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

on the estimation of a probability density function by the maximum penalized likelihood method abstract a class of probability density estimates can be obtained by penalizing the likelihood by a functional which depends on the roughness of the logarithm of the density the limiting case of the estimates as the amount of smoothing increasing has a natural form which makes the method attractive for data analysis and which provides a rationale for a particular choice of roughness penalty the estimates are shown to be the solution of an unconstrained convex optimization problem and mild natural conditions are given for them to exist rates of consistency in various norms and conditions for asymptotic normality and approximation by a gaussian process are given thus breaking new ground in the theory of maximum penalized likelihood density estimation author

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

mice multiple peak identification characterization and estimation summary mice multiple peak identification characterization and estimation is a procedure for estimating a lower bound of the number of frequency peaks and for estimating the frequency peak parameters the leading application is protein structure determination using nuclear magnetic resonance nmr experiments nmr frequency data are multiple peak data where each frequency peak corresponds to two connected atoms in the three dimensional protein structure we analyze the nmr frequency data through a series of steps a preliminary step for separating the signal from the background followed by identification of local maxima up to a noise level dependent threshold estimation of the frequency peak parameters using an iterative algorithm and detection of mixtures of peaks using hypothesis testing

percolation transitions in two dimensions we investigate bond and site percolation models on several two dimensional lattices numerically by means of transfer matrix calculations and monte carlo simulations the lattices include the square triangular honeycomb kagome and diced lattices with nearest neighbor bonds and the square lattice with nearest and next nearest neighbor bonds results are presented for the bond percolation thresholds of the kagome and diced lattices and the site percolation thresholds of the square honeycomb and diced lattices we also include the bond and site percolation thresholds for the square lattice with nearest and next nearest neighbor bonds we find that corrections to scaling behave according to the second temperature dimension xt2 4 predicted by the coulomb gas theory and the theory of conformal invariance in several cases there is evidence for an additional term with the same exponent but modified by a logarithmic factor only for the site percolation problem on the triangular lattice does such a logarithmic term appear to be small or absent the amplitude of the power law correction associated with xt2 4 is found to be dependent on the orientation of the lattice with respect to the cylindrical geometry of the finite systems

moment based oscillation properties of mixture models consider finite mixture models of the form g x q f f x dq where f is a parametric density and q is a discrete probability measure an important and difficult statistical problem concerns the determination of the number of support points usually known as components of q from a sample of observations from g for an important class of exponential family models we have the following result if p has more than p components and q is an appropriately chosen p component approximation of p then g x p g x q demonstrates a prescribed sign change behavior as does the corresponding difference in the distribution functions these strong structural properties have implications for diagnostic plots for the number of components in a finite mixture

quasi concave density estimation maximum likelihood estimation of a log concave probability density is formulated as a convex optimization problem and shown to have an equivalent dual formulation as a constrained maximum shannon entropy problem closely related maximum renyi entropy estimators that impose weaker concavity restrictions on the fitted density are also considered notably a minimum hellinger discrepancy estimator that constrains the reciprocal of the square root of the density to be concave a limiting form of these estimators constrains solutions to the class of quasi concave densities

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

asymptotic normality of the l_1 error of the grenander estimator groeneboom introduced a jump process that can be used among other things to study the asymptotic properties of the grenander estimator of a monotone density in this paper we derive the asymptotic normality of a suitably rescaled version of the l 1 error of the grenander estimator using properties of this jump process

sizer for exploration of structures in curves abstract in the use of smoothing methods in data analysis an important question is which observed features are really there as opposed to being spurious sampling artifacts an approach is described based on scale space ideas originally developed in the computer vision literature assessment of significant zero crossings of derivatives results in the sizer map a graphical device for display of significance of features with respect to both location and scale here scale means level of resolution that is bandwidth

strong approximation of density estimators from weakly dependent observations by density estimators from independent observations we derive an approximation of a density estimator based on weakly dependent random vectors by a density estimator built from independent random vectors we construct on a sufficiently rich probability space such a pairing of the random variables of both experiments that the set of observations x 1 x n from the time series model is nearly the same as the set of observations y 1 y n from the i i d model with a high probability all sets of the form x 1 x n delta y 1 y n boolean and a 1 b 1 x x a d b d contain no more than o n 1 2 pi b i a i 1 log n elements respectively although this does not imply very much for parametric problems it has important implications in nonparametric statistics it yields a strong approximation of a kernel estimator of the stationary density by a kernel density estimator in the i i d model moreover it is shown that such a strong approximation is also valid for the standard bootstrap and the smoothed bootstrap using these results we derive simultaneous confidence bands as well as supremum type nonparametric tests based on reasoning for the i i d model

testing uniformity versus a monotone density the paper is concerned with testing uniformity versus a monotone density this problem arises in two important contexts after transformations testing whether a sample is a simple random sample or a biased sample and testing whether the intensity function of a nonhomogeneous poisson process is constant against monotone alternatives a penalized likelihood ratio test p test and a dip likelihood test d test are developed the d test is analogous to hartigan and hartigan s 1985 dip test for bump hunting problems while nonparametric both the p and d tests are quite efficient in comparison to the most powerful mp tests for some simple alternatives and also the laplace test developed for nonhomogeneous poisson process the p and d tests have higher power than the above mp tests under different sets of monotone alternatives and so have greater applicability moderate sample size performance and applications of our tests are illustrated via simulations and examination of an air conditioning equipment data set

cube root asymptotics we establish a new functional central limit theorem for empirical processes indexed by classes of functions in a neighborhood of a fixed parameter point an n 1 3 rescaling of the parameter is compensated for by an n 2 3 rescaling of the empirical measure resulting in a limiting gaussian process by means of a modified continuous mapping theorem for the location of the maximizing value we deduce limit theorems for several statistics defined by maximization or constrained minimization of a process derived from the empirical measure these statistics include the short rousseeuw s least median of squares estimator manski s maximum score estimator and the maximum likelihood estimator for a monotone density the limit theory depends on a simple new sufficient condition for a gaussian process to achieve its maximum almost surely at a unique point

a reliable data based bandwidth selection method for kernel density estimation we present a new method for data based selection of the bandwidth in kernel density estimation which has excellent properties it improves on a recent procedure of park and marron which itself is a good method in various ways first the new method has superior theoretical performance second it also has a computational advantage third the new method has reliably good performance for smooth densities in simulations performance that is second to none in the existing literature these methods are based on choosing the bandwidth to approximately minimize good quality estimates of the mean integrated squared error the key to the success of the current procedure is the reintroduction of a non stochastic term which was previously omitted together with use of the bandwidth to reduce bias in estimation without inflating variance

multiscale testing of qualitative hypotheses suppose that one observes a process y on the unit interval where dy t n1 2f t dt dw t with an unknown function parameter f given scale parameter n 1 sample size and standard brownian motion w we propose two classes of tests of qualitative nonparametric hypotheses about f such as monotonicity or concavity these tests are asymptotically optimal and adaptive in a certain sense they are constructed via a new class of multiscale statistics and an extension of levy s modulus of continuity of brownian motion

errors in variables in linear systems this paper extends the simple errors in variable bound to the setting of systems of equations both diagonal and nondiagonal measurement error covariance matrices are considered in the nondiagonal case the analogue of the simple errors in variable interval of estimates is an ellipsoid with diagonal equal to the line segment connecting the direct least squares with a two stage least squares estimate for the diagonal case the set of estimates under some conditions must lie within the convex hull of 2k points

universal formulas for percolation thresholds a power law is postulated for both site and bond percolation thresholds the formula is mathit p mathit c mathit p 0 d 1 q 1 mathrm ensuremath mathit a mathit d mathit b where d is the space dimension and q the coordination number all thresholds up to d ensuremath rightarrow ensuremath infty are found to belong to only three universality classes for two classes b 0 for site dilution while b a for bond dilution the remaining class associated with high dimensions is characterized by b 2a 1 for both sites and bonds classes are defined by a set of value for mathit p 0 a deviations from available numerical estimates at d ensuremath le 7 are within ifmmode pm else textpm fi 0 008 and ifmmode pm else textpm fi 0 0004 for high dimensional hypercubic expansions at d ensuremath ge 8 the formula is found to be also valid for ising critical temperatures textcopyright 1996 the american physical society

critical binder cumulant in two dimensional anisotropic ising models the binder cumulant at the phase transitions of ising models on square lattices with various ferromagnetic nearest and next nearest neighbour couplings is determined using mainly monte carlo techniques we discuss the possibility of relating the value of the critical cumulant in the isotropic nearest neighbour and in the anisotropic cases to each other by means of a scale transformation in rectangular geometry to pinpoint universal and nonuniversal features

universal formulas for percolation thresholds ii extension to anisotropic and aperiodic lattices in a recent paper we reported a universal power law for both site and bond percolation thresholds in any bravais lattice with q equivalent nearest neighbors in dimension d we now extend it to three different classes of lattices which are respectively anisotropic lattices without equivalent nearest neighbors non bravais lattices with two atom unit cells and quasicrystals the investigation is focused on d 2 and 3 due to the lack of experimental data at higher dimensions the extension to these lattices requires the substitution of q by an effective non integer value q mathrm eff in the universal law for each of the 17 lattices which constitute our sample we argue for the existence of one q mathrm eff which reproduces both the site and the percolation threshold with a deviation with respect to numerical estimates which does not exceed ensuremath mp 0 01

instrumental variable estimator for the nonlinear errors in variables model abstract estimation for the nonlinear errors in variables model is considered it isassumed that additional information is available in the form of observations on instrumental variables an estimation procedure is presented for the parameters of the model asymptotic properties of the estimator are investigated

residual diagnostics for mixture models abstract a sample is commonly modeled by a mixture distribution if the observations follow a common distribution but the parameter of interest differs between observations for example we observe the lengths but not the ages of a sample offish it may be reasonable to assume that length is normally distributed about an unknown mean that depends on the age of the fish provided there is more than one age class in the sample then the data are distributed as a mixture of normals in this article we assume that the data are a random sample from a mixture of exponential family distributions and that for each observation the parameter of interest is sampled independently from an unknown mixing distribution q the adequacy of a fitted mixture model can be assessed by examining residuals based on the ratio of the observed to expected fit residuals based on the homogeneity model in which q is a one point distribution display a convexity property when the data follow a mixture model this becomes the basis fo

monte carlo study of percolation on disordered triangular lattices abstract a simple model for amorphous solids consisting of a mixed bond triangular lattice with a fraction of attenuated bonds randomly distributed which simulate the presence of defects in the surface is studied here by using computational simulation the degree of disorder of the surface is tunable by selecting the values of 1 the fraction of regular attenuated bonds 1 0 1 and 2 the factor r which is defined as the ratio between the value of the conductivity associated to an attenuated bond and that corresponding to a regular bond 0 r 1 the results obtained show how the percolation properties of the disordered system are modified with respect to the standard random bond percolation problem r 0

a simple estimator for nonlinear error in variable models we propose a simple estimator for nonlinear method of moment models with measurement error of the classical type when no additional data such as validation data or double measurements are available we assume that the marginal distributions of the measurement errors are laplace double exponential with zero means and unknown variances and the measurement errors are independent of the latent variables and are independent of each other under these assumptions we derive simple revised moment conditions in terms of the observed variables they are used to make inference about the model parameters and the variance of the measurement error the results of this paper show that the distributional assumption on the measurement errors can be used to point identify the parameters of interest our estimator is a parametric method of moments estimator that uses the revised moment conditions and hence is simple to compute our estimation method is particularly useful in situations where no additional data are available which is the case in many economic data sets simulation study demonstrates good finite sample properties of our proposed estimator we also examine the performance of the estimator in the case where the error distribution is misspecified

semiparametric estimation in logistic measurement error models summary we describe semiparametric estimation and inference in a logistic regression model with measurement error in the predictors the particular measurement error model consists of a primary data set in which only the response y and a fallible surrogate w of the true predictor x are observed plus a smaller validation data set for which y x w are observed except for the underlying assumption of a logistic model in the true predictor no parametric distributional assumption is made about the true predictor or its surrogate we develop a semiparametric parameter estimate of the logistic regression parameter which is asymptotically normally distributed and computationally feasible the estimate relies on kernel regression techniques for scalar predictors by a detailed analysis of the mean squared error of the parameter estimate we obtain a representation for an optimal bandwidth

monte carlo simulation of the percolation process caused by the random sequential adsorption of k mers on heterogeneous triangular lattices abstract mixed site bond percolation is studied for a random sequential adsorption process of k mers on heterogeneous lattices with variable connectivity by means of monte carlo simulation the percolation phase diagrams are built to render evidence about complex structures critical exponents are also calculated to show that the universality class corresponding to ordinary percolation in two dimensions is preserved

percolation thresholds on two dimensional voronoi networks and delaunay triangulations the site percolation threshold for the random voronoi network is determined numerically with the result pc 0 714 10 0 000 02 using monte carlo simulation on periodic systems of up to 40 000 sites the result is very close to the recent theoretical estimate pc approximately 0 7151 of neher for the bond threshold on the voronoi network we find pc 0 666 931 0 000 005 implying that for its dual the delaunay triangulation pc 0 333 069 0 000 005 these results rule out the conjecture by hsu and huang that the bond thresholds are 2 3 and 1 3 respectively but support the conjecture of wierman that for fully triangulated lattices other than the regular triangular lattice the bond threshold is less than 2 sin pi 18 approximately 0 3473

estimation of a convex function characterizations and asymptotic theory we study nonparametric estimation of convexregression and density functions by methods of least squares in the regression and density cases and maximum likelihood in the density estimation case we provide characterizations of these estimators prove that they are consistent and establish their asymptotic distributions at a fixed point of positive curvature of the functions estimated the asymptotic distribution theory relies on the existence of an invelope function for integrated two sided brownian motion t 4 which is established in a companion paper by groeneboom jongbloed and wellner

bond percolation for homogeneous two dimensional lattices abstract bond percolation is studied for the three homogeneous two dimensional lattices square lattice sl triangular lattice tl and honeycomb lattice hl an expanding cell technique is used to obtain percolation thresholds and other relevant information for different cell sizes we extend the analysis as to include slightly asymmetric cells in addition to the usual symmetric cells to get more points in the scaling analysis exact percolation functions are obtained for each size then the percolation threshold is obtained by means of two complementary methods one based on the well known renormalization techniques and the other one introduced here which is based upon determining the inflection point of the percolation curves a comparison of the results obtained by these two methods is performed the study includes iterations to extrapolate numerical results towards the thermodynamic limit critical exponents and are obtained values are compared with numerical results and expected theoretical estimations present results show agreement and even improvement in the case of with respect to some numeric values available in the literature comparison tables are provided

mcmc convergence diagnosis via multivariate bounds on log concave densities we begin by showing how piecewise linear bounds may be devised which bound both above and below any concave log density in general dimensions we then show how these bounds may be used to gain an upper bound to the volume in the tails outside the convex hull of the sample path in order to assess how well the sampler has explored the target distribution this method can be used as a stand alone diagnostic to determine when the sampler output provides a reliable basis for inference on the stationary density or in conjunction with existing convergence diagnostics to ensure that they are based upon good sampler output we provide an example and briefly discuss possible extensions to the method and alternative applications of the bounds 1 introduction the use of markov chain monte carlo mcmc methods has increased dramatically within the statistical community since their

brownian motion with a parabolic drift and airy functions let w t t be two sided brownian motion originating from zero and let v a be defined by v a sup t w t t a 2 is maximal then v a a is a markovian jump process running through the locations of maxima of two sided brownian motion with respect to the parabolas fa t t a 2 we give an analytic expression for the infinitesimal generators of the processes a in terms of airy functions in theorem 4 1 this makes it possible to develop asymptotics for the global behavior of a large class of isotonic estimators i e estimators derived under order restrictions an example of this is given in groeneboom 1985 where the asymptotic distribution of the standardized l1 distance between a decreasing density and the grenander maximum likelihood estimator of this density is determined on our way to theorem 4 1 we derive some other results for example we give an analytic expression for the joint density of the maximum and the location of the maximum of the process w t ct2 t where c is an aribrary positive constant we also determine the laplace transform of the integral over a brownian excursion these last results also have recently been derived by several other authors using a variety of methods

minireview new results for old percolation abstract percolation theory deals with the formation of clusters in disordered media at the percolation threshold for the first time an infinite cluster appears however research of the last years showed that at this phase transition some old ideas were wrong there can also be two or three clusters spanning from top to bottom even in large two dimensional lattices and the probability of spanning is not given by simple real space renormalization ideas

consistent estimation for some nonlinear errors in variables models abstract the issues of identification and estimation of nonlinear errors in variables models are explored the deficiencies of the conventional definition of identification are discussed and an alternative definition in terms of observed samples is suggested the consistency and asymptotic normality of minimum distance estimators are studied to simplify the computation a two step estimation procedure is also suggested in which a consistent estimate of a subset of parameters is first obtained and treated as if they were known in the second step estimation of the rest of the parameters conditions for the consistency of this two step estimator and its asymptotic variance covariance matrix are also derived

dependence of the percolation threshold on the size of the percolating species abstract site and bond percolation of k mers of different structures and forms deposited on 2 d regular lattices is studied in addition the percolation threshold for percolating k mers on a bethe lattice is analytically obtained by using finite size scaling theory the analysis of the results is performed in order to determine the behavior of the percolation threshold which exhibits an exponential decrease with the k mer size characteristic parameters of that function are dependent not only on the form and structure of k mers but also on the properties of the lattice where they are deposited an expression for the percolation threshold as a function of the parameters of the problem is proposed and discussed

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

a graphical technique for determining the number of components in a mixture of normals abstract when a population is assumed to be composed of a finite number of subpopulations a natural model to choose is the finite mixture model it will often be the case however that the number of component distributions is unknown and must be estimated this problem can be difficult for instance the density of two mixed normals is not bimodal unless the means are separated by at least 2 standard deviations hence modality of the data per se can be an insensitive approach to component estimation we demonstrate that a mixture of two normals divided by a normal density having the same mean and variance as the mixed density is always bimodal this analytic result and other related results form the basis for a diagnostic and a test for the number of components in a mixture of normals the density is estimated using a kernel density estimator under the null hypothesis the proposed diagnostic can be approximated by a stationary gaussian process under the alternative hypothesis components in the mixtu

the number of incipient spanning clusters in two dimensional percolation using methods of conformal field theory we conjecture an exact form for the probability that n distinct clusters span a large rectangle or open cylinder of aspect ratio k i n the limit when k is large the study of the structure of large clusters at the percolation threshold continues to pose interesting problems whose solution sheds light on the nature of the critical state in general recently some attention has been paid to incipient spanning clusters iscs these are clusters which connect two disjoint segments of the boundary of a macroscopically large region langlands et al 1 conjectured that the probability that at least one such cluster exists that is that the segments are connected is invariant under conformal transformations this statement was placed in the context of conformal field theory in 2 where an explicit formula was given for this crossing probability for a review of the status of conformal invariance in critical percolation see 3 more recently aizenman 4 has considered among other things the probability that there exist n distinct iscs connecting the two segmentsy in the case of a rectangular region 0 kl 0 l he has proved that the probability p n k l that the strip is traversed in the direction of length kl b y nindependent clusters satisfies the bounds a e n 2 k