detecting the presence of mixing with multiscale maximum likelihood a test of homogeneity tries to decide whether observations come from a single distribution or from a mixture of several distributions a powerful theory has been developed for the case where the component distributions are members of an exponential family when no parametric assumptions are appropriate the standard approach is to test for bimodality which is known not to be very sensitive for detecting heterogeneity to develop a more sensitive procedure this article builds on an approach employed in sampling literature and models the component distributions as logarithmically concave densities it is shown how this leads to a special semiparametric model in which the homogeneity problem is equivalent to testing whether a parameter c equals zero versus the alternative that c 0 this setup leads naturally to a novel multiscale maximum likelihood procedure where the multiscale character reflects the desirable property of adaptivity to the unknown value of the parameter c under the alternative to en

convergence of fluctuation splitting schemes for two dimensional scalar conservation laws with a kinetic solver the fluctuation splitting schemes fss in short have been introduced by roe and sildikover to solve advection equations on rectangular grids and then extended to triangular grids by roe deconinck struij for a two dimensional nonlinear scalar conservation law we consider the case of a triangular grid and of a kinetic approach to reduce the discretization of the nonlinear equation to a linear equation and apply a particular fss called n scheme we show that the resulting scheme converges strongly in l 2 loc in a finite volume sense

stability versus accuracy for an explicit wavefield extrapolation operator wavefield extrapolation by recursive depth by depth application of a convolutional operator in the frequency space domain commonly used for depth migration in a laterally varying earth has interesting accuracy and stability properties we analyze these properties by investigating the operator and its spatial fourier transform in particular we show that the instability caused by spatially truncating the operator can be remedied unconditionally by applying an appropriately chosen spatial taper however unconditional stability is gained only at the expense of accuracy we also identify frequencies and depth extrapolation step sizes for which the problems of accuracy or stability are the most pronounced

inference for mixtures of symmetric distributions this article discusses the problem of estimation of parameters in finite mixtures when the mixture components are assumed to be symmetric and to come from the same location family we refer to these mixtures as semiparametric because no additional assumptions other than symmetry are made regarding the parametric form of the component distributions because the class of symmetric distributions is so broad identifiability of parameters is a major issue in these mixtures we develop a notion of identifiability of finite mixture models which we call k identifiability where k denotes the number of components in the mixture we give sufficient conditions for k identifiability of location mixtures of symmetric components when k 2 or 3 we propose a novel distance based method for estimating the location and mixing parameters from a k identifiable model and establish the strong consistency and asymptotic normality of the estimator in the specific case of l 2 distance we show that our estimator generalizes the hodges lehmann estimator we discuss the numerical implementation of these procedures along with an empirical estimate of the component distribution in the two component case in comparisons with maximum likelihood estimation assuming normal components our method produces somewhat higher standard error estimates in the case where the components are truly normal but dramatically outperforms the normal method when the components are heavy tailed

maximum likelihood estimation of a multi dimensional log concave density fn is shown to have smaller mean integrated squared error compared with kernel based methods even when we allow the use of a theoret ical optimal fixed bandwidth for the kernel estimator that would not be available in practice we also present a real data clustering example which shows that our methodology can be used in conjunction with the expectation maximization algorithm to fit finite mixtures of log concave densities

stable explicit depth extrapolation of seismic wavefields stable explicit filters for depth extrapolation of seismic wavefields may be derived through a modification of the conventional taylor series method the modified taylor series method described here yields extrapolators with maximally flat amplitude spectra in their passband while ensuring that no spectral components in the wavefield are amplified by any factor greater than one the price for stability is increased phase error for low normalized frequencies implicit extrapolators are more accurate than the 39 coefficient stable explicit extrapolator described here however the small spatial sampling intervals required to obtain high phase accuracy in implicit extrapolation imply that this accuracy is rarely achieved in practice over the wide range of normalized frequencies likely to be encountered in practice stable explicit extrapolators outperform implicit ones the method presented here for deriving stable explicit extrapolators is in no formal sense optimal it is only guaranteed to yield stable extrapolators in my limited experience with alternative methods for designing stable extrapolators the method presented here produced the least phase error while ensuring stability nevertheless a simple method for designing optimal in some sense stable explicit extrapolators would be preferred over the method presented here 7 refs 15 figs

multiscale maximum likelihood analysis of a semiparametric model with applications a special semiparametric model for a univariate density is introduced that allows analyzing a number of problems via appropriate transformations two problems treated in some detail are testing for the presence of a mixture and detecting a wear out trend in a failure rate the analysis of the semiparametric model leads to an approach that advances the maximum likelihood theory of the grenander estimator to a multiscale analysis the construction of the corresponding test statistic rests on an extension of a result on a two sided brownian motion with quadratic drift to the simultaneous control of excursions under parabolas at various scales of a brownian bridge the resulting test is shown to be asymptotically optimal in the minimax sense regarding both rate and constant and adaptive with respect to the unknown parameter in the semiparametric model the performance of the method is illustrated with a simulation study for the failure rate problem and with data from a flow cytometry experiment for the mixture analysis

2 d depth migration in transversely isotropic media using explicit operators stable explicit depth extrapolation filters can be used to propagate plane waves corresponding to the qp and qsv quasi p and quasi sv propagation modes for transversely isotropic ti media here i discuss and compare results of two different methods for obtaining the filters for ti media with a vertical axis of symmetry vti the first a modified taylor series method is used to calculate the n coefficients of a finite length filter such that the taylor expansion around vertical propagation matches the spatial fourier transform of the downward continuation operator for vti media second a least squares method is used to calculate the filter coefficients such that the amplitude and phase departures from the ideal response of the downward continuation operator for vti media are minimized over a range of frequencies and propagation angles in both methods the amplitude response of the filter is forced to be less than unity in the evanescent region to achieve stability in general as exemplified in

improving explicit seismic depth migration with a stabilizing wiener filter and spatial resampling we present a new approach to the design and implementation of explicit wavefield extrapolation for seismic depth migration in the space frequency domain instability of the wavefield extrapolation operator is addressed by splitting the operator into two parts one to control phase accuracy and a second to improve stability the first partial operator is simply a windowed version of the exact operator for a half step the second partial operator is designed using the wiener filter method as a band limited least squares inverse of the first the final wavefield extrapolation operator for a full step is formed as a convolution of the first partial operator with the complex conjugate of the second this resulting wavefield extrapolation operator can be designed to have any desired length and is generally more stable and more accurate than a simple windowed operator of similar length additional stability is gained by reducing the amount of evanescent filtering and by spatially downsampling the lower tempor

estimators related to u processes with applications to multivariate medians asymptotic normality if a criterion function g x 1 x m depends on m 1 samples then a natural estimator of arg max p m g x 1 x m is the arg max of a uprocess it is observed that under suitable conditions these estimators are asymptotically normal this is then applied to prove asymptotic normality of liu s simplicial median and of oja s medians in r d

convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions we prove convergence of a class of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions the result is applied to the discontinuous galerkin method due to cockburn hou and shu

globally optimized finite difference extrapolator for strongly vti media implicit finite difference fd migration is unconditionally stable and is popular in handling strong velocity variations but its extension to strongly transversely anisotropic media with vertical symmetric axis media is difficult traditional local optimizations generate the optimized coefficients for each pair of thomsen anisotropy parameters independently which can degrade results substantially for large anisotropy variations and lead to a huge table we developed an implicit fd method using the analytic taylor series expansion and used a global optimization scheme to improve its accuracy for wide phase angles we first extended the number of the constant coefficients then we relaxed the coefficient of the time delay extrapolation term by tuning a small factor such that error is less than 0 1 finally we optimized the constant coefficients using a simulated annealing algorithm by constraining that all the error functions on a fine grid of the whole anisotropic region did not exceed 0 5 simultaneously the extended number of the constant coefficients and the relaxed coefficient greatly enhanced the flexibility of matching the dispersion relation and significantly improved the ability of handling strong anisotropy over a much wider range compared with traditional local optimization our scheme does not need any table and table lookup for each order of the fd method only one group of optimized coefficients is enough to handle strong variations in velocity and anisotropy more importantly our global optimization scheme guarantees the accuracy for various possible ranges of anisotropy parameters no matter how strong the anisotropy is for the globally optimized second order fd method the accurate phase angle is up to 58 and the increase is about 18 22 for the globally optimized fourth order fd method the accurate phase angle is up to 77 and the increase is about 22 27

an error estimate for finite volume methods for multidimensional conservation laws in this paper an l ll error estimate for a class of finite volume methods for the approximation of scalar multidimensional conservation laws is obtained these methods can be formally high order accurate and are defined on general triangulations the error is proven to be of order ft 4 where h represents the size of the mesh via an extension of kuznetsov approximation theory for which no estimate of the total variation and of the modulus of con tinuity in time are needed the result is new even for the finite volume method constructed from monotone numerical flux functions

convergence of the finite volume method for multidimensional conservation laws we establish the convergence of the finite volume method applied to multidimensional hyperbolic conservation laws and based on monotone numerical flux functions our technique applies with a fairly unrestrictive assumption on the triangulations flat elements are allowed and to lipschitz continuous flux functions we treat the initial and boundary value problem and obtain the strong convergence of the scheme to the unique entropy discontinuous solution in the sense of kruzkov the proof of convergence is based on a convergence framework coquel and lefloch math comp 57 1991 pp 169 210 and j numer anal 30 1993 pp 675 700 from a convex decomposition of the scheme we derive a new estimate for the rate of entropy dissipation and a new formulation of the discrete entropy inequalities these estimates are shown to be sufficient for the passage to the limit in the discrete equation convergence follows from diperna s uniqueness result in the class of entropy measure valued solutions

non parametric estimation of finite mixtures from repeated measurements this paper provides methods to estimate finite mixtures from data with repeated measurements non parametrically we present a constructive identification argument and use it to develop simple two step estimators of the component distributions and all their functionals we discuss a computationally efficient method for estimation and derive asymptotic theory simulation experiments suggest that our theory provides confidence intervals with good coverage in small samples

semiparametric estimation of a two component mixture model suppose that univariate data are drawn from a mixture of two distributions that are equal up to a shift parameter such a model is known to be nonidentifiable from a nonparametric viewpoint however if we assume that the unknown mixed distribution is symmetric we obtain the identifiability of this model which is then defined by four unknown parameters the mixing proportion two location parameters and the cumulative distribution function of the symmetric mixed distribution we propose estimators for these four parameters when no training data is available our estimators are shown to be strongly consistent under mild regularity assumptions and their convergence rates are studied their finite sample properties are illustrated by a monte carlo study and our method is applied to real data

maximum smoothed likelihood for multivariate mixtures summary we introduce an algorithm for estimating the parameters in a finite mixture of completely unspecified multivariate components in at least three dimensions under the assumption of conditionally independent coordinate dimensions we prove that this algorithm based on a majorization minimization idea possesses a desirable descent property just as any em algorithm does we discuss the similarities between our algorithm and a related one the so called nonlinearly smoothed em algorithm for the non mixture setting we also demonstrate via simulation studies that the new algorithm gives very similar results to another algorithm that has been shown empirically to be effective but that does not satisfy any descent property we provide code for implementing the new algorithm in a publicly available r package

recursive integral time extrapolation methods for waves a comparative review abstractwe compared several families of algorithms for recursive integral time extrapolation rite algorithms for waves in isotropic and anisotropic media these methods allow simulating accurate wave extrapolation with little numerical dispersion even when using larger time steps than are usually possible for conventional finite difference methods these various rite algorithms all share the use of mixed space wavenumber domain operators derived from fourier integral solutions of single mode wave equations we evaluated a taxonomy for rite methods based on how they approximated the influence of medium heterogeneity one family of methods uses mixed domain series expansions to provide accurate approximations to heterogeneous extrapolators even for large time steps we compared several methods for deriving coefficients for such series approximations another family of methods uses interpolation between different homogeneous extrapolations to approximate heterogeneous time extrapolation such methods can b

nonparametric estimation of component distributions in a multivariate mixture suppose k variate data are drawn from a mixture of two distributions each having independent components it is desired to estimate the univariate marginal distributions in each of the products as well as the mixing proportion this is the setting of two class fully parametrized latent models that has been proposed for estimating the distributions of medical test results when disease status is unavailable the problem is one of inference in a mixture of distributions without training data and until now it has been tackled only in a fully parametric setting we investigate the possibility of using nonparametric methods of course when k 1 the problem is not identifiable from a nonparametric viewpoint we show that the problem is almost identifiable when k 2 there the set of all possible representations can be expressed in terms of any one of those representations as a two parameter family furthermore it is proved that when k geq3 the problem is nonparametrically identifiable under particularly mild regularity conditions in this case we introduce root n consistent nonparametric estimators of the 2k univariate marginal distributions and the mixing proportion finite sample and asymptotic properties of the estimators are described

optimized implicit finite difference and fourier finite difference migration for vti media propagation velocity of seismic waves in heterogeneous vti media depends not only on spatial location but also on their propagation direction which leads to a much more complex dispersion relation than in isotropic media as a result designing implicit finite difference fd schemes for wavefield extrapolation in anisotropic media through analytic taylor series expansion is more difficult implicit fd and fourier finite difference ffd schemes are developed for vertical transversely isotropic vti media based on function fitting the dispersion relation of vti media is approximated with a rational function and its coefficients are estimated by weighted least squares optimization because these coefficients are functions of thomsen anisotropy parameters e and and vary laterally in heterogeneous vti media they are calculated before wavefield extrapolation and stored in a table implicit fd and ffd schemes for vti media are almost the same as for isotropic media except that coefficients are looke

a kinetic construction of global solutions of first order quasilinear equations on considere le probleme de cauchy pour des equations quasilineaires d ordre 1 de type conservation dans la region t 0 u t i 1 u a i u x i 0 u x o u o x ou x x 1 x n u u x t et a i i 1 n sont des fonctions continument differentiables a valeurs reelles d une seule variable reelle

3 d depth migration via mcclellan transformations three dimensional seismic wavefields may be extrapolated in depth one frequency at a time by two dimensional convolution with a circularly symmetric frequency and velocity dependent filter this depth extrapolation performed for each frequency independently lies at the heart of 3 d finite difference depth migration the computational efficiency of 3 d depth migration depends directly on the efficiency of this depth extrapolation mcclellan transformations provide an efficient method for both designing and implementing two dimensional digital filters that have a particular form of symmetry such as the circularly symmetric depth extrapolation filters used in 3 d depth migration given the coefficients of one dimensional frequency and velocity dependent filters used to accomplish 2 d depth migration mcclellan transformations lead to a simple and efficient algorithm for 3 d depth migration 3 d depth migration via mcclellan transformations is simple because the coefficients of two dimensional depth

estimation of a two component mixture model with applications to multiple testing we consider a two component mixture model with one known component we develop methods for estimating the mixing proportion and the other unknown distribution nonparametrically given i i d data from the mixture model we use ideas from shape restricted function estimation and develop tuning parameter free estimators that are easily implementable and have good finite sample performance we establish the consistency of our procedures distribution free finite sample lower confidence bounds are developed for the mixing proportion the identifiability of the model and the estimation of the density of the unknown mixing distribution are also addressed we discuss the connection with the problem of multiple testing and compare our procedure with some of the existing methods in that area through simulation studies we also analyse two data sets one arising from an application in astronomy and the other from a microarray experiment

migration of seismic data by phase shift plus interpolation under the horizontally layered velocity assumption migration is defined by a set of independent ordinary differential equations in the wavenumber frequency domain the wave components are extrapolated downward by rotating their phases this paper shows that one can generalize the concepts of the phase shift method to media having lateral velocity variations the wave extrapolation procedure consists of two steps in the first step the wave field is extrapolated by the phase shift method using l laterally uniform velocity fields the intermediate result is l reference wave fields in the second step the actual wave field is computed by interpolation from the reference wave fields the phase shift plus interpolation pspi method is unconditionally stable and lends itself conveniently to migration of three dimensional data the performance of the methods is demonstrated on synthetic examples the pspi migration results are then compared with those obtained from a finite difference method

a kinetic equation with kinetic entropy functions for scalar conservation laws we construct a nonlinear kinetic equation and prove that it is welladapted to describe general multidimensional scalar conservation laws in particular we prove that it is well posed uniformly in e the microscopic scale we also show that the proposed kinetic equation is equipped with a family of kinetic entropy functions analogous to boltzmann s microscopich function such that they recover krushkov type entropy inequality on the macroscopic scale finally we prove by both bv compactness arguments in the multidimensional case and by compensated compactness arguments in the one dimensional case that the local density of kinetic particles admits a continuum limit as it converges strongly with e 0 to the unique entropy solution of the corresponding conservation law

design of one way wavefield extrapolation operators using smooth functions in wlsq optimization many depth migration methods use one way frequency space depth extrapolation methods these methods are generally considered to be expensive so it is important to find the most efficient way of implementing them this usually means making spatial convolution operators that are as short as possible applying the extrapolation operators in a recursive way using small depth steps also demands that the operators do not amplify the wavefield at every depth step weighted least squares is an appropriate method to use for designing extrapolation operators that are accurate and efficient and that remain stable in a recursive algorithm the extrapolated wavefields calculated with these operators are comparable with the extrapolation results obtained with other known operator design techniques as the remez exchange method and nonlinear optimization in this paper the weighted least squares technique is refined by using different model functions by smoothing the phase and amplitude transition at the evanescent c

almost nonparametric inference for repeated measures in mixture models summary we consider ways to estimate the mixing proportions in a finite mixture distribution or to estimate the number of components of the mixture distribution without making parametric assumptions about the component distributions we require a vector of observations on each subject this vector is mapped into a vector of os and 1 s and summed the resulting distribution of sums can be modelled as a mixture of binomials we then work with the binomial mixture the efficiency and robustness of this method are compared with the strategy of assuming multivariate normal mixtures when typically the true underlying mixture distribution is different it is shown that in many cases the approach based on simple binomial mixtures is superior let pi x and p2 x be probability distributions with ai and a2 positive real numbers that sum to 1 then p x al pi x a2 p2 x defines a two component mixture distribution with com ponents pl x and p2 x and mixture weights al and a2 mixture distributions arise when observations come from different populations each with a different distribution and the population sampled is unobserved interesting examples include those in the social sciences in which important variables are latent and unobserved as an example consider the problem of trying to infer the solution strategies which young children employ in solving a cognitive task when different solution strategies can be employed a group of same age children can be considered a sample from a mixture distribution in which the components correspond to the various strategies see thomas and horton 1997 recently there has been an expanding interest in mixtures and there is now a large literature examples and theory may be found in many sources including lindsay 1995 mclachlan and basford 1988 titterington et al 1985 and everitt and hand 1981 computational issues in estimating the model parameters rely mainly on the em algorithm see mclachlan and krishnan 1997 for a detailed account two main issues remain unresolved in applications of mixtures one is the specification of the number of components although there may sometimes be substantive theory to guide this specification in exploratory settings