estimates of population proportions are obtained from maximum likelihood a method for handling nonrandomly missing data arrayed in contingency tables is discussed sensitivity analyses incorporating parameters related to the missing data mechanism are recommended for estimation and testing

optimal rates of convergence for degenerate parabolic problems in two dimensions fully discrete implicit schemes for the solution of degenerate parabolic equations in two dimensions are shown to satisfy error estimates of the form o tau ln 1 h alpha h where tau is the time step size and h is the mesh size these results apply to both the stefan problem and the porous medium equation

mixed finite elements for the richards equation linearization procedure we consider mixed finite element discretization for a class of degenerate parabolic problems including the richards equation after regularization time discretization is achieved by an euler implicit scheme while mixed finite elements are employed for the discretization in space based on the results obtained in radu et al rana preprint 02 06 eindhoven university of technology 2002 this paper considers a simple iterative scheme to solve the emerging nonlinear elliptic problems

solution of nonlinear diffusion problems by linear approximation schemes an approximation of degenerate parabolic problems of porous medium and stefan type is studied the enthalpy formulation and the variational technique are used nonstandard semidiscretization in time is used and newton like iterations are applied to solve the corresponding elliptic problems some numerical experiments are discussed and compared to other methods and to analytic solutions

error estimates for a mixed finite element discretization of some degenerate parabolic equations we consider a numerical scheme for a class of degenerate parabolic equations including both slow and fast diffusion cases a particular example in this sense is the richards equation modeling the flow in porous media the numerical scheme is based on the mixed finite element method mfem in space and is of one step implicit in time the lowest order raviart thomas elements are used here we extend the results in radu et al siam j numer anal 42 1452 1478 2004 schneid et al numer math 98 353 370 2004 to a more general framework by allowing for both types of degeneracies we derive error estimates in terms of the discretization parameters and show the convergence of the scheme the features of the mfem especially of the lowest order raviart thomas elements are now fully exploited in the proof of the convergence the paper is concluded by numerical examples

finite element approximation of the fast diffusion and the porous medium equations 1the fast diffusion equation u t delta u m 1 u 0 m 1 and the porous medium equation 1 m infty are studied in a parabolic cylinder omega times 0 t a fully discrete galerkin approximation is considered using c 0 piecewise linear finite elements in space and the backward euler time discretization a priori error estimates in quasi norms and rates of convergence are proved

geometric representation of high dimension low sample size data summary high dimension low sample size data are emerging in various areas of science we find a common structure underlying many such data sets by using a non standard type of asymptotics the dimension tends to while the sample size is fixed our analysis shows a tendency for the data to lie deterministically at the vertices of a regular simplex essentially all the randomness in the data appears only as a random rotation of this simplex this geometric representation is used to obtain several new statistical insights

a numerical approach to degenerate parabolic equations in this work we propose a numerical approach to solve some kind of degenerate parabolic equations the underlying idea is based on the maximum principle more precisely we locally perturb the initial and boundary data instead of the nonlinear diffusion coefficients so that the resulting problem is not degenerate the efficiency of this method is shown analytically as well as numerically the numerical experiments show that this new approach is comparable with the existing ones

a method of moments estimation procedure for categorical quality of life data with nonignorable missingness abstract quality of life outcomes collected during clinical trials often have considerable amounts of missing data which if not appropriately accounted for may lead to bias in inferences we introduce a method of moments mm estimating procedure for a model designed to handle nonignorable missingness arising in categorical data measured on independent populations the missingness mechanism is assumed to be the same across the populations we derive necessary and sufficient conditions for the identifiability of the model and fit the model to quality of life data collected as part of a breast cancer clinical trial we compare the mm estimator to the maximum likelihood estimator in a simulation study

approximation of degenerate parabolic problems using numerical integration a class of multidimensional degenerate parabolic equations is considered the two phase stefan problem and the porous medium equation are analyzed as models of singular parabolic equations nonstat

error estimates for a class of degenerate parabolic equations the degenerate parabolic equation u t delta beta u is considered in the cylinder omega times 0 t where omega subset r n is bounded beta 0 beta 0 0 and beta geq0 e g beta u u u m 1 m 1 a fully discrete scheme is given using c0 piecewise linear finite elements in space and a backward difference scheme in time l2 error estimates for the approximation of u and of beta u are proven for that purpose we employ the fact that the free boundary has a finite speed of propagation