measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

misspecified proportional hazard models summary the properties of an estimator based on a proportional hazards model are investigated when the model is incorrect the estimator from the partial likelihood is shown to be consistent for a parameter that is defined implicitly the results are used to investigate the effects on estimation if the true model is accelerated failure time or if covariates are omitted from the proportional hazards model

estimating dynamic models of imperfect competition we describe a two step algorithm for estimating dynamic games under the assumption that behavior is consistent with markov perfect equilibrium in the first step the policy functions and the law of motion for the state variables are estimated in the second step the remaining structural parameters are estimated using the optimality conditions for equilibrium the second step estimator is a simple simulated minimum distance estimator the algorithm applies to a broad class of models including industry competition models with both discrete and continuous controls such as the ericson and pakes 1995 model we test the algorithm on a class of dynamic discrete choice models with normally distributed errors and a class of dynamic oligopoly models similar to that of pakes and mcguire 1994

survival models for heterogeneous populations derived from stable distributions summary a new three parameter family of distributions on the positive numbers is proposed it includes the stable distributions on the positive numbers the gamma the degenerate and the inverse gaussian distributions the family is characterized by the laplace transform from which moments convolutions infinite divisibility unimodality and other properties are derived the density is complicated but a simple saddlepoint approximation is provided weibull and gompertz distributions are naturally mixed over some of the distributions the family is natural exponential in one of the parameters the distributions are relevant for application as frailty distributions in life table methods for heterogeneous populations desirable properties of such distributions are discussed as an example survival after myocardial infarction is considered

a deterministic geometric representation of temporal rainfall results for a storm in boston the use of a deterministic fractal multifractal fm representation to model high resolution rainfall time series via projections of fractal interpolating functions weighed by multifractal measures is reported it is shown that the intrinsic shape and variability of an 8 hour boston storm recorded every 15 s on october 25 1980 may be encoded wholistically employing the fractal geometric methodology it is illustrated that the fm methodology provides very faithful descriptions of both major trends and small noisy fluctuations for this storm resulting in preservation of not only classical statistical characteristics of the records but also multifractal and chaotic properties present in them these results and those for other storms suggest that a stochastic framework for rainfall may be bypassed in favor of a deterministic representation based on projections

generalized gamma frailty model in this article we present a frailty model using the generalized gamma distribution as the frailty distribution it is a power generalization of the popular gamma frailty model it also includes other frailty models such as the lognormal and weibull frailty models as special cases the flexibility of this frailty distribution makes it possible to detect a complex frailty distribution structure which may otherwise be missed due to the intractable integrals in the likelihood function and its derivatives we propose to approximate the integrals either by monte carlo simulation or by a quadrature method and then determine the maximum likelihood estimates of the parameters in the model we explore the properties of the proposed frailty model and the computation method through a simulation study the study shows that the proposed model can potentially reduce errors in the estimation and that it provides a viable alternative for correlated data the merits of proposed model are demonstrated in analysing the effects of sublingual nitroglycerin and oral isosorbide dinitrate on angina pectoris of coronary heart disease patients based on the data set in danahy et al sustained hemodynamic and antianginal effect of high dose oral isosorbide dinitrate circulation 1977 55 381 387

effect of frailty on marginal regression estimates in survival analysis unexplained heterogeneity in univariate survival data and association in multivariate survival can both be modelled by the inclusion of frailty effects this paper investigates the consequences of ignoring frailty in analysis fitting misspecified cox proportional hazards models to the marginal distributions regression coefficients are biased towards 0 by an amount which depends in magnitude on the variability of the frailty terms and the form of frailty distribution the bias is reduced when censoring is present fitted marginal survival curves can also differ substantially from the true marginals

statistical analysis of repeated events forming renewal processes for each of several individuals a sequence of repeated events forming a renewal process is observed up to some censoring time the object is to estimate the average interevent time over the population of individuals as well as the variation of interevent times within and between individuals medical motivation comes from gastroenterology and concerns the occurrence of certain cyclic movements in the small bowel during the fasting state two statistical models are considered one is the standard variance component model adapted to censored data and the other is a recent intensity based model with a random proportionality factor representing interindividual variation these models are applied to the motility data and their advantages are discussed the intensity based model allows simple empirical bayes estimation of the expected interevent times for an individual in the presence of censoring

chaos in rainfall the question is posed regarding possible chaotic dynamics in the temporal rainfall of storm events chaotic dynamics implies a nonlinear deterministic system very sensitive to initial conditions which yields outputs indistinguishable from a stochastic process by standard techniques the trajectories of these systems in the phase space are characterized by being contained in a strange attractor of fractal dimension a time series of 1990 points of 15 s rainfall for the storm of october 25 1980 in boston is analyzed in detail toward this objective it is found that both the characteristics of the correlation integral and the lyapunov exponents of the historical data give preliminary support to the presence of chaotic dynamics with a strange attractor the implications of these findings for the modeling of storm rainfall and for the limits of predictability of the process are discussed jointly with the research challenges lying ahead in this field

a class of multivanate failure time distributions summary a class of continuous multivariate lifetime distributions is proposed the dependence between individuals in a group is modelled by a group specific quantity which can be interpreted as an unobserved covariate common to the individuals in the group and assumed to follow a positive stable distribution it is possible to include covariates in the model and discuss whether the dependence is still present after specific covariates are taken into account if the conditional hazards are proportional then the hazards in the marginal distributions are also proportional but with different constants of proportionality also the hazard for the minimum in a group is proportional to the marginal hazards if the conditional distributions given the group quantity are weibull then the marginal distributions are also weibull this class can be used to test the hypothesis of independence of litter mates in the proportional hazards model

bayesian estimation of dynamic discrete choice models we propose a new methodology for structural estimation of infinite horizon dynamic discrete choice models we combine the dynamic programming dp solution algorithm with the bayesian markov chain monte carlo algorithm into a single algorithm that solves the dp problem and estimates the parameters simultaneously as a result the computational burden of estimating a dynamic model becomes comparable to that of a static model another feature of our algorithm is that even though the number of grid points on the state variable is small per solution estimation iteration the number of effective grid points increases with the number of estimation iterations this is how we help ease the curse of dimensionality we simulate and estimate several versions of a simple model of entry and exit to illustrate our methodology we also prove that under standard conditions the parameters converge in probability to the true posterior distribution regardless of the starting values

sequential estimation of dynamic discrete games a comment recursive procedures which are based on iterating on the best response mapping have difficulties converging to all equilibria in multi player games we illustrate these difficulties by revisiting the asymptotic properties of the iterative nested pseudo maximum likelihood method for estimating dynamic games introduced by aguirregabiria and mira 2007 an example shows that the iterative method may not be consistent

bivariate extreme value theory models and estimation summary bivariate extreme value distributions arise as the limiting distributions of renormalized componentwise maxima no natural parametric family exists for the dependence between the marginal distributions but there are considerable restrictions on the dependence structure we consider modelling the dependence function with parametric models for which two new models are presented tests for independence and discriminating between models are also given the estimation procedure and the flexibility of the new models are illustrated with an application to sea level data extreme value theory has recently been an area of much theoretical and practical work univariate theory is a well documented area whereas bivariate multivariate extreme value theory has until recently received surprisingly little attention in the multivariate case no natural parametric family exists for the dependence structure so this must be modelled in some way in the analysis of environmental extreme value data there is a need for models of dependence between extremes from different sources for example at various sea ports or at various points of a river in this paper we consider bivariate extreme value distributions we assume without loss of generality that we have exponential marginal distributions with unit means the class of bivariate exponential distributions in which we are interested satisfy a strong stability relation exponential variables x y satisfy the stability relation if and only if w min ax by is also exponentially distributed for all a b 0 pickands 1981 therefore the models we will consider have particular application in reliability and survival analysis one approach to modelling the dependence structure is via parametric models this requires a flexible family of models which satisfy certain constraints models are of two kinds either differentiable or nondifferentiable all nondifferentiable models give distri butions which are singular with nonzero probability concentrated on a certain subspace the differentiable models have densities but the existing models are symmetric which leads to the variables being exchangeable here we present two new asymmetric differenti able models which have increased flexibility properties of the differentiable models are examined estimation of the parametric models has previously been by ad hoc methods because there is a nonregular estimation problem when the margins are independent for the

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

marginal maximum likelihood estimation of item parameters application of an em algorithm maximum likelihood estimation of item parameters in the marginal distribution integrating over the distribution of ability becomes practical when computing procedures based on an em algorithm are used by characterizing the ability distribution empirically arbitrary assumptions about its form are avoided the em procedure is shown to apply to general item response models lacking simple sufficient statistics for ability this includes models with more than one latent dimension

proportional hazards models with frailties and random effects we discuss some of the fundamental concepts underlying the development of frailty and random effects models in survival one of these fundamental concepts was the idea of a frailty model where each subject has his or her own disposition to failure their so called frailty additional to any effects we wish to quantify via regression although the concept of individual frailty can be of value when thinking about how data arise or when interpreting parameter estimates in the context of a fitted model we argue that the concept is of limited practical value individual random effects frailties whenever detected can be made to disappear by elementary model transformation in consequence unless we are to take some model form as unassailable beyond challenge and carved in stone and if we are to understand the term frailty as referring to individual random effects then frailty models have no value random effects models on the other hand in which groups of individuals share some common effect can be used to advantage even in this case however if we are prepared to sacrifice some efficiency we can avoid complex modelling by using the considerable power already provided by the stratified proportional hazards model stratified models and random effects models can both be seen to be particular cases of partially proportional hazards models a view that gives further insight the added structure of a random effects model viewed as a stratified proportional hazards model with some added distributional constraints will for group sizes of five or more provide no more than modest efficiency gains even when the additional assumptions are exactly true on the other hand for moderate to large numbers of very small groups of sizes two or three the study of twins being a well known example the efficiency gains of the random effects model can be far from negligible for such applications the case for using random effects models rather than the stratified model is strong this is especially so in view of the good robustness properties of random effects models nonetheless the simpler analysis based upon the stratified model remains valid albeit making a less efficient use of resources

swapping the nested fixed point algorithm a class of estimators for discrete markov decision models this paper proposes a new nested algorithm npl for the estimation of a class of discrete markov decision models and studies its statistical and computational properties our method is based on a representation of the solution of the dynamic programming problem in the space of conditional choice probabilities when the npl algorithm is initialized with consistent nonparametric estimates of conditional choice probabilities successive iterations return a sequence of estimators of the structural parameters which we call k stage policy iteration estimators we show that the sequence includes as extreme cases a hotz miller estimator for k 1 and rust s nested fixed point estimator in the limit when k furthermore the asymptotic distribution of all the estimators in the sequence is the same and equal to that of the maximum likelihood estimator we illustrate the performance of our method with several examples based on rust s bus replacement model monte carlo experiments reveal a trade off between finite sample precision and computational cost in the sequence of policy iteration estimators

the natural variability of vital rates and associated statistics the first concern of this work is the development of approximations to the distributions of crude mortality rates age specific mortality rates age standardized rates standardized mortality ratios and the like for the case of a closed population or period study it is found that assuming poisson birthtimes and independent lifetimes implies that the number of deaths and the corresponding midyear population have a bivariate poisson distribution it is suggested that situations in which explanatory variables are present may be modelled via a doubly stochastic poisson distribution for the number of deaths with mean proportional to the population size and an exponential function of a linear combination of the explanatories such a model is fit to mortality data for canadian females classified by age and year a dynamic variant of the model is further fit to the time series of total female deaths alone by year the models with extra poisson variation are found to lead to substantially improved fits excerpt

on multiple eigenvalues of matrices depending on several parameters the eigenvalues and eigenvectors as well as their derivatives of matrices depending on one or several parameters have been studied by sun j comput math 3 1985 pp 351 364 using the implicit function theorems and others but not for the more general multiple eigenvalues in this paper the mean of the multiple eigenvalues in terms of the trace operator tr the corresponding invariant subspaces and their derivatives are studied using a generalization of sun s approach the numerical aspects concerning the computation of such derivatives by direct and iterative methods e g simultaneous and inverse simultaneous iterations will be discussed briefly the main results in this paper apply to clusters of nonmultiple eigenvalues as well the implications e g on simultaneous iteration will be explored

an attractor in a solar time series a time series analysis of observed solar radio pulsations suggests that there must be a low dimensional attractor the power spectrum cannot be interpreted as a superposition of periodic components estimates of the maximum lyapunov exponent and of the kolmogorov entropy give some indications for a deterministic chaos in order to study the limitations inherent in small data samples we include data from the lorenz model and an artificial noise record consequences for the physical modelling of the pulsation event are discussed

a comparison of frailty models for multivariate survival data this paper reviews some of the main approaches to the analysis of multivariate censored survival data such data typically have correlated failure times the correlation can be a consequence of the observational design for example with clustered sampling and matching or it can be a focus of interest as in genetic studies longitudinal studies of recurrent events and other studies involving multiple measurements we assume that the correlation between the failure or survival times can be accounted for by fixed or random frailty effects we then compare the performance of conditional and mixture likelihood approaches to estimating models with these frailty effects in censored bivariate survival data we find that the mixture methods are surprisingly robust to misspecification of the frailty distribution the paper also contains an illustrative example on the times to onset of chest pain brought on by three endurance exercise tests during a drug treatment trial of heart patients

stabilization of unstable procedures the recursive projection method fixed point iterative procedures for solving nonlinear parameter dependent problems can converge for some interval of parameter values and diverge as the parameter changes the recursive projection method rpm which stabilizes such procedures by computing a projection onto the unstable subspace is presented on this subspace a newton or special newton iteration is performed and the fixed point iteration is used on the complement as continuation in the parameter proceeds the projection is efficiently updated possibly increasing or decreasing the dimension of the unstable subspace the method is extremely effective when the dimension of the unstable subspace is small compared to the dimension of the system convergence proofs are given and pseudo arclength continuation on the unstable subspace is introduced to allow continuation past folds examples are presented for an important application of the rpm in which a black box time integration scheme is stabilized enabling it to compute unstable steady states the rpm can also be used to accelerate iterative procedures when slow convergence is due to a few slowly decaying modes

predictive uncertainty of chaotic daily streamflow using ensemble wavelet networks approach 1 perfect or even mediocre weather predictions over a long period are almost impossible because of the ultimate growth of a small initial error into a significant one even though the sensitivity of initial conditions limits the predictability in chaotic systems an ensemble of prediction from different possible initial conditions and also a prediction algorithm capable of resolving the fine structure of the chaotic attractor can reduce the prediction uncertainty to some extent all of the traditional chaotic prediction methods in hydrology are based on single optimum initial condition local models which can model the sudden divergence of the trajectories with different local functions conceptually global models are ineffective in modeling the highly unstable structure of the chaotic attractor this paper focuses on an ensemble prediction approach by reconstructing the phase space using different combinations of chaotic parameters i e embedding dimension and delay time to quantify the uncertainty in initial conditions the ensemble approach is implemented through a local learning wavelet network model with a global feed forward neural network structure for the phase space prediction of chaotic streamflow series quantification of uncertainties in future predictions are done by creating an ensemble of predictions with wavelet network using a range of plausible embedding dimensions and delay times the ensemble approach is proved to be 50 more efficient than the single prediction for both local approximation and wavelet network approaches the wavelet network approach has proved to be 30 50 more superior to the local approximation approach compared to the traditional local approximation approach with single initial condition the total predictive uncertainty in the streamflow is reduced when modeled with ensemble wavelet networks for different lead times localization property of wavelets utilizing different dilation and translation parameters helps in capturing most of the statistical properties of the observed data the need for taking into account all plausible initial conditions and also bringing together the characteristics of both local and global approaches to model the unstable yet ordered chaotic attractor of a hydrologic series is clearly demonstrated

choice of conditional models in bivariate survival we consider bivariate survival problems in which interest is in the conditional distribution of one survival variable given an uncensored observation of the other the work is motivated by an analysis of time to cancer diagnosis then subsequent survival amongst a group of organ transplant recipients the effect of conditioning is illustrated for five standard bivariate models the consequences of adopting a misspecified marginal approach in which the conditioning variable is considered to be a fixed covariate are investigated

nested frailty models using maximum penalized likelihood estimation the frailty model is a random effect survival model which allows for unobserved heterogeneity or for statistical dependence between observed survival data the nested frailty model accounts for the hierarchical clustering of the data by including two nested random effects nested frailty models are particularly appropriate when data are clustered at several hierarchical levels naturally or by design in such cases it is important to estimate the parameters of interest as accurately as possible by taking into account the hierarchical structure of the data we present a maximum penalized likelihood estimation mpnle to estimate non parametrically a continuous hazard function in a nested gamma frailty model with right censored and left truncated data the estimators for the regression coefficients and the variance components of the random effects are obtained simultaneously the simulation study demonstrates that this semi parametric approach yields satisfactory results in this complex setting in order to illustrate the mpnle method and the nested frailty model we present two applications one is for modelling the effect of particulate air pollution on mortality in different areas with two levels of geographical regrouping the other application is based on recurrent infection times of patients from different hospitals we illustrate that using a shared frailty model instead of a nested frailty model with two levels of regrouping leads to inaccurate estimates with an overestimation of the variance of the random effects we show that even when the frailty effects are fairly small in magnitude they are important since they alter the results in a systematic pattern

modelling marginal hazards in multivariate failure time data we consider the problem of estimating relative risk parameters in the marginal distribution of failure times from a sample in which individuals are grouped into clusters and failure times for individuals within clusters may be dependent we assume that the marginal hazard of failure for individuals within a cluster is specified by a relative risk regression model and the structure of dependence between individuals within a cluster is left unspecified estimating equations for the regression parameters and an estimator for the cumulative base line marginal hazard are described

pseudo likelihood estimation and bootstrap inference for structural discrete markov decision models abstract this paper analyzes the higher order properties of the estimators based on the nested pseudo likelihood npl algorithm and the practical implementation of such estimators for parametric discrete markov decision models we derive the rate at which the npl algorithm converges to the mle and provide a theoretical explanation for the simulation results in aguirregabiria and mira aguirregabiria v mira p 2002 swapping the nested fixed point algorithm a class of estimators for discrete markov decision models econometrica 70 1519 1543 in which iterating the npl algorithm improves the accuracy of the estimator we then propose a new npl algorithm that can achieve quadratic convergence without fully solving the fixed point problem in every iteration and apply our estimation procedure to a finite mixture model we also develop one step npl bootstrap procedures for discrete markov decision models the monte carlo simulation evidence based on a machine replacement model of rust rust j 1987 optimal replacement of gmc bus engines an empirical model of harold zurcher econometrica 55 999 1033 shows that the proposed one step bootstrap test statistics and confidence intervals improve upon the first order asymptotics even with a relatively small number of iterations

a distributional model for repeated failure time measurements summary a standard model for repeated measurements is based on the multivariate normal distribution with equicorrelated variables a corresponding analysis is developed for weibull failure times the resulting distribution has some attractive properties for practical application to multivariate failure times

biases of correlation dimension estimates of streamflow data in the canadian prairies the correlation dimension d2 of the unregulated streamflow data of six rivers in the canadian prairies were estimated using the hill estimate and grassberger procaccia algorithms from both procedures and using an embedding phase space up to 10 the d2 obtained for all cases are found to saturate at 3 0 however the saturated d2 generally increased to between 4 and 6 when the data were first randomly resampled prior to estimating the correlation dimension since the highest embedding dimension d used in this study is 10 theoretically randomly resampled data should have a d2 approximately equal to 10 therefore there is a consistent underestimation of d2 by an amount of 4 6 by accounting for the combined clustering effect of the streamflow data s underlying distribution which is gamma the effect of sample size and possibly the choice of time delay the d2 of the prairie streamflow data should be 7 9 the hypothesis that the clustering effect of unevenly distributed data contributed to the underestimation of d2 was confirmed from the different d2 obtained for random numbers of uniform gamma and poisson distributions in addition analysis of the uniformly distributed random numbers and streamflow data showed that increasing the sample size from say 6500 to 17 000 marginally improved the estimated d2 however for the range of time delay chosen for this study 40 180 days its effect on d2 is less obvious this may be partly because all the streamflow data tested exhibit long term persistence

nonparametric identification of finite mixture models of dynamic discrete choices in dynamic discrete choice analysis controlling for unobserved heterogeneity is an important issue and finite mixture models provide flexible ways to account for it this paper studies nonparametric identifiability of type probabilities and type specific component distributions in finite mixture models of dynamic discrete choices we derive sucient conditions for nonparametric identification for various finite mixture models of dynamic discrete choices used in applied work under dierent assumptions on the markov property stationarity and type invariance in the transition process three elements emerge as the important determinants of identification the time dimension of panel data the number of values the covariates can take and the heterogeneity of the response of dierent types to changes in the co

sequential estimation of structural models with a fixed point constraint this paper considers the estimation problem of structural models for which empirical restrictions are characterized by a fixed point constraint such as structural dynamic discrete choice models or models of dynamic games we analyze a local condition under which the nested pseudo likelihood npl algorithm converges to a consistent estimator and derive its convergence rate we find that the npl algorithm may not necessarily converge to a consistent estimator when the fixed point mapping does not have a local contraction property to address the issue of divergence we propose alternative sequential estimation procedures that can converge to a consistent estimator even when the npl algorithm does not

a semiparametric recurrent events model with time varying coefficients we consider a recurrent events model with time varying coefficients motivated by two clinical applications a random effects gaussian frailty model is used to describe the intensity of recurrent events the model can accommodate both time varying and time constant coefficients the penalized spline method is used to estimate the time varying coefficients laplace approximation is used to evaluate the penalized likelihood without a closed form the smoothing parameters are estimated in a similar way to variance components we conduct simulations to evaluate the performance of the estimates for both time varying and time independent coefficients we apply this method to analyze two data sets a stroke study and a child wheeze study