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

multiscale inference and long run variance estimation in non parametric regression with time series errors we develop new multiscale methods to test qualitative hypotheses about the function m in the non parametric regression model yt t m t t t with time series errors t in time series applications m represents a non parametric time trend practitioners are often interested in whether the trend m has certain shape properties for example they would like to know whether m is constant or whether it is increasing or decreasing in certain time intervals our multiscale methods enable us to test for such shape properties of the trend m to perform the methods we require an estimator of the long run error variance 2 l cov e0 el we propose a new difference based estimator of 2 for the case that t belongs to the class of auto regressive ar processes in the technical part of the paper we derive asymptotic theory for the proposed multiscale test and the estimator of the long run error variance the theory is complemented by a simulation study and an empirical application to climate data

testing for monotonicity of a regression mean by calibrating for linear functions a new approach to testing for monotonicity of a regression mean not requiring computation of a curve estimator or a bandwidth is suggested it is based on the notion of running gradients over short intervals although from some viewpoints it may be regarded as an analogue for monotonicity testing of the dip excess mass approach for testing modality hypotheses about densities like the latter methods the new technique does not suffer difficulties caused by almost flat parts of the target function in fact it is calibrated so as to work well for flat response curves and as a result it has relatively good power properties in boundary cases where the curve exhibits shoulders in this respect as well as in its construction the running gradients approach differs from alternative techniques based on the notion of a critical bandwidth

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

unified approach to testing functional hypotheses in semiparametric contexts abstract we suggest a new method with very wide applicability for testing semiparametric hypotheses about functions such as regression means and probability densities the technique is based on characterising hypotheses in terms of functionals which can be estimated root n consistently and constructing test statistics in terms of estimators of the functionals since the tests are semiparametric it is appropriate to assess them on the basis of their ability to detect departures of size n 1 2 from the null hypothesis we show that they do indeed have this property unlike tests constructed in a nonparametric setting their power does not depend critically on choice of a bandwidth and in particular smoothing parameter selection is not an issue that has to be addressed by users of the tests bootstrap methods are suggested for calibrating the tests in a regression setting applications include tests of specification such as partial linear and index models against nonparametric or semiparametric alternatives and tests of monotonicity concavity separability equality of regression functions and base independence of equivalence scales in a density setting they include tests of radial symmetry and stochastic dominance

estimating smooth monotone functions many situations call for a smooth strictly monotone function f of arbitrary flexibility the family of functions defined by the differential equation d 2 f w df where w is an unconstrained coefficient function comprises the strictly monotone twice differentiable functions the solution to this equation is f c 0 c 1 d 1 exp d 1 w where c 0 and c 1 are arbitrary constants and d 1 is the partial integration operator a basis for expanding w is suggested that permits explicit integration in the expression of f in fitting data it is also useful to regularize f by penalizing the integral of w 2 since this is a measure of the relative curvature in f applications are discussed to monotone nonparametric regression to the transformation of the dependent variable in non linear regression and to density estimation

estimating a treatment effect with repeated measurements accounting for varying effectiveness duration to assess treatment efficacy in clinical trials certain clinical outcomes are repeatedly measured over time for the same subject the difference in their means may characterize a treatment effect since treatment effectiveness lag and saturation times may exist erosion of treatment effect often occurs during the observation period instead of using models based on ad hoc parametric or purely nonparametric time varying coefficients we model the treatment effectiveness durations which are the time intervals between the lag and saturation times then we use some mean response models to include such treatment effectiveness durations our methodology is demonstrated by simulations and analysis of a landmark hiv aids clinical trial of short course nevirapine against mother to child hiv vertical transmission during labour and delivery

testing for monotonicity under endogeneity an application to the reservation wage function abstract this paper develops a test for monotonicity of nonparametric regression models under endogeneity which in its generality is novel in the literature the test statistic which is built upon a second order u process introduces correction terms based on control functions that purge the endogeneity the test has a non standard asymptotic distribution from which asymptotic critical values can directly be derived furthermore the test statistic is extended to accommodate multivariate exogenous regressors consistency against general alternatives is proved and the finite sample properties of the test are examined in a monte carlo experiment the test is used to formally assess the monotonicity of the reservation wage as a declining function of elapsed unemployment duration which has implications for underlying job search models this relationship is difficult to measure due to the simultaneity of both variables results for uk data indicate that reservation wage functions do in fact not decline monotonically thereby contradicting some partial equilibrium job search models

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

nonparametric estimation of residual variance revisited summary several difference based estimators of residual variance are compared for finite sample size since the introduction of a rather simple estimator by gasser sroka jennensteinmetz 1986 other proposals have been made here the one given by hall kay titterington 1990 is of particular interest it minimizes the asymptotic variance unfortunately it has severe problems with finite sample bias and the estimator of gasser et al 1986 proves still to be a good choice a new estimator is introduced compromising between bias and variance

on multiple regression models with nonstationary correlated errors summary we consider the estimation of parameters of a multiple regression model with non stationary errors we assume the nonstationary errors satisfy a time dependent auto regressive process and describe a method for estimating the parameters of the regressors and the time dependent autoregressive parameters the parameters are rescaled as in nonparametric regression to obtain the asymptotic sampling properties of the estimators the method is illustrated with an example taken from global temperature anomalies

estimating the variance in nonparametric regression what is a reasonable choice the exact mean squared error mse of estimators of the variance in nonparametric regression based on quadratic forms is investigated in particular two classes of estimators are compared hall kay and titterington s optimal difference based estimators and a class of ordinary difference based estimators which generalize methods proposed by rice and gasser sroka and jennen steinmetz for small sample sizes the mse of the first estimator is essentially increased by the magnitude of the integrated first two squared derivatives of the regression function it is shown that in many situations ordinary difference based estimators are more appropriate for estimating the variance because they control the bias much better and hence have a much better overall performance it is also demonstrated that rice s estimator does not always behave well data driven guidelines are given to select the estimator with the smallest mse

distribution free tests of stochastic monotonicity this article proposes an omnibus test for monotonicity of nonpara metric conditional distributions and its moments unlike previous proposals our method does not require smooth estimation of the derivatives of nonparametric curves and it can be implemented even when probability densities do not exist in fact we only require continuity of the marginal distributions under the null and xed alternatives distinguishing features of our approach are that critical values are pivotal under the null in nite samples and the test is invariant to any monotonic continuous transformation of the explanatory variable the test sta tistic is the sup norm of the dierence between the empirical copula function and its least concave majorant with respect to the explanatory variable coordinate the resulting test is able to detect local alternatives converging to the null at the parametric rate n 1 2 with n the sample size the article also discusses several applications and extensions of the proposal these include testing monotonicity of general conditional moments and the extension to multivariate explanatory variables the nite sample performance of the test is examined by means of a monte carlo experiment

nearly isotonic regression we consider the problem of approximating a sequence of data points with a nearly isotonic or nearly monotone function this is formulated as a convex optimization problem that yields a family of solutions with one extreme member being the standard isotonic regression fit we devise a simple algorithm to solve for the path of solutions which can be viewed as a modified version of the well known pool adjacent violators algorithm and computes the entire path in o n log n operations n being the number of data points in practice the intermediate fits can be used to examine the assumption of monotonicity nearly isotonic regression admits a nice property in terms of its degrees of freedom at any point along the path the number of joined pieces in the solution is an unbiased estimate of its degrees of freedom we also extend the ideas to provide nearly convex approximations

surveillance of a simple linear regression this article considers an important aspect of the general sequential analysis problem where a process is in control up to some unknown point i 1 after which the distribution from which the observations are generated changes an extensive sequential analytic literature assumes that the change in distribution is abrupt for example from n 0 1 to n 1 there is also an extensive literature that deals with a gradual change in the case where the decision whether or not a change has occurred is based on a fixed set of observations rather than an ongoing process of decision making every time a new observation is obtained however there is virtually no literature on the practical case of sequentially detecting a gradual change in distribution visualize a machine deteriorating gradually this article considers solutions to this problem as a first approximation the gradual change problem can be modeled as a change from a fixed distribution to a model of simple linear regression with respect to ti

a self normalized confidence interval for the mean of a class of nonstationary processes we construct an asymptotic confidence interval for the mean of a class of nonstationary processes with constant mean and time varying variances due to the large number of unknown parameters traditional approaches based on consistent estimation of the limiting variance of sample mean through moving block or non overlapping block methods are not applicable under a block wise asymptotically equal cumulative variance assumption we propose a self normalized confidence interval that is robust against the nonstationarity and dependence structure of the data we also apply the same idea to construct an asymptotic confidence interval for the mean difference of nonstationary processes with piecewise constant means the proposed methods are illustrated through simulations and an application to global temperature series

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

mode identification of volatility in time varying autoregression in many applications time series exhibit nonstationary behavior that might reasonably be modeled as a time varying autoregressive ar process in the context of such a model we discuss the problem of testing for modality of the variance function we propose a test of modality that is local and when used iteratively can be used to identify the total number of modes in a given series this problem is closely related to peak detection and identification which has applications in many fields we propose a test that under appropriate assumptions is asymptotically distribution free under the null hypothesis even though nonparametric estimation of the ar parameter functions is involved simulation studies and applications to real datasets illustrate the behavior of the test

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

nonparametric kernel regression subject to monotonicity constraints we suggest a method for monotonizing general kernel type estimators for example local linear estimators and nadaraya watson estimators attributes of our approach include the fact that it produces smooth estimates indeed with the same smoothness as the unconstrained estimate the method is applicable to a particularly wide range of estimator types it can be trivially modified to render an estimator strictly monotone and it can be employed after the smoothing step has been implemented therefore an experimenter may use his or her favorite kernel estimator and their favorite bandwidth selector to construct the basic nonparametric smoother and then use our technique to render it monotone in a smooth way implementation involves only an off the shelf programming routine the method is based on maximizing fidelity to the conventional empirical approach subject to monotonicity we adjust the unconstrained estimator by tilting the empirical distribution so as to make the least possible change in the sense of a distance measure subject to imposing the constraint of monotonicity

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

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

on the degrees of freedom in shape restricted regression for the problem of estimating a regression function say subject to shape constraints like monotonicity or convexity it is argued that the divergence of the maximum likelihood estimator provides a useful measure of the effective dimension of the model inequalities are derived for the expected mean squared error of the maximum likelihood estimator and the expected residual sum of squares these generalize equalities from the case of linear regression as an application it is shown that the maximum likelihood estimator of the error variance 2 is asymptotically normal with mean 2 and variance 2 2 n for monotone regression it is shown that the maximum likelihood estimator of attains the optimal rate of convergence and a bias correction to the maximum likelihood estimator of 2 is derived

updating the inverse of a matrix the sherman morrison woodbury formulas relate the inverse of a matrix after a small rank perturbation to the inverse of the original matrix the history of these fomulas is presented and various applications to statistics networks structural analysis asymptotic analysis optimization and partial differential equations are discussed the sherman morrison woodbury formulas express the inverse of a matrix after a small rank perturbation in terms of the inverse of the original matrix this paper surveys the history of these formulas and we examine some applications where these formulas are helpful

a comparison of variance estimators in nonparametric regression summary we compare two estimators of error variance both based on quadratic forms in the residuals about smoothing spline fits to data the estimators are compared over the whole range of values of the smoothing parameter as well as for data based choices of the smoothing parameter we show that the commonly used estimator of variance has the serious drawback of underestimating the error variance for small choices of the smoothing parameter this drawback is not shared by a simple but more computationally intensive alternative

isotonic regression and isotonic projection abstract the note describes the cones in the euclidean space admitting isotonic metric projection with respect to the coordinate wise ordering as a consequence it is shown that the metric projection onto the isotonic regression cone the cone defined by the general isotonic regression problem admits a projection which is isotonic with respect to the coordinate wise ordering

inference of trends in time series summary we consider statistical inference of trends in mean non stationary models a test statistic is proposed for the existence of structural breaks in trends on the basis of a strong invariance principle of stationary processes we construct simultaneous confidence bands with asymptotically correct nominal coverage probabilities the results are applied to global warming temperature data and nile river flow data our confidence band of the trend of the global warming temperature series supports the claim that the trend is increasing over the last 150 years

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

a note on lattices in semi stable representations let p be a prime k a finite extension over mathbb q p and g rm gal overline k k we extend kisin s theory on varphi modules of finite e u height to give a new classification of g stable mathbb z 1 p lattices in semi stable representations

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

residual variance and residual pattern in nonlinear regression summary a nonparametric estimator of residual variance in nonlinear regression is proposed it is based on local linear fitting asymptotically the estimator has a small bias but a larger variance compared with the parametric estimator in linear regression finite sample properties are investigated in a simulation study including a comparison with other nonparametric estimators the method is also useful for spotting heteroscedasticity and outliers in the residuals at an early stage of the data analysis a further application is checking the fit of parametric models this is illustrated for longitudinal growth data

modularity of 2 adic barsotti tate representations we prove a modularity lifting theorem for two dimensional 2 adic potentially barsotti tate representations this proves hypothesis h of khare wintenberger and completes the proof of serre s conjecture the main new ingredient is a classification of connected finite flat group schemes over rings of integers of finite extensions of 2

stochastic model for dielectric breakdown we discuss a model for the development of discharge patterns in dielectric breakdown based on the laplace equation associated with a probability field the model gives rise to random fractals with well defined hausdorff dimensions the relations of this model with the diffusion limited aggregation are discussed in detail the possibility of application to other stochastic phenomena like fracture propagation is proposed

a discrete model for the lightning discharge for a lightning discharge model presented by the author j s nisbet and j r kasha j comput phys 82 193 1989 we simulate lightning by letting the conductivity tend to infinity wherever the electric field reaches the breakdown threshold here we show that for this discharge model and for a one parameter family of integration schemes the backward euler scheme is the only one that leads to the equilibration of the electric potential along the discharge channel moreover the potential obtained by letting the conductivity tend to infinity in the continuous equation is identical to the potential obtained in the backward euler approximation when conductivity tends to infinity connections to diffusion limited aggregation dla to more recent schemes for simulating the lightning discharge and to experiments of williamset al are discussed

conjecture de l inertie moderee de serre on considere k un corps complet pour une valuation discrete de caracteristique nulle et dont le corps residuel est suppose parfait de caracteristique p on appelle e l indice de ramification absolue de k mathcal o k son anneau des entiers et overline k une cloture algebrique soit xk un schema propre et lisse sur k admettant un modele propre et semi stable x sur mathcal o k generalisant les resultats de breuil valables pour e 1 on demontre dans cet article un isomorphisme de periodes reliant le r ieme groupe de cohomologie etale de x overline k a coefficients dans pn et un r ieme groupe de cohomologie log cristalline de la fibre speciale de x nous avons toutefois les restrictions er 1

the dip test of unimodality the dip test measures multimodality in a sample by the maximum difference over all sample points between the empirical distribution function and the unimodal distribution function that minimizes that maximum difference the uniform distribution is the asymptotically least favorable unimodal distribution and the distribution of the test statistic is determined asymptotically and empirically when sampling from the uniform

the evolution and discharge of electric fields within a thunderstorm a 3 dimensional electrical model for a thunderstorm is developed and finite difference approximations to the model are analyzed if the spatial derivatives are approximated by a method akin to the scheme and if the temporal derivative is approximated by either a backward difference or the crank nicholson scheme we show that the resulting discretization is unconditionally stable the forward difference approximation to the time derivative is stable when the time step is sufficiently small relative to the ratio between the permittivity and the conductivity max norm error estimates for the discrete approximations are established to handle the propagation of lightning special numerical techniques are devised based on the inverse matrix modification formula and cholesky updates numerical comparisons between the model and theoretical results of wilson and holzer saxon are presented we also apply our model to a storm observed at the kennedy space center on july 11 1978

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

boundary crossing probabilities and statistical applications abstract this paper surveys recent results involving boundary crossing probabilities and related statistical applications the first part is concerned with problems of sequential analysis especially repeated significance tests and their application to sequential clinical trials involving survival data the second part develops the probability theory motivated by the problems of part 1 a method for computing first passage distributions of brownian motion to linear boundaries is introduced and then modified to handle problems in discrete time and those involving nonlinear boundaries the third part is concerned with fixed sample statistical problems especially change point problems which involve boundary crossing probabilities examples are given of problems for which methods of part 2 appear adequate and of problems which require new methods

on variance estimation in nonparametric regression a simple explicit estimator of variance in nonparametric regression is suggested based on the mean square of a sequence of residuals the estimator is valid for general error distributions has elementary bias and variance properties and is optimal to both first and second orders

inference for monotone functions under short and long range dependence confidence intervals and new universal limits abstractwe introduce new point wise confidence interval estimates for monotone functions observed with additive dependent noise our methodology applies to both short and long range dependence regimes for the errors the interval estimates are obtained via the method of inversion of certain discrepancy statistics this approach avoids the estimation of nuisance parameters such as the derivative of the unknown function which previous methods are forced to deal with the resulting estimates are therefore more accurate stable and widely applicable in practice under minimal assumptions on the trend and error structure the dependence of the errors especially long range dependence leads to new phenomena where new universal limits based on convex minorant functionals of drifted fractional brownian motion emerge some extensions to uniform confidence bands are also developed supplementary materials for this article are available online

lattices in the cohomology of shimura curves we prove the main conjectures of breuil j reine angew math 2012 including a generalisation from the principal series to the cuspidal case and dembele j reine angew math 2012 subject to a mild global hypothesis that we make in order to apply certain r mathbb t theorems more precisely we prove a multiplicity one result for the mod p cohomology of a shimura curve at iwahori level and we show that certain apparently globally defined lattices in the cohomology of shimura curves are determined by the corresponding local p adic galois representations we also indicate a new proof of the buzzard diamond jarvis conjecture in generic cases our main tools are the geometric breuil mezard philosophy developed in emerton and gee j inst math jussieu 2012 and a new and more functorial perspective on the taylor wiles kisin patching method along the way we determine the tamely potentially barsotti tate deformation rings of generic two dimensional mod p representations generalising a result of breuil and mezard bull soc math de france 2012 in the principal series case

heteroscedasticity and autocorrelation robust structural change detection the assumption of weak stationarity is crucial for the validity of most of the conventional tests of structure change in time series under complicated nonstationary temporal dynamics we argue that traditional testing procedures result in mixed structural change signals of the first and second order and hence could lead to biased testing results the article proposes a simple and unified bootstrap testing procedure that provides consistent testing results under general forms of smooth and abrupt changes in the temporal dynamics of the time series monte carlo experiments are performed to compare our testing procedure with various traditional tests our robust bootstrap test is applied to testing changes in an environmental and a financial time series and our procedure is shown to provide more reliable results than the conventional tests

longcrisp a test for bump hunting in longitudinal data we propose an extension of the harezlak and heckman j comput graph statist 2001 10 4 713 729 test for detecting local extrema to the longitudinal data setting we use penalized spline regression techniques statist sci 1996 11 89 102 to provide a computationally efficient method of testing for relatively large data sets we estimate the p values of our test longcrisp with a smoothed bootstrap our simulation studies indicate that the test is generally conservative and has power exceeding 70 per cent at the alpha 0 1 nominal level in most considered settings finally we apply our testing procedure to the longitudinal measurements of body mass index of former prisoners of war in vietnam and conclude that the mean population curve exhibits non monotone behaviour

testing noisy numerical data for monotonic association rank correlation measures are intended to measure to which extent there is a monotonic association between two observables while they are mainly designed for ordinal data they are not ideally suited for noisy numerical data in order to better account for noisy data a family of rank correlation measures has previously been introduced that replaces classical ordering relations by fuzzy relations with smooth transitions thereby ensuring that the correlation measure is continuous with respect to the data the given paper briefly repeats the basic concepts behind this family of rank correlation measures and investigates it from the viewpoint of robust statistics then on this basis we introduce a framework of novel rank correlation tests an extensive experimental evaluation using a large number of simulated data sets is presented which demonstrates that the new tests indeed outperform the classical variants in terms of type ii error rates without sacrificing good performance in terms of type i error rates this is mainly due to the fact that the new tests are more robust to noise for small samples the gaussian rank correlation estimator turned out to be the best choice in situations where no prior knowledge is available about the data whereas the new family of robust gamma test provides an advantage in situations where information about the noise distribution is available an implementation of all robust rank correlation tests used in this paper is available as an r package from the cran repository

testing convex hypotheses on the mean of a gaussian vector application to testing qualitative hypotheses on a regression function in this paper we propose a general methodology based on multiple testing for testing that the mean of a gaussian vector in r n belongs to a convex set we show that the test achieves its nominal level and characterize a class of vectors over which the tests achieve a prescribed power in the functional regression model this general methodology is applied to test some qualitative hypotheses on the regression function for example we test that the regression function is positive increasing convex or more generally satisfies a differential inequality uniform separation rates over classes of smooth functions are established and a comparison with other results in the literature is provided a simulation study evaluates some of the procedures for testing monotonicity

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

excess mass estimates and tests for multimodality abstract we propose and study a method for analyzing the modality of a distribution the method is based on the excess mass functional this functional measures excessive empirical mass in comparison with multiples of uniform distribution by the excess mass approach we separate the investigation about the number of modes from questions concerning their location for distributions on the line the excess mass functional can be estimated at a square root rate the asymptotic behavior of estimators is studied and tests for multimodality based on the excess mass are derived

testing for stochastic monotonicity we propose a test of the hypothesis of stochastic monotonicity this hypothesis is of interest in many applications in economics our test is based on the supremum of a rescaled u statistic we show that its asymptotic distribution is gumbel the proof is difficult because the approximating gaussian stochastic process contains both a stationary and a nonstationary part and so we have to extend existing results that only apply to either one or the other case we also propose a refinement to the asymptotic approximation that we show works much better in finite samples we apply our test to the study of intergenerational income mobility

consistent detection of a monotonic trend superposed on a stationary time series summary consider a time series made up of a signal and a stationary autocorrelated error series a statistic is proposed for examining the hypothesis that the signal term is constant versus the hypothesis that it is monotonic in time the statistic is the ratio of a linear combination of the time series values with coefficients introduced by abelson tukey 1963 to an estimate of the standard error of the linear combination the statistic has asymptotic power 1 for a broad class of monotonic alternatives the procedure is illustrated for the series of river heights at a location on the rio negro in brazil where there is concern that the height is rising due to deforestation of the amazon basin the significance level obtained is 0025

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

asymptotically optimal difference based estimation of variance in nonparametric regression summary we define and compute asymptotically optimal difference sequences for estimating error variance in homoscedastic nonparametric regression our optimal difference sequences do not depend on unknowns such as the mean function and provide substantial improvements over the suboptimal sequences commonly used in practice for example in the case of normal data the usual variance estimator based on symmetric second order differences is only 64 efficient relative to the estimator based on optimal second order differences the efficiency of an optimal mth order difference estimator relative to the error sample variance is 2m 2m 1 again this is for normal data and increases as the tails of the error distribution become heavier

nonparametric variable selection the earth algorithm we consider regression experiments involving a response variable y and a large number of predictor variables x1 xd many of which may be irrelevant for the prediction of y and thus must be removed before y can be predicted from the x s we consider two procedures that select variables by using importance scores that measure the strength of the relationship between predictor variables and a response and keep those variables whose importance scores exceed a threshold in the first of these procedures scores are obtained by randomly drawing subregions tubes of the predictor space that constrain all but one predictor and in each subregion computing a signal to noise ratio efficacy based on a nonparametric univariate regression of y on the unconstrained variable the subregions are adapted to boost weak variables iteratively by searching hunting for the subregions in which the efficacy is maximized the efficacy can be viewed as an approximation to a one to one function of the probability of identif

some asymptotics for multimodality tests based on kernel density estimates a test due to b w silverman for modality of a probability density is based on counting modes of a kernel density estimator and the idea of critical smoothing an asymptotic formula is given for the expected number of modes this together with other methods establishes the rate of convergence of the critically smoothed bandwidth these ideas are extended to provide insight concerning the behaviour of the test based on bootstrap critical values

asymptotic normality of statistics based on the convex minorants of empirical distribution functions let f be the uniform empirical distribution function write fn for the least concave majorant of f and let an denote the corresponding density it is shown that n fl fn t 1 2 dt is asymptotically standard normal when centered at log n and normalized by 3 log n 1i2 a similar result is obtained in the 2 sample case in which fn is replaced by the slope of the convex minorant of fm fm hn1

isotonic regression another look at the changepoint problem a test based on isotonic regression is developed for monotonic trends in short range dependent sequences and is applied to argentina rainfall data and global warming data this test provides another perspective for changepoint problems the isotonic test is shown to be more powerful than some existing tests for trend

stochastic non convex envelopment of data applying isotonic regression to frontier estimation abstract isotonic nonparametric least squares inls is a regression method for estimating a monotonic function by fitting a step function to data in the literature of frontier estimation the free disposal hull fdh method is similarly based on the minimal assumption of monotonicity in this paper we link these two separately developed nonparametric methods by showing that fdh is a sign constrained variant of inls we also discuss the connections to related methods such as data envelopment analysis dea and convex nonparametric least squares cnls further we examine alternative ways of applying isotonic regression to frontier estimation analogous to corrected and modified ordinary least squares cols mols methods known in the parametric stream of frontier literature we find that inls is a useful extension to the toolbox of frontier estimation both in the deterministic and stochastic settings in the absence of noise the corrected inls cinls has a higher discriminating power than fdh in the case of noisy data we propose to apply the method of non convex stochastic envelopment of data non convex stoned which disentangles inefficiency from noise based on the skewness of the inls residuals the proposed methods are illustrated by means of simulated examples

tests for monotonicity of a regression mean with guaranteed level in this paper a nonparametric procedure for testing for monotonicity of a regression mean with guaranteed level is proposed the procedure is based on signs of differences of observations from the response variable the test is calibrated against the most difficult null hypothesis when the regression function is constant and produces an exact test in this context in general the test is conservative the power of the test is good and comparable with that of other nonparametric tests it is shown that the testing procedure has asymptotic power 1 against certain local alternatives the method is also robust against heavy tailed error distributions and even maintains good power when the errors are for example cauchy distributed a simulation study is provided to demonstrate finite sample behaviour of the testing procedure

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

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 graphical method for estimating the residual variance in nonparametric regression summary if one wishes to make inferences about a regression then an accurate estimator of the residual variance is required however except in the simplest cases all estimators of variance are biased by an amount which depends on the unknown regression here a graphical procedure is proposed which indicates whether the bias in a particular estimate is small this procedure is studied in some detail for estimators which are weighted sums of squares of divided differences of the data and leads to a practical method of variance estimation

moduli of finite flat group schemes and modularity we prove that under some mild conditions a two dimensional p adic galois representation which is residually modular and potentially barsotti tate at p is modular this provides a more conceptual way of establishing the shimura taniyama weil conjecture especially for elliptic curves which acquire good reduction over a wildly ramified extension of 3 the main ingredient is a new technique for analyzing flat deformation rings it involves resolving them by spaces which parametrize finite flat group scheme models of galois representations introduction