on a class of m out of n bootstrap confidence intervals it is widely known that bootstrap failure can often be remedied by using a technique known as the m out of n bootstrap by which a smaller number m say of observations are resampled from the original sample of size n in successful cases of the bootstrap the m out of n bootstrap is often deemed unnecessary we show that the problem of constructing nonparametric confidence intervals is an exceptional case by considering a new class of m out of n bootstrap confidence limits we develop a computationally efficient approach based on the double bootstrap to construct the optimal m out of n bootstrap intervals we show that the optimal intervals have a coverage accuracy which is comparable with that of the classical double bootstrap intervals and we conduct a simulation study to examine their performance the results are in general very encouraging alternative approaches which yield even higher order accuracy are also discussed

multilevel time series models with applications to repeated measures data the analysis of repeated measures data can be conducted efficiently using a two level random coefficients model a standard assumption is that the within individual level 1 residuals are uncorrelated in some cases especially where measurements are made close together in time this may not be reasonable and this additional correlation structure should also be modelled a time series model for such data is proposed which consists of a standard multilevel model for repeated measures data augmented by an autocorrelation model for the level 1 residuals first and second order autoregressive models are considered in detail together with a seasonal component both discrete and continuous time are considered and it is shown how the autocorrelation parameters can themselves be structured in terms of further explanatory variables the models are fitted to a data set consisting of repeated height measurements on children

bayesian computation for nonhomogeneous poisson processes in software reliability abstract a unified approach to the nonhomogeneous poisson process in software reliability models is given this approach models the epochs of failures according to a general order statistics model or to a record value statistics model their corresponding point processes can be related to the nonhomogeneous poisson processes for example the goel okumoto the musa okumoto the duane and the cox lewis processes bayesian inference for the nonhomogeneous poisson processes is studied the gibbs sampling approach sometimes with data augmentation and with the metropolis algorithm is used to compute the bayes estimates of credible sets mean time between failures and the current system reliability model selection based on a predictive likelihood is studied a numerical example with a real software failure data set is given

confidence sets for a multivariate distribution on etudie une version multivariable alternative de la bande de confiance unidimensionnelle de kolmogorov smirnov qui preserve l invariance affine

some asymptotic theory for the bootstrap efron s bootstrap method of distribution approximation is shown to be asymptotically valid in a large number of situations including t statistics the empirical and quantile processes and von mises functionals some counter examples are also given to show that the approximation does not always succeed

tests for an increasing trend in the intensity of a poisson process a power study abstract this article concerns a comparison of several tests for testing the hypothesis of a constant intensity against the alternative of an increasing intensity function in a nonhomogeneous poisson process nhpp the study includes the well known laplace test statistic the most powerful test for the shape parameter in a poisson process with weibull intensity the likelihood ratio test against arbitrary nhpp alternatives two nonparametric tests for trends based on kendall s tau and spearman s rho and a test based on an f statistic the powers of the tests are determined by monte carlo simulation against alternatives that are increasing at an exponential rate a power rate weibull intensity and a logarithmic rate alternatives that are step functions with one jump are also considered in a few cases the exact powers are also obtained analytically

confidence interval procedures for the weibull process with applications to reliability growth the weibull process a nonhomogeneous poisson process with intensity r t t 1 is considered as a stochastic model for the duane 1964 reliability growth postulate under this model the mean time between failure mtbf for the system at time t is given by m t r t 1 small sample and asymptotic confidence intervals on m t are discussed for failure and time truncated testing tabled values to compute the confidence intervals and numercial examples illustrating these procedures are presented

the jacobian conjecture linear triangularization for homogeneous polynomial maps in dimension three let k be a field of characteristic zero and f k3 k3 a polynomial map of the form f x h where h is homogeneous of degree d 2 we show that the jacobian conjecture is true for such mappings more precisely we show that if jh is nilpotent there exists an invertible linear map t such that t 1ht 0 h2 x1 h3 x1 x2 where the hi are homogeneous of degree d as a consequence of this result we show that all generalized druzkowski mappings f x h x1 l1d xn lnd where li are linear forms for all i and d 2 are linearly triangularizable if jh is nilpotent and rkjh 3

iterative simulation methods abstract the standard methods of generating sample from univariate distributions often become hopelessly inefficient when applied to realizations of stochastic processes iterative methods are not widely known amongst statisticians but some are standard practice in statistical physics and chemistry the methods are surveyed and compared with particular reference to their convergence properties

the jacobian conjecture reduction of degree and formal expansion of the inverse introduction i the jacobian conjecture 1 statement of the jacobian problem first observations 2 some history of the jacobian conjecture 3 faulty proofs 4 the use of stabilization and of formal methods ii the reduction theorem 1 notation 2 statement of the reduction theorem 3 reduction to degree 3 4 proof of the reduction theorem 5 r linearization and unipotent reduction 6 nilpotent rank 1 jacobians iii the formal inverse 1 notation 2 abhyankar s inversion formula 3 the terms gj 4 the tree expansion g lt a t lfptf 5 calculations references

estimation in generalized linear models with random effects summary a conceptually very simple but general algorithm for the estimation of the fixed effects random effects and components of dispersion in generalized linear models with random effects is proposed conditions are described under which the algorithm yields approximate maximum likelihood or quasi maximum likelihood estimates of the fixed effects and dispersion components and approximate empirical bayes estimates of the random effects the algorithm is applied to two data sets to illustrate the estimation of components of dispersion and the modelling of overdispersion

a scalable bootstrap for massive data summary the bootstrap provides a simple and powerful means of assessing the quality of estimators however in settings involving large data sets which are increasingly prevalent the calculation of bootstrap based quantities can be prohibitively demanding computationally although variants such as subsampling and the m out of n bootstrap can be used in principle to reduce the cost of bootstrap computations these methods are generally not robust to specification of tuning parameters such as the number of subsampled data points and they often require knowledge of the estimator s convergence rate in contrast with the bootstrap as an alternative we introduce the bag of little bootstraps blb which is a new procedure which incorporates features of both the bootstrap and subsampling to yield a robust computationally efficient means of assessing the quality of estimators the blb is well suited to modern parallel and distributed computing architectures and furthermore retains the generic applicability and statistical efficiency of the bootstrap we demonstrate the blb s favourable statistical performance via a theoretical analysis elucidating the procedure s properties as well as a simulation study comparing the blb with the bootstrap the m out of n bootstrap and subsampling in addition we present results from a large scale distributed implementation of the blb demonstrating its computational superiority on massive data a method for adaptively selecting the blb s tuning parameters an empirical study applying the blb to several real data sets and an extension of the blb to time series data

calibrating prediction regions abstract suppose that the variable x to be predicted and the learning sample yn that was observed have a joint distribution which depends on an unknown parameter the parameter can be finite or infinite dimensional a prediction region dn for x is a random set depending on yn that contains x with prescribed probability this article studies methods for controlling simultaneously the conditional coverage proability of dn given yn and the overall unconditional coverage probability of dn the basic construction yields a prediction region dn which has the following properties in regular models both the conditional and overall coverage probabilities of dn converge to as the size n of the learning sample increases the convergence of the former is in probability moreover the asymptotic distribution of the conditional coverage probability about is typically normal and the overall coverage probability tends to at rate n 1 can one reduce the dispersion of the conditional coverage pr

an introduction to stochastic modeling stochastic modeling probability review the major discrete distributions rtant continuous distributions some elementary exercises useful functions integrals and sums conditional probability and conditional expectation the discrete case the dice game craps random sums conditioning on a continuous random variable markov chains introduction definitions transition probability matrices of a markov chain some markov chain models first step analysis some special markov chains functionals of random walks and success runs another look at first step analysis the long run behavior of markov chains regular transition probability matrices examples the classification of states the basic limit theorem of markov chains reducible markov chains sequential decisions and markov chains poisson processes the poisson distribution and the poisson processes the law of rare events distributions associated with the poisson process the uniform distribution and poisson processes spatial poisson processes compound and marked poisson processes continuous time markov chains pure birth processes ptire death processes birth and death processes the limiting behavior of birth and death processes birth and death processes with absorbing states finite state continuous time markov chains set valued processes renewal phenomena definition of a renewal process and related concepts some examples of renewal processes the poisson process viewed as a renewal process the asymptotic 3ehavior as renewal process

approximate inference in generalized linear mixed models statistical approaches to overdispersion correlated errors shrinkage estimation and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model glmm given an unobserved vector of random effects observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function known prior weights and a scale factor the random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components for problems involving time series spatial aggregation and smoothing the dispersion may be specified in terms of a rank deficient inverse covariance matrix approximation of the marginal quasi likelihood using laplace s method leads eventually to estimating equations based on penalized quasilikelihood or pql for the mean parameters and pseudo likelihood for the variances im

the linear dependence problem for power linear maps abstract let bl l 1 k be m nl complex matrices and let x l c n l l 1 k be complex vector variables we show that the components of the map h b 1 x 1 d 1 b k x k d k are linearly dependent over c if and only if det b 1 b 1 d 1 b k b k d k 0 where means the hadamard product x and x d denote the conjugate transpose and the dth hadamard power of a matrix x respectively connections are established between the homogenous dependence problem hdp n d which arises in the study of the jacobian conjecture and the dependence problem for power linear maps pldp n d an algorithm is given to compute counterexamples to pldp n d from those to hdp n d and counterexamples to pldp n 3 are obtained for all n 67

correlated binary regression with covariates specific to each binary observation regression methods are considered for the analysis of correlated binary data when each binary observation may have its own covariates it is argued that binary response models that condition on some or all binary responses in a given block are useful for studying certain types of dependencies but not for the estimation of marginal response probabilities or pairwise correlations fully parametric approaches to these latter problems appear to be unduly complicated except in such special cases as the analysis of paired binary data hence a generalized estimating equation approach is advocated for inference on response probabilities and correlations illustrations involving both small and large block sizes are provided

dynamic generalized linear models and bayesian forecasting abstract dynamic bayesian models are developed for application in nonlinear non normal time series and regression problems providing dynamic extensions of standard generalized linear models a key feature of the analysis is the use of conjugate prior and posterior distributions for the exponential family parameters this leads to the calculation of closed standard form predictive distributions for forecasting and model criticism the structure of the models depends on the time evolution of underlying state variables and the feedback of observational information to these variables is achieved using linear bayesian prediction methods data analytic aspects of the models concerning scale parameters and outliers are discussed and some applications are provided dynamic bayesian models are developed for application in nonlinear non normal time series and regression problems providing dynamic extensions of standard generalized linear models a key feature of the analysis is the use of conjugate prior and

richardson extrapolation and the bootstrap abstract simulation methods particularly efron s 1979 bootstrap are being applied more and more frequently in statistical inference given data x 1 xn distributed according to p which belongs to a hypothesized model p the basic goal is to estimate the distribution l p of a function tn x 1 xn p the bootstrap presupposes the existence of an estimate and involves estimating l p by the distribution l n of where x 1 xn is distributed according to the method is of particular interest when ln though known in principle can realistically only be computed by simulation such computation can be expensive if n is large and tn is complex e g see the multivariate goodness of fit tests of beran and millar 1986 even when bootstrap application to a single data set is not excessively expensive monte carlo studies of the bootstrap are another matter we propose a method based on the classical ideas of richardson extrapolation for reducing the computational cost inherent in bootstra

injectivity and almost global asymptotic stability of hurwitz vector fields abstract we present in dimension n 3 a survey of examples to the jacobian conjecture the weak markus yamabe conjecture furthermore we show and construct new examples of vector fields where the origin is almost globally asymptotic stable by using the novel concept of density functions introduced by rantzer

quadratic homogeneous keller maps of rank two abstract let h be a quadratic homogeneous polynomial map of dimension n over an infinite field in which 2 is invertible such that its jacobian jh is nilpotent meisters and olech have shown that jh is strongly nilpotent if n 4 they also proved that it is not true when n 5 we show that if rank j h 2 and n arbitrary then jh is strongly nilpotent we also give examples to show that this is no longer true for any rank and dimension as long as the rank is greater than 2

some remarks on the jacobian conjecture and druzkowski mappings abstract in this paper we first show that the jacobian conjecture is true for non homogeneous power linear mappings under some conditions secondly we prove an equivalent statement about the jacobian conjecture in dimension r 1 and give some partial results for r 2 finally for a homogeneous power linear keller map f x h of degree d 2 we give the inverse polynomial map under the condition that j h 3 0 we shall show that deg f 1 d k if k 2 and j h k 1 0 but also give an example with d 2 and j h 4 0 such that deg f 1 d 3

some simple robust methods for the analysis of recurrent events nelson discussed a method of estimating the cumulative mean function for identically distributed processes of recurrent events we show that a similar approach can be used with more general models including regression the key idea is to use point estimates based on poisson models and to develop robust variance estimates that are valid more generally the methods are illustrated on reliability and warranty data

generalized linear models with random effects a gibbs sampling approach abstract generalized linear models have unified the approach to regression for a wide variety of discrete continuous and censored response variables that can be assumed to be independent across experimental units in applications such as longitudinal studies genetic studies of families and survey sampling observations may be obtained in clusters responses from the same cluster cannot be assumed to be independent with linear models correlation has been effectively modeled by assuming there are cluster specific random effects that derive from an underlying mixing distribution extensions of generalized linear models to include random effects has thus far been hampered by the need for numerical integration to evaluate likelihoods in this article we cast the generalized linear random effects model in a bayesian framework and use a monte carlo method the gibbs sampler to overcome the current computational limitations the resulting algorithm is flexible to easily accommodate changes in the number

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

consistency and asymptotic normality of the maximum likelihood estimator in generalized linear models generalized linear models are used for regression analysis in a number of cases including categorical responses where the classical assumptions are violated the statistical analysis of such models is based on the asymptotic properties of the maximum likelihood estimator we present mild general conditions which respectively assure weak or strong consistency or asymptotic normality most of the previous work has been concerned with natural link functions in this case our normality condition though obtained by a different approach is closely related to a condition of haberman 1977a examples show how the general conditions reduce to weak requirements for special exponential families further for regressors with a compact range sufficient conditions are given which do not involve the unknown parameter and are therefore easy to check in practice responses with a bounded range e g categorical responses and stochastic regressors also are treated

bayesian computations for a class of reliability growth models in this article we consider the development and analysis of both attribute and variable data reliability growth models from a bayesian perspective we begin with an overview of a bayesian attribute data reliability growth model and illustrate how this model can be extended to cover the variable data growth models as well bayesian analysis of these models requires inference over ordered regions and even though closed form results for posterior quantities can be obtained in the attribute data case variable data models prove difficult in general when the number of test stages gets large computations become burdensome and more importantly the results may become inaccurate due to computational difficulties we illustrate how the difficulties in the posterior and predictive analyses can be overcome using markov chain monte carlo methods we illustrate the implementation of the proposed models by using examples from both attribute and variable reliability growth data

a finite element splitting extrapolation for second order hyperbolic equations splitting extrapolation is an efficient technique for solving large scale scientific and engineering problems in parallel this article discusses a finite element splitting extrapolation for second order hyperbolic equations with time dependent coefficients this method possesses a higher degree of parallelism less computational complexity and more flexibility than richardson extrapolation while achieving the same accuracy by means of domain decomposition and isoparametric mapping some grid parameters are chosen according to the problem the multiparameter asymptotic expansion of the d quadratic finite element error is also established the splitting extrapolation formulas are developed from this expansion an approximation with higher accuracy on a globally fine grid can be computed by solving a set of smaller discrete subproblems on different coarser grids in parallel some a posteriori error estimates are also provided numerical examples show that this method is efficient for solving discontinuous problems and nonlinear hyperbolic equations

bayesian software reliability models based on martingale processes this article proposes new bayesian models for software reliability based on a piecewise constant failure rate a martingale process prior is assumed on the failure rate three different hyperprior models on the martingale process are considered the first two models assume that the conditional variance is dependent on the mean whereas the third model assumes constant conditional variance markov chain sampling based posterior analysis and prequential predictions are developed for these models and are illustrated in a software failure dataset in addition model comparison is studied via the bayes factor criterion general techniques are described for estimating the marginal likelihood of the proposed models as well as of many existing software reliability models and these are illustrated in two datasets

testing for monotone trend in recurrent event processes tests for the presence of monotone trends in recurrent event processes have been proposed and studied by many authors however a general concept of trend is elusive and the dependence of the behavior of tests on the assumed definitions of no trend and trend and on the observation periods for the processes has been largely ignored we discuss these issues and study them through analytical results and simulation studies the article also presents robust tests for trend across multiple processes extends them to deal with interval censored event times and compares them and other well known trend tests with respect to the issues mentioned

robust estimation in heteroscedastic linear models we consider a heteroscedastic linear model in which the variances are given by a parametric function of the mean responses and a parameter theta we propose robust estimates for the regression parameter beta and show that as long as a reasonable starting estimate of theta is available our estimates of beta are asymptotically equivalent to the natural estimate obtained with known variances a particular method for estimating theta is proposed and shown by monte carlo to work quite well especially in power and exponential models for the variances we also briefly discuss a feedback estimate of beta

models for longitudinal data a generalized estimating equation approach this article discusses extensions of generalized linear models for the analysis of longitudinal data two approaches are considered subject specific ss models in which heterogeneity in regression parameters is explicitly modelled and population averaged pa models in which the aggregate response for the population is the focus we use a generalized estimating equation approach to fit both classes of models for discrete and continuous outcomes when the subject specific parameters are assumed to follow a gaussian distribution simple relationships between the pa and ss parameters are available the methods are illustrated with an analysis of data on mother s smoking and children s respiratory disease

modelling memory processes and internet response times weibull or power law abstract the weibull distribution is proposed as a model for response times theoretical support is offered by classical results for extreme value distributions fits of the weibull distribution to response time data in different contexts show that this distribution and the exponential distribution on small time scales perform better than the often suggested power law and logarithmic function this study suggests that the power law can be viewed as an approximation at neural level for the aggregate strength of superposed memory traces that have different decay rates in distinct parts of the brain as we predict this view does not find support at the level of induced response processes the distinction between underlying and induced processes might also be considered in other fields such as engineering biology and physics

orthogonal decomposition of finite population statistics and its applications to distributional asymptotics we study orthogonal decomposition of symmetric statistics based on samples drawn without replacement from finite populations several applications to finite population statistics are given we establish one term edge worth expansions for general asymptotically normal symmetric statistics prove an efron stein inequality and the consistency of the jackknife estimator of variance our expansions provide second order a s approximations to wu s jackknife histogram

on sample reuse methods for dependent data we suggest a sample reuse method for dependent data based on a cross between the block bootstrap and richardson extrapolation instead of simulating a same size resample by resampling blocks and placing them end to end it analyses the blocks directly and employs a variant of richardson extrapolation to adjust for block size a simple empirical rule also based on richardson extrapolation is suggested for empirically selecting the block size performance in the contexts of distribution and bias estimation is discussed via theoretical analysis and numerical simulation

estimation of current reliability in a duane based reliability growth model in this article i investigate the statistical inference concerning the current field stage reliability of a reliability growth model the model assuming a step intensity structure evolves from the physical consideration of the duane learning curve property and incorporates the effect of a testanalyze and fix program that is typically undertaken in a developmental testing program both exact and large sample distributional results are derived for the maximum likelihood and the least squares estimators of the current intensity under the assumption that the step intensity model represents the reality i provide an assessment of the extent of misspecification when the widely used power law process model is fit to the failure data of a system experiencing recurrent failures extensive simulation results are carried out to supplement the theoretical findings an illustration with a dataset is provided as a demonstration of the application of the inference results

geometrizing rates of convergence ii consider estimating a functional t f of an unknown distribution f in mathbf f from data x 1 cdots x n i i d f let omega varepsilon denote the modulus of continuity of the functional t over mathbf f computed with respect to hellinger distance for well behaved loss functions l t we show that inf t n sup mathbf f e fl t n t f is equivalent to l omega n 1 2 to within constants whenever t is linear and mathbf f is convex the same conclusion holds in three nonlinear cases estimating the rate of decay of a density estimating the mode and robust nonparametric regression we study the difficulty of testing between the composite infinite dimensional hypotheses h 0 t f leq t and h 1 t f geq t delta our results hold in the cases studied because the difficulty of the full infinite dimensional composite testing problem is comparable to the difficulty of the hardest simple two point testing subproblem

nonlinear mixed effects models for repeated measures data we propose a general nonlinear mixed effects model for repeated measures data and define estimators for its parameters the proposed estimators are a natural combination of least squares estimators for nonlinear fixed effects models and maximum likelihood or restricted maximum likelihood estimators for linear mixed effects models we implement newton raphson estimation using previously developed computational methods for nonlinear fixed effects models and for linear mixed effects models two examples are presented and the connections between this work and recent work on generalized linear mixed effects models are discussed

power linear keller maps with ditto triangularizations abstract we show that power linear keller maps f x 1 a 1 x d x 2 a 2 x d x n a n x d are linearly triangularizable if 1 rk a 2 or 2 cork a 2 and d 3 or 3 cork a 3 d 5 and the diagonal of a is nonzero furthermore we show that the triangularizations can be chosen power linear as well