a proportional hazards model for interval censored failure time data this paper develops a method for fitting the proportional hazards regression model when the data contain left right or interval censored observations results given for testing the hypothesis of a zero regression coefficient lead to a generalization of the log rank test for comparison of several survival curves the method is used to analyze data from an animal tumorigenicity study and also a clinical trial

efficient estimation for the proportional hazards model with interval censoring the maximum likelihood estimator mle for the proportional hazards model with case 1 interval censored data is studied it is shown that the mle for the regression parameter is asymptotically normal with n convergence rate and achieves the information bound even though the mle for the baseline cumulative hazard function only converges at n 1 3 rate estimation of the asymptotic variance matrix for the mle of the regression parameter is also considered to prove our main results we also establish a general theorem showing that the mle of the finite dimensional parameter in a class of semiparametric models is asymptotically efficient even though the mle of the infinite dimensional parameter converges at a rate slower than n the results are iliustrated by applying them to a data set from a tumorigenicity study

cox s regression model for counting processes a large sample study the cox regression model for censored survival data specifies that covariates have a proportional effect on the hazard function of the life time distribution of an individual in this paper we discuss how this model can be extended to a model where covariate processes have a proportional effect on the intensity process of a multivariate counting process this permits a statistical regression analysis of the intensity of a recurrent event allowing for complicated censoring patterns and time dependent covariates furthermore this formulation gives rise to proofs with very simple structure using martingale techniques for the asymptotic properties of the estimators from such a model finally an example of a statistical analysis is included

semiparametric analysis of zero inflated count data medical and public health research often involve the analysis of count data that exhibit a substantially large proportion of zeros such as the number of heart attacks and the number of days of missed primary activities in a given period a zero inflated poisson regression model which hypothesizes a two point heterogeneity in the population characterized by a binary random effect is generally used to model such data subjects are broadly categorized into the low risk group leading to structural zero counts and high risk or normal group so that the counts can be modeled by a poisson regression model the main aim is to identify the explanatory variables that have significant effects on i the probability that the subject is from the low risk group by means of a logistic regression formulation and ii the magnitude of the counts given that the subject is from the high risk group by means of a poisson regression where the effects of the covariates are assumed to be linearly related to the natural logarithm of the mean of the counts in this article we consider a semiparametric zero inflated poisson regression model that postulates a possibly nonlinear relationship between the natural logarithm of the mean of the counts and a particular covariate a sieve maximum likelihood estimation method is proposed asymptotic properties of the proposed sieve maximum likelihood estimators are discussed under some mild conditions the estimators are shown to be asymptotically efficient and normally distributed simulation studies were carried out to investigate the performance of the proposed method for illustration purpose the method is applied to a data set from a public health survey conducted in indonesia where the variable of interest is the number of days of missed primary activities due to illness in a 4 week period

stochastic balancing and approximation stability and minimality a new method of stochastic model reduction has recently been introduced by desai et al 1 2 the stability of the reduced order model has not previously been considered in this paper we show that the stability of the reduced order model follows directly from the results of pernebo and silverman 3 it is also shown that the reduced order model is minimal in the controllability observability sense the relevance of this notion of minimality to stochastic minimality is made clear

a multiple imputation approach to cox regression with interval censored data we propose a general semiparametric method based on multiple imputation for cox regression with interval censored data the method consists of iterating the following two steps first from finite interval censored but not right censored data exact failure times are imputed using tanner and wei s poor man s or asymptotic normal data augmentation scheme based on the current estimates of the regression coefficient and the baseline survival curve second a standard statistical procedure for right censored data such as the cox partial likelihood method is applied to imputed data to update the estimates through simulation we demonstrate that the resulting estimate of the regression coefficient and its associated standard error provide a promising alternative to the nonparametric maximum likelihood estimate our proposal is easily implemented by taking advantage of existing computer programs for right censored data

an error bound for a discrete reduced order model of a linear multivariable system the design of feasible controllers for high dimension multivariable systems can be greatly aided by a method of model reduction in order for the design based on the order reduction to include a guarantee of stability it is sufficient to have a bound on the model error previous work has provided such a bound for continuous time systems for algorithms based on balancing in this note an linftybound is derived for model error for a method of order reduction of discrete linear multivariable systems based on balancing

smooth inference for survival functions with arbitrarily censored data we propose a procedure for estimating the survival function of a time to event random variable under arbitrary patterns of censoring the method is predicated on the mild assumption that the distribution of the random variable and hence the survival function has a density that lies in a class of smooth densities whose elements can be represented by an infinite hermite series truncation of the series yields a parametric expression that can well approximate any plausible survival density and hence survival function provided the degree of truncation is suitably chosen the representation admits a convenient expression for the likelihood for the parameters in the approximation under arbitrary censoring truncation that is straightforward to compute and maximize a test statistic for comparing two survival functions which is based on an integrated weighted difference of estimates of each under this representation is proposed via simulation studies and application to a number of data sets we demonstrate that the approach yields reliable inferences and can result in gains in efficiency over traditional nonparametric methods

some numerical experiments with variable storage quasi newton algorithms this paper describes some numerical experiments with variable storage quasi newton methods for the optimization of some large scale models coming from fluid mechanics and molecular biology in addition to assessing these kinds of methods in real life situations we compare an algorithm of a buckley with a proposal by j nocedal the latter seems generally superior provided that careful attention is given to some nontrivial implementation aspects which concern the general question of properly initializing a quasi newton matrix in this context we find it appropriate to use a diagonal matrix generated by an update of the identity matrix so as to fit the rayleigh ellipsoid of the local hessian in the direction of the change in the gradient also a variational derivation of some rank one and rank two updates in hilbert spaces is given

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

inference based on imputed failure times for the proportional hazards model with interval censored data abstract we propose an approach to the proportional hazards model for interval censored data in which parameter estimates are obtained by solving estimating equations that are the partial likelihood score equations for the full data proportional hazards model averaged over all rankings of imputed failure times consistent with the observed censoring intervals imputed failure times are generated using a parametric estimate of the baseline distribution the parameters of the baseline distribution are estimated simultaneously with the proportional hazards regression parameters although a parametric form for the baseline distribution must be specified simulation studies show that the method performs well even when the baseline distribution is misspecified the estimating equations are solved using monte carlo techniques we present a recursive stochastic approximation scheme that converges to the zero of the estimating equations the solution has a random error that is asymptotically normally distributed w

multiplicative error bound for balanced stochastic truncation model reduction the multiplicative error for the balanced stochastic truncation model reduction method is discussed and an h sub infinity norm error bound is presented the result together with a previous result of the authors syst contr lett vol 14 p 307 14 1990 establishes that the same error bound applies to the relative error and the multiplicative error

hellinger consistency of certain nonparametric maximum likelihood estimators consider a class mathscr p p theta theta in theta of probability measures on a measurable space mathscr x mathscr a dominated by a sigma finite measure mu let f theta dp theta d mu theta in theta and let theta n be a maximum likelihood estimator based on n independent observations from p theta 0 theta 0 in theta we use results from empirical process theory to obtain convergence for the hellinger distance h f hat theta n f theta 0 under certain entropy conditions on the class of densities f theta theta in theta the examples we present are a model with interval censored observations smooth densities monotone densities and convolution models in most examples the convexity of the class of densities is of special importance

diagnosis and impacts of non gaussianity of innovations in data assimilation most of the atmospheric and oceanic data assimilation da schemes rely on the best linear unbiased estimator blue which is sub optimal if errors of assimilated data are non gaussian thus calling for a full bayesian data assimilation this paper contributes to the study of the non gaussianity of errors in the observational space possible sources of non gaussianity range from the inherent statistical skewness and positiveness of some physical observables e g moisture chemical species the nonlinearity both of the data assimilation models and of the observation operators among others deviations from gaussianity can be justified from a priori hypotheses or inferred from statistical diagnostics of innovations observation minus background leading to consistency relationships between the error statistics from samples of observations and backgrounds as well as their specified error variances we evaluate some measures of the innovation non gaussianity such as the skewness kurtosis and negentropy under the assumption of additive errors and by relating statistical moments from both data errors and innovations we identify potential sources of the innovation non gaussianity these sources range from 1 univariate error non gaussianity 2 nonlinear correlations between errors 3 spatio temporal variability of error variances heteroscedasticity and 4 multiplicative noise observational and background errors are often assumed independent this leads to variance dependent bounds for the skewness and the kurtosis of errors from innovation statistics we assess the potential da impact of some scenarios of non gaussian errors this impact is measured through the mean square difference between the blue and the minimum variance unbiased estimator mvue obtained with univariate observations and background estimates in order to accomplish this we compute maximum entropy probability density functions pdfs of the errors constrained by the first four order moments these pdfs are then used to compute the bayesian posterior pdf and the mvue the referred impact is studied for a large range of statistical moments being higher for skewed innovations and growing in average with the skewness of data errors specially if the skewnesses have the same sign an application has been performed to the quality accepted ecmwf innovations of brightness temperatures of a set of high resolution infrared sounder hirs channels in this context the mvue has led in some extreme cases to a potential reduction of 20 60 of the posterior error variance as compared to the blue specially for extreme values of the innovations

relative error bound for discrete balanced stochastic truncation fundamental differences between discrete time balanced stochastic truncation bst and its continuous time counterpart are shown to preclude a straightforward application of bilinear mappings for directly generalizing existing continuous time error bounds nevertheless discrete time bst is shown to satisfy precisely the same lk error bound as continuous time bst although the discrete time derivations are somewhat more complex

a nonparametric test for panel count data summary panel count data arise when a recurrent event is under investigation and each study subject is observed only at discrete time points in this situation observed data include only the numbers of occurrences of the event of interest between observation time points and no information is available on subjects between their observation time points we propose a nonparametric test for comparing the point processes characterising the recurrent event when only panel count data are available the asymptotic distribution of the test statistic is derived and a simulation study is conducted to evaluate its performance the method is illustrated using data from a medical follow up study

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

stability preserving rational approximation subject to interpolation constraints a quite comprehensive theory of analytic interpolation with degree constraint dealing with rational analytic interpolants with an a priori bound has been developed in recent years in this paper we consider the limit case when this bound is removed and only stable interpolants with a prescribed maximum degree are sought this leads to weighted h 2 minimization where the interpolants are parameterized by the weights the inverse problem of determining the weight given a desired interpolant profile is considered and a rational approximation procedure based on the theory is proposed this provides a tool for tuning the solution to specifications the basic idea could also be applied to the case with bounded analytic interpolants

a local likelihood proportional hazards model for interval censored data we discuss the use of local likelihood methods to fit proportional hazards regression models to right and interval censored data the assumed model allows for an arbitrary smoothed baseline hazard on which a vector of covariates operates in a proportional manner and thus produces an interpretable baseline hazard function along with estimates of global covariate effects for estimation we extend the modified em algorithm suggested by betensky lindsey ryan and wand we illustrate the method with data on times to deterioration of breast cosmeses and hiv 1 infection rates among haemophiliacs

functional anova modeling for proportional hazards regression the logarithm of the relative risk function in a proportional hazards model involving one or more possibly time dependent covariates is treated as a specified sum of a constant term main effects and selected interaction terms maximum partial likelihood estimation is used where the maximization is taken over a suitably chosen finite dimensional estimation space whose dimension increases with the sample size and which is constructed from linear spaces of functions of one covariate and their tensor products the l 2 rate of convergence for the estimate and its anova components is obtained an adaptive numerical implementation is discussed whose performance is compared to full likelihood hazard regression both with and without the restriction to proportional hazards

discrete model reduction preserving bounded realness abstract in this paper we propose a discrete model reduction method that preserves the bounded realness we define the discrete bounded real br balancing and obtain the reduced order discrete models from the generalized singular perturbation approximation gspa and direct truncation dt of the br balanced system we show the properties of the reduced order systems and derive the h norm error bounds for the cases of the gspa at 1 of the complex plane and the dt

hazard regression with interval censored data in a recent paper kooperberg stone and truong 1995a introduced hazard regression hare in which linear splines and their tensor products are used to estimate the conditional log hazard function based on possibly censored positive response data and one or more covariates model selection is carried out in an adaptive fashion using maximum likelihood estimation of the unknown coefficients rao and wald statistics to carry out stepwise addition and deletion of basis functions and the bayesian information criterion bic to select the final model in the present paper the hare methodology is extended to accommodate interval censored data time dependent covariates and cubic splines the presence of interval censored data means that the log likelihood function may no longer be concave presenting additional numerical challenges the extended methodology is applied to a data set containing both interval censoring and time dependent covariates the new software will be available in a future release of s plus

a unified approach to the calculation of information operators in semiparametric models the infinite dimensional information operator for the nuisance parameter plays a key role in semiparametric inference as it is closely related to the regular estimability of the target parameter calculation of information operators has traditionally proceeded in a case by case manner and has often entailed lengthy derivations with complicated arguments we develop a unified framework for this task by exploiting commonality in the form of semiparametric likelihoods the general formula developed allows one to derive information operators with simple calculus and if necessary at all a minimal amount of probabilistic evaluation this streamlined approach shows its simplicity and versatility in application to a number of existing models as well as a new model of practical interest

balanced stochastic realizations abstract the balanced stochastic realization is introduced as a balanced solution to the continuous time positive real equations and the dual positive real equations the structure of the associated balanced spectral factors is derived the main result being a product decomposition of the spectral factors the properties of the stochastic model reduction technique of balanced stochastic truncation are considered in particular it is shown that balanced stochastic truncation preserves the right half plane zeros of the spectral factors leading to a transformation approach to the approximation of non minimum phase stochastic systems

smooth estimation of the survival function for interval censored data interval censored data arise naturally in large scale panel studies where subjects can only be followed periodically and the event of interest can only be observed in some time intervals to estimate the survival function the non parametric maximum likelihood estimator npmle is commonly used which is a step function having some large jumps however in many applications the underlying survival function can be reasonably assumed to be smooth and then the npmle does not efficiently use this information two smooth estimators have been proposed in the literature one is based on the kernel smoothing of the npmle another is the logspline density model however to our knowledge there is no finite sample study yet to assess their performance in the literature in this paper we first show by simulation that both smooth estimators improve over the npmle for smooth survival functions we then apply these estimators to compare two survival curves the test statistics based on the maximum difference between two survival functions kolmogorov smirnov test and on the integrated weighted difference of two survival functions iwdb test are investigated via the bootstrap from our simulation studies the iwdb test seems particularly promising for some stochastically ordered survival functions that do not satisfy the proportional hazards model the methods are illustrated by reanalysing the breast cosmesis study data set copyright 2000 john wiley sons ltd

nonparametric estimation for a form of doubly censored data with application to two problems in aids abstract in many epidemiologic studies of human immunodeficiency virus hiv disease interest focuses on the distribution of the length of the interval of time between two events two such problems are considered here estimation of the distribution of time or number of sexual contacts between infection of an individual an index case and transmission of hiv to their sexual partner and estimation of the distribution of time between infectiousness as a blood donor and the development of detectable antibody data regarding these two problems are available from certain partner studies and the hiv lookback study in both cases the statistical development is complicated by the fact that the times of both events are interval censored so that the length of time between the events is never observed exactly nonparametric methods for estimation of the interval length distribution are developed by casting the problem in terms of nonparametric estimation of a mixing distribution particular attention is paid to

on maximum likelihood estimation in infinite dimensional parameter spaces an approximate maximum likelihood estimate is known to be consistent under some compactness and integrability conditions in this paper we study its convergence rate and its asymptotic efficiency in estimating smooth functionals of the parameter we provide conditions under which the rate of convergence can be established this rate is essentially governed by the size of the space of score functions as measured by an entropy index we also show that for a large class of smooth functionals the plug in maximum likelihood estimate is asymptotically efficient that is it achieves the minimal fisher information bound the theory is illustrated by several nonparametric or semiparametric examples

flexible estimation of survival curves conditional on non linear and time dependent predictor effects prognostic studies often estimate survival curves for patients with different covariate vectors but the validity of their results depends largely on the accuracy of the estimated covariate effects to avoid conventional proportional hazards and linearity assumptions flexible extensions of cox s proportional hazards model incorporate non linear nl and or time dependent td covariate effects however their impact on survival curves estimation is unclear our primary goal is to develop and validate a flexible method for estimating individual patients survival curves conditional on multiple predictors with possibly nl and or td effects we first obtain maximum partial likelihood estimates of nl and td effects and use backward elimination to select statistically significant effects into a final multivariable model we then plug the selected nl and td estimates in the full likelihood function and estimate the baseline hazard function and the resulting survival curves conditional on individual covariate vectors the td and nl functions and the log hazard are modeled with unpenalized regression b splines in simulations our flexible survival curve estimates were unbiased and had much lower mean square errors than the conventional estimates in real life analyses of mortality after a septic shock our model improved significantly the deviance likelihood ratio test 84 8 df 20 p 0 0001 and changed substantially the predicted survival for several subjects

data generation for the cox proportional hazards model with time dependent covariates a method for medical researchers the proliferation of longitudinal studies has increased the importance of statistical methods for time to event data that can incorporate time dependent covariates the cox proportional hazards model is one such method that is widely used as more extensions of the cox model with time dependent covariates are developed simulations studies will grow in importance as well an essential starting point for simulation studies of time to event models is the ability to produce simulated survival times from a known data generating process this paper develops a method for the generation of survival times that follow a cox proportional hazards model with time dependent covariates the method presented relies on a simple transformation of random variables generated according to a truncated piecewise exponential distribution and allows practitioners great flexibility and control over both the number of time dependent covariates and the number of time periods in the duration of follow up measurement within this framework an additional argument is suggested that allows researchers to generate time to event data in which covariates change at integer valued steps of the time scale the purpose of this approach is to produce data for simulation experiments that mimic the types of data structures applied that researchers encounter when using longitudinal biomedical data validity is assessed in a set of simulation experiments and results indicate that the proposed procedure performs well in producing data that conform to the assumptions of the cox proportional hazards model

state space approach to approximation by phase matching a state space realization of the stable part of the phase function of a proper spectral factor is provided the phase matching problem jonckheere and helton 1985 harshavardhana and jonckheere 1985 can now be solved completely in the state space framework using existing numerically reliable procedures further a connection is made between the stochastic balancing approach and the phase approximation procedure for the stochastic realization approximation

joint estimation of time dependent and non linear effects of continuous covariates on survival in order to yield more flexible models the cox regression model lambda t x lambda 0 t exp betax has been generalized using different non parametric model estimation techniques one generalization is the relaxation of log linearity in x lambda t x lambda 0 t exp r x another is the relaxation of the proportional hazards assumption lambda t x lambda 0 t exp beta t x these generalizations are typically considered independently of each other we propose the product model lambda t x lambda 0 t exp beta t r x which allows for joint estimation of both effects and investigate its properties the functions describing the time dependent beta t and non linear r x effects are modelled simultaneously using regression splines and estimated by maximum partial likelihood likelihood ratio tests are proposed to compare alternative models simulations indicate that both the recovery of the shapes of the two functions and the size of the tests are reasonably accurate provided they are based on the correct model by contrast type i error rates may be highly inflated and the estimates considerably biased if the model is misspecified applications in cancer epidemiology illustrate how the product model may yield new insights about the role of prognostic factors

smooth semiparametric regression analysis for arbitrarily censored time to event data a general framework for regression analysis of time to event data subject to arbitrary patterns of censoring is proposed the approach is relevant when the analyst is willing to assume that distributions governing model components that are ordinarily left unspecified in popular semiparametric regression models such as the baseline hazard function in the proportional hazards model have densities satisfying mild smoothness conditions densities are approximated by a truncated series expansion that for fixed degree of truncation results in a parametric representation which makes likelihood based inference coupled with adaptive choice of the degree of truncation and hence flexibility of the model computationally and conceptually straightforward with data subject to any pattern of censoring the formulation allows popular models such as the proportional hazards proportional odds and accelerated failure time models to be placed in a common framework provides a principled basis for choosing among them and renders useful extensions of the models straightforward the utility and performance of the methods are demonstrated via simulations and by application to data from time to event studies

calculation of maximum entropy densities with application to income distribution the maximum entropy approach is a flexible and powerful tool for density approximation this paper proposes a sequential updating method to calculate the maximum entropy density subject to known moment constraints instead of imposing the moment constraints simultaneously the sequential updating method incorporates the moment constraints into the calculation from lower to higher moments and updates the density estimates sequentially the proposed method is employed to approximate the size distribution of u s family income empirical evidence demonstrates the efficiency of this method

an improved error estimate for reduced order models of discrete time systems the authors derive under weaker conditions a discrete time counterpart of k glover s 1984 error estimate for reduced order models following the same lines a simple proof of glover s result for continuous time systems can be given

a transformation approach to stochastic model reduction a new and direct approach to stochastic model reduction is developed the order reduction algorithm is obtained by establishing an equivalence between canonical correlation analysis and solutions to algebraic riccati equations also the concept of balanced stochastic realization bsr plays a fundamental role asymptotic stability of the reduced order realization is established and spectral domain interpretations for the bsr are given

a general framework for parametric survival analysis parametric survival models are being increasingly used as an alternative to the cox model in biomedical research through direct modelling of the baseline hazard function we can gain greater understanding of the risk profile of patients over time obtaining absolute measures of risk commonly used parametric survival models such as the weibull make restrictive assumptions of the baseline hazard function such as monotonicity which is often violated in clinical datasets in this article we extend the general framework of parametric survival models proposed by crowther and lambert journal of statistical software 53 12 2013 to incorporate relative survival and robust and cluster robust standard errors we describe the general framework through three applications to clinical datasets in particular illustrating the use of restricted cubic splines modelled on the log hazard scale to provide a highly flexible survival modelling framework through the use of restricted cubic splines we can derive the cumulative hazard function analytically beyond the boundary knots resulting in a combined analytic numerical approach which substantially improves the estimation process compared with only using numerical integration user friendly stata software is provided which significantly extends parametric survival models available in standard software

a multiple imputation approach to regression analysis for doubly censored data with application to aids studies sun liao and pagano 1999 proposed an interesting estimating equation approach to cox regression with doubly censored data here we point out that a modification of their proposal leads to a multiple imputation approach where the double censoring is reduced to single censoring by imputing for the censored initiating times for each imputed data set one can take advantage of many existing techniques and software for singly censored data under the general framework of multiple imputation the proposed method is simple to implement and can accommodate modeling issues such as model checking which has not been adequately discussed previously in the literature for doubly censored data here we illustrate our method with an application to a formal goodness of fit test and a graphical check for the proportional hazards model for doubly censored data we reanalyze a well known aids data set

relative error h sub infinity identification from autocorrelation data a stochastic realization method a variant of the balanced stochastic truncation bst method for approximated realization of power spectrum matrices is shown to form the basis for an identification procedure that is well suited to the task of determining relative error bounded approximate plant models for use in control design from input output cross correlation data central to the theory is a novel l sub infinity norm bound on the relative error between an exact realization of the data and bst approximate realization

entropy densities with an application to autoregressive conditional skewness and kurtosis abstract the entropy principle yields for a given set of moments a density that involves the smallest amount of prior information we first show how entropy densities may be constructed in a numerically efficient way as the minimization of a potential next for the case where the first four moments are given we characterize the skewness kurtosis domain for which densities are defined this domain is found to be much larger than for hermite or edgeworth expansions last we show how this technique can be used to estimate a garch model where skewness and kurtosis are time varying we find that there is little predictability of skewness and kurtosis for weekly data

a semiparametric model for regression analysis of interval censored failure time data left right and interval censored response time data arise in a variety of settings including the analyses of data from laboratory animal carcinogenicity experiments clinical trials and longitudinal studies for such incomplete data the usual regression techniques such as the cox 1972 journal of the royal statistical society series b 34 187 220 proportional hazards model are inapplicable in this paper we present a method for regression analysis which accommodates interval censored data we present applications of this methodology to data sets from a study of breast cancer patients who were followed for cosmetic response to therapy a small animal tumorigenicity study and a clinical trial

semiparametric regression analysis of interval censored data summary we propose a semiparametric approach to the proportional hazards regression analysis of interval censored data an em algorithm based on an approximate likelihood leads to an m step that involves maximizing a standard cox partial likelihood to estimate regression coefficients and then using the breslow estimator for the unknown baseline hazards the e step takes a particularly simple form because all incomplete data appear as linear terms in the complete data log likelihood the algorithm of turnbull 1976 journal of the royal statistical society series b38 290 295 is used to determine times at which the hazard can take positive mass we found multiple imputation to yield an easily computed variance estimate that appears to be more reliable than asymptotic methods with small to moderately sized data sets in the right censored survival setting the approach reduces to the standard cox proportional hazards analysis while the algorithm reduces to the one suggested by clayton and cuzick 1985 applied statistics34 148 156 the method is illustrated on data from the breast cancer cosmetics trial previously analyzed by finkelstein 1986 biometrics42 845 854 and several subsequent authors