asymptotic normality of a nonparametric estimator of sample coverage this paper establishes a necessary and sufficient condition for the asymptotic normality of the nonparametric estimator of sample coverage proposed by good biometrica 40 1953 237 264 this new necessary and sufficient condition extends the validity of the asymptotic normality beyond the previously proven cases

discontinuous galerkin approximations of the stokes and navier stokes equations numerical schemes to compute approximate solutions of the evolutionary stokes and navier stokes equations are studied the schemes are discontinuous in time and conforming in space and of arbitrarily high order fully discrete error estimates are derived and dependence of the viscosity constant is carefully tracked it is shown that the errors are bounded by projection errors of the exact solution which exhibit optimal rates when the solutions are smooth

discontinuous galerkin methods for ordinary differential equations a class of galerkin methods derived from discontinuous piecewise polynomial spaces is analyzed for polynomials of degree k these methods lead to a family of one step schemes generating approximations up to order 2k 2 for the solution of an ordinary differential equation

a note on the convergence of the discontinuous galerkin method for a scalar hyperbolic equation a numerical example is presented which shows that the known l 2 error estimate left u u h right leqq ch k 1 2 left u right k 1 for the discontinuous galerkin method cannot be improved within the class of quasi uniform meshes not even for smooth exact solutions u

estimation of sums of random variables examples and information bounds this paper concerns the estimation of sums of functions of observable and unobservable variables lower bounds for the asymptotic variance and a convolution theorem are derived in general finite and infinite dimensional models an explicit relationship is established between efficient influence functions for the estimation of sums of variables and the estimation of their means certain plug in estimators are proved to be asymptotically efficient in finite dimensional models while u v estimators of robbins are proved to be efficient in infinite dimensional mixture models examples include certain species network and data confidentiality problems

rejoinder to modification of the computational procedure in parker and bregman s method of calculating sample size from matched case control studies with a dichotomous exposure

convergence of the discontinuous galerkin method for discontinuous solutions we consider linear first order scalar equations of the form rho t rm div rho v a rho f with appropriate initial and boundary conditions it is shown that approximate solutions computed using the discontinuous galerkin method will converge in ltwoltwo when the coefficients v and a and data f satisfy the minimal assumptions required to establish existence and uniqueness of solutions in particular v need not be lipschitz so characteristics of the equation may not be defined and the solutions being approximated my not have bounded variation

discontinuous galerkin method for the solution of a transport level set problem the subject of the paper is the numerical analysis of the transport level set problem discretized by the discontinuous galerkin method without the assumption that the first order nonstationary transport equation contains a reaction term which is used in a standard literature we prove error estimates in the l l 2 norm in the case of the space semidiscretization method of lines and in the case of the space time discontinuous galerkin method in the l 2 l 2 norm numerical experiments support the derived error estimates and show that they are not sharp in the case of the space time discontinuous galerkin method

numerical solution of parabolic integro differential equations by the discontinuous galerkin method the numerical solution of a parabolic equation with memory is considered the equation is first discretized in time by means of the discontinuous galerkin method with piecewise constant or piecewise linear approximating functions the analysis presented allows variable time steps which as will be shown call then efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term or by nonsmooth initial data the combination with finite element discretizat in space is also studied

numerical analysis of a second order pure lagrange galerkin method for convection diffusion problems part ii fully discretized scheme and numerical results we analyze a second order pure lagrange galerkin method for variable coefficient convection diffusion possibly degenerate diffusion equations with mixed dirichlet robin boundary conditions in a previous paper the proposed second order pure lagrangian time discretization scheme was introduced and analyzed for the same problem more precisely the l infty h 1 stability and l infty h 1 error estimates of order o delta t 2 had been obtained moreover for the particular case of incompressible flows stability inequalities with constants independent of the final time have been stated in the present paper l infty h 1 error estimates of order o delta t 2 o h k are obtained for the fully discretized pure lagrange galerkin method to prove these results we use some properties obtained in the previous paper finally numerical tests are presented that confirm the theoretical results

a poisson model for the coverage problem with a genomic application summary suppose a population has infinitely many individuals and is partitioned into unknown n disjoint classes the sample coverage of a random sample from the population is the total proportion of the classes observed in the sample this paper uses a nonparametric poisson mixture model to give new understanding and results for inference on the sample coverage the poisson mixture model provides a simplified framework for inferring any general abundance f coverage the sum of the proportions of those classes that contribute exactly k individuals in the sample for some k in with being a set of nonnegative integers a new moment based derivation of the well known turing estimators is presented as an application a gene categorisation problem in genomic research is addressed since turing s approach is a moment based method maximum likelihood estimation and minimum distance estimation are indicated as alternatives for the coverage problem finally it will be shown that any turing estimator is asymptotically fully efficient

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

error estimates for discontinuous galerkin approximations of implicit parabolic equations we analyze the classical discontinuous galerkin method for implicit parabolic equations symmetric error estimates for schemes of arbitrary order are presented the ideas developed allow certain assumptions frequently required in previous work to be relaxed for example different discrete spaces are allowed at each time step and the spatial operator is not required to be self adjoint or independent of time error estimates are posed in terms of projections of the exact solution onto the discrete spaces and are valid under the minimal regularity guaranteed by the natural energy estimate these projections are local and enjoy optimal approximation properties when the solution is sufficiently regular

estimating a probability mass function with unknown labels in the context of a species sampling problem we discuss a nonparametric maximum likelihood estimator for the underlying probability mass function the estimator is known in the computer science literature as the high profile estimator we prove strong consistency and derive the rates of convergence for an extended model version of the estimator we also study a sieved estimator for which similar consistency results are derived numerical computation of the sieved estimator is of great interest for practical problems such as forensic dna analysis and we present a computational algorithm based on the stochastic approximation of the expectation maximisation algorithm as an interesting byproduct of the numerical analyses we introduce an algorithm for bounded isotonic regression for which we also prove convergence less

sample size for individually matched case control studies the standard formulas used to calculate sample size for an individually matched case control study assume a constant probability of exposure throughout the pool of possible controls we propose new formulas that allow for heterogeneity in the probability of exposure among controls in different matched sets since matching factors are suspected of being confounders they are expected to divide the total population into subgroups with different proportions exposed thus the assumption of homogeneity of exposure among controls made by the currently used formulas is inconsistent with the assumptions used to design a matched study the proposed formulas avoid this inconsistency we present an example to illustrate how heterogeneity can affect the required sample size

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

uniform error analysis for lagrange galerkin approximations of convection dominated problems in this paper we present a rigorous error analysis for the lagrange galerkin method applied to convection dominated diffusion problems we prove new error estimates in which the constants depend on norms of the data and not of the solution and do not tend to infinity in the hyperbolic limit this is in contrast to other results in this field for the time discretization uniform convergence with respect to the diffusion parameter of order o k tn is shown for initial values in l2 and o k for initial values in h2 for the spatial discretization with linear finite elements we verify uniform convergence of order o h2 min h h2 k for data in h2 by interpolation of banach spaces suboptimal convergence rates are derived under less restrictive assumptions the analysis is heavily based on a priori estimates uniform in the diffusion parameter for the solution of the continuous and the semidiscrete problem they are derived in a lagrangian framework by transforming the eulerian coordinates completely into subcharacteristic coordinates finally we illustrate the error estimates by some numerical results

did shakespeare write a newly discovered poem summary the consistency of the word usage in a previously unknown nine stanza poem attributed to shakespeare with that of the shakespearean canon is examined using a nonparametric empirical bayes model we consider also poems by jonson marlowe and donne as well as four poems definitely attributed to shakespeare on balance the poem is found to fit previous shakespearean usage reasonably well

modification of the computational procedure in parker and bregman s method of calculating sample size from matched case control studies with a dichotomous exposure summary several authors have addressed the problem of calculating sample size for a matched case control study with a dichotomous exposure the approach of parker and bregman 1986 biometrics42 919 926 is in our view one of the most satisfactory since it requires specification of quantities that are often easily available to the investigator however its recommended implementation involves a computational approximation we show here that the approximation performs poorly in extreme situations and can be easily replaced with a more exact calculation

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

discrete maximal parabolic regularity for galerkin finite element methods for nonautonomous parabolic problems the main goal of the paper is to establish time semidiscrete and space time fully discrete maximal parabolic regularity for the lowest order time discontinuous galerkin solution of linear parabolic

a normal limit law for a nonparametric estimator of the coverage of a random sample the coverage of a multinomial random sample is the sum of the probabilities of the observed classes a normal limit law is rigorously proved for good s 1953 coverage estimator the result is valid under very general conditions and all terms except the coverage itself are observable nevertheless the implied confidence intervals are not much wider than those developed under restrictive assumptions such as in the classical occupancy problem the asymptotic variance is somewhat unexpected the proof utilizes a method of holst 1979

on the smoothing property of the galerkin method for parabolic equations we analyze the galerkin approximation of the general second order parabolic initial boundary value problem for the second order continuous time method with initial data only in l 2 we prove an l 2 error estimate of order o h 2 t analogous results are shown to hold for the error in negative sobolev norms and for the time derivative of the error our analysis uses only elementary energy techniques and the same techniques are shown to give a simple analysis of the backward euler time discretization

compactness properties of the dg and cg time stepping schemes for parabolic equations for a broad class of parabolic equations it is shown that numerical solutions computed using the discontinuous galerkin or the continuous galerkin time stepping schemes of arbitrary order will inherit the compactness properties of the underlying equation convergence of numerical schemes for a phase field approximation of the flow of two fluids with surface tension is presented to illustrate these results

numerical analysis of a second order pure lagrange galerkin method for convection diffusion problems part i time discretization we propose and analyze a second order pure lagrangian method for variable coefficient convection possibly degenerate diffusion equations with mixed dirichlet robin boundary conditions first the method is rigorously introduced for exact and approximate characteristics next l infty h 1 stability is proved and l infty h 1 error estimates of order o delta t 2 are obtained moreover l infty l 2 stability and l infty l 2 error estimates of order o delta t 2 with constants bounded in the hyperbolic limit are shown for the particular case of dirichlet boundary conditions diffusion tensor a epsilon i and right hand side f 0 the l infty h 1 stability estimate is independent of epsilon moreover for incompressible flows the constants in the stability inequalities are independent of the final time in a second part of this work the pure lagrangian scheme will be combined with galerkin discretization using finite elements spaces and numerical examples will be presented

entropy estimation in turing s perspective a new nonparametric estimator of shannon s entropy on a countable alphabet is proposed and analyzed against the well known plug in estimator the proposed estimator is developed based on turing s formula which recovers distributional characteristics on the subset of the alphabet not covered by a size n sample the fundamental switch in perspective brings about substantial gain in estimation accuracy for every distribution with finite entropy in general a uniform variance upper bound is established for the entire class of distributions with finite entropy that decays at a rate of o ln n n compared to o ln n 2 n for the plug in in a wide range of subclasses the variance of the proposed estimator converges at a rate of o 1 n and this rate of convergence carries over to the convergence rates in mean squared errors in many subclasses specifically for any finite alphabet the proposed estimator has a bias decaying exponentially in n several new bias adjusted estimators are also discussed

adaptive finite element methods for parabolic problems ii optimal error estimates in l l 2 and l l optimal error estimates are derived for a complete discretization of linear parabolic problems using space time finite elements the discretization is done first in time using the discontinuous galerkin method and then in space using the standard galerkin method the underlying partitions in time and space need not be quasi uniform and the partition in space may be changed from time step to time step the error bounds show in particular that the error may be controlled globally in time on a given tolerance level by controlling the discretization error on each individual time step on the same given level i e without error accumulation effects the derivation of the estimates is based on the orthogonality of the galerkin procedure and the use of strong stability estimates the particular and precise form of these error estimates makes it possible to design efficient adaptive methods with reliable automatic error control for parabolic problems in the norms under consideration

symmetric error estimates for moving mesh mixed methods for advection diffusion equations a mixed method allowing a general class of mesh movements is proposed for an advection diffusion equation in either conservative or nonconservative form several symmetric error estimates are derived for the method under certain conditions in one space dimension optimal order l2 convergence and superconvergence are proved as corollaries of the symmetric estimates

the efficiency of good s nonparametric coverage estimator the asymptotic efficiency of good s nonparametric coverage estimator is obtained relative to the best estimator derived under the assumption that all classes are equally likely even when that assumption is true good s estimator is quite efficient with an asymptotic relative efficiency of greater than 85 in all cases and greater than 95 if the expected coverage is less than one half

error estimates for discontinuous galerkin time stepping schemes for robin boundary control problems constrained to parabolic pdes we consider fully discrete finite element approximations of a robin optimal boundary control problem constrained by linear parabolic pdes with rough initial data conforming finite element methods for spatial discretization combined with discontinuous time stepping galerkin schemes are being used for the space time discretization error estimates are proved under weak regularity hypotheses for the state adjoint and control variables computational examples validating our expected rates of convergence are also provided

anisotropic hp adaptive discontinuous galerkin method for the numerical solution of time dependent pdes we deal with the numerical solution of time dependent problems with the aid of anisotropic hp grids we present an algorithm which generates a sequence of anisotropic triangular grids and the corresponding polynomial approximation degrees in such a way that the interpolation error measured in the discrete l 0 t l q norm q 1 and r 2 is under a given tolerance and the number of degrees of freedom is as small as possible the efficiency of the algorithm is demonstrated by numerical experiments

a forward approach to numerical data assimilation variational data assimilation problems are concerned with computing unknown initial values for the simulation and prediction of natural phenomena most notably in weather prediction and are usually solved via an ill posed optimal control problem for the initial state at the time of the first available measurements an alternative forward approach focuses on computation of the final state after this interval which is just as suitable for prediction purposes and is well posed without additional regularization specifically it is possible to compute projections of the unknown final state on all elements of an orthonormal basis which theoretically allows for the complete reconstruction of the final state in this paper an efficient numerical method for linear evolution equations of diffusive type is presented and convergence of the numerical approximation based on a discontinuous galerkin discretization is proved the key of this method is the computation of an adaptively ordered orthonormal basis using proper orthogonal decomposition numerical examples for a scalar convection diffusion equation in two and three dimensions show the effectiveness of the method

error estimates for the discontinuous galerkin methods for parabolic equations the classical discontinuous galerkin method for a general parabolic equation is analyzed symmetric error estimates for schemes of arbitrary order are presented the ideas developed below relax many assumptions required in previous work for example different discrete spaces may be used at each time step and the spatial operator need not be self adjoint or independent of time our error estimates are posed in terms of projections of the exact solution onto the discrete spaces and are valid under the minimal regularity guaranteed by the natural energy estimate these projections are local and enjoy optimal approximation properties when the solution is sufficiently regular