proportional means regression for censored medical costs summary the semiparametric proportional means model specifies that the mean function for the cumulative medical cost over time conditional on a set of covariates is equal to an arbitrary baseline mean function multiplied by an exponential regression function we demonstrate how to estimate the vector valued regression parameter using possibly censored lifetime costs the estimator is consistent and asymptotically normal with an easily estimable covariance matrix simulation studies show that the proposed methodology is appropriate for practical use an application to aids is provided

risk sensitive and risk neutral multiarmed bandits for the multiarmed bandit the classic result is probabilistic each state of each bandit markov chain with rewards has an index that is determined by an optimal stopping time for that state s bandit and expected discounted income is maximized by playing at each epoch a bandit whose current state has the largest index our approach is analytic not probabilistic it uses pairwise comparison in place of stopping times a simple recursion assigns to each state of each bandit a utility and an amplification of future utility that depend solely on the data for that state s bandit these utilities and amplifications determine whether or not one state dominates another we show that it is optimal to play at each epoch any bandit whose current state is not dominated by the current states of the other bandits we obtain this result by a coherent analysis that encompasses three models one with risk averse exponential utility one with risk seeking exponential utility and one with linear utility and either stopping or discounting we also show that the risk seeking case and a model of nash nash p 1980 a generalized bandit problem j roy statist soc b42 165 169 are equivalent to each other

estimating medical costs with censored data incompleteness of follow up data is a common problem in estimating medical costs naive analysis using summary statistics on the collected data can result in severely misleading statistical inference this paper focuses on the problem of estimating the mean medical cost from a sample of individuals whose medical costs may be right censored a class of weighted estimators which account appropriately for censoring are introduced our estimators are shown to be consistent and asymptotically normal with easily estimated variances the efficiency of these estimators is studied with the goal of finding as efficient an estimator for the mean medical cost as is feasible extensive simulation studies are used to show that our estimators perform well in finite samples even with heavily censored data for a variety of circumstances the methods are applied to a set of cost data from a cardiology trial conducted by the duke university medical center extensions to other censored data problems are also discussed

the circulant operator in the banach algebra of matrices abstract we study an operator c which maps every n by n matrix a n to a circulant matrix c a n that minimizes the frobenius norm a n c n f over all n by n circulant matrices c n the circulant matrix c a n called the optimal circulant preconditioner has proved to be a good preconditioner for a general class of toeplitz systems in this paper we give different formulations of the operator discuss its algebraic and geometric properties and compute its operator norms in different banach algebras of matrices using these results we are able to give an efficient algorithm for finding the superoptimal circulant preconditioner which is defined to be the minimizer of i c n 1 a n f over all nonsingular circulant matrices c n

block preconditioners with circulant blocks for general linear systems block preconditioner with circulant blocks bpcb has been used for solving linear systems with block toeplitz structure since 1992 r chan x jin a family of block preconditioners for block systems siam j sci statist comput 13 1992 1218 1235 in this new paper we use bpcbs to general linear systems with no block structure usually the bpcbs are constructed by partitioning a general matrix into a block matrix with blocks of the same size and then applying t chan s optimal circulant preconditioner t chan an optimal circulant preconditioner for toeplitz systems siam j sci statist comput 9 1988 766 771 to each block these bpcbs can be viewed as a generalization of t chan s preconditioner it is well known that the optimal circulant preconditioner works well for solving some structured systems such as toeplitz systems by using the preconditioned conjugate gradient pcg method but it is usually not efficient for solving general linear systems unlike t chan s preconditioner bpcbs used here are efficient for solving some general linear systems by the pcg method several basic properties of bpcbs are studied the relations of the block partition with the cost per iteration and the convergence rate of the pcg method are discussed numerical tests are given to compare the cost of the pcg method with different bpcbs

index policies for stochastic search in a forest with an application to r d project management this paper concerns a stochastic search problem in a forest as motivation consider the issue of investing in a research and development project each activity that could be undertaken in the project is represented as an edge in a forest each edge has a cost of being attempted and a probability of success an edge can be attempted if each of its predecessors has been attempted and if each of those attempts has succeeded the overall project succeeds if a path is found from a stem to a leaf of the forest all of whose edges are successful overall success yields an economic benefit the problem is to find an investment strategy that maximizes expected utility either with a linear or an exponential utility function this problem will be shown to have a simple solution each edge will be assigned an index such that expected utility is maximized by attempting at each opportunity an edge whose index is most positive terminating the search when no edges remain whose indices are positive these indices are nested in a way that makes them quick to compute

m preconditioner for m matrices linear systems with m matrices often appear in a wide variety of areas of biological physical and social sciences see a berman r plemmons nonnegative matrices in mathematical sciences siam press philadelphia 1994 in this paper we propose a preconditioner for solving the system with symmetric nonsingular m matrix by the preconditioned conjugate gradient pcg method we show that our proconditioner increases the convergence rate of the pcg method and reduces the operation cost numerical results are also given

the best circulant preconditioners for hermitian toeplitz systems ii the multiple zero case in 10 14 circulant type preconditioners have been proposed for ill conditioned hermitian toeplitz systems that are generated by nonnegative continuous functions with a zero of even order the proposed circulant preconditioners can be constructed without requiring explicit knowledge of the generating functions it was shown that the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers and that all eigenvalues are uniformly bounded away from zero therefore the conjugate gradient method converges linearly when applied to solving the circulant preconditioned systems in 10 14 it was claimed that this result can be the case where the generating functions have multiple zeros the main aim of this paper is to give a complete convergence proof of the method in 10 14 for this class of generating functions

a note on preconditioned block toeplitz matrices in siam j sci statist comput 13 1992 pp 948 966 ku and kuo proposed and analysed a block circulant preconditioner r mn for solving a family of block toeplitz systems t mn v b for a special class of block matrices called the quadrantally symmetric toeplitz matrices they proved that the eigenvalues of r mn 1 t mn are clustered around one except at most o m n outliers with t mn generated by a two dimensional rational function the superior convergence rate of the preconditioned conjugate gradient pcg method is explained by the clustering property of the spectrum of r mn 1 t mn however in their analysis there is no discussion on the positive definiteness of the matrix t mn and the preconditioner r mn is assumed to be invertible in this paper we give some results on these two aspects under the assumptions in the above referenced paper we prove that if the generating function f x y of t mn is positive then t mn is posit

the generalized superoptimal preconditioner abstract for any given n by n matrix a a specific circulant preconditioner t f a introduced by tyrtyshnikov e tyrtyshnikov optimal and super optimal circulant preconditioners siam j matrix anal appl 13 1992 459 473 is defined to be the solution of min c i c 1 a f over all n by n nonsingular circulant matrices c the preconditioner t f a called the superoptimal circulant preconditioner has been proved to be a good preconditioner for a large class of structured systems in this paper we study this preconditioner in the general case by using the moore penrose inverse we give a formula for the superoptimal preconditioner and discuss the stability properties of this preconditioner a spectral relation between the optimal and superoptimal preconditioned matrices in the general case is also given

an econometric model of network formation with degree heterogeneity i introduce a model of undirected dyadic link formation which allows for assortative matching on observed agent characteristics homophily as well as unrestricted agent level heterogeneity in link surplus degree heterogeneity like in fixed effects panel data analyses the joint distribution of observed and unobserved agent level characteristics is left unrestricted two estimators for the common homophily parameter 0 are developed and their properties studied under an asymptotic sequence involving a single network growing large the first tetrad logit tl estimator conditions on a sufficient statistic for the degree heterogeneity the second joint maximum likelihood jml estimator treats the degree heterogeneity a i0 i 1 super n as additional incidental parameters to be estimated the tl estimate is consistent under both sparse and dense graph sequences whereas consistency of the jml estimate is shown only under dense graph sequences

bootstrap confidence intervals for medical costs with censored observations medical costs data with administratively censored observations often arise in cost effectiveness studies of treatments for life threatening diseases mean of medical costs incurred from the start of a treatment until death or a certain time point after the implementation of treatment is frequently of interest in many situations due to the skewed nature of the cost distribution and non uniform rate of cost accumulation over time the currently available normal approximation confidence interval has poor coverage accuracy in this paper we propose a bootstrap confidence interval for the mean of medical costs with censored observations in simulation studies we show that the proposed bootstrap confidence interval had much better coverage accuracy than the normal approximation one when medical costs had a skewed distribution when there is light censoring on medical costs 25 per cent we found that the bootstrap confidence interval based on the simple weighted estimator is preferred due to its simplicity and good coverage accuracy for heavily censored cost data censoring rate 30 per cent with larger sample sizes n 200 the bootstrap confidence intervals based on the partitioned estimator has superior performance in terms of both efficiency and coverage accuracy we also illustrate the use of our methods in a real example

efficient estimation of the distribution of quality adjusted survival time quality of life is an important aspect in evaluation of clinical trials of chronic diseases such as cancer and aids quality adjusted survival analysis is a method that combines both the quantity and quality of a patient s life into one single measure in this paper we discuss the efficiency of weighted estimators for the distribution of quality adjusted survival time using the general representation theorem for missing data processes we are able to derive an estimator that is more efficient than the one proposed in zhao and tsiatis 1997 biometrika 84 339 348 simulation experiments are conducted to assess the small sample properties of this estimator and to compare it with the semiparametric efficiency bound the value of this estimator is demonstrated from an application of the method to a data set obtained from a breast cancer clinical trial

a generalized oil exploration problem abstract in this paper we consider a general version of the oil exploration problem model previously investigated by the authors and others we assume that a region is hypothesized to contain an unknown number of oilfields represented by a prior the discovery of oilfields is governed by a probability law which is a function of the number of undiscovered sources drilling a well costs c and the value of a field is fixed at v rewards are additive and discounted by a factor 0

semi parametric regression models for cost effectiveness analysis improving the efficiency of estimation from censored data in a cost effectiveness analysis using clinical trial data estimates of the between treatment difference in mean cost and mean effectiveness are needed several methods for handling censored data have been suggested one of them is inverse probability weighting and has the advantage that it can also be applied to estimate the parameters from a linear regression of the mean such regression models can potentially estimate the treatment contrast more precisely since some of the residual variance can be explained by baseline covariates the drawback however is that inverse probability weighting may not be efficient using existing results on semi parametric efficiency this paper derives the semi parametric efficient parameter estimates for regression of mean cost mean quality adjusted survival time and mean survival time the performance of these estimates is evaluated through a simulation study applying both the new estimators and the inverse probability weighted estimators to the results of the evaluate trial showed that the new estimators achieved a halving of the variance of the estimated treatment contrast for cost some practical suggestions for choosing an estimator are offered

gittins indices and oil exploration summary an earlier study by benkherouf and bather considered the question of when the results of oil exploration in a single area were sufficiently discouraging to render it optimal to stop drilling altogether this stopping problem is now considerably elaborated by the inclusion of a choice between several areas in which to explore our oil exploration problem is modelled as a bayesian sequential decision problem with a markovian structure optimal drilling strategies are determined by gittins indices these indices also play an important part in the evaluation of heuristics

estimating medical costs from incomplete follow up data estimation of the average total cost for treating patients with a particular disease is often complicated by the fact that the survival times are censored on some study subjects and their subsequent costs are unknown the naive sample average of the observed costs from all study subjects or from the uncensored cases only can be severely biased and the standard survival analysis techniques are not applicable to minimize the bias induced by censoring we partition the entire time period of interest into a number of small intervals and estimate the average total cost either by the sum of the kaplan meier estimator for the probability of dying in each interval multiplied by the sample mean of the total costs from the observed deaths in that interval or by the sum of the kaplan meier estimator for the probability of being alive at the start of each interval multiplied by an appropriate estimator for the average cost over the interval conditional on surviving to the start of the interval the resultant estimators are consistent if censoring occurs solely at the boundaries of the intervals in addition the estimators are asymptotically normal with easily estimated variances extensive numerical studies show that the asymptotic approximations are adequate for practical use and the biases of the proposed estimators are small even when censoring may occur in the interiors of the intervals an ovarian cancer study is provided

sensitivity analysis for stochastic scheduling problems we seek to evaluate the effect which departures from assumed parameter values have upon the performance of scheduling strategies most of the material is presented in terms of a single machine scheduling model denoted rj j 1 j n the theoretical basis of the sensitivity analysis is developed from previous work by the author and gittins

approximating the inverse of a symmetric positive definite matrix abstract it is shown for an n n symmetric positive definite matrix t t i j with negative off diagonal elements positive row sums and satisfying certain bounding conditions that its inverse is well approximated uniformly to order l n 2 by a matrix s s i j where s i j i j t i j 1 t i j being the kronecker delta function and t being the sum of the elements of t an explicit bound on the approximation error is provided

quantifying the totality of treatment effect with multiple event time observations in the presence of a terminal event from a comparative clinical study to evaluate the totality of one treatment s benefit risk profile relative to an alternative treatment via a longitudinal comparative clinical study the timing and occurrence of multiple clinical events are typically collected during the patient s follow up these multiple observations reflect the patient s disease progression burden over time the standard practice is to create a composite endpoint from the multiple outcomes the timing of the occurrence of the first clinical event to evaluate the treatment via the standard survival analysis techniques by ignoring all events after the composite outcome this type of assessment may not be ideal various parametric or semiparametric procedures have been extensively discussed in the literature for the purposes of analyzing multiple event time data many existing methods were developed based on extensive model assumptions when the model assumptions are not plausible the resulting inferences for the treatment effect may be misleading in this article we propose a simple nonparametric inference procedure to quantify the treatment effect which has an intuitive clinically meaningful interpretation we use the data from a cardiovascular clinical trial for heart failure to illustrate the procedure a simulation study is also conducted to evaluate the performance of the new proposal

a circulant preconditioner for the systems of lmf based ode codes in this paper a recently introduced block circulant preconditioner for the linear systems of the codes for ordinary differential equations odes is investigated most ode codes based on implicit formulas at each integration step need the solution of one or more unsymmetric linear systems that are often large and sparse here the boundary value methods a class of implicit methods for the numerical integration of odes based on linear multistep formulas are considered more in detail for initial value problems theoretical and practical arguments are given to show that the block circulant preconditioner can give fast preconditioned iterations for various classes of differential problems moreover the p circulants a recently introduced circulant approximation for unsymmetric toeplitz matrices are shown to be more suitable sometimes than other circulant matrices for the underlying block preconditioner

survival models for cost data the forgotten additive approach the usage of the aalen additive approach is proposed to model cost data using a monte carlo simulation in a wide set of scenarios we showed that the aalen model is performing well and can be a reasonable alternative to the standard gamma regression models in addition with reference to the costami trial data we highlighted the ability of the aalen model to offer additional information about the relationships between costs and specific covariates as compared with standard regression techniques

regression analysis of incomplete medical cost data the accumulation of medical cost over time for each subject is an increasing stochastic process defined up to the instant of death the stochastic structure of this process is complex in most applications the process can only be observed at a limited number of time points furthermore the process is subject to right censoring so that it is unobservable after the censoring time these special features of the medical cost data especially the presence of death and censoring pose major challenges in the construction of plausible statistical models and the development of the corresponding inference procedures in this paper we propose several classes of regression models which formulate the effects of possibly time dependent covariates on the marginal mean of cost accumulation in the presence of death or on the conditional means of cost accumulation given specific survival patterns we then develop estimating equations for these models by combining the approach of generalized estimating equations for longitudinal data with the inverse probability of censoring weighting technique the resultant estimators are shown to be consistent and asymptotically normal with simple variance estimators simulation studies indicate that the proposed inference procedures behave well in practical situations an application to data taken from a large cancer study reveals that the medicare enrollees who are diagnosed with less aggressive ovarian cancer tend to accumulate medical cost at lower rates than those with more aggressive disease but tend to have higher lifetime costs because they live longer

estimating mean cost using auxiliary covariates we study the estimation of mean medical cost when censoring is dependent and a large amount of auxiliary information is present under missing at random assumption we propose semiparametric working models to obtain low dimensional summarized scores an estimator for the mean total cost can be derived nonparametrically conditional on the summarized scores we show that when either the two working models for cost survival process or the model for censoring distribution is correct the estimator is consistent and asymptotically normal small sample performance of the proposed method is evaluated via simulation studies finally our approach is applied to analyze a real data set in health economics

semiparametric estimation of marginal mark distribution in many applications the outcome of interest is a mark such that its observation is contingent upon occurrence of an event with incomplete follow up data the marginal mark distribution is however nonparametrically nowhere identifiable in many practical situations to address this problem we suggest a semiparametric model that postulates a normal copula for the association between the mark and survival time but leaves the marginals unspecified we show identifiability of the marginal mark distribution under this model and propose an inference procedure the estimated marginal distribution function is consistent and asymptotically normal and it provides a basis for estimating summaries of the mark furthermore we propose graphical model checking methods and kolmogorov smirnov type goodness of fit tests simulation studies demonstrate that the inference procedure performs well in practical settings the method is applied to the estimation of lifetime medical cost in a lung cancer trial

joint regression analysis of survival and quality adjusted survival summary in this paper a semiparametric bivariate linear regression model for survival and quality adjusted survival is investigated even with a parametric specification for the joint distribution maximum likelihood is not applicable because of induced informative censoring we propose inference procedures based on estimating functions the estimators are consistent and asymptotically normal hypothesis tests and confidence intervals may be constructed with easy to implement resampling techniques simultaneous regression modeling of survival and quality adjusted survival has not been studied formally our methodology gives parameter estimates that are highly interpretable in the context of a cost effectiveness analysis the usefulness of the proposal is illustrated with a breast cancer dataset

stochastic scheduling with release dates and due dates we consider stochastic scheduling problems in which the processing times of jobs are independent exponentially distributed random variables the release dates are random variables with an arbitrary joint distribution and the due dates are random variables with a joint distribution that satisfies certain conditions our development establishes simple policies that minimize such criteria as the expected weighted sum of completion times and the expected weighted number of late jobs these results contrast markedly with the deterministic counterparts of these models for which no polynomial time algorithms are known

the best circulant preconditioners for hermitian toeplitz systems in this paper we propose a new family of circulant preconditioners for ill conditioned hermitian toeplitz systems a x b the preconditioners are constructed by convolving the generating function f of a with the generalized jackson kernels for an n by n toeplitz matrix a the construction of the preconditioners requires only the entries of a and does not require the explicit knowledge of f when f is a nonnegative continuous function with a zero of order 2p the condition number of a is known to grow as o n2p we show however that our preconditioner is positive definite and the spectrum of the preconditioned matrix is uniformly bounded except for at most 2p 1 outliers moreover the smallest eigenvalue is uniformly bounded away from zero hence the conjugate gradient method when applied to solving the preconditioned system converges linearly the total complexity of solving the system is therefore of o n log n operations in the case when f is positive we show that the convergence is superlinear numerical results are included to illustrate the effectiveness of our new circulant preconditioners

a generalization of t chan s preconditioner abstract in this paper we propose a new preconditioner for solving linear systems by the preconditioned conjugate gradient pcg method the preconditioner can be thought of as a generalization of the well known t chan s preconditioner for hermitian positive definite matrix we observe that our preconditioner is also hermitian positive definite the operation cost and convergence of the pcg method are discussed numerical experiments have been performed on structured problems to show the competitiveness of this preconditioner

a survey and some extensions of t chan s preconditioner abstract for any given n by n matrix an t chan s circulant preconditioner cf an proposed by t chan t chan an optimal circulant preconditioner for toeplitz systems siam j sci statist comput 9 1988 766 771 is defined to be the solution of min c n a n c n f over all n by n circulant matrices cn the cf an called the optimal circulant preconditioner by t chan has been proved to be a good preconditioner for a large class of structured systems a generalization of t chan s circulant preconditioner was given by huckle t huckle circulant and skew circulant matrices for solving toeplitz matrix problems siam j matrix anal appl 13 1992 767 777 which can be used for solving some general linear systems in this paper we review some old and develop some new properties of t chan s circulant preconditioner

smooth quantile ratio estimation in a study of health care expenditures attributable to smoking we seek to compare the distribution of medical costs for persons with lung cancer or chronic obstructive pulmonary disease cases to those without controls using a national survey which includes hundreds of cases and thousands of controls the distribution of costs is highly skewed toward larger values making estimates of the mean from the smaller sample dependent on a small fraction of the biggest values one approach to deal with the smaller sample is to rely on a simple parametric model such as the log normal but this makes the undesirable assumption that the distribution of the log expenditures is symmetric we propose a novel approach to estimate the mean difierence of two highly skewed distributions which we call smooth quantile ratio estimation square square is obtained by smoothing over percentiles the ratio of the cost quantiles of the cases and controls square deflnes a large class of estimators of including 1 the sample mean difference 2 the maximum likelihood estimate under log normal samples and 3 l estimates we detail asymptotic properties of square such as consistency and asymptotic normality and also provide a closed form expression for the asymptotic variance through a simulation study we show that square has lower mean squared error than several competitors including the sample mean difierence and log normal parametric estimates in several realistic situations we apply square to the 1987 national medicare expenditure survey to estimate the difierence in medical expenditures between persons sufiering from the smoking attributable diseases lung cancer and chronic obstructive pulmonary disease and persons without these diseases software in r ihaka and gentleman 1996 for the implementation of square and of all its special cases and the cost data used in this paper are available at http biostat jhsph edu fdominic square html

a note on t chan s preconditioner abstract a matrix is said to be stable if the real part of all the eigenvalues are negative in this note for some class of matrices we show that t chan s preconditioner is stable by using this result we prove the invertibility of some preconditioners proposed in numerical odes recently

empirical likelihood based confidence intervals for mean medical cost with censored data in this paper we propose empirical likelihood methods based on influence function and jackknife techniques for constructing confidence intervals for mean medical cost with censored data we conduct a simulation study to compare the coverage probabilities and interval lengths of our proposed confidence intervals with that of the existing normal approximation based confidence intervals and bootstrap confidence intervals the proposed methods have better finite sample performances than existing methods finally we illustrate our proposed methods with a relevant example

nonparametric estimation of the joint distribution of survival time and mark variables summary in many applications variables of interest are marks of the endpoint which are not observed when the survival time is censored this paper focuses on nonparametric estimation of the joint distribution and summaries of survival time and mark variables we establish a representation of the joint distribution function through the cumulative markspecific hazard function which is analogous to the product integral representation of univariate survival function we identify a basic data structure common to various applications propose nonparametric estimators and show that they maximise the likelihood we formulate the problem in the marked point process framework and study both finite and large sample properties of the estimators we show that the joint distribution function estimator is nearly unbiased uniformly strongly consistent and asymptotically normal we also derive asymptotic variances for the estimators and propose sample based variance estimates numerical studies demonstrate that both the estimators and their variance estimates perform well for practical sample sizes we outline an application strategy

some stability properties of t chan s preconditioner abstract a matrix is said to be stable if the real parts of all the eigenvalues are negative in this paper for any matrix a n we give some sufficient and necessary conditions for the stability of t chan s preconditioner c u a n

approximation algorithms for restless bandit problems the restless bandit problem is one of the most well studied generalizations of the celebrated stochastic multi armed bandit mab problem in decision theory in its ultimate generality the restless bandit problem is known to be pspace hard to approximate to any nontrivial factor and little progress has been made on this problem despite its significance in modeling activity allocation under uncertainty in this article we consider the feedback mab problem where the reward obtained by playing each of n independent arms varies according to an underlying on off markov process whose exact state is only revealed when the arm is played the goal is to design a policy for playing the arms in order to maximize the infinite horizon time average expected reward this problem is also an instance of a partially observable markov decision process pomdp and is widely studied in wireless scheduling and unmanned aerial vehicle uav routing unlike the stochastic mab problem the feedback mab problem does not admit to greedy index based optimal policies we develop a novel duality based algorithmic technique that yields a surprisingly simple and intuitive 2 approximate greedy policy to this problem we show that both in terms of approximation factor and computational efficiency our policy is closely related to the whittle index which is widely used for its simplicity and efficiency of computation subsequently we define a multi state generalization that we term monotone bandits which remains subclass of the restless bandit problem we show that our policy remains a 2 approximation in this setting and further our technique is robust enough to incorporate various side constraints such as blocking plays switching costs and even models where determining the state of an arm is a separate operation from playing it our technique is also of independent interest for other restless bandit problems and we provide an example in nonpreemptive machine replenishment interestingly in this case our policy provides a constant factor guarantee whereas the whittle index is provably polynomially worse by presenting the first o 1 approximations for nontrivial instances of restless bandits as well as of pomdps our work initiates the study of approximation algorithms in both these contexts

evaluating strategies for markov decision processes in parallel the authors work on the use of gittins indices in the evaluation of strategies for families of alternative bandit processes has found many applications among these are procedures for sensitivity analysis in stochastic scheduling this theoretical paper aims at developing results which will form the basis of an approach to strategy evaluation for a class of processes of greater complexity these are markov decision processes in parallel satisfying a condition first enunciated by whittle the theory of gittins indices forms the basis of the analysis

a confidence interval for the median survival time a nonparametric asymptotic confidence interval for the median survival time is developed for the case where data are subject to arbitrary right censoring this is accomplished by inverting a generalization of the sign test for censored data a simulation study shows that this nonparametric confidence interval performs well for a variety of underlying survival functions the procedure is applied to data from a clinical trial that compared f our dosage regimens of 5 uorouracil

index policies and a novel performance space structure for a class of generalised branching bandit problems our object of study is a new class of controlled stochastic systems called generalised branching bandits which include discounted branching bandits and generalised bandit problems as special cases these models allow us to study queueing scheduling and project scheduling problems in which the reward earned from processing a particular job is influenced by other job types present in the system our mode of analysis is the achievable region approach what is novel here is that the full system may be partitioned into two subsystems each of which satisfies its own set of conservation laws and has its own highly structured performance space an optimal policy for the full system is constructed from two sets of gittins indices derived from the conservation laws governing the two subsystems

submodular returns and greedy heuristics for queueing scheduling problems we consider a range of controlled stochastic systems satisfying conservation laws together with a reducibility property that says that appropriate laws continue to hold when access to the system is restricted to a subset of all possible demand job customer types we show that for linear objectives the optimum system wide performance is a nondecreasing submodular or supermodular function of the subset chosen and that these properties are inherited from the geometry of the performance space concerned these results are of considerable interest in their own right but they also motivate the use of greedy heuristics for the solution of a range of job selection and scheduling problems which have hitherto seemed intractable computational experience suggests that such heuristics perform very well

design and analysis of toeplitz preconditioners the solution of symmetric positive definite toeplitz systems ax b by the preconditioned conjugate gradient pcg method was recently proposed by strang 1986 and analyzed by chan 1989 and chan and strang 1989 the convergence rate of the pcg method depends on the choice of preconditioners for the given toeplitz matrices the authors present a general approach to the design of toeplitz preconditioners based on the idea to approximate a partially characterized linear deconvolution with circular deconvolutions all resulting preconditioners can therefore be inverted via various fast transform algorithms with o n log n operations for a wide class of problems the pcg method converges in a finite number of iterations independent of n so that the computational complexity for solving these toeplitz systems is o n log n

two armed restless bandits with imperfect information stochastic control and indexability we present a two armed bandit model of decision making under uncertainty where the expected return to investing in the risky arm increases when choosing that arm and decreases when choosing the safe arm these dynamics are natural in applications such as human capital development job search and occupational choice using new insights from stochastic control along with a monotonicity condition on the payoff dynamics we show that optimal strategies in our model are stopping rules that can be characterized by an index which formally coincides with gittins index our result implies the indexability of a new class of restless bandit models

the two sample problem with induced dependent censorship summary induced dependent censorship is a general phenomenon in health service evaluation studies in which a measure such as quality adjusted survival time or lifetime medical cost is of interest we investigate the two sample problem and propose two classes of nonparametric tests based on consistent estimation of the survival function for each sample the two classes of test statistics examine the cumulative weighted difference in hazard functions and in survival functions we derive a unified asymptotic null distribution theory and inference procedure the tests are applied to trial v of the international breast cancer study group and show that long duration chemotherapy significantly improves time without symptoms of disease and toxicity of treatment as compared with the short duration treatment simulation studies demonstrate that the proposed tests with a wide range of weight choices perform well under moderate sample sizes

circulant preconditioned wr bvm methods for ode systems we consider the solution of a system of ordinary differential equations odes by waveform relaxation wr iterations in conjunction with boundary value methods bvms the wr method is a continuous in time analogue of the stationary method and it iterates with functions in each wr iteration we use bvms to discretize systems of odes bvms are relatively new ode solvers based on linear multistep formulae in this paper we discuss the use of the generalized minimal residual gmres method with block circulant circulant block preconditioners for solving the linear systems arising from the application of bvms in each wr iteration these preconditioners are effective in speeding up the convergence rate of the gmres method numerical experiments are presented to illustrate the effectiveness of our methods

a linear regression model for the analysis of life times a linear model is suggested for the influence of covariates on the intensity function this approach is less vulnerable than the cox model to problems of inconsistency when covariates are deleted or the precision of covariate measurements is changed a method of non parametric estimation of regression functions is presented this results in plots that may give information on the change over time in the influence of covariates a test method and two goodness of fit plots are also given the approach is illustrated by simulation as well as by data from a clinical trial of treatment of carcinoma of the oropharynx

further results on the non parametric linear regression model in survival analysis this paper gives further developments of a non parametric linear regression model in survival analysis three subjects are studied first martingale residuals originally developed for the cox model are introduced for our linear model their theory is developed and they are shown to be useful for judging goodness of fit the second focus of the paper is on the use of bootstrap replications to judge which features of the cumulative regression plots are likely to reflect real phenomena and not merely random variation in particular this is applied to judging whether the effect of a covariate disappears over time a problem for which no formal test exists the third subject is density type or kernel estimation of the regression functions themselves this might give more direct information than the cumulative plots the approaches are illustrated by data from a clinical trial of carcinoma of the oropharynx and by survival times of grafts in renal patients

statistical methods for multivariate interval censored recurrent events multi type recurrent event data arise when two or more different kinds of events may occur repeatedly over a period of observation the scientific objectives in such settings are often to describe features of the marginal processes and to study the association between the different types of events interval censored multi type recurrent event data arise when the precise event times are unobserved but intervals are available during which the events are known to have occurred this type of data is common in studies of patients with advanced cancer for example where the events may represent the development of different types of metastatic lesions which are only detectable by conducting bone scans of the entire skeleton in this setting it is of interest to characterize the incidence of the various types of bone lesions to estimate the impact of treatment and other covariate effects on the development of new lesions and to understand the relationship between the processes generating the bone lesions we develop joint models for multi type interval censored recurrent events which accommodate dependencies between different types of events and enable one to examine the covariate effects via regression however since the marginal likelihood resulting from the multivariate random effect model is intractable we describe a gibbs sampling algorithm to facilitate model fitting and inference we use generalized estimating equations for estimation and inference based on marginal models the finite sample properties of the marginal approach are studied via simulation the estimates of both the regression coefficients and the variance covariance parameters are shown to have negligible bias and 95 per cent confidence intervals based on the asymptotic variance formula are shown to have excellent empirical coverage probabilities in all of the settings considered the application of these methods to data from a trial of women with advanced breast cancer provides insight into the clinical course of bone metastases in this population

median regression with censored cost data because of the skewness of the distribution of medical costs we consider modeling the median as well as other quantiles when establishing regression relationships to covariates in many applications the medical cost data are also right censored in this article we propose semiparametric procedures for estimating the parameters in median regression models based on weighted estimating equations when censoring is present numerical studies are conducted to show that our estimators perform well with small samples and the resulting inference is reliable in circumstances of practical importance the methods are applied to a dataset for medical costs of patients with colorectal cancer

calibration regression of censored lifetime medical cost analysis of lifetime medical cost is challenged by its irregular distribution and by incomplete follow up particularly the latter with limited study duration as is typical in medical research this article proposes the so called calibration regression to address these issues by modeling not only lifetime medical cost but also survival time in a semiparametric fashion it is postulated that both outcomes on possibly transformed scales linearly relate to the covariates however the bivariate model error distribution is completely unspecified whereas this conceptually simple model is a natural multivariate generalization of the accelerated failure time model this article proposes an inference procedure by extending the weighted log rank estimating function to the marked point process framework the resulting estimators are consistent and asymptotically normal in addition robustness and efficiency issues are discussed furthermore a novel sample based variance estimation procedure is proposed for es

stochastic scheduling with priority classes we consider controlled stochastic systems in which multiple job types customers projects jobs etc may require processing by one or more servers the job types are all members of priority classes and admissible controls must respect the constraints imposed by these for systems which when unconstrained satisfy generalised conservation laws we obtain the performance space and show that linear objectives are optimised by admissible controls which choose job types within each priority class according to a set of gittins indices various degrees of decomposition of the problem are described and illustrated in the special case of discounted and undiscounted branching bandits the theory is developed further to yield index based suboptimality bounds for general policies

asymptotics when the number of parameters tends to infinity in the bradley terry model for paired comparisons we are concerned here with establishing the consistency and asymptotic normality for the maximum likelihood estimator of a merit vector u 0 u t representing the merits of t 1 teams players treatments objects under the bradley terry model as t this situation contrasts with the well known neyman scott problem under which the number of parameters grows with t the amount of sampling and for which the maximum likelihood estimator fails even to attain consistency a key feature of our proof is the use of an effective approximation to the inverse of the fisher information matrix specifically under the bradley terry model when teams i and j with respective merits u i and u j play each other the probability that team i prevails is assumed to be u i u i u j suppose each pair of teams play each other exactly n times for some fixed n the objective is to estimate the merits u i s based on the outcomes of the nt t 1 2 games clearly the model depends on the u i s only through their ratios under some condition on the growth rate of the largest ratio u i u j 0 i j t as t the maximum likelihood estimator of u 1 u 0 u t u 0 is shown to be consistent and asymptotically normal some simulation results are provided

optimal control of two interacting service stations optimal controls described by switching curves in the two dimensional state space are shown to exist for the optimal control of a markov network with two service stations and linear cost the controls govern routing and service priorities finite horizon and long run average cost problems are considered and value iteration is a key tool nonconvex value functions are shown to exist for slightly more general networks nonconvex value functions are also shown to arise for a simple single station control problem in which the instantaneous cost is convex but not monotone nevertheless optimality of threshold policies is established for the single station problem the proof is based on a novel use of stochastic coupling and policy iteration

convergence analysis of superoptimal pcg algorithm for toeplitz systems with a fisher hartwig singularity recently lu and hurvich y lu c hurvich on the complexity of the preconditioned conjugate gradient algorithm for solving toeplitz systems with a fisher hartwig singularity siam j matrix anal appl 27 2005 638 653 used the preconditioned conjugate gradient method with the optimal circulant preconditioner proposed in chan t chan an optimal circulant preconditioner for toeplitz systems siam j sci statist comput 9 1988 766 771 for solving the toeplitz system t n f x b where the generating function f is given by f 1 e i 2d h with d e 7 7 0 the function h is positive continuous on and differentiable on 0 in this paper we will use the superoptimal circulant preconditioner proposed by tyrtyshnikov e tyrtyshnikov optimal and superoptimal circulant preconditioners siam j matrix anal appl 13 1992 459 473 to solve the same problem when 0 d 1 2 our convergence analysis shows that the number of iterations is bounded by o log 3 n and therefore the complexity of our algorithm is o n log 4 n we notice that the numerical performance of superoptimal preconditioner is almost the same as that of the optimal preconditioner

optimal preconditioners for functions of matrices abstract the optimal preconditioner c u a of a given matrix a was proposed in 1988 by t chan 6 since then it has been proved to be efficient for solving a large class of structured systems in this paper we construct the optimal preconditioners for different functions of matrices more precisely let f be a function of matrices from c n n to c n n given a c n n there are two possible optimal preconditioners for f a c u f a and f c u a in the paper we study properties of both c u f a and f c u a for different functions of matrices numerical experiments are given to illustrate the efficiency of the optimal preconditioners when they are used to solve f a x b

monotonic and insensitive optimal policies for control of queues with undiscounted costs we consider the problem of controlling the service and or arrival rates in queues with the objective of minimizing the total expected cost to reach state zero we present a unified simple method for proving that an optimal policy is monotonic in the number of customers in the system applications to average cost minimization over an infinite horizon are given both exponential and nonexponential models are considered the essential characteristic is a left skip free transition structure and a nondecreasing not necessarily convex holding cost function some of our results are insensitive to service time distributions

estimating the mean of an increasing stochastic process at a censored stopping time abstract data often come in the form of a longitudinal series say n whose observation has been stopped either by a terminal event u e g death or censoring event c e g lost to follow up when n is not deterministic it is generally not possible to use standard survival analysis tools e g the kaplan meier estimator to consistently estimate functionals of the distribution of n u because of induced informative censoring various seemingly unrelated estimators recently have been proposed to handle this difficulty within the context of specific problems ranging from the analysis of quality adjusted survival data to recurrent event data to lifetime medical cost data in this article i consider a general framework containing each of these specific examples as a special case propose a class of estimators for the mean of the stopped longitudinal process and establish the asymptotic properties of this class of estimators connections between the aforementioned estimators are established