survival analysis using a scale change random effects model abstract frailty models are effective in broadening the class of survival models and inducing dependence in multivariate survival distributions in proportional hazards the random effect multiplies the hazard function the scale change model incorporates unobserved heterogeneity through a random effect that enters the baseline hazard function to change the time scale we interpret this random effect as frailty or other unobserved risks that create heterogeneity in the population this model produces a wide range of shapes for univariate survival and hazard functions we extend this model to multivariate survival data by assuming that members of a group share a common random effect this structure induces association among the survival times in a group and provides alternative association structures to the proportional hazards frailty model we present parametric and semiparametric estimation techniques and illustrate these methods with an example

high dimensional classification via nonparametric empirical bayes and maximum likelihood inference we propose new nonparametric empirical bayes methods for high dimensional classification our classifiers are designed to approximate the bayes classifier in a hypothesized hierarchical model where the prior distributions for the model parameters are estimated nonparametrically from the training data as is common with nonparametric empirical bayes the proposed classifiers are effective in high dimensional settings even when the underlying model parameters are in fact nonrandom we use nonparametric maximum likelihood estimates of the prior distributions following the elegant approach studied by kiefer wolfowitz in the 1950s however our implementation is based on a recent convex optimization framework for approximating these estimates that is well suited for large scale problems we derive new theoretical results on the accuracy of the approximate estimator which help control the misclassification rate of one of our classifiers we show that our methods outperform several existing methods in simulations and perform well when gene expression microarray data is used to classify cancer patients

approximate standard errors in semiparametric models we consider semiparametric models with p regressor terms and q smooth terms we obtain an explicit expression for the estimate of the regression coefficients given by the back fitting algorithm the calculation of the standard errors of these estimates based on this expression is a considerable computational exercise we present an alternative approximate method of calculation that is less demanding with smoothing splines the method is exact while with loess it gives good estimates of standard errors we assess the adequacy of our approximation and of another approximation with the help of two examples

estimating normal means with a dirichlet process prior abstract in this article the dirichlet process prior is used to provide a nonparametric bayesian estimate of a vector of normal means in the past there have been computational difficulties with this model this article solves the computational difficulties by developing a gibbs sampler algorithm the estimator developed in this article is then compared to parametric empirical bayes estimators peb and nonparametric empirical bayes estimators npeb in a monte carlo study the monte carlo study demonstrates that in some conditions the peb is better than the npeb and in other conditions the npeb is better than the peb the monte carlo study also shows that the estimator developed in this article produces estimates that are about as good as the peb when the peb is better and produces estimates that are as good as the npeb estimator when that method is better

interval censored data a note on the nonparametric maximum likelihood estimator of the distribution function summary gentleman geyer 1994 discuss the analysis of interval censored data and present results based on standard convex optimisation theory here this problem is viewed from the perspective of a mixing problem of indicator functions using an analogy with the problem of mixture distributions a variety of results are easily derived including a characterisation theorem for the maximum likelihood estimator and various reliably convergent algorithms software for the analysis of mixture distributions can be used to find the nonparametric estimate of the distribution function of the interval censored survival time examples are provided to demonstrate the theory

an algorithm for computing the nonparametric mle of a mixing distribution abstract a fast algorithm for calculating the nonparametric maximum likelihood estimate mle of a mixing distribution g in a mixture model is discussed the literature contains several methods for computing the nonparametric mle of g but these methods are slow in this article we develop an algorithm for maximizing the log likelihood l g over the family of all distribution functions that yields the nonparametric mle of g in some semiparametric problems a structural or fixed parameter is of interest and we are interested in computing profile likelihood sup gl g for a grid of values of the algorithm that we propose is fast enough for this purpose examples illustrate and compare the algorithms one taken from an article by laird the common mean problem and one taken from the literature on optimal experimental design it is also noted that routines from the literature on semi infinite programming may be used to compute the profile log likelihood

meta analysis of diagnostic accuracy and roc curves with covariate adjusted semiparametric mixtures many screening tests dichotomize a measurement to classify subjects typically a cut off value is chosen in a way that allows identification of an acceptable number of cases relative to a reference procedure but does not produce too many false positives at the same time thus for the same sample many pairs of sensitivities and false positive rates result as the cut off is varied the curve of these points is called the receiver operating characteristic roc curve one goal of diagnostic meta analysis is to integrate roc curves and arrive at a summary roc sroc curve holling bohning and bohning psychometrika 77 106 126 2012a demonstrated that finite semiparametric mixtures can describe the heterogeneity in a sample of lehmann roc curves well this approach leads to clusters of sroc curves of a particular shape we extend this work with the help of the t alpha transformation a flexible family of transformations for proportions a collection of sroc curves is constructed that approximately contains the lehmann family but in addition allows the modeling of shapes beyond the lehmann roc curves we introduce two rationales for determining the shape from the data using the fact that each curve corresponds to a natural univariate measure of diagnostic accuracy we show how covariate adjusted mixtures lead to a meta regression on sroc curves three worked examples illustrate the method

asymptotic optimality of data driven neyman s tests for uniformity data driven neyman s tests resulting from a combination of neyman s smooth tests for uniformity and schwarz s selection procedure are investigated asymptotic intermediate efficiency of those tests with respect to the neyman pearson test is shown to be 1 for a large set of converging alternatives the result shows that data driven neyman s tests contrary to classical goodness of fit tests are indeed omnibus tests adapting well to the data at hand

testing lack of fit in multiple regression summary we study lack of fit tests based on orthogonal series estimators a common feature of these tests is that they are functions of score statistics that employ data driven model dimensions the criteria used to select the dimension are score based versions of aic and bic the tests can be applied in a wide variety of settings including both continuous and discrete data with two or more covariates a model sequence i e a path in the alternative models space has to be chosen critical points and p values of the lack of fit tests can be obtained via asymptotic distribution theory or by use of the bootstrap data examples and a simulation study illustrate the applicability of the tests

generalized additive models some applications abstract generalized additive models have the form x fj x j where might be the regression function in a multiple regression or the logistic transformation of the posterior probability pr y 1 x in a logistic regression in fact these models generalize the whole family of generalized linear models x x where x g x is some transformation of the regression function we use the local scoring algorithm to estimate the functions fj xj nonparametrically using a scatterplot smoother as a building block we demonstrate the models in two different analyses a nonparametric analysis of covariance and a logistic regression the procedure can be used as a diagnostic tool for identifying parametric transformations of the covariates in a standard linear analysis a variety of inferential tools have been developed to aid the analyst in assessing the relevance and significance of the estimated functions these include confidence curves degrees of freedom estimates and approximat

linear smoothers and additive models we study linear smoothers and their use in building nonparametric regression models in the first part of this paper we examine certain aspects of linear smoothers for scatterplots examples of these are the running mean and running line kernel and cubic spline smoothers the eigenvalue and singular value decompositions of the corresponding smoother matrix are used to describe qualitatively a smoother and several other topics such as the number of degrees of freedom of a smoother are discussed in the second part of the paper we describe how linear smoothers can be used to estimate the additive model a powerful nonparametric regression model using the backfitting algorithm we show that backfitting is the gauss seidel iterative method for solving a set of normal equations associated with the additive model we provide conditions for consistency and nondegeneracy and prove convergence for the backfitting and related algorithms for a class of smoothers that includes cubic spline smoothers

statistical models for intercepted data abstract consider a population of replaceable items of different ages and suppose that for each item there is a certain measurable attribute that is a monotone function perhaps random of the item s age to fix ideas we refer to this attribute as deterioration and we assume that as items age their level of deterioration increases we further assume that as the level of deterioration hits a certain threshold value perhaps unknown the item dies and disappears from the population so it is not available for sampling any more since items that deteriorate at a slower rate live longer any sample obtained over a short period of time would overrepresent slowly deteriorating items consequently under such circumstances longitudinal studies are often needed to statistically study deterioration as a function of age this article presents a statistical model that enables us to study the deterioration function from a single cross section snapshot of the population or some random parts of it specifically

computer assisted analysis of mixtures c a man statistical algorithms summary this paper presents various algorithmic approaches for computing the maximum likelihood estimator of the mixing distribution of a one parameter family of densities and provides a unifying computeroriented concept for the statistical analysis of unobserved heterogeneity i e observations stemming from different subpopulations in a univariate sample the case with unknown number of population subgroups as well as the case with known number of population subgroups with emphasis on the first is considered in the computer package c a man computer assisted mixture analysis it includes an algorithmic menu with choices of the em algorithm the vertex exchange algorithm a combination of both as well as the vertex direction method to ensure reliable convergence a steplength menu is provided for the three latter methods each achieving monotonicity for the direction of choice c a man has the option to work with restricted support size that is the case when the number of components is known a priori in the latter case the em algorithm is used applications of mixture modelling in medical problems are discussed

mixtures of exponential distributions arbitrary nonparametric mixtures of exponential and weibull fixed shape distributions are considered as possible models for a lifetime distribution a characterization of such distributions is given by the well known characterization of laplace transforms the maximum likelihood estimate of the mixing distribution is investigated and found to be supported on a finite number of points it is shown to be unique and weakly convergent to the true mixing measure with probability one a practical algorithm for computing the maximum likelihood estimate is described its performance is briefly discussed and some illustrative examples given

triple goal estimates in two stage hierarchical models the beauty of the bayesian approach is its ability to structure complicated models inferential goals and analyses to take full advantage of it methods should be linked to an inferential goal via a loss function for example in the two stage compound sampling model the posterior means are optimal under squared error loss however they can perform poorly in estimating the histogram of the parameters or in ranking them triple goal estimates are motivated by the desire to have a set of estimates that produce good ranks a good parameter histogram and good co ordinate specific estimates no set of estimates can simultaneously optimize these three goals and we seek a set that strikes an effective trade off we evaluate and compare three candidate approaches the posterior means the constrained bayes estimates of louis and ghosh and a new approach that optimizes estimation of the histogram and the ranks mathematical and simulation based analyses support the superiority of the new approach and document its excellent performance for the three inferential goals

nonparametric mle under two closed capture recapture models with heterogeneity we conduct nonparametric maximum likelihood estimation under two common heterogeneous closed population capture recapture models our models specify mixture models as did previous researchers models which have a common generating distribution say f for the capture probabilities using lindsay and roeder s 1992 journal of the american statistical association 87 785 794 mixture model results and the em algorithm a nonparametric maximum likelihood estimator mle of f for any specified population size n is obtained then the nonparametric mle of the n f pair and thus for n is determined perhaps most importantly since our mle pair maximizes the likelihood under the entire nonparametric probability model it provides an excellent foundation for estimating properties of estimators conducting a goodness of fit test and performing a likelihood ratio test these are illustrated in the paper

semiparametric estimation in the rasch model and related exponential response models including a simple latent class model for item analysis abstract the rasch model for item analysis is an important member of the class of exponential response models in which the number of nuisance parameters increases with the number of subjects leading to the failure of the usual likelihood methodology both conditional likelihood methods and mixture model techniques have been used to circumvent these problems in this article we show that these seemingly unrelated analyses are in fact closely linked to each other despite dramatic structural differences between the classes of models implied by each approach we show that the finite mixture model for j dichotomous items having t latent classes gives the same estimates of item parameters as conditional likelihood on a set whose probability approaches one if t j 1 2 unconditional maximum likelihood estimators for the finite mixture model can be viewed as keifer wolfowitz estimators for the random effects version of the rasch model latent class versions of the model are especially attractive when t is

the geometry of mixture likelihoods a general theory in this paper certain fundamental properties of the maximum likelihood estimator of a mixing distribution are shown to be geometric properties of the likelihood set the existence support size likelihood equations and uniqueness of the estimator are revealed to be directly related to the properties of the convex hull of the likelihood set and the support hyperplanes of that hull it is shown using geometric techniques that the estimator exists under quite general conditions with a support size no larger than the number of distinct observations analysis of the convex dual of the likelihood set leads to a dual maximization problem a convergent algorithm is described the defining equations for the estimator are compared with the usual parametric likelihood equations for finite mixtures sufficient conditions for uniqueness are given part ii will deal with a special theory for exponential family mixtures

a simple nonparametric approach to estimating the distribution of random coefficients in structural models abstract we explore least squares and likelihood nonparametric mixtures estimators of the joint distribution of random coefficients in structural models the estimators fix a grid of heterogeneous parameters and estimate only the weights on the grid points an approach that is computationally attractive compared to alternative nonparametric estimators we provide conditions under which the estimated distribution function converges to the true distribution in the weak topology on the space of distributions we verify most of the consistency conditions for three discrete choice models we also derive the convergence rates of the least squares nonparametric mixtures estimator under additional restrictions we perform a monte carlo study on a dynamic programming model

binomial mixtures geometric estimation of the mixing distribution given a mixture of binomial distributions how do we estimate the unknown mixing distribution we build on earlier work of lindsay and further elucidate the geometry underlying this question exploring the approximating role played by cyclic polytopes convergence of a resulting maximum likelihood fitting algorithm is proved and numerical examples given problems over the lack of identifiability of the mixing distribution in part disappear

duration dependence and nonparametric heterogeneity a monte carlo study we examine the behaviour of the nonparametric maximum likelihood estimator npmle for a discrete duration model with unobserved heterogeneity and unknown duration dependence we find that a nonparametric specification of either the duration dependence or unobserved heterogeneity when the other feature of the hazard is known to be absent leads to estimators that are well behaved even in modestly sized samples in contrast there is a large and systematic bias in the parameters of these components when both are specified nonparametrically as well as a complementary bias in the coefficients on observed heterogeneity furthermore these biases diminish very gradually as sample size increases we find that a minor modification of the quasilikelihood that penalizes specifications with many points of support leads to a dramatic improvement

consistency and monte carlo simulation of a data driven version of smooth goodness of fit tests the data driven method of selecting the number of components in neyman s smooth test for uniformity introduced by ledwina is extended the resulting tests consist of a combination of schwarz s bayesian information criterion bic procedure and smooth tests the upper bound of the dimension of the exponential families in applying schwarz s rule is allowed to grow with the number of observations to infinity simulation results show that the data driven version of neyman s test performs very well for a wide range of alternatives and is competitive with other recently introduced data driven procedures it is shown that the data driven smooth tests are consistent against essentially all alternatives in proving consistency new results on schwarz s selection rule are derived which may be of independent interest

a general maximum likelihood analysis of variance components in generalized linear models this paper describes an em algorithm for nonparametric maximum likelihood ml estimation in generalized linear models with variance component structure the algorithm provides an alternative analysis to approximate mql and pql analyses mcgilchrist and aisbett 1991 biometrical journal 33 131 141 breslow and clayton 1993 journal of the american statistical association 88 9 25 mcgilchrist 1994 journal of the royal statistical society series b 56 61 69 goldstein 1995 multilevel statistical models and to gee analyses liang and zeger 1986 biometrika 73 13 22 the algorithm first given by hinde and wood 1987 in longitudinal data analysis 110 126 is a generalization of that for random effect models for overdispersion in generalized linear models described in aitkin 1996 statistics and computing 6 251 262 the algorithm is initially derived as a form of gaussian quadrature assuming a normal mixing distribution but with only slight variation it can be used for a completely unknown mixing distribution giving a straightforward method for the fully nonparametric ml estimation of this distribution this is of value because the ml estimates of the glm parameters can be sensitive to the specification of a parametric form for the mixing distribution the nonparametric analysis can be extended straightforwardly to general random parameter models with full npml estimation of the joint distribution of the random parameters this can produce substantial computational saving compared with full numerical integration over a specified parametric distribution for the random parameters a simple method is described for obtaining correct standard errors for parameter estimates when using the em algorithm several examples are discussed involving simple variance component and longitudinal models and small area estimation

penalized partially linear models using sparse representations with an application to fmri time series in this paper we consider modeling the nonparametric component in partially linear models plms using linear sparse representations e g wavelet expansions two types of representations are investigated namely orthogonal bases complete and redundant overcomplete expansions for bases we introduce a regularized estimator of the nonparametric part the important contribution here is that the nonparametric part can be parsimoniously estimated by choosing an appropriate penalty function for which the hard and soft thresholding estimators are special cases this allows us to represent in an effective manner a broad class of signals including stationary and or nonstationary signals and avoids excessive bias in estimating the parametric component we also give a fast estimation algorithm the method is then generalized to handle the case of overcomplete representations a large scale simulation study is conducted to illustrate the finite sample properties of the estimator the estimator is finally applied to real neurophysiological functional magnetic resonance imaging mri data sets that are suspected to contain both smooth and transient drift features

on fast computation of the non parametric maximum likelihood estimate of a mixing distribution summary a fast algorithm for computing the non parametric maximum likelihood estimate of a mixing distribution is presented at each iteration the algorithm adds new important points to the support set as guided by the gradient function updates all mixing proportions via a quadratically convergent method and discards redundant support points straightaway with its convergence being theoretically established numerical studies show that it is very fast and stable compared with several other algorithms that are available in the literature

data driven smooth tests when the hypothesis is composite in recent years several authors have recommended smooth tests for testing goodness of fit however the number of components in the smooth test statistic should be chosen well otherwise considerable loss of power may occur schwarz s selection rule provides one such good choice earlier results on simple null hypotheses are extended here to composite hypotheses which tend to be of more practical interest for general composite hypotheses consistency of the data driven smooth tests holds at essentially any alternative monte carlo experiments on testing exponentiality and normality show that the data driven version of neyman s test compares well to other even specialized tests over a wide range of alternatives

data driven smooth tests for composite hypotheses the classical problem of testing goodness of fit of a parametric family is reconsidered a new test for this problem is proposed and investigated the new test statistic is a combination of the smooth test statistic and schwarz s selection rule more precisely as the sample size increases an increasing family of exponential models describing departures from the null model is introduced and schwarz s selection rule is presented to select among them schwarz s rule provides the right dimension given by the data while the smooth test in the right dimension finishes the job theoretical properties of the selection rules are derived under null and alternative hypotheses they imply consistency of data driven smooth tests for composite hypotheses at essentially any alternative

likelihood inference for mixtures geometrical and other constructions of monotone step length algorithms summary this paper considers algorithms for the maximum likelihood estimator of the mixing distribution of a family of parametric densities specific interest is devoted to the monotonicity of the step length of the solution in 2 we introduce the idea of estimating the area above the second derivative curve and in theorem 1 we relate area overestimation to the monotonicity of the associated algorithm in 3 we apply theorem 1 to the monotonicity analysis of well known algorithms as well as to the construction of a new class of monotone step length choices which is particularly simple in the mixture setting numerical comparisons and refinements in keeping the monotonic step length from becoming too conservative are given theorem 3 characterizes conditions under which the newton raphson step is monotonic and if these conditions fail to hold states that the regula falsi step is monotonic

testing goodness of fit with multinomial data abstract several new test procedures are proposed for assessing the goodness of fit of a postulated multinomial distribution the new tests are neyman smooth type tests with orders selected adaptively from the data they are shown through fourier and large sample analyses to provide potential improvements over classical methods in terms of their ability to detect certain types of alternatives simulation results and a real example illustrate the finite sample validity of the large sample theory and the practical utility of the proposed methods

optimal calibration for multiple testing against local inhomogeneity in higher dimension based on two independent samples x 1 x m and x m 1 x n drawn from multivariate distributions with unknown lebesgue densities p and q respectively we propose an exact multiple test in order to identify simultaneously regions of significant deviations between p and q the construction is built from randomized nearest neighbor statistics it does not require any preliminary information about the multivariate densities such as compact support strict positivity or smoothness and shape properties the properly adjusted multiple testing procedure is shown to be sharp optimal for typical arrangements of the observation values which appear with probability close to one the proof relies on a new coupling bernstein type exponential inequality reflecting the non subgaussian tail behavior of a combinatorial process for power investigation of the proposed method a reparametrized minimax set up is introduced reducing the composite hypothesis p q to a simple one with the multivariate mixed density m n p 1 m n q as infinite dimensional nuisance parameter within this framework the test is shown to be spatially and sharply asymptotically adaptive with respect to uniform loss on isotropic holder classes the exact minimax risk asymptotics are obtained in terms of solutions of the optimal recovery

a continuation ratio random effects model for repeated ordinal responses this paper presents methods of analysis for ordinal repeated measures we use a generalization of the continuation ratio model with a random effect we handle frailty by using a normal distribution and also a non parametric distribution the methodology is readily implemented in existing software and flexible enough to treat large data sets with uncommon time measurements as well as different numbers of repeated measures for each individual the models are used to investigate how a group of explanatory variables influences the overall condition of patients treated for breast cancer

semiparametric estimation of normal mixture densities a semiparametric method for estimating densities of normal mean mixtures is presented this consistent data driven method of estimation is based on probability spacings the estimation technique involves iteratively fixing the standard deviation of the normal kernel that serves as a smoothing parameter and then maximizing a function of the probability spacings over all mixing distributions based on the distribution of uniform spacings a distribution free goodness of fit criterion is developed to guide the selection of the smoothing parameter the result is a set of consistent estimators indexed by a range of smoothing parameters empirical process results are used to prove consistency

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

smooth goodness of fit tests for composite hypothesis in hazard based models consider a counting process n t t j with compensator process a t t j where a t 0 t y s 0 s ds y t t j is an observable predictable process and 0 is an unknown hazard rate function a general procedure for extending neyman s smooth goodness of fit test for the composite null hypothesis h 0 0 c 0 r q is proposed and developed the extension is obtained by embedding c in the class a k whose members are of the form 0 exp t x r k where 1 k t is a vector of observable random processes satisfying certain regularity conditions the tests are based on quadratic forms of the statistic 0 t s dm s where m t n t 0 t y s 0 s ds and is a restricted maximum likelihood estimator of asymptotic properties of the test statistics are obtained under a sequence of local alternatives and the asymptotic local powers of the tests are examined the effect of estimating by is ascertained and the problem of choosing the process is discussed the procedure is illustrated by developing tests for testing that 0 belongs to i the class of constant hazard rates and ii the class of weibull hazard rates with particular emphasis on the random censorship model simulation results concerning the achieved levels and powers of the tests are presented and the procedures are applied to three data sets that have been considered in the literature

estimating the number of classes estimating the unknown number of classes in a population has numerous important applications in a poisson mixture model the problem is reduced to estimating the odds that a class is undetected in a sample the discontinuity of the odds prevents the existence of locally unbiased and informative estimators and restricts confidence intervals to be one sided confidence intervals for the number of classes are also necessarily one sided a sequence of lower bounds to the odds is developed and used to define pseudo maximum likelihood estimators for the number of classes

nonparametric estimation of a mixing density via the kernel method abstract editor s note this manuscript was first submitted in may 1996 and a first review with a request for revision was completed in september 1996 costas goutis lost his life in a scuba diving accident in july 1996 the associate editor jianqing fan graciously offered to carry out the required revisions this article presents a method for estimating the latent distribution of a mixture model the method is motivated by the standard kernel density estimation but instead of using an estimate based on the unobserved latent variables it takes the expectation with respect to their distribution conditional on the data the resulting estimator is continuous and hence appropriate when there is a strong belief in the continuity of the mixing distribution an asymptotic justification is presented and the associated computational problems are discussed the method is illustrated by an example of fission track analysis in which the density of the age of crystals is estimated

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

predicting the conditional probability of discovering a new class consider a population comprising disjoint classes an important problem arising from various fields is prediction of the random conditional probability of discovering a new class the asymptotic normality of the discovery probability is established in a poisson model where the number of individuals from each class is a poisson process with a class specific rate a new derivation is presented for the well known good toulmin predictor as a moment based estimator for the asymptotic limit of the discovery probability the good toulmin predictor is also shown to be a nonparametric empirical bayes estimator for the expectation of the discovery probability given the rates of the poisson processes and an approximation to an unbiased estimator only for the identifiable part of the expectation of the discovery probability in a multinomial model the properties of the moment based estimator are investigated so that confidence and prediction intervals can be constructed the good toulmin predictor and the discovery

global power functions of goodness of fit tests usseldorf it is shown that the global power function of any nonparametric test is flat on balls of alternatives except for alternatives coming from a finite dimensional subspace the present benchmark is here the upper one sided or two sided envelope power function every choice of a test fixes a priori a finite dimensional region with high power it turns out that also the level points are far away fromthe corresponding neym an pearson test level points except for a finite number of orthogonal directions of alternatives for certain submodels the result is independent of the underlying sample size in the last section the statistical consequences and special goodness of fit tests are discussed 1 introduction omnibus tests are commonly used if the specific structure of certain nonparametric alternatives is unknown among other justifications it turns out that they typically are consistent against fixed alternatives and n consistent under sequences of local alternatives of sample size n f or these reasons people often trust in goodness of fit tests and these are frequently applied to data of finite sample size on the other hand every asymptotic approximation should be understood as an approximation of the underlying finite sample case thus the statistician likes to distinguish and to compare the power of different competing tests the present paper offers a concept for the comparison and justification of different tests by their power functions and level points it is shown that under certain circumstances every test has a preference for a finite dimensional space of alternatives apart fromthis space the power function is almost flat on balls of alternatives there exists no test which pays equal attention to an infinite number of orthogonal alternatives the results do not only hold for asymptotic models but they also hold for concrete alternatives on the real line at finite sample size and their level points uniformly for the sample size the results are not surprising every statistician knows that it is impossible to separate an infinite sequence of different parameters simultaneously if only a finite number of observations is available the conclusions of the results are two fold 1 the statistician should analyze the goodness of fit tests of his computer package in order to get some knowledge and an impression about their preferences

convex optimization shape constraints compound decisions and empirical bayes rules estimation of mixture densities for the classical gaussian compound decision problem and their associated empirical bayes rules is considered from two new perspectives the first motivated by brown and greenshtein introduces a nonparametric maximum likelihood estimator of the mixture density subject to a monotonicity constraint on the resulting bayes rule the second motivated by jiang and zhang proposes a new approach to computing the kiefer wolfowitz nonparametric maximum likelihood estimator for mixtures in contrast to prior methods for these problems our new approaches are cast as convex optimization problems that can be efficiently solved by modern interior point methods in particular we show that the reformulation of the kiefer wolfowitz estimator as a convex optimization problem reduces the computational effort by several orders of magnitude for typical problems by comparison to prior em algorithm based methods and thus greatly expands the practical applicability of the resulting method

a maximum likelihood estimation method for random coefficient regression models summary a method for estimating the distribution of the parameters of a random coefficient regression model is proposed this distribution accounting for interindividual variability is assumed to lie in a wide class of probability distributions rather than in a given parametric class estimation is based on observations from a sample of individuals and likelihood is the estimation criterion experimental designs may be different among individuals allowing the method to apply to routinely collected data the problem has strong connections with the theory of optimum design of experiments conditions are given under which the problem has a unique solution which then corresponds to a discrete distribution a simple pharmacokinetic model involving two parameters is used as an example these parameters have a bimodal distribution as statistical specification moreover only one observation is available per individual thus the method applies even when the model is not identifiable

diagnosing misspecification of the random effects distribution in mixed models it is traditionally assumed that the random effects in mixed models follow a multivariate normal distribution making likelihood based inferences more feasible theoretically and computationally however this assumption does not necessarily hold in practice which may lead to biased and unreliable results we introduce a novel diagnostic test based on the so called gradient function proposed by verbeke and molenberghs 2013 to assess the random effects distribution we establish asymptotic properties of our test and show that under a correctly specified model the proposed test statistic converges to a weighted sum of independent chi squared random variables each with one degree of freedom the weights which are eigenvalues of a square matrix can be easily calculated we also develop a parametric bootstrap algorithm for small samples our strategy can be used to check the adequacy of any distribution for random effects in a wide class of mixed models including linear mixed models generalized linear mixed models and non linear mixed models with univariate as well as multivariate random effects both asymptotic and bootstrap proposals are evaluated via simulations and a real data analysis of a randomized multicenter study on toenail dermatophyte onychomycosis

adaptive goodness of fit tests based on signed ranks within the nonparametric regression model with unknown regression function l and independent symmetric errors a new multiscale signed rank statistic is introduced and a conditional multiple test of the simple hypothesis l 0 against a nonparametric alternative is proposed this test is distribution free and exact for finite samples even in the heteroscedastic case it adapts in a certain sense to the unknown smoothness of the regression function under the alternative and it is uniformly consistent against alternatives whose sup norm tends to zero at the fastest possible rate the test is shown to be asymptotically optimal in two senses it is rate optimal adaptive against holder classes furthermore its relative asymptotic efficiency with respect to an asymptotically minimax optimal test under sup norm loss is close to 1 in case of homoscedastic gaussian errors within a broad range of holder classes simultaneously

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

nonparametric em algorithms for estimating prior distributions consider a sequence of repetitive experiments e 1 e 2 e n each e i is represented by a pair of random vectors y i q i such that y i is observed q i is not observed and the conditional density of y i given q i is known the q i s are independent and identically distributed with common but unknown distribution function f q the problem is to estimate f given the data y 1 y 2 y n two nonparametric maximum likelihood methods based on the em algorithm are presented motivation for this problem comes from the analysis of the distribution of parameters in pharmacokinetic models based on population data

from image deblurring to optimal investments maximum likelihood solutions for positive linear inverse problems the problem of recovering an input signal from a blurred output in an input output system with linear distortion is ubiquitous in science and technology when the blurred output is not degraded by statistical noise the problem is entirely deterministic and amounts to a mathematical inversion of a linear system with positive parameters subject to positivity constraints on the solution we show that all such linear inverse problems with positivity restrictions lininpos problems for short can be interpreted as statistical estimation problems from incomplete data based on infinitely large samples and that maximum likelihood ml estimation and the em algorithm provide a straightforward method of solution for such problems

a semiparametric mixture approach to case control studies with errors in covariables abstract methods are devised for estimating the parameters of a prospective logistic model in a case control study with dichotomous response d that depends on a covariate x for a portion of the sample both the gold standard x and a surrogate covariate w are available however for the greater portion of the data only the surrogate covariate w is available by using a mixture model the relationship between the true covariate and the response can be modeled appropriately for both types of data the likelihood depends on the marginal distribution of x and the measurement error density w x d the latter is modeled parametrically based on the validation sample the marginal distribution of the true covariate is modeled using a nonparametric mixture distribution in this way we can improve the efficiency and reduce the bias of the parameter estimates the results also apply when there is no validation data provided the error distribution is known or estimated from an independent data source many of the

seasonal confounding in air pollution and health time series studies effect on air pollution effect estimates a major statistical challenge in air pollution and health time series studies is to adequately control for confounding effects of time varying covariates daily health outcome counts are most commonly analysed by poisson regression models adjusted for overdispersion with air pollution levels included as a linear predictor and smooth functions for calendar time and weather variables to adjust for time varying confounders various smoothers have been used so far but the optimal strategy for choosing smoothers and their degree of smoothing remains controversial in this work we evaluate the performance of various smoothers with different criteria for choosing the degree of smoothing in terms of bias and efficiency of the air pollution effect estimate in a simulation study the evaluated approaches were also applied to real mortality data from 22 european cities the simulation study imitated a multi city study data were generated from a fully parametric model model selection methods which optimize prediction may lead to increased biases in the air pollution effect estimate minimization of the absolute value of the sum of the partial autocorrelation function of the model s residuals pacf as a criterion to choose the degree of smoothness gave the smallest biases the penalized splines ps method with a large number of effective dfs e g 8 12 per year could be used as the basic relatively conservative analysis whereas the ps and natural splines in combination with pacf could be applied to provide a reasonable range of the effect estimate

a data driven smooth test of symmetry in this paper we propose a data driven smooth test of symmetry we first transform the raw data via the probability integral transformation according to a symmetrized empirical distribution and show that under the null hypothesis of symmetry the transformed data has a limiting uniform distribution reducing testing for symmetry to testing for uniformity employing neyman s smooth test of uniformity we show that only odd ordered orthogonal moments of the transformed data are required in constructing the test statistic we present a standardized smooth test that is distribution free asymptotically and derive the asymptotic behavior of the test and establish its consistency extension to dependent data case is discussed we investigate the finite sample performance of the proposed tests on both homogeneous and mixed distributions with unobserved heterogeneity an empirical application on testing symmetry of wage adjustment process based on heterogeneous wage contracts with different durations is provided

testing for lack of fit in inverse regression with applications to biophotonic imaging summary we propose two test statistics for use in inverse regression problems y d k c where k is a given linear operator which cannot be continuously inverted thus only noisy indirect observationsy for the function are available both test statistics have a counterpart in classical hypothesis testing where they are called the order selection test and the data driven neyman smooth test we also introduce two model selection criteria which extend the classical akaike information criterion and bayes information criterion to inverse regression problems in a simulation study we show that the inverse order selection and neyman smooth tests outperform their direct counterparts in many cases the theory is motivated by data arising in confocal fluorescence microscopy here images are observed with blurring modelled as convolution and stochastic error at subsequent times the aim is then to reduce the signal to noise ratio by averaging over the distinct images in this context it is relevant to decide whether the images are still equal or have changed by outside influences such as moving of the object table

recent developments in computer assisted analysis of mixtures this paper reviews recent developments in the area of computer assisted analysis of mixture distributions c a man given a biometric situation of interest in which under homogeneity assumptions a certain parametric density occurs such as the poisson the binomial the geometric the normal and so forth then it is argued that this situation can easily be enlarged to allow a variation of the scalar parameter in the population this situation is called unobserved heterogeneity this naturally leads to a specific form of nonparametric mixture distribution that can then be assumed to be the standard model in the biometric application of interest since it also incorporates the homogeneous situations as a special case besides developments in theory and algorithms the work focuses on developments in biometric applications such as meta analysis fertility studies estimation of prevalence under clustering and estimation of the distribution function of survival time under interval censoring the approach is nonparametric for the mixing distribution including leaving the number of components subpopulations of the mixing distribution unknown

an automatic portmanteau test for serial correlation this article introduces a data driven box pierce test for serial correlation the proposed test is very attractive compared to the existing ones in particular implementation of this test is extremely simple for two reasons rst the researcher does not need to specify the order of the autocorrelation tested since the test automatically chooses this number second its asymptotic null distribution is chi square with one degree of freedom so there is no need of using a bootstrap procedure to estimate the critical values in addition the test is consistent and robust to the presence of conditional heteroskedasticity of unknown form finally the proposed test presents higher power in simulations than the existing ones for models commonly employed in empirical nance

towards data driven selection of a penalty function for data driven neyman tests the data driven neyman statistic consists of two elements a score statistic in a finite dimensional submodel and a selection rule to determine the best fitted submodel for instance schwarz bic and akaike aic rules are often applied in such constructions for moderate sample sizes aic is sensitive in detecting complex models while bic works well for relatively simple structures when the sample size is moderate the choice of selection rule for determining a best fitted model from a number of models has a substantial influence on the power of the related data driven neyman test this paper proposes a new solution in which the type of penalty aic or bic is chosen on the basis of the data the resulting refined data driven test combines the advantages of these two selection rules

nonparametric estimation of convex models via mixtures we present a general approach to estimating probability measures constrained to lie in a convex set we represent constrained measures as mixtures of simple known extreme measures and so the problem of estimating a constrained measure becomes one of estimating an unconstrained mixing measure convex constraints arise in many modeling situations such as estimation of the mean and estimation under stochastic ordering constraints we describe mixture representation techniques for these and other situations and discuss applications to maximum likelihood and bayesian estimation

nonparametric maximum likelihood methods for binary response models with random coefficients the venerable method of maximum likelihood has found numerous recent applications in nonparametric estimation of regression and shape constrained densities for mixture models the nonparametric max

data driven rank tests for independence we introduce new rank tests for testing independence the new testing procedures are sensitive not only for grade linear correlation but also for grade correlations of higher order polynomials the number of polynomials involved is determined by the data model selection is combined with application of the score test in the selected model whereas well known tests as spearman s test or hoeffding s test may completely break down for alternatives that are dependent but have low grade linear correlation the new tests have greater power stability monte carlo results clearly show this behavior theoretical support is obtained by proving consistency of the new tests

optimal stopping of sequential size dependent search in many areas of application one searches within finite populations for items of interest where the probability of sampling an item is proportional to a random size attribute from an i i d superpopulation of attributes which may or may not be observable upon discovery here we treat the problem of asymptotically optimal stopping rules for size dependent searches of this type as the size of the underlying population grows where the loss function includes an asymptotically smooth time dependent cost a constant cost per item sampled and a cost per undiscovered item which may depend on the size attribute of the undiscovered item under some regularity and convexity conditions related to the asymptotic expected loss we characterize asymptotically optimal rules even when the initial population size and the distribution of size attributes are unknown we direct especial attention to applications in software reliability where the items of interest are software faults bugs in this setting the size attributes will not be observable when faults are found and in addition our search model allows new bugs to be introduced into the software when faults are detected imperfect debugging our results extend those of dalal and mallows and kramer and starr and are illustrated in the perfect debugging case on a previously analyzed dataset of musa

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

sensitivity of dynamical systems to banach space parameters we consider general nonlinear dynamical systems in a banach space with dependence on parameters in a second banach space an abstract theoretical framework for sensitivity equations is developed an application to measure dependent delay differential systems arising in a class of hiv models is presented

kernel smoothing in partial linear models on considere deux methodes d estimation l une reliee aux splines de lissage partiels l autre motivee par une analyse de residus partielle

optimal design in random effects regression models summary an approach is proposed to optimal design of experiments for estimating randomeffects regression models the population designs are defined by the number of subjects and the individual designs to be performed cost functions associated with individual designs are incorporated for a given maximal cost an algorithm is proposed for finding the statistical population design that maximises the determinant of the fisher information matrix of the population parameters the fisher information matrix is formulated for linear models and normal distributions the approach is applied to the design of an optimal experiment in toxicokinetics using a first order linearisation of the model several cost functions and designs of various orders are studied an example illustrates the optimal population designs and the increased efficiency of some optimal designs over more standard designs

structural tests in additive regression we consider the component analysis problem for a regression model with an additive structure the problem is to test whether some of the additive components are of polynomial structure e g linear without specifying the structure of the remaining components a particular case is the problem of selecting the significant covariates the method that we present is based on the wavelet transform using the haar basis which allows for applications under mild conditions on the design and smoothness of the regression function the results demonstrate that each component of the model can be tested with the rate corresponding to the case if all of the remaining components were known the proposed procedure is also computationally straightforward simulation results and a real data example about female labor supply demonstrate the test s good performance

data driven rank tests for classes of tail alternatives tail alternatives describe the frequent occurrence of a non constant shift in the two sample problem with a shift function increasing in the tail the classes of shift functions can be built up using legendre polynomials it is important to rightly choose the number of polynomials involved here this choice is based on the data using a modification of schwarz s selection rule given the data driven choice of the model appropriate rank tests are applied simulations show that the new data driven rank tests work very well while other tests for detecting shift alternatives as wilcoxon s test may completely break down for important classes of tail alternatives the new tests have high and stable power the new tests have also higher power than data driven rank tests for the unconstrained two sample problem theoretical support is obtained by proving consistency of the new tests against very large classes of alternatives including all common tail alternatives a simple but accurate approximation of the null distribution makes application of the new tests easy

diagnostic analysis and computational strategies for estimating discrete time duration models a monte carlo study abstract this paper uses monte carlo analysis to study important and contentious issues in estimating single spell discrete time duration models we find simulated annealing dominates gradient methods for recovering true models we recommend a partially flexible step function for duration dependence combined with likelihood ratio tests for determining support points of unobserved heterogeneity we find that ignoring time changing features of explanatory variables introduces substantial biases in model coefficient and average partial effect estimates these biases do not diminish as sample size increases

nonignorable models for intermittently missing categorical longitudinal responses a class of nonignorable models is presented for handling nonmonotone missingness in categorical longitudinal responses this class of models includes the traditional selection models and shared parameter models this allows us to perform a broader than usual sensitivity analysis in particular instead of considering variations to a chosen nonignorable model we study sensitivity between different missing data frameworks an appealing feature of the developed class is that parameters with a marginal interpretation are obtained while algebraically simple models are considered specifically marginalized mixed effects models heagerty 1999 biometrics 55 688 698 are used for the longitudinal process that model separately the marginal mean and the correlation structure for the correlation structure random effects are introduced and their distribution is modeled either parametrically or non parametrically to avoid potential misspecifications

bayesian model selection in finite mixtures by marginal density decompositions we consider the problem of estimating the number of components d and the unknown mixing distribution in a finite mixture model in which d is bounded by some fixed finite number n our approach relies on the use of a prior over the space of mixing distributions with at most n components by decomposing the resulting marginal density under this prior we discover a weighted bayes factor method for consistently estimating d that can be implemented by an iid generalized weighted chinese restaurant gwcr monte carlo algorithm we also discuss a gibbs sampling method the blocked gibbs sampler for estimating d and also the mixing distribution we show that our resulting posterior is consistent and achieves the frequentist optimal op n 1 4 rate of estimation we compare the performance of the new gwcr model selection procedure with that of the akaike information criterion and the bayes information criterion implemented through an em algorithm applications of our methods to five real datasets and simulation

minimum volume ellipsoids containing compact sets application to parameter bounding abstract the problem of finding the minimum volume ellipsoid containing a compact set is shown to be equivalent to the determination of an optimal distribution of weights over several equivalent conditions for optimality of this distribution are obtained and used to construct algorithms guaranteed to converge to the optimum these algorithms can be used to approximate the posterior feasible set for the parameters of a model in the bounded error context linear and nonlinear illustrative examples are treated

efficient greedy learning of gaussian mixture models this article concerns the greedy learning of gaussian mixtures in the greedy approach mixture components are inserted into the mixture one after the other we propose a heuristic for searching for the optimal component to insert in a randomized manner a set of candidate new components is generated for each of these candidates we find the locally optimal new component and insert it into the existing mixture the resulting algorithm resolves the sensitivity to initialization of state of the art methods like expectation maximization and has running time linear in the number of data points and quadratic in the final number of mixture components due to its greedy nature the algorithm can be particularly useful when the optimal number of mixture components is unknown experimental results comparing the proposed algorithm to other methods on density estimation and texture segmentation are provided