on the estimation of the large deviations spectrum we propose an estimation algorithm for large deviations spectra of measures and functions the algorithm converges for natural examples of multifractals

lacunarity and multifractal analysis of the large dla mass distribution abstract we show the methodology used to analyze fractal and mass multifractal properties of very large diffusion limited aggregation dla clusters with a maximum of 10 9 particles for 2d aggregates and 10 8 particles for 3d clusters to support our main result the scaling behavior obtained by our experimental results corresponds to the expected performance of monofractal objects in order to estimate lacunarity measures for large dla clusters we develop a variant of the gliding box algorithm which reduces the computer time needed to obtain experimental results we show how our mass multifractal data have a tendency to present monofractal behavior for the mass distribution of the cases presented in this paper in the limit of very large clusters lacunarity analysis shows provided we study small clusters mass distributions data which might be interpreted as two different values of fractal dimensions while the cluster grows however this effect tends to vanish when the cluster size increases further in such a way that monofractality is achieved the outcomes of this paper lead us to conclude that the previously reported mass multifractality behavior vicsek et al 1990 13 detected for dla clusters is a consequence of finite size effects and floating point precision limitations and not an intrinsic feature of the phenomena since the scaling behavior of our dla clusters space corresponds to monofractal objects being this situation remarkably noticeable in the limit of very large clusters

orbits and phase transitions in the multifractal spectrum we consider the one dimensional classical ising model in a symmetric dichotomous random field the problem is reduced to a random iterated function system rifs for an effective field the dq spectrum of the invariant measure of this effective field exhibits a sharp drop of all dq with q 0 at some critical strength of the random field we introduce the concept of orbits which naturally group the points of the support of the invariant measure we then show that the pointwise dimension at all points of an orbit has the same value and calculate it for a class of periodic orbits and their so called offshoots as well as for generic orbits in the non overlapping case the sharp drop in the dq spectrum is analytically explained by a drastic change of the scaling properties of the measure near the points of a certain periodic orbit at a critical strength of the random field which is explicitly given a similar drastic change near the points of a special family of periodic orbits explains a second hitherto unnoticed transition in the dq spectrum as it turns out a decisive role in this mechanism is played by a specific offshoot we furthermore give rigorous upper and or lower bounds on all dq in a wide parameter range in most cases the numerically obtained dq coincide with either the upper or the lower bound the results in this paper are relevant for the understanding of rifss in the case of moderate overlap in which periodic orbits with weak singularity can play a decisive role

an approximate generalized linear model with random effects for informative missing data this paper develops a class of models to deal with missing data from longitudinal studies we assume that separate models for the primary response and missingness e g number of missed visits are linked by a common random parameter such models have been developed in the econometrics heckman 1979 econometrica 47 153 161 and biostatistics wu and carroll 1988 biometrics 44 175 188 literature for a gaussian primary response we allow the primary response conditional on the random parameter to follow a generalized linear model and approximate the generalized linear model by conditioning on the data that describes missingness the resultant approximation is a mixed generalized linear model with possibly heterogeneous random effects an example is given to illustrate the approximate approach and simulations are performed to critique the adequacy of the approximation for repeated binary data

estimation and comparison of changes in the presence of information right censoring by modeling the censoring process abstract in estimating and comparing the rates of change of a continuous variable between two groups the unweighted averages of individual simple least squares estimates from each group are often used under a linear random effects model when all individuals have completed observations at identical time points these statistics are maximum likelihood estimates for the expected rates of change however with censored of missing data these estimates are no longer efficient when compared to generalized least squares estimates when in addition the right censoring process is dependent upon the individual rates of change i e informative right censoring the generalized least squares estimates will be biased likelihood ratio test for informativeness of the censoring process and maximum likelihood estimates for the expected rates of change and the parameters of the right censoring process are developed under a linear random effect models with a probit model for the right censoring process in realistic situations we illustrate that the bias in estimating group rate of change and the reduction of power in comparing group difference could be substantial when strong dependency of the right censoring process on individual rates of change is ignored author

model for the analysis of binary longitudinal pain data subject to informative dropout through remedication abstract we address the problem of accounting for informative dropout in the form of rescue medication when comparing pain relievers with respect to longitudinal binary pain relief outcomes we present a selection model approach for binary longitudinal data that accommodates informative dropout the relationship between dropout or remedication and the binary pain relief response is assumed to be characterized by a random effect that is conditional on this random effect response and dropout are independent unlike previous approaches to this problem which rely on numerical or approximation methods we obtain a closed form expression for the marginal log likelihood of response and dropout by specifying a complementary log log link function for both components and a conjugate log gamma random effect distribution a data analysis supported by simulation results suggest that the model fits reasonably well results are compared to those obtained from conventional but somewhat inappropriate analyses

convolution of multifractals and the local magnetization in a random field ising chain the local magnetization in the one dimensional random field ising model is essentially the sum of two effective fields with multifractal probability measure the probability measure of the local magnetization is thus the convolution of two multifractals in this paper we prove relations between the multifractal properties of two measures and the multifractal properties of their convolution the pointwise dimension at the boundary of the support of the convolution is the sum of the pointwise dimensions at the boundary of the support of the convoluted measures and the generalized box dimensions of the convolution are bounded from above by the sum of the generalized box dimensions of the convoluted measures the generalized box dimensions of the convolution of cantor sets with weights can be calculated analytically for certain parameter ranges and illustrate effects we also encounter in the case of the measure of the local magnetization returning to the study of this measure we apply the general inequalities and present numerical approximations of the dq spectrum for the first time we are able to obtain results on multifractal properties of a physical quantity in the one dimensional random field ising model which in principle could be measured experimentally the numerically generated probability densities for the local magnetization show impressively the gradual transition from a monomodal to a bimodal distribution for growing random field strength h

relation between spectral classes of a self similar cantor set a self similar cantor set is completely decomposed as a class of the lower upper distribution sets we give a relationship between the distribution sets in the distribution class and the subsets in a spectral class generated by the lower upper local dimensions of a self similar measure in particular we show that each subset of a spectral class is exactly a distribution set having full measure of a self similar measure related to the distribution set using the strong law of large numbers this gives essential information of its hausdorff and packing dimensions in fact the spectral class by the lower upper local dimensions of every self similar measure except for a singular one is characterized by the lower or upper distribution class finally we compare our results with those of other authors

diverging moments and parameter estimation heavy tailed distributions are enjoying increased popularity and are becoming more readily applicable as the arsenal of analytical and numerical tools grows they play key roles in modeling approaches in networking finance and hydrology to name but a few areas the tail parameter is of central importance because it governs both the existence of moments of positive order and the thickness of the tails of the distribution some of the best known tail estimators such as those of koutrouvelis and hill are either parametric or show a lack of robustness or accuracy this article develops a shift and scale invariant nonparametric estimator for both upper and lower bounds for orders with finite moments the estimator builds on the equivalence between tail behavior and the regularity of the characteristic function at the origin and achieves its goal by deriving a simplified wavelet analysis that is particularly suited to characteristic functions

multifractal analysis of stock exchange crashes abstract we analyze the complexity of rare events of the djia index we reveal that the returns of the time series exhibit strong multifractal properties meaning that temporal correlations play a substantial role the effect of major stock market crashes can be best illustrated by the comparison of the multifractal spectra of the time series before and after the crash aftershock periods compared to foreshock periods exhibit richer and more complex dynamics compared to an average crash calculated by taking into account the larger 5 crashes of the djia index the 1929 event exhibits significantly more increase in multifractality than the 1987 crisis

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

modelling progression of cd4 lymphocyte count and its relationship to survival time the purpose of this article is to model the progression of cd4 lymphocyte count and the relationship between different features of this progression and survival time the complicating factors in this analysis are that the cd4 lymphocyte count is observed only at certain fixed times and with a high degree of measurement error and that the length of the vector of observations is determined in part by the length of survival if probability of death depends on the true unobserved cd4 lymphocyte count then the survival process must be modelled wu and carroll 1988 biometrics 44 175 188 proposed a random effects model for two sample longitudinal data in the presence of informative censoring in which the individual effects included only slopes and intercepts we propose methods for fitting a broad class of models of this type in which both the repeated cd4 lymphocyte counts and the survival time are modelled using random effects these methods permit us to estimate parameters describing the progression of cd4 lymphocyte count as well as the effect of differences in the cd4 trajectory on survival we apply these methods to results of aids clinical trials

markov chain monte carlo convergence diagnostics a comparative review abstract a critical issue for users of markov chain monte carlo mcmc methods in applications is how to determine when it is safe to stop sampling and use the samples to estimate characteristics of the distribution of interest research into methods of computing theoretical convergence bounds holds promise for the future but to date has yielded relatively little of practical use in applied work consequently most mcmc users address the convergence problem by applying diagnostic tools to the output produced by running their samplers after giving a brief overview of the area we provide an expository review of 13 convergence diagnostics describing the theoretical basis and practical implementation of each we then compare their performance in two simple models and conclude that all of the methods can fail to detect the sorts of convergence failure that they were designed to identify we thus recommend a combination of strategies aimed at evaluating and accelerating mcmc sampler convergence including ap

on the l q spectrum of planar self affine measures the author was supported by the epsrc grant ep j013560 1 this work was started whilst the author was an epsrc funded phd student at the university of st andrews and he expresses his gratitude for the support he found there

on gibbs sampling for state space models summary we show how to use the gibbs sampler to carry out bayesian inference on a linear state space model with errors that are a mixture of normals and coefficients that can switch over time our approach simultaneously generates the whole of the state vector given the mixture and coefficient indicator variables and simultaneously generates all the indicator variables conditional on the state vectors the states are generated efficiently using the kalman filter we illustrate our approach by several examples and empirically compare its performance to another gibbs sampler where the states are generated one at a time the empirical results suggest that our approach is both practical to implement and dominates the gibbs sampler that generates the states one at a time

the global statistics of return times return time dimensions versus generalized measure dimensions we investigate return times in dynamical systems i e the time required by a trajectory to complete a return journey to a neighborhood of the initial position in particular we study the relations holding between the scaling exponents of phase space moments of return times in balls of diminishing radius on the one side and the generalized dimensions of invariant measures on the other because of a heuristic use of kac theorem the former have been used in place of the latter in numerical and experimental investigations to mark the distinction we call them return time dimensions we derive a full set of inequalities linking generalized dimensions of invariant measures and return time dimensions we comment on their optimality with the aid of two maps due to von neumann kakutani and to gaspard wang we conjecture a formula for the return time dimensions in a typical system we only assume that the dynamical system under investigation is ergodic and that motion takes place in a compact finite dimensional space

multifractality in time series we apply the concepts of multifractal physics to financial time series in order to characterize the onset of crash for the standard poor 500 s p500 stock index x t it is found that within the framework of multifractality the analogous specific heat of the s p500 discrete price index displays a shoulder to the right of the main peak for low time lag values for decreasing t the presence of the shoulder is a consequence of the peaked temporal x t t x t fluctuations in this regime for large time lags t 80 we have found that cq displays typical features of a classical phase transition at a critical point an example of such dynamic phase transition in a simple economic model system based on a mapping with multifractality phenomena in random multiplicative processes is also presented by applying former results obtained with a continuous probability theory for describing scaling measures

a bayesian model for detecting acute change in nonlinear profiles we propose a model for longitudinal data with random effects that includes model based smoothing of measurements over time this research is motivated by experiments evaluating the hemodynamic effects of various agents in tumor bearing rats in one set of experiments the rats breathed room air followed by carbogen a mixture of pure oxygen and carbon dioxide the experimental responses are longitudinal measurements of oxygen pressure measured in tissue tumor blood flow and mean arterial pressure the nature of the recorded responses does not allow any meaningful parametric form to model these profiles over time additionally response patterns differ widely across individuals therefore we propose a nonparametric regression to model the profile data over time we propose a dynamic state space model to smooth the data at the profile level using the state parameters we formally define change in the measured responses a hierarchical extension allows inference to include a regression on covariates

a random pattern mixture model for ordinal outcomes with informative dropouts we extend a random pattern mixture joint model for longitudinal ordinal outcomes and informative dropouts the patients are generalized to pattern groups based on known covariates that are potential surrogated for the severity of the underlying condition the random pattern effects are defined as the latent effects linking the dropout process and the ordinal longitudinal outcome conditional on the random pattern effects the longitudinal outcome and the dropout times are assumed independent estimates are obtained via the em algorithm we applied the model to the end stage renal disease esrd data anemia was found to be significantly affected by baseline iron treatment when the dropout information was adjusted via the study model as opposed to an independent or shared parameter model simulations were performed to evaluate the performance of the random pattern mixture model under various assumptions

numerical estimates of the generalized dimensions of the henon attractor for negative q the usual fixed size box counting algorithms are inefficient for computing generalized fractal dimensions in the range of q 0 in this letter we describe a new numerical algorithm specifically devised to estimate generalized dimensions for large negative q providing evidence of its better performance we compute the complete spectrum of the henon attractor and interpret our results in terms of a phase transition between different multiplicative laws

a multifractal analysis of equilibrium measures for conformal expanding maps and moran like geometric constructions in this paper we establish the complete multifractal formalism for equilibrium measures for holder continuous conformal expanding maps andexpanding markov moran like geometric constructions examples include markov maps of an interval beta transformations of an interval rational maps with hyperbolic julia sets and conformal toral endomorphisms we also construct a holder continuous homeomorphism of a compact metric space with an ergodic invariant measure of positive entropy for which the dimension spectrum is not convex and hence the multifractal formalism fails

mixed effects logistic regression models for longitudinal ordinal functional response data with multiple cause drop out from the longitudinal study of aging summary in the context of analyzing ordinal functional limitation responses from the longitudinal study of aging we investigate the association between current functional limitation and previous year s limitation and its modification by physical activity and multiple causes of drop out we accommodate the longitudinal nature of the multiple causes of informative drop out death and unknown loss to follow up with a mixed effects logistic model under the proposed model with a random intercept and slope the ordinal functional outcome and multiple discrete time survival profiles share a common random effect structure this shared parameter selection model assumes that the multiple causes of drop out are conditionally independent of the functional limitation outcome given the underlying random effect representing an individual s trajectory of general health status across time although it is not possible to fully assess the adequacy of this assumption we assess the robustness of the approach by varying the assumptions underlying the proposed model such as the random effects distribution and the drop out component it appears that between subject differences in initial functional limitation are strongly associated with future functional limitation and that this association is stronger for those who do not have physical activity regardless of the random effects and informative dropout specifications in contrast the association between current functional limitation and previous trajectory of functional status within an individual is weaker and more sensitive to changes in the random effects and drop out assumptions

on the properties of random multiplicative measures with the multipliers exponentially distributed under the formalism of annealed averaging of the partition function a type of random multifractal measures with the multipliers exponentially distributed is investigated in detail branching emerges in the curve of generalized dimensions and negative values of generalized dimensions arise three equivalent methods of classification of the random multifractal measures are proposed which is based on i the properties of the curves of generalized dimensions ii the solution properties of equation q 0 and iii the relative position of the curve f and the diagonal f in the first quadrant these classes of measures correspond to 0 1 0 1 1 and 0 1 0 respectively phase diagram is introduced to illustrate the diverse performance of the random measures that is multiplicatively generated

mixed effects logistic regression models for longitudinal binary response data with informative drop out a shared parameter model with logistic link is presented for longitudinal binary response data to accommodate informative drop out the model consists of observed longitudinal and missing response components that share random effects parameters to our knowledge this is the first presentation of such a model for longitudinal binary response data comparisons are made to an approximate conditional logit model in terms of a clinical trial dataset and simulations the naive mixed effects logit model that does not account for informative drop out is also compared the simulation based differences among the models with respect to coverage of confidence intervals bias and mean squared error mse depend on at least two factors whether an effect is a between or within subject effect and the amount of between subject variation as exhibited by variance components of the random effects distributions when the shared parameter model holds the approximate conditional model provides confidence intervals with good coverage for within cluster factors but not for between cluster factors the converse is true for the naive model under a different drop out mechanism when the probability of drop out is dependent only on the current unobserved observation all three models behave similarly by providing between subject confidence intervals with good coverage and comparable mse and bias but poor within subject confidence intervals mse and bias the naive model does more poorly with respect to the within subject effects than do the shared parameter and approximate conditional models the data analysis which entails a comparison of two pain relievers and a placebo with respect to pain relief conforms to the simulation results based on the shared parameter model but not on the simulation based on the outcome driven drop out process this comparison between the data analysis and simulation results may provide evidence that the shared parameter model holds for the pain data

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

the nonlinear mixed effects model with a smooth random effects density summary the fixed parameters of the nonlinear mixed effects model and the density of the random effects are estimated jointly by maximum likelihood the density of the random effects is assumed to be smooth but is otherwise unrestricted the method uses a series expansion that follows from the smoothness assumption to represent the density and quadrature to compute the likelihood standard algorithms are used for optimization empirical bayes estimates of random coefficients are obtained by computing posterior modes the method is applied to data from pharmacokinetics and properties of the method are investigated by application to simulated data

a multifractal wavelet model with application to network traffic we develop a new multiscale modeling framework for characterizing positive valued data with long range dependent correlations 1 f noise using the haar wavelet transform and a special multiplicative structure on the wavelet and scaling coefficients to ensure positive results the model provides a rapid o n cascade algorithm for synthesizing n point data sets we study both the second order and multifractal properties of the model the latter after a tutorial overview of multifractal analysis we derive a scheme for matching the model to real data observations and to demonstrate its effectiveness apply the model to network traffic synthesis the flexibility and accuracy of the model and fitting procedure result in a close fit to the real data statistics variance time plots and moment scaling and queuing behavior although for illustrative purposes we focus on applications in network traffic modeling the multifractal wavelet model could be useful in a number of other areas involving positive data including image processing finance and geophysics

multifractal measures and a weak separation condition abstract we define a new separation property on the family of contractive similitudes that allows certain overlappings this property is weaker than the open set condition of hutchinson it includes the well known class of infinite bernoulli convolutions associated with the p v numbers and the solutions of the two scale dilation equations our main purpose in this paper is to prove the multifractal formalism under such condition

a generalized multifractal spectrum of the general sierpinski carpets in this paper we study a class of subsets of the general sierpinski carpets for which the limiting frequency of a horizontal fibre falls into a prescribed closed interval we obtain the explicit expression for the hausdorff dimension of these subsets in terms of the parameters of the construction and give necessary and sufficient conditions for the corresponding hausdorff measure to be positive finite

pattern mixture models for multivariate incomplete data consider a random sample on variables x1 xv with some values of xv missing selection models specify the distribution of x1 xv over respondents and nonrespondents to xv and the conditional distribution that xv is missing given x1 xv in contrast pattern mixture models specify the conditional distribution of x 1 xv given that xv is observed or missing respectively and the marginal distribution of the binary indicator for whether or not xv is missing for multivariate data with a general pattern of missing values the literature has tended to adopt the selection modeling approach see for example little and rubin here pattern mixture models are proposed for this more general problem pattern mixture models are chronically underidentified in particular for the case of univariate nonresponse mentioned above there are no data on the distribution of xv given x1 xv 1 in the stratum with xv missing thus the models require restrictions or prior information to identify the paramet

a statistical test for the time constancy of scaling exponents a statistical test is described for determining if scaling exponents vary over time it is applicable to diverse scaling phenomena including long range dependence and exactly self similar processes in a uniform framework without the need for prior knowledge of the type in question it is based on the special properties of wavelet based estimates of the scaling exponent strongly motivating an idealized inference problem the equality or otherwise of means of independent gaussian variables with known variances a uniformly most powerful invariant test exists for this problem and is described a separate uniformly most powerful invariant test is also given for when the scaling exponent undergoes a level change the power functions of both tests are given explicitly and compared using simulation the effect in practice of deviations from the idealizations made of the statistical properties of the wavelet detail coefficients are analyzed and found to be small the tests inherit the significant robustness and computational advantages of the underlying wavelet based estimator a detailed methodology is given describing its use in practical situations the use and benefits of the test are illustrated on the bellcore ethernet data sets

inside singularity sets of random gibbs measures we evaluate the scale at which the multifractal structure of some random gibbs measures becomes discernible the value of this scale is obtained through what we call the growth speed in holder singularity sets of a borel measure this growth speed yields new information on the multifractal behavior of the rescaled copies involved in the structure of statistically self similar gibbs measures our results are useful to understand the multifractal nature of various heterogeneous jump processes

estimation and comparison of changes in the presence of informative right censoring conditional linear model a general linear regression model for the usual least squares estimated rate of change slope on censoring time is described as an approximation to account for informative right censoring in estimating and comparing changes of a continuous variable in two groups two noniterative estimators for the group slope means the linear minimum variance unbiased lmvub estimator and the linear minimum mean squared error lmmse estimator are proposed under this conditional model in realistic situations we illustrate that the lmvub and lmmse estimators derived under a simple linear regression model are quite competitive compared to the pseudo maximum likelihood estimator pmle derived by modeling the censoring probabilities generalizations to polynomial response curves and general linear models are also described

a mixed effects model for the analysis of ordinal longitudinal pain data subject to informative drop out we extend the model of pulkstenis et al that models binary longitudinal data subject to informative drop out through remedication to the ordinal response case we present a selection model shared parameter approach that specifies mixed models for both ordinal response and discrete survival time to remedication in this fashion the random parameter present in both models completely characterizes the relationship between response and time to remedication inducing their conditional independence with a log log link function for both response and study survival as well as specification of a log gamma distribution for the random effect we obtain a closed form expression for the marginal log likelihood of response and time to remedication that does not require approximation or numerical integration techniques a data analysis is performed and simulation results presented which support the consistency of parameter and standard error estimates

numerical investigations of discrete scale invariance in fractals and multifractal measures fractals and multifractals and their associated scaling laws provide a quantification of the complexity of a variety of scale invariant complex systems here we focus on lattice multifractals which exhibit complex exponents associated with observable log periodicity we perform detailed numerical analyses of lattice multifractals and explain the origin of three different scaling regions found in the moments a novel numerical approach is proposed to extract the log frequencies in the non lattice case there is no visible log periodicity i e no preferred scaling ratio since the set of complex exponents spreads irregularly within the complex plane a non lattice multifractal can be approximated by a sequence of lattice multifractals so that the sets of complex exponents of the lattice sequence converge to the set of complex exponents of the non lattice one an algorithm for the construction of the lattice sequence is proposed explicitly

some matrix variate distribution theory notational considerations and a bayesian application summary we introduce and justify a convenient notation for certain matrix variate distributions which by its emphasis on the important underlying parameters and the theory on which it is based eases greatly the task of manipulating such distributions important examples include the matrix variate normal t f and beta and the wishart and inverse wishart distributions the theory is applied to compound matrix distributions and to bayesian prediction in the multivariate linear model

a random effects ordinal regression model for multilevel analysis a random effects ordinal regression model is proposed for analysis of clustered or longitudinal ordinal response data this model is developed for both the probit and logistic response functions the threshold concept is used in which it is assumed that the observed ordered category is determined by the value of a latent unobservable continuous response that follows a linear regression model incorporating random effects a maximum marginal likelihood mml solution is described using gauss hermite quadrature to numerically integrate over the distribution of random effects an analysis of a dataset where students are clustered or nested within classrooms is used to illustrate features of random effects analysis of clustered ordinal data while an analysis of a longitudinal dataset where psychiatric patients are repeatedly rated as to their severity is used to illustrate features of the random effects approach for longitudinal ordinal data

generalized dimensions large deviations and the distribution of rare events abstract generalized dimensions of multifractal measures are usually seen as static objects related to the scaling properties of suitable partition functions or moments of measures of cells when these measures are invariant for the flow of a chaotic dynamical system generalized dimensions take on a dynamical meaning as they provide the rate function for the large deviations of the first hitting time which is the average time required to connect any two different regions in phase space we prove this result rigorously under a set of stringent assumptions as a consequence the statistics of hitting times provides new algorithms for the computation of the spectrum of generalized dimensions numerical examples presented along with the theory suggest that the validity of this technique reaches far beyond the range covered by the theorem we state our result within the framework of extreme value theory this approach reveals that hitting times are also linked to dynamical indicators such as stability of the motion and local dimensions of the invariant measure this suggests that one can use local dynamical indicators from finite time series to gather information on the multifractal spectrum of generalized dimension we show an application of this technique to experimental data from climate dynamics

automatic bayesian curve fitting summary a method of estimating a variety of curves by a sequence of piecewise polynomials is proposed motivated by a bayesian model and an appropriate summary of the resulting posterior distribution a joint distribution is set up over both the number and the position of the knots defining the piecewise polynomials throughout we use reversible jump markov chain monte carlo methods to compute the posteriors the methodology has been successful in giving good estimates for smooth functions i e continuous and differentiable as well as functions which are not differentiable and perhaps not even continuous at a finite number of points the methodology is extended to deal with generalized additive models

smoothing spline models for the analysis of nested and crossed samples of curves abstract we introduce a class of models for an additive decomposition of groups of curves stratified by crossed and nested factors generalizing smoothing splines to such samples by associating them with a corresponding mixed effects model the models are also useful for imputation of missing data and exploratory analysis of variance we prove that the best linear unbiased predictors blups from the extended mixed effects model correspond to solutions of a generalized penalized regression where smoothing parameters are directly related to variance components and we show that these solutions are natural cubic splines the model parameters are estimated using a highly efficient implementation of the em algorithm for restricted maximum likelihood reml estimation based on a preliminary eigenvector decomposition variability of computed estimates can be assessed with asymptotic techniques or with a novel hierarchical bootstrap resampling scheme for nested mixed effects models our methods are applied to me

thick points for spatial brownian motion multifractal analysis of occupation measure let t x r denote the total occupation measure of the ball of radius r centered at x for brownian motion in ir we prove that sup x 1 t x r r log r 16 a s as r 0 thus solving a problem posed by taylor in 1974 furthermore for any a 0 16 the hausdorff dimension of the set of thick points x for which lim sup r 0 t x r r log r a is almost surely 2 a 8 this is the correct scaling to obtain a nondegenerate multifractal spectrum for brownian occupation measure analogous results hold for brownian motion in any dimension d 3 these results are related to the lil of ciesielski and taylor 1962 for the brownian occupation measure of small balls in the same way that levy s uniform modulus of continuity and the formula of orey and taylor 1974 for the dimension of fast points are related to the usual lil we also show that the lim inf scaling of t x r is quite different we exhibit non random c1 c2 0 such that c1 supx lim infr 0 t x r r 2 c2 a s in the course of our work we provide a general framework for obtaining lower bounds on the hausdorff dimension of random fractals of limsup type

multifractality tests using bootstrapped wavelet leaders multifractal analysis which mostly consists of measuring scaling exponents is becoming a standard technique available in most empirical data analysis toolboxes making use of the most recent theoretical results it is based here on the estimation of the cumulants of the log of the wavelet leaders an elaboration on the wavelet coefficients these log cumulants theoretically enable discrimination between mono and multifractal processes as well as between simple log normal multifractal models and more advanced ones the goal of the present contribution is to design nonparametric bootstrap hypothesis tests aiming at testing the nature of the multifractal properties of stochastic processes and empirical data bootstrap issues together with six declinations of test designs are analyzed their statistical performance significances powers and p values are assessed and compared by means of monte carlo simulations performed on synthetic stochastic processes whose multifractal properties and log cumulants are known theoretically a priori we demonstrate that the joint use of wavelet leaders log cumulants and bootstrap procedures enable us to obtain a powerful tool for testing the multifractal properties of data this tool is practically effective and can be applied to a single observation of data with finite length

an improved multifractal formalism and self similar measures abstract to characterize the geometry of a measure its generalized dimensions dq have been introduced recently the mathematically precise definition given by falconer fractal geometry 1990 turns out to be unsatisfactory for reasons of convergence as well as of undesired sensitivity to the particular choice of coordinates in the negative q range a new definition is introduced which is also based on box counting but which carries relevant information about for negative q also in particular the legendre connection between generalized dimensions and the multifractal spectrum is established using large deviations and rigorous proofs are provided for the implicit formula giving the generalized dimensions of selfsimilar measures which was until now known only for positive q

semiparametric stochastic mixed models for longitudinal data abstract we consider inference for a semiparametric stochastic mixed model for longitudinal data this model uses parametric fixed effects to represent the covariate effects and an arbitrary smooth function to model the time effect and accounts for the within subject correlation using random effects and a stationary or nonstationary stochastic process we derive maximum penalized likelihood estimators of the regression coefficients and the nonparametric function the resulting estimator of the nonparametric function is a smoothing spline we propose and compare frequentist inference and bayesian inference on these model components we use restricted maximum likelihood to estimate the smoothing parameter and the variance components simultaneously we show that estimation of all model components of interest can proceed by fitting a modified linear mixed model we illustrate the proposed method by analyzing a hormone dataset and evaluate its performance through simulations