fourier chebyshev spectral method for the two dimensional navier stokes equations a fourier chebyshev spectral scheme is proposed for the two dimensional navier stokes equations to avoid the difficulty of choosing the trial function space satisfying the incompressible condition the pressure is evaluated by solving a poisson equation the generalized stability and convergence are analyzed strictly the numerical results show the high accuracy of this method

a semiparametric extension of the mann whitney test for randomly truncated data in many applications statistical data are frequently observed subject to a retrospective sampling criterion resulting in pure right truncated data in classical testing problems the mann whitney test is used for testing the equality of two distributions a semiparametric extension of this test is developed for the case when truncation is present we consider a model in which the truncation distribution is parameterized while the lifetime distribution is left as a nonparametric component the method is seen to be applicable to many patterns of truncation including left truncation right truncation and doubly truncated data for which no other nonparametric or semiparametric test is currently available applications of the semiparametric method are given simulation results indicate that for pure right truncated data the semiparametric test is more powerful than a recent nonparametric test

higher order nonlinear degenerate parabolic equations this paper is concerned with nonlinear degenerate parabolic equations of the form ut 1 m 1 d f u d2m 1u 0 with f u u n n 1 near u 0 and d x under appropriate boundary conditions it is shown that there exists a weak solution u some of the main results of the paper are that u 0 if u0 0 and that the support of u t when u0 0 increases with t for the last property we require that n 2 and m 1

the accelerated failure time model under biased sampling chen 2009 biometrics studies the semi parametric accelerated failure time model for data that are size biased chen considers only the uncensored case and uses hazard based estimation methods originally developed for censored observations however for uncensored data a simple linear regression on the log scale is more natural and provides better estimators

empirical likelihood ratio confidence intervals for a single functional summary the empirical distribution function based on a sample is well known to be the maximum likelihood estimate of the distribution from which the sample was taken in this paper the likelihood function for distributions is used to define a likelihood ratio function for distributions it is shown that this empirical likelihood ratio function can be used to construct confidence intervals for the sample mean for a class of m estimates that includes quantiles and for differentiable statistical functionals the results are nonparametric extensions of wilks s 1938 theorem for parametric likelihood ratios the intervals are illustrated on some real data and compared in a simulation to some bootstrap confidence intervals and to intervals based on student s t statistic a hybrid method that uses the bootstrap to determine critical values of the likelihood ratio is introduced

marginal likelihood conditional likelihood and empirical likelihood connections and applications marginal likelihood and conditional likelihood are often used for eliminating nuisance parameters for a parametric model it is well known that the full likelihood can be decomposed into the product of a conditional likelihood and a marginal likelihood this property is less transparent in a nonparametric or semiparametric likelihood setting in this paper we show that this nice parametric likelihood property can be carried over to the empirical likelihood world we discuss applications in case control studies genetical linkage analysis genetical quantitative traits analysis tuberculosis infection data and unordered paired data all of which can be treated as semiparametric finite mixture models we consider the estimation problem in detail in the simplest case of unordered paired data where we can only observe the minimum and maximum values of two random variables the identities of the minimum and maximum values are lost the profile empirical likelihood approach is used for maximum semiparametric likelihood estimation we present some large sample results along with a simulation study

transformations of the explanatory variables in the logistic regression model for binary data summary some results are presented on improving the fit of the logistic regression model for binary data by transforming the vector of explanatory variables the methods are based on consideration of the distributions of these variables conditional on outcome group the transformations required are the functions of the explanatory variables which appear in the log density ratio of the conditional distributions

the lubrication approximation for thin viscous films regularity and long time behavior of weak solutions we consider the fourth order degenerate diffusion equation in one space dimension this equation derived from a lubrication approximation models the surface tension dominated motion of thin viscous films and spreading droplets 15 the equation with f h h also models a thin neck of fluid in the hele shaw cell 10 11 23 in such problems h x t is the local thickness of the the film or neck this paper considers the properties of weak solutions that are more relevant to the droplet problem than to hele shaw for simplicity we consider periodic boundary conditions with the interpretation of modeling a periodic array of droplets we consider two problems the first has initial data h0 0 and f h h n 0 n 3 we show that there exists a weak nonnegative solution for all time also we show that this solution becomes a strong positive solution after some finite time t and asymptotically approaches its means as t the weak solution is in the classical sense of distributions for 3 8 n 3 and in a weaker sense introduced in 1 for the remaining 0 n 3 8 furthermore the solutions have high enough regularity to just include the unique source type solutions 2 with zero slope at the edge of the support they do not include any of the less regular solutions with positive slope at the edge of the support second we consider strictly positive initial data h0 m 0 and f h h n 0 n for this problem we show that even if a finite time singularity of the form h 0 does occur there exists a weak nonnegative solution for all time t this weak solution becomes strong and positive again after some critical time t as in the first problem we show that the solution approaches its mean as t the main technical idea is to introduce new classes of dissipative entropies to prove existence and higher regularity we show that these entropies are related to norms of the difference between the solution and its mean to prove the relaxation result 1996 john wiley sons inc

combined finite element and spectral approximation of the navier stokes equations we present a method for the numerical approximation of navier stokes equations with one direction of periodicity in this direction a fourier pseudospectral method is used in the two others a standard f e m is applied we prove optimal rate of convergence where the two parameters of discretization intervene independently on presente une methode d approximation numerique des equations de navier stokes possedant une direction de periodicite dans cette direction une methode pseudospectrale basee sur des developpements en serie de fourier est utilisee dans les deux autres on applique une methode d elements finis standard on montre que la convergence est optimale et que les deux parametres de discretisation peuvent etre choisis de facon independante

long wave instabilities and saturation in thin film equations hocherman and rosenau conjectured that long wave unstable cahn hilliard type interface models develop nite time singularities when the nonlinearity in the destabilizing term grows faster at large amplitudes than the nonlinearity in the stabilizing term phys d 67 113 125 1993 we consider this conjecture for a class of equations often used to model thin lms in a lubrication context by showing that if the solutions are uniformly bounded above or below e g are nonnegative then the destabilizing term can be stronger than previously conjectured yet the solution still remains globally bounded for example they conjecture that for the long wave unstable equation ht h n hxxx x h m hx x m n leads to blow up using a conservation of volume constraint for the case m n 0 we conjecture a dierent critical exponent for possible singularities of nonnegative solutions we prove that nonlinearities with exponents below the conjectured critical exponent yield globally bounded solutions specically for the above equation solutions are bounded if m n 2 the bound is proved using energy methods and is then used to prove the existence of non negative weak solutions in the sense of distributions we present preliminary numerical evidence suggesting that m n 2 can allow blow up

nonparametric estimation from cross sectional survival data abstract in many follow up studies survival data are often observed according to a cross sectional sampling scheme data of this type are subject to left truncation in addition to the usual right censoring a number of characteristics and properties of the product limit estimate for left truncated and right censored data have been explored and found to be similar to those of the kaplan meier estimate under the stationarity assumption however it is believed that an alternative estimate has much better efficiency in this article the conditional maximum likelihood estimate mle property of the product limit estimate is visited the non parametric mle of the truncation distribution is derived use of this estimate includes testing the stationarity assumption estimating the proportion of truncated data and other applications in prevalent cohort studies the analysis of the estimation is based on a working data approach the asymptotic properties of the proposed estimates are developed through nonpar

multiplicative censoring renewal processes deconvolution and decreasing density nonparametric estimation on donne un estimateur du maximum de vraisemblance non parametrique pour une distribution de durees de vie il est base sur un modele d observations censurees de facon multiplicative et il generalise des problemes statistiques courants tels que l estimation d une fonction de repartition sous une contrainte de densite decroissante la deconvolution non parametrique d une variable aleatoire exponentielle ou un probleme d estimation dans les processus de renouvellement

symmetric singularity formation in lubrication type equations for interface motion fourth order degenerate diffusion equations arise in a lubrication approximation of a thin film or neck driven by surface tension numerical studies of the lubrication equation le h t h n h xxx x 0 with various boundary conditions indicate that singularity formation in which h x t t to 0 occurs for small enough n with anomalous or second type scaling inconsistent with usual dimensional analysis this paper considers locally symmetric or even singularities in the le and in the modified lubrication equation mle h t h n h xxxx 0 both equations have the property that entropy bounds forbid finite time singularities when n is sufficiently large power series expansions for local symmetric similarity solutions are proposed for equation le with n 1 and mle for all n in mathbb r in the latter case special boundary conditions that force singularity formation are required to produce singularities when n is large matching conditions at higher order terms in the

positivity preserving numerical schemes for lubrication type equations lubrication equations are fourth order degenerate diffusion equations of the form h t nabla cdot f h nabla delta h 0 describing thin films or liquid layers driven by surface tension recent studies of singularities in which h to 0 at a point describing rupture of the fluid layer show that such equations exhibit complex dynamics which can be difficult to simulate accurately in particular one must ensure that the numerical approximation of the interface does not show a false premature rupture generic finite difference schemes have the potential to manifest such instabilities especially when underresolved we present new numerical methods in one and two space dimensions that preserve positivity of the solution regardless of the spatial resolution whenever the pde has such a property we also show that the schemes can preserve positivity even when the pde itself is only known to be nonnegativity preserving we prove that positivity preserving finite difference schemes have unique positive solutions at all times we prove stability and convergence of both positivity preserving and generic methods in one and two space dimensions to positive solutions of the pde showing that the generic methods also preserve positivity and have global solutions for sufficiently fine meshes we generalize the positivity preserving property to a finite element framework and show via concrete examples how this leads to the design of other positivity preserving schemes

semiparametric regression in size biased sampling size biased sampling arises when a positive valued outcome variable is sampled with selection probability proportional to its size in this article we propose a semiparametric linear regression model to analyze size biased outcomes in our proposed model the regression parameters of covariates are of major interest while the distribution of random errors is unspecified under the proposed model we discover that regression parameters are invariant regardless of size biased sampling following this invariance property we develop a simple estimation procedure for inferences our proposed methods are evaluated in simulation studies and applied to two real data analyses

a boundary integral formulation of quasi steady fluid wetting this paper considers the motion of a liquid droplet on a solid surface when capillary relaxation is much faster than the motion of the contact line the fluid geometry and its dynamical evolution can be characterized in terms of the contact line alone this problem can be cast in terms of boundary integral equations involving a dirichlet neumann map coupled to a volume conservation constraint a computational method for this formulation is described which has two principal advantages over approaches which track the entire free surface 1 only the curve which describes the contact line is computed and 2 the resulting method exhibits only mild numerical stiffness obviating the need for implicit timestepping effects of both capillary and body forces are considered computational examples include surface inhomogeneities topological transitions and cusp formation

nonnegative solutions of a fourth order nonlinear degenerate parabolic equation elena beretta michiel bertsch roberta dal passo communicated by m guatin o introduction in this paper we study nonnegative solutions of the fourth order equation 0 1 ut u uxxx x o where n is a real positive constant equation 0 1 is a nonlinear parabolic partial differential equation which is degenerate at points at which u vanishes it arises in several applications see the references in 2 4 5 6 in which u x t stands for a nonnegative quantity for example in the study of the dynamics of thin liquid films u denotes the thickness of the film or another relevant nonnegative quantity the case n 1 models the flow in a hele shaw cell while n 3 corresponds to viscous flow on a solid surface without slip driven by surface tension see also 6 7 8 for a related equation it is well known that fourth order equations do not satisfy a maximum principle in particular classical solutions of the linear equation ut u 0 may become negative in finite time bernis 8 friedman 2 have observed that if n is positive equation 0 1 has generalized solutions which remain nonnegative for all times in addition they show that the qualitative positivity of these solutions strongly depends on the value of n in more recent papers by kadanoff et al a combination of numerical calculations and matched asymptotics 4 and the analysis of travelling wave solutions 5 leads to results which suggest a very rich structure of the solutions of 0 1 it is the purpose of the present paper to clarify several aspects of the nonlinear equation 0 1 and its dependence on the parameter n our results lead to several conjectures and open problems as was pointed out in 4 for sufficiently small values of n solutions of 0 1 which are initially strictly positive may vanish at some point x 0 after a finite time to but to the best of our knowledge there is no rigorous proof of this phenomenon in the literature on the other hand it is of particular interest for applications for example if u denotes the thickness of a liquid film it implies that at time to the film breaks at xo

likelihood approaches for the invariant density ratio model with biased sampling data the full likelihood approach in statistical analysis is regarded as the most efficient means for estimation and inference for complex length biased failure time data computational algorithms and theoretical properties are not readily available especially when a likelihood function involves infinite dimensional parameters relying on the invariance property of length biased failure time data under the semiparametric density ratio model we present two likelihood approaches for the estimation and assessment of the difference between two survival distributions the most efficient maximum likelihood estimators are obtained by the em algorithm and profile likelihood we also provide a simple numerical method for estimation and inference based on conditional likelihood which can be generalized to k arm settings unlike conventional survival data the mean of the population failure times can be consistently estimated given right censored length biased data under mild regularity conditions to check the semiparametric density ratio model assumption we use a test statistic based on the area between two survival distributions simulation studies confirm that the full likelihood estimators are more efficient than the conditional likelihood estimators we analyse an epidemiological study to illustrate the proposed methods

consistency in a proportional hazards model incorporating a random effect supported in part by the north atlantic treaty organization under a grant awarded in 1990 conducted at the mathematical institute of the university of utrecht the netherlands

length biased sampling with right censoring when survival data arise from prevalent cases ascertained through a cross sectional study it is well known that the survivor function corresponding to these data is length biased and different from the survivor function derived from incident cases length biased data have been treated both unconditionally and conditionally in the literature in the latter case where length bias is viewed as being induced by random left truncation of the survival times the truncating distribution is assumed to be unknown conditioning on the observed truncation times hence causes very little loss of information in many instances however it can be supposed that the truncating distribution is uniform and it has been pointed out that under these circumstances an unconditional analysis will be more informative there are no results in the current literature that give the asymptotic properties of the unconditional nonparametric maximum likelihood estimator npmle of the unbiased survivor function in the presence of cen

numerical investigation of the generalized lubrication equation abstract explicit implicit explicit and crank nicolson implicit explicit numerical schemes for solving the generalized lubrication equation are derived we prove that the implicit explicit and crank nicolson implicit explicit numerical schemes are unconditionally stable numerical solutions obtained from both schemes are compared initial curves with both zero and finite contact angles are considered

a semiparametric two component compound mixture model and its application to estimating malaria attributable fractions summary malaria remains a major epidemiologic problem in many developing countries malaria is defined as the presence of parasites and symptoms usually fever due to the parasites in endemic areas an individual may have symptoms attributable either to malaria or to other causes from a clinical viewpoint it is important to correctly diagnose an individual who has developed symptoms so that the appropriate treatments can be given from an epidemiologic and economic viewpoint it is important to determine the proportion of malaria affected cases in individuals who have symptoms so that policies on intervention program can be developed once symptoms have developed in an individual the diagnosis of malaria can be based on the analysis of the parasite levels in blood samples however even a blood test is not conclusive as in endemic areas many healthy individuals can have parasites in their blood slides therefore data from this type of study can be viewed as coming from a mixture distribution with the components corresponding to malaria and nonmalaria cases a unique feature in this type of data however is the fact that a proportion of the nonmalaria cases have zero parasite levels therefore one of the component distributions is itself a mixture distribution in this article we propose a semiparametric likelihood approach for estimating the proportion of clinical malaria using parasite level data from a group of individuals with symptoms our approach assumes the density ratio for the parasite levels in clinical malaria and nonclinical malaria cases can be modeled using a logistic model we use empirical likelihood to combine the zero and nonzero data the maximum semiparametric likelihood estimate is more efficient than existing nonparametric estimates using only the frequencies of zero and nonzero data on the other hand it is more robust than a fully parametric maximum likelihood estimate that assumes a parametric model for the nonzero data simulation results show that the performance of the proposed method is satisfactory the proposed method is used to analyze data from a malaria survey carried out in tanzania

smoothing of crank nicolson scheme for the two dimensional diffusion with an integral condition the parabolic partial differential equations pdes with nonlocal boundary conditions model various physical phenomena e g chemical diffusion thermoelasticity heat conduction process control theory and medicine science this paper deals with the smoothing of the crank nicolson numerical scheme for two dimensional parabolic pdes with nonlocal boundary conditions we use the numerical scheme based on pade approximations of the matrix exponential the graphs of numerical results demonstrate the successful smoothing of the crank nicolson numerical scheme

asymptotic theory for the correlated gamma frailty model the frailty model is a generalization of cox s proportional hazard model where a shared unobserved quantity in the intensity induces a positive correlation among the survival times murphy showed consistency and asymptotic normality of the nonparametric maximum likelihood estimator npmle for the shared gamma frailty model without covariates in this paper we extend this result to the correlated gamma frailty model and we allow for covariates we discuss the definition of the nonparametric likelihood function in terms of a classical proof of consistency for the maximum likelihood estimator which goes back to wald our proof of the consistency for the npmle is essentially the same as the classical proof for the maximum likelihood estimator a new central limit theorem for processes of bounded variation is given furthermore we prove that a consistent estimator for the asymptotic variance of the npmle is given by the inverse of a discrete observed information matrix

a positivity preserving numerical scheme for a nonlinear fourth order parabolic system noindent a positivity preserving numerical scheme for a fourth order nonlinear parabolic system arising in quantum semiconductor modeling is studied the system is numerically treated by introducing an additional nonlinear potential and a subsequent semidiscretization in time the resulting sequence of nonlinear second order elliptic systems admits at each time level strictly positive solutions which is proved by an exponential transformation of variables the stability of the scheme is shown and convergence is proved in one space dimension numerical results concerning the switching behavior of a resonant tunneling diode are presented

spectral approximation of the periodic nonperiodic navier stokes equations in order to approximate the navier stokes equations with periodic boundary conditions in two directions and a no slip boundary condition in the third direction by spectral methods we justify by theoretical arguments an appropriate choice of discrete spaces for the velocity and the pressure the compatibility between these two spaces is checked via an infsup condition we analyze a spectral and a collocation pseudo spectral method for the stokes problem and a collocation pseudo spectral method for the navier stokes equations we derive error bounds of spectral type i e which behave likem wherem depends on the number of degrees of freedom of the method and represents the regularity of the data

counting the stationary states and the convergence to equilibrium for the 1 d thin film equation abstract this paper is concerned with the one dimensional thin film equation u t x m u x 2 u x 2 p u 0 p u 1 u n m n u m 0 n m 0 in 0 l r with the homogeneous neumann boundary conditions u x x p u x x 0 l 0 u x x 0 l 0 for all t 0 we prove that for any given positive initial datum the number of positive stationary states is at most infinitely countable furthermore we prove that the solution of the evolution problem converges to an equilibrium as time tends to infinity

maximum likelihood estimation in semiparametric regression models with censored data summary semiparametric regression models play a central role in formulating the effects of covariates on potentially censored failure times and in the joint modelling of incomplete repeated measures and failure times in longitudinal studies the presence of infinite dimensional parameters poses considerable theoretical and computational challenges in the statistical analysis of such models we present several classes of semiparametric regression models which extend the existing models in important directions we construct appropriate likelihood functions involving both finite dimensional and infinite dimensional parameters the maximum likelihood estimators are consistent and asymptotically normal with efficient variances we develop simple and stable numerical techniques to implement the corresponding inference procedures extensive simulation experiments demonstrate that the inferential and computational methods proposed perform well in practical settings applications to three medical studies yield important new insights we conclude that there is no reason theoretical or numerical not to use maximum likelihood estimation for semiparametric regression models we discuss several areas that need further research

a goodness of fit test for logistic regression models based on case control data summary we test the logistic regression assumption under a case control sampling plan after reparameterisation the assumed logistic regression model is equivalent to a two sample semiparametric model in which the log ratio of two density functions is linear in data by identifying this model with a biased sampling model we propose a kolmogorov smirnovtype statistic to test the validity of the logistic link function moreover we point out that this test statistic can also be used in mixture sampling we present a bootstrap procedure along with some results on simulation and on analysis of two real datasets