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

properties of hazard based residuals and implications in model diagnostics abstract model diagnostic procedures in failure time models using hazard based residuals rely on the assumption that the residual vector closely resembles a random sample from a unit exponential distribution when the model holds this article formally investigates the validity of this critical assumption by deriving and examining the properties of parametrically semiparametrically and nonparametrically estimated residuals for complete and right censored data the joint distribution of the residual vector is characterized and the behavior of some tests for exponentiality when applied to the residuals is examined analytically and through monte carlo methods findings reveal that the critical assumption of approximate unit exponentiality of the residual vector may not be viable and consequently the model diagnostic procedures considered which revolve on checking the approximate unit exponentiality of the residual vector specifically hazard plotting and the use of spacings and total time on test stati

on some new contiguous relations for the gauss hypergeometric function with applications contiguous relations for hypergeometric series contain an enormous amount of hidden information applications of contiguous relations range from the evaluation of hypergeometric series to the derivation of summation and transformation formulas for such series in this paper a new set of contiguous function relations are established applications of such relations to hypergeometric summation formulas and the theory of jacobi polynomials are presented

combining one sample confidence procedures for inference in the two sample case we present a simple general method for combining two one sample confidence procedures to obtain inferences in the two sample problem some applications give striking connections to established methods for example combining exact binomial confidence procedures gives new confidence intervals on the difference or ratio of proportions that match inferences using fisher s exact test and numeric studies show the associated confidence intervals bound the type i error rate combining exact one sample poisson confidence procedures recreates standard confidence intervals on the ratio and introduces new ones for the difference combining confidence procedures associated with one sample t tests recreates the behrens fisher intervals other applications provide new confidence intervals with fewer assumptions than previously needed for example the method creates new confidence intervals on the difference in medians that do not require shift and continuity assumptions we create a new confidence interval for the difference between two survival distributions at a fixed time point when there is independent censoring by combining the recently developed beta product confidence procedure for each single sample the resulting interval is designed to guarantee coverage regardless of sample size or censoring distribution and produces equivalent inferences to fisher s exact test when there is no censoring we show theoretically that when combining intervals asymptotically equivalent to normal intervals our method has asymptotically accurate coverage importantly all situations studied suggest guaranteed nominal coverage for our new interval whenever the original confidence procedures themselves guarantee coverage

optimal control of harvesting in a nonlinear elliptic system arising from population dynamics the aim of this article is to study an optimal control problem for a system of nonlinear elliptic equations with dirichlet boundary conditions u x a x x c x u x eu x u x x x 1 1 x b x u x d x x f x u x x x u x x 0 on 1 2 the authors were supported in part by cicyt ministry of education and culture spain under grant pb98 1343 by junta de andalucia fqm116 and by eec contract human capital and mobility program erbchrxct 940494 also they thank o arino for suggesting the study of this control problem

stability of nonlinear elliptic systems with distributed parameters and variable boundary data in this paper nonlinear partial differential equations of the elliptic type with the dirichlet boundary data are investigated some sufficient conditions under which the solutions of considered equations depend continuously on parameters and boundary conditions are proved the proofs of main results are based on variational methods in the final part of the paper we give a short survey of the results and methods related to the question of stability of the boundary value problems

efficiency of age adjusted tests in animal carcinogenicity experiments based on asymptotic relative efficiency calculations ryan 1985 biometrics 41 525 531 concludes that in the analysis of animal carcinogenicity experiments age adjusted tests of tumor rates should be routinely preferred to simple proportions tests for both lethal and nonlethal tumors we recalculate the asymptotic efficiencies of the simple proportions test relative to the log rank test for the lethal tumor case for a simplified model it is shown that the relative efficiency may be easily computed as a function of the crude tumor rate and the survival rate at the time of terminal sacrifice more generally we calculate by numerical quadrature the asymptotic relative efficiency for all models considered by ryan and using simulations examine the relevance of asymptotic efficiencies to typical sample sizes contrary to the numerical results of ryan we find for experiments with good survival and typical tumor rates that the relative efficiencies are greater than 95 usually about 99 in the nonlethal tumor case similar results follow from ryan for tumor rates and survival rates typically encountered in practice as it is often difficult to determine whether or not a tumor is lethal we conclude for equal interim mortality rates that the simple proportions test is usually adequate in evaluating animal carcinogenicity experiments

extended weighted log rank estimating functions in censored regression abstract in the censored regression model some regression estimators are introduced using weighted integrals of the log rank estimating functions their limiting covariance matrices do not involve the error density and can be reliably estimated inference can then be easily obtained some of these estimators have high asymptotic efficiency at some important submodels some lack of fit tests are derived that require little extra computing time these tests are asymptotically normal under the model and consistent against certain monotone or convex model misspecifications numerical studies show that for some weight functions the estimators and tests perform well implementation of the proposed procedures is discussed and illustrated in a real data example

a generalization of the product limit estimator with an application to censored regression the product limit estimator ple and weighted empirical processes are two important ingredients of almost any censored regression analysis a link between them is provided by the generalized ples introduced in this paper these generalized ples are the product limit integrals of the empirical cumulative hazard function estimators in which weighted empirical processes are used to replace the standard empirical processes the weak convergence and some large sample approximations of the generalized ples are established as an application these generalized ples are used to define some minimum distance estimators which are shown to be asymptotically normal these estimators are qualitatively robust in some submodels an optimal choice of the weight matrix is the covariate matrix and some of these estimators are quite efficient at a few common survival distributions to implement these estimators some computational aspects are discussed and an algorithm is given from a real data example and some preliminary simulation results these estimators seem to be very competitive to and more robust than some more traditional estimators such as the buckley james estimator

properties of generalized univariate hypergeometric functions based on spiridonov s analysis of elliptic generalizations of the gauss hypergeometric function we develop a common framework for 7 parameter families of generalized elliptic hyperbolic and trigonometric univariate hypergeometric functions in each case we derive the symmetries of the generalized hypergeometric function under the weyl group of type e 7 elliptic hyperbolic and of type e 6 trigonometric using the appropriate versions of the nassrallah rahman beta integral and we derive contiguous relations using fundamental addition formulas for theta and sine functions the top level degenerations of the hyperbolic and trigonometric hypergeometric functions are identified with ruijsenaars relativistic hypergeometric function and the askey wilson function respectively we show that the degeneration process yields various new and known identities for hyperbolic and trigonometric special functions we also describe an intimate connection between the hyperbolic and trigonometric theory which yields an expression of the hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric hypergeometric functions

spectral analysis and the haar functional on the quantum su 2 group the haar functional on the quantumsu 2 group is the analogue of invariant integration on the groupsu 2 if restricted to a subalgebra generated by a self adjoint element the haar functional can be expressed as an integral with a continuous measure or with a discrete measure or by a combination of both these results by woronowicz and koornwinder have been proved by using the corepresentation theory of the quantumsu 2 group and schur s orthogonality relations for matrix elements of irreducible unitary corepresentations these results are proved here by using a spectral analysis of the generator of the subalgebra the spectral measures can be described in terms of the orthogonality measures of orthogonal polynomials by using the theory of jacobi matrices

some scale estimators and lack of fit tests for the censored two sample accelerated life model some new scale estimators for the censored two sample accelerated life model are introduced they are zeros of some integrated weighted difference between the two cumulative hazard estimators these estimators are asymptotically normal the weight is chosen to result in estimators whose asymptotic variances do not involve the destiny functions and can be easily estimated this provides a fast and simple means of statistical inference in the censored two sample accelerated life model through investigating the asymptotic relative efficiency at some important censoring submodels and the finite example behaviors in various numerical studies we obtain some estimators with very competitive performance from the new class of scale estimators some lack of fit tests for the accelerated life model are also derived these tests are related to gill schumacher type tests and require little extra computing time once the estimator is obtained the estimators and tests are illustrated in two applications for a vaginal cancer data set for rats the effect of pretreatment regime was found to be well described by the two sample accelerated life model for a data set on progression of ovarian cancer it was found that the effect of grade of disease could not be described either by the two sample proportional hazards model or the two sample accelerated life model

effective sample sizes for confidence intervals for survival probabilities we examine various methods to estimate the effective sample size for construction of confidence intervals for survival probabilities we compare the effective sample sizes of cutler and ederer1 and peto et al 2 as well as a modified cutler ederer effective sample size we investigate the use of these effective sample sizes in the common situation of many censored observations that intervene between the time point of interest and the last death before this time we note that there is no a priori reason to treat upper and lower confidence intervals in a symmetric fashion since censored survival data are by nature asymmetric we recommend the use of the cutler ederer effective sample size in construction of upper confidence intervals and the peto effective sample size in construction of lower confidence intervals two examples with real data demonstrate the differences between confidence intervals formed with different effective sample sizes this study also illustrates the need for caution in the application of simulation studies to real problems

a comparison of the bias in four versions of the product limit estimator summary the bias of the product limit estimator depends entirely on the method used to complete the estimator beyond the last observation when it is censored the bias has been investigated for the upper gill 1980 and lower efron 1967 bounds on these methods rather than using a completion method investigators in practice commonly truncate the estimator using data driven rules the bias under two common truncation rules are considered i truncating at the last observation if it is censored and ii truncating at the n l th order statistic a number of inequalities between these biases are derived and properties of the bias functions are proved this investigation shows

criteria for bayesian model choice with application to variable selection in objective bayesian model selection no single criterion has emerged as dominant in defining objective prior distributions indeed many criteria have been separately proposed and utilized to propose differing prior choices we first formalize the most general and compelling of the various criteria that have been suggested together with a new criterion we then illustrate the potential of these criteria in determining objective model selection priors by considering their application to the problem of variable selection in normal linear models this results in a new model selection objective prior with a number of compelling properties

extensions of two q series expansions with applications to biorthogonal rational functions in this paper we derive two most general possible q hypergeometric expansion formulae for 12 11 q and r s z respectively the results are unique in the sense that no such results are available in the literature beyond 4 3 q so far and have the advantage that these results are also applicable to the top level 10 9 q biorthogonal rational functions

finite sample pointwise confidence intervals for a survival distribution with right censored data we review and develop pointwise confidence intervals for a survival distribution with right censored data for small samples assuming only independence of censoring and survival when there is no censoring at each fixed time point the problem reduces to making inferences about a binomial parameter in this case the recently developed beta product confidence procedure bpcp gives the standard exact central binomial confidence intervals of clopper and pearson additionally the bpcp has been shown to be exact gives guaranteed coverage at the nominal level for progressive type ii censoring and has been shown by simulation to be exact for general independent right censoring in this paper we modify the bpcp to create a mid p version which reduces to the mid p confidence interval for a binomial parameter when there is no censoring we perform extensive simulations on both the standard and mid p bpcp using a method of moments implementation that enforces monotonicity over time all simulated scenarios suggest that the standard bpcp is exact the mid p bpcp like other mid p confidence intervals has simulated coverage closer to the nominal level but may not be exact for all survival times especially in very low censoring scenarios in contrast the two asymptotically based approximations have lower than nominal coverage in many scenarios this poor coverage is due to the extreme inflation of the lower error rates although the upper limits are very conservative both the standard and the mid p bpcp methods are available in our bpcp r package published 2016 this article is us government work and is in the public domain in the usa

a partially parametric estimator of survival in the presence of randomly censored data many biological or medical experiments have as their goal to estimate the survival function of a specified population of subjects when the time to the specified event may be censored due to loss to follow up the occurrence of another event that precludes the occurrence of the event of interest or the study being terminated before the event of interest occurs this paper suggests an improvement of the kaplan meier product limit estimator when the censoring mechanism is random the proposed estimator treats the uncensored observations nonparametrically and uses a parametric model only for the censored observations one version of this proposed estimator always has a smaller bias and mean squared error than the product limit estimator an example estimating the survival function of patients enrolled in the ohio state university bone marrow transplant program is presented

what price kaplan meier summary the asymptotic efficiency of the kaplan meier product limit estimator relative to the maximum likelihood estimator of a parametric survival function is examined under a random censoring model

on the small sample performance of efron s and of gill s version of the product limit estimator under nonproportional hazards various proposals have been made to extend the product limit estimator to survival times beyond the largest observation in case that observation is censored two extreme extensions are examined with respect to bias and mean squared error mse their quality depends considerably on the censoring constellation the mse of one extension appears to be robust against a wide variety of nonproportionalities of the hazard rates of the distributions of lifelength and time to censoring

bootstrapping the kaplan meier estimator abstract randomly censored data consist of iid pairs of observations xi i i 1 n if i 0 xi denotes a censored observation and if i 1 xi denotes an exact survival time which is the variable of interest for estimating the distribution f of the survival times the product limit estimator proposed by kaplan and meier 1958 has been studied extensively and it has been shown to enjoy a number of optimality properties wellner 1982 see gill 1980 chaps 1 4 1983 for a modern treatment and for references with censored data bootstrapping can be carried out using two different resampling plans introduced by efron 1981 and reid 1981 respectively with efron s plan one takes a random sample with replacement from x 1 xn n whereas with reid s plan one takes a random sample from the kaplan meier estimator the purpose of this article is to study the asymptotic behavior of the bootstrapped kaplan meier estimator with both resampling plans the approach adopted uses the

contiguous relations basic hypergeometric functions and orthogonal polynomials iii associated continuous dual q hahn polynomials explicit solutions for the three term recurrence satisfied by associated continuous dual q hahn polynomials are obtained a minimal solution is identified and an explicit expression for the related continued fraction is derived the absolutely continuous component of the spectral measure is obtained eleven limit cases are discussed in some detail these include associated big q laguerre associated wall associated al salam chihara associated al salam carlitz i and associated continuous q hermite polynomials

covariates dependent confidence intervals for the difference or ratio of two median survival times in this paper we are concerned with the estimation of the discrepancy between two treatments when right censored survival data are accompanied with covariates conditional confidence intervals given the available covariates are constructed for the difference between or ratio of two median survival times under the unstratified and stratified cox proportional hazards models respectively the proposed confidence intervals provide the information about the difference in survivorship for patients with common covariates but in different treatments the results of a simulation study investigation of the coverage probability and expected length of the confidence intervals suggest the one designed for the stratified cox model when data fit reasonably with the model when the stratified cox model is not feasible however the one designed for the unstratified cox model is recommended the use of the confidence intervals is finally illustrated with a hiv data set

generalized fiducial inference for ultrahigh dimensional regression in recent years the ultrahigh dimensional linear regression problem has attracted enormous attention from the research community under the sparsity assumption most of the published work is devoted to the selection and estimation of the predictor variables with nonzero coefficients this article studies a different but fundamentally important aspect of this problem uncertainty quantification for parameter estimates and model choices to be more specific this article proposes methods for deriving a probability density function on the set of all possible models and also for constructing confidence intervals for the corresponding parameters these proposed methods are developed using the generalized fiducial methodology which is a variant of fisher s controversial fiducial idea theoretical properties of the proposed methods are studied and in particular it is shown that statistical inference based on the proposed methods will have correct asymptotic frequentist property in terms of empirical perform

small sample results for the kaplan meier estimator abstract whereas much is known about the asymptotic properties of the kaplan meier 1958 estimator kme of a survival function exact results for small samples have been difficult to obtain in this article we obtain an exact expression for the th moment 0 of the kme under a model of proportional hazards this enables us under proportional hazards to a study the bias of the kme and b compare the exact variance of the kme to its asymptotic variance

a k sample median test for censored data abstract a k sample median test for censored data is proposed the test is more sensitive to differences in median survival times than to differences in shape of the survival curves the asymptotic distribution of the test statistic is derived under both the null and the alternative hypotheses and its variance is shown to take a simple form the median test is compared to several other tests for censored data by computing asymptotic relative efficiencies and performing a monte carlo simulation under various conditions the procedure described is applied to data from a colorectal cancer clinical trial

contiguous relations basic hypergeometric functions and orthogonal polynomials i abstract the second order q difference equation of a 2 1 and the contiguous relations of 2 1 s are used to derive continued fraction expansions for quotients of 2 1 s we identify the cases when these continued fractions are positive definite j fractions i e the cases when the denominators of the approximants of the j fractions are orthogonal polynomials the measures with respect to which the denominator polynomials are orthogonal are found and the orthogonal polynomials are studied in some detail

contiguous relations basic hypergeometric functions and orthogonal polynomials ii associated big q jacobi polynomials abstract explicit solutions to the recurrence relation for the associated big q jacobi polynomials are obtained using 3 2 contiguous relations a new continued fraction is derived with the help of the minimal solution the orthogonality for polynomial solutions in some special cases is discussed two exceptional cases yield the basic analogues of entry 25 and entry 33 in chapter 12 of ramanujan s second notebook connections of the general case with other associated cases are indicated

inference for cumulative incidence quantiles via parametric and nonparametric approaches in survival analysis a point estimate and confidence interval for median survival time have been frequently used to summarize the survival curve however such quantile analyses on competing risks data have not been widely investigated in this paper we propose parametric inferences for quantiles from the cumulative incidence function and develop parametric confidence intervals for quantiles in addition we study a simplified method of inference for the nonparametric approach we compare the parametric and nonparametric inferences in empirical studies simulation studies show that the procedures perform well with parametric analyses yielding smaller mean square error when the model is not too badly misspecified we illustrate the methods with data from a breast cancer clinical trial copyright 2011 john wiley sons ltd

estimating the median survival time summary two methods are proposed for setting confidence intervals for the median survival time it is shown how these are related to bootstrap methods for censored data and some numerical comparisons are presented in many medical follow up studies the survival experience of a group of patients is summarized by the product limit or kaplan meier estimate of the survival distribution this may follow an analysis of the relation between survival time and some explanatory variables by for example the proportional hazards model the median of the estimated survival distribution is sometimes a convenient summary statistic and may be preferred to the mean because it is less sensitive to long tails in the estimated survival distribution this paper is concerned with estimating the distribution of the estimated median and providing confidence intervals for the true median two methods are considered in 2 they are both based in different ways on generalizations of the solution for the uncensored case these methods turn out to be connected to two ways of extending bootstrap methods for use with censored data this connexion is discussed in 3 thinking about extensions of the bootstrap to censored data highlights the need for careful consideration of the role of the censoring mechanism and how it should be incorporated into the data analysis section 4 provides some examples and monte carlo simulations the work is closely related to that of efron 1981 and emerson 1981

contiguous relations and their computations for f12 hypergeometric series the hypergeometric function f12 a 1 a 2 a 3 z plays an important role in mathematical analysis and its application gauss defined two hypergeometric functions to be contiguous if they have the same power series variable if two of the parameters are pairwise equal and if the third pair differs by 1 he showed that a hypergeometric function and any two other contiguous to it are linearly related in this paper we present an interesting formula as a linear relation of three shifted gauss polynomials in the three parameters a 1 a 2 and a 3 more precisely we obtained a recurrence relation including f12 a email protected 1 a 2 a 3 z f12 a 1 a email protected 2 a 3 z andf12 a 1 a 2 a email protected 3 z for any arbitrary integers a 1 a 2 and a 3

accurate bootstrap confidence limits for the cumulative hazard and survivor functions under random censoring abstract in clinical trials and other settings confidence intervals for survival probabilities at a fixed point in time are often required typically such intervals are constructed on the cumulative hazard scale the quantiles required for computing these intervals are generally obtained using a first order normal theory approximation the appropriateness of such intervals for small sample sizes or under moderate to heavy censoring is suspect we demonstrate that significant improvement is possible for randomly censored data using the nonparametric bootstrap in particular using edgeworth expansion it is shown that the percentile t and bc a bootstrap methodologies based on the studentized nelson aalen estimator yield second order correct confidence limits for the cumulative hazard function at a fixed time point an explicit formula for the correction term of the edgeworth expansion is found easy to compute estimators for the components of the correction term are also found and lead to simple analytic

stability in semilinear problems abstract in the paper we derive results concerning a continuous dependence of solutions on the right hand side for a semilinear operator equation lu g u by assuming that l d l h h h a hilbert space is self adjont with a closed range and g h r is continuous convex on h and g teaux differentiable on d l using these results we obtain theorems on the continuous dependence of solutions on functional parameters for a semilinear problem of the second order u au d u f t u t 0 a e with the dirichlet boundary conditions u 0 u 0 where a 1 and is a functional parameter

on the calibration problem extended to nonlinear models the traditional calibration problem is concerned with estimating the unknown value of an explanatory variable x corresponding to an observed value of a response variable y that is assumed to be functionally related to x the problem for which y is related to x by a simple linear model has been discussed in the literature this paper is concerned with obtaining a confidence interval for an unknown value of x corresponding to a specified or fixed value of y where y and x are related through a known nonlinear function given a specified value y of a response variable y that is related to x through a known function linear or nonlinear a point estimate of xc can be found through the inverse function if the inverse exists or through an iterative search procedure two approaches are presented for determining a confidence interval for xc given a value y when the inverse function exists a confidence interval about xc can be obtained from the approximate distribution of kc which provides a symmetric confidence interval about xc confidence intervals based on the asymptotic distribution of f3 are not symmetric about kc which is particularly appropriate when considering nonlinear functions that have asymptotes the technique based on the distribution of f3 is more general because it does not depend on having a closed form of the inverse function a simulation study for sample sizes of 20 involving an exponential decay function indicates that both approaches attained the desired confidence coefficient a test procedure comparing two calibration points from different populations or treatment groups is presented a simulation study of sample sizes of 20 involving an exponential decay function indicates that the test procedure attains the desired level of significance with good power additional simulation studies may be necessary to determine the appropriateness of these techniques for different model types and or sample sizes an example from meat sciences research is presented demonstrating the techniques discussed

simultaneous non parametric confidence intervals for survival probabilities from censored data in medical studies with censored data kaplan and meier s product limit estimator has frequent use as the estimate of the survival function simultaneous confidence intervals for the survival function at various time points constitute a useful addition to the analysis this study compares several such methods we consider in a simulation investigation two whole curve confidence bands and four methods based on the bonferroni inequality the results show that three bonferroni type methods are essentially equivalent all being better than the other methods when the number of time points is small 3 or 5

symmetric tests and confidence intervals for survival probabilities and quantiles of censored survival data we describe existing tests and introduce two new tests concerning the value of a survival function these tests may be used to construct a confidence interval for the survival probability at a given time or for a quantile of the survival distribution simulation studies show that error rates can differ substantially from their nominal values particularly at survival probabilities close to zero or one we recommend our new constrained bootstrap test for its good overall performance

non parametric estimation for the difference or ratio of median failure times for paired observations we propose a non parametric method to calculate a confidence interval for the difference or ratio of two median failure times for paired observations with censoring the new method is simple to calculate does not involve non parametric density estimates and is valid asymptotically even when the two underlying distribution functions differ in shape the method also allows missing observations we report numerical studies to examine the performance of the new method for practical sample sizes

the bayesian bootstrap the bayesian bootstrap is the bayesian analogue of the bootstrap instead of simulating the sampling distribution of a statistic estimating a parameter the bayesian bootstrap simulates the posterior distribution of the parameter operationally and inferentially the methods are quite similar because both methods of drawing inferences are based on somewhat peculiar model assumptions and the resulting inferences are generally sensitive to these assumptions neither method should be applied without some consideration of the reasonableness of these model assumptions in this sense neither method is a true bootstrap procedure yielding inferences unaided by external assumptions

a class of rank test procedures for censored survival data summary a class of linear rank statistics is proposed for the k sample problem with rightcensored survival data the class contains as special cases the log rank test mantel 1966 cox 1972 and a test essentially equivalent to peto peto s 1972 generalization of the wilcoxon test martingale theory is used to establish asymptotic normality of test statistics under the null hypotheses considered and to derive expressions for asymptotic relative efficiencies under contiguous sequences of alternative hypotheses a class of distributions is presented which corresponds to the class of rank statistics in the sense that for each distribution there is a statistic with some optimal properties for detecting location alternatives from that distribution some monte carlo results are displayed which present small sample behaviour

generalized fiducial inference a review and new results abstractr a fisher the father of modern statistics proposed the idea of fiducial inference during the first half of the 20th century while his proposal led to interesting methods for quantifying uncertainty other prominent statisticians of the time did not accept fisher s approach as it became apparent that some of fisher s bold claims about the properties of fiducial distribution did not hold up for multi parameter problems beginning around the year 2000 the authors and collaborators started to reinvestigate the idea of fiducial inference and discovered that fisher s approach when properly generalized would open doors to solve many important and difficult inference problems they termed their generalization of fisher s idea as generalized fiducial inference gfi the main idea of gfi is to carefully transfer randomness from the data to the parameter space using an inverse of a data generating equation without the use of bayes theorem the resulting generalized fiducial distribution gfd can

the associated askey wilson polynomials we derive some contiguous relations for very well poised 8 7 series and use them to construct two linearly independent solutions of the three term recurrence relation of the associated askey wilson polynomials we then use these solutions to find explicit representations of two families of associated askey wilson polynomials we identify the corresponding continued fractions as quotients of two very well poised 807 series and find the weight functions

a note on confidence limits for quartiles with right censored data two new methods are proposed for constructing the confidence limits for quartiles the small sample behaviour of these two methods is compared with the jennison and turnbull modified brookmeyer crowley method at three quartiles the simulation study indicates that one of the new methods the smoothed modified reflected method is preferred over the other methods when the censoring rate is less than 40 per cent while the jennison and turnbull method is preferred for higher censoring the results for the upper quartile are similar to the median for the lower quartile with small sample and high censoring semi infinite intervals may often occur the correct practice is to permit the semi infinite intervals without modifying them since it is more informative and gives closer to nominal level coverage

nonparametwic confidence intervals for the median in the presence of right censoriilg a method is proposed for constructing nonpararnetric confidence intervals for the median of a survival distribution the method applies when survival data are subject to random right censoring the confidence interval is constructed by inverting a hypothesis test the test used being one that generalizes the binomial test to accommodate censoring a nonparametric estimate of the eventual failure time for each censored observation modifies the usual binomial test statistic the estimates are computed using the kaplan meier product limit estimate of the survival function a large simulation study indicates that the nonparametric method performs well for various survival distributions of different shapes two existing techniques that assume exponentiality perform well for exponential data they give good coverage probabilities with intervals that are shorter than those provided by the nonparametric approach the nonparametric intervals exhibit superior coverage probabilities when data are generated from nonexponential distributions this is the case for weibull data with both increasing and decreasing hazard functions the nonparametric construction is appropriate for samples of size 10 or greater with up to 50 censoring a monte carlo study supports the use of a binomial distribution to approximate the exact discrete distribution of the test statistic median survival time is often used for evaluation of the efficacy of a treatment for a chronic disease one would like to provide a confidence interval for the median as well as a point estimate when little is known about the shape of the survival distribution an interval is needed for which no particular parametric form for the distribution is assumed a nonparametric confidence interval for the median is easily formed when all observations are complete the problem increases in subtlety and complexity when incomplete or censored observations exist in this paper a method is proposed for constructing nonparametric confidence intervals for the median when observations may be right censored the distribution of survival time may be either discrete or continuous and it need not have finite moments the method does assume however that the censoring mechanism is noninformative in that censoring acts independently of survival time see lagakos 1979 in the absence of censoring the usual construction of a nonparametric confidence interval for the median inverts the sign test this test is based upon the number of failures that occur beyond the hypothesized value of the median the present paper generalizes the procedure to allow for right censoring by using the kaplan meier estimate of survival to approximate the number of failures beyond the hypothesized median confidence interval estimation when data are censored has received recent attention in

a simple hybrid variance estimator for the kaplan meier survival function in this paper we propose a hybrid variance estimator for the kaplan meier survival function this new estimator approximates the true variance by a binomial variance formula where the proportion parameter is a piecewise non increasing function of the kaplan meier survival function and its upper bound as described below also the effective sample size equals the number of subjects not censored prior to that time in addition we consider an adjusted hybrid variance estimator that modifies the regular estimator for small sample sizes we present a simulation study to compare the performance of the regular and adjusted hybrid variance estimators to the greenwood and peto variance estimators for small sample sizes we show that on average these hybrid variance estimators give closer variance estimates to the true values than the traditional variance estimators and hence confidence intervals constructed with these hybrid variance estimators have more nominal coverage rates indeed the greenwood and peto variance estimators can substantially underestimate the true variance in the left and right tails of the survival distribution even with moderately censored data finally we illustrate the use of these hybrid and traditional variance estimators on a data set from a leukaemia clinical trial

the formal definition of reference priors reference analysis produces objective bayesian inference in the sense that inferential statements depend only on the assumed model and the available data and the prior distribution used to make an inference is least informative in a certain information theoretic sense reference priors have been rigorously defined in specific contexts and heuristically defined in general but a rigorous general definition has been lacking we produce a rigorous general definition here and then show how an explicit expression for the reference prior can be obtained under very weak regularity conditions the explicit expression can be used to derive new reference priors both analytically and numerically

nonparametric quantile inference with competing risks data a conceptually simple quantile inference procedure is proposed for cause specific failure probabilities with competing risks data the quantiles are defined using the cumulative incidence function which is intuitively meaningful in the competing risks set up we establish the uniform consistency and weak convergence of a nonparametric estimator of this quantile function these results form the theoretical basis for extensions of standard one sample and two sample quantile inference for independently censored data this includes the construction of confidence intervals and bands for the quantile function and two sample tests simulation studies and a real data example illustrate the practical utility of the methodology

supremum versions of the log rank and generalized wilcoxon statistics abstract considerable progress has been made in generalizing to censored data hypothesis tests based on rank statistics the most commonly used generalizations are the log rank test statistic which extends the savage exponential scores statistic cox 1972 mantel 1966 and the generalized wilcoxon statistics gehan 1965 peto and peto 1972 gill 1980 showed that in the two sample problem the asymptotic efficiency properties of the savage and wilcoxon statistics are maintained in censored data by the log rank and peto and peto wilcoxon statistics respectively when censoring is the same in both samples heuristics and simulations have implied that some recently proposed supremum statistics closely related to these two popular linear rank statistics can be used to construct tests that provide good power against a much wider range of nonlocal alternatives see e g fleming and harrington 1981 fleming o brien o fallon and harrington 1980 gill 1980 schumacher 1984 in this article we consider n

existence of optimal controls via continuous dependence on parameters abstract we consider a system of elliptic equations and derive sufficient conditions for solutions of the system to depend continuously on parameters these conditions are employed to obtain an existence result for an optimal control problem

spectral transformation chains and some new biorthogonal rational functions a discrete time chain associated with the generalized eigenvalue problem for two jacobi matrices is derived various discrete and continuous symmetries of this integrable equation are revealed a class of its rational elementary and elliptic functions solutions appearing from a similarity reduction are constructed the latter lead to large families of biorthogonal rational functions based upon the very well poised balanced hypergeometric series of three types the standard hypergeometric series 9 f 8 basic series 10 9 and its elliptic analogue 10 e 9 for an important subclass of the elliptic biorthogonal rational functions the weight function and normalization constants are determined explicitly

zeros and orthogonality of the askey wilson polynomials for q a root of unity we study some properties of the askey wilson polynomials awp when q is a primitive nth root of unity for general four parameter awp zeros of the nth polynomial and the orthogonality measure are found explicitly special subclasses of the awp e g the continuous q jacobi and big q jacobi polynomials are considered in detail a set of discrete weight functions positive on a real interval is described some new trigonometric identities related to the awp are obtained normalization conditions of some polynomials are expressed in terms of the gauss sums

symmetry groups of bailey s transformations for 10 φ 9 series although most of the symmetry groups or invariance groups associated with two term transformations between basic hypergeometric series have been studied and identified this is not the case for the most general transformation formulae in the theory of basic hypergeometric series namely bailey s transformations for f910 series first we show that the invariance group for both bailey s two term transformations for terminating f910 series and bailey s four term transformations for non terminating f910 series rewritten as a two term transformation of a so called f series is isomorphic to the weyl group of type e 6 we continue our recent research concerning the group structure underlying three term transformations s lievens j van der jeugt invariance groups of three term transformations for basic hypergeometric series j comput appl math 197 2006 1 14 and demonstrate that the group associated with a three term transformation between these f series each admitting bailey s two term transformation is the weyl group of type e 7 we do this by giving a description of the root system of type e 7 that allows to find a transformation between equivalent three term identities in an easy way a computation shows that there are five essentially different three term transformations between these f series we give an explicit form of each of these five transformations in an elegant way to our knowledge only one of these transformations has appeared in the literature

weighted kaplan meier statistics large sample and optimality considerations summary we propose a cumulative weighted difference in the kaplan meier estimates as a test statistic for equality of distributions in the two sample censored data survival analysis problem for stability of such a statistic the absolute value of the possibly random weight function must be bounded above by a multiple of c 1 2 6 where 1 c is the left continuous censoring distribution function and 6 0 for these weighted kaplan meier wkm statistics asymptotic distribution theory is presented along with expressions for the efficacy under a sequence of local alternatives a simple censored data generalization of the two sample difference in means test z test is a member of this class and in large samples is seen to be quite efficient relative to the popular log rank test under a range of alternatives including the proportional hazards alternative optimal weight functions are also calculated the optimal wkm statistic is as efficient as the optimal weighted log rank statistic for any particular sequence of local alternatives stratified statistics and trend statistics are also presented

repeated confidence intervals for the median survival time summary we describe methods for the construction of a confidence interval for median survival time based on right censored data these methods are extended to the construction of repeated confidence intervals for the median based on accumulating data here the overall probability that all intervals contain the true median is guaranteed at a fixed level the use of repeated confidence intervals for median survival time in post marketing surveillance is discussed a confidence interval for median survival time provides a useful summary of the survival experience of a group of patients if confidence intervals are calculated repeatedly as data accumulates the probability that at least one interval fails to contain the median may be much higher than the error rate for a single interval and if these confidence intervals are used in a decision making process the probability of an incorrect decision increases accordingly jennison turnbull 1984 have proposed methods for calculating repeated confidence intervals appropriate to such situations similar ideas have also been discussed by lai 1984 in 2 we propose a new form of single sample nonparametric confidence interval this interval has asymptotically correct coverage probability and monte carlo simulations suggest it is superior to its competitors for small sample sizes repeated confidence intervals for the median are presented in 3 and their stnall sample size performance is assessed by monte carlo simulation an example of their use is given in 4 all the methods considered can easily be modified to give confidence intervals for other quantiles or for the survival probability at a fixed time

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

approximate confidence intervals for probabilities of survival and quantiles in life table analysis for survival probabilities with censored data rothman 1978 journal of chronic diseases 31 557 560 has recommended the use of quadratic confidence limits based on the assumption that the product of the effective sample size at time t and the life table estimate of the survival probability past time t follows a binomial distribution this paper shows that the proposed confidence limits are asymptotically correct for continuous survival data these intervals as well as those based on the arcsine transformation the logit transformation and the log log transformation are compared by simulation to those based on greenwood s formula the usual method of interval estimation in life table analysis with large amounts of data the alternatives to the greenwood method all produce acceptable intervals on the basis of overall performance the intervals suggested by rothman are preferred for smaller samples any of these methods may be used to generate confidence sets for the median survival time or for any other quantile

accurate confidence limits for quantiles under random censoring in survival analysis estimates of median survival times in homogeneous samples are often based on the kaplan meier estimator of the survivor function confidence intervals for quantiles such as median survival are typically constructed via large sample theory or the bootstrap the former has suspect accuracy for small sample sizes under moderate censoring and the latter is computationally intensive in this paper improvements on so called test based intervals and reflected intervals cf slud byar and green 1984 biometrics 40 587 600 are sought using the edgeworth expansion for the distribution of the studentized nelson aalen estimator derived in strawderman and wells 1997 journal of the american statistical association 92 we propose a method for producing more accurate confidence intervals for quantiles with randomly censored data the intervals are very simple to compute and numerical results using simulated data show that our new test based interval outperforms commonly used methods for computing confidence intervals for small sample sizes and or heavy censoring especially with regard to maintaining specified coverage

a sample size formula for the supremum log rank statistic un avantage du test du log rank supremum sur le test du log rank standard est une sensibilite accrue a une plus large variete d alternatives stochastiques ordonnees dans ce papier nous etablissons une formule de calcul de la taille de l echantillon pour des etudes utilisant le test du log rank supremum l idee est de fonder la puissance sur l alternative de hasards proportionnels si bien que le log rank supremum aura la meme puissance que le log rank standard dans la situation ou le log rank standard est optimal cela a pour consequence une legere augmentation de la taille de l echantillon par rapport a celle requise pour le log rank standard par exemple cette augmentation est de 5 733 pour un test bilateral avec une erreur de type i de 0 05 et une puissance de 0 80 cette legere augmentation est compensee par les gains significatifs de puissance que permet le test du log rank supremum pour une large famille d alternatives a hazards non proportionnels une petite etude de simulation est utilisee a titre d illustration ces resultats devraient faciliter une utilisation plus large du test du log rank supremum dans les essais cliniques

a comparison of reflected versus test based confidence intervals for the median survival time based on censored data the small sample performance of some recently proposed nonparametric methods of constructing confidence intervals for the median survival time based on randomly right censored data is compared with that of two new methods most of these methods are equivalent for large samples all proposed intervals are either test based or reflected intervals in the sense defined in the paper coverage probabilities for the interval estimates were obtained by exact calculation for uncensored data and by stimulation for three life distributions and four censoring patterns in the range of situations studied test based methods often have less than nominal coverage while the coverage of the new reflected confidence intervals is closer to nominal although somewhat conservative and these intervals are easy to compute

survival estimation using splines a nonparametric maximum likelihood procedure is given for estimating the survivor function from right censored data it approximates the hazard rate by a simple function such as a spline with different approximations yielding different estimators a special case is that proposed by nelson 1969 journal of quality technology 1 27 52 and altshuler 1970 mathematical biosciences 6 1 11 the estimators are uniformly consistent and have the same asymptotic weak convergence properties as the kaplan meier 1958 journal of the american statistical association 53 457 481 estimator however in small and in heavily censored samples the simplest spline estimators have uniformly smaller mean squared error than do the kaplan meier and nelson altshuler estimators the procedure is extended to estimate the baseline hazard rate and regression coefficients in the cox 1972 journal of the royal statistical society series b 34 187 220 proportional hazards model and is illustrated using experimental carcinogenesis data

two sample quantile tests under general conditions summary a simple nonparametric two sample test for equality of a given collection of quantiles is developed which can be applied to a variety of empirical distribution functions including the kaplan meier estimator a self consistent estimator for doubly censored data and an estimator for repeated measures data the null hypothesis tested is that the quantiles are equal but other aspects of the distributions may diver between the two samples this procedure can also be applied to quantile testing in group sequential clinical trials with staggered patient entry a simple simulation study demonstrates that the moderate sample size properties of this procedure are reasonable

generalized fiducial inference for logistic graded response models samejima s graded response model grm has gained popularity in the analyses of ordinal response data in psychological educational and health related assessment obtaining high quality point and interval estimates for grm parameters attracts a great deal of attention in the literature in the current work we derive generalized fiducial inference gfi for a family of multidimensional graded response model implement a gibbs sampler to perform fiducial estimation and compare its finite sample performance with several commonly used likelihood based and bayesian approaches via three simulation studies it is found that the proposed method is able to yield reliable inference even in the presence of small sample size and extreme generating parameter values outperforming the other candidate methods under investigation the use of gfi as a convenient tool to quantify sampling variability in various inferential procedures is illustrated by an empirical data analysis using the patient reported emotional distress data

estimates and confidence intervals for median and mean life in the proportional hazard model summary estimation procedures and confidence intervals are given for the median and mean survival time in the proportional hazard regression model for median survival the methods apply to censored data the procedures are based on cox s partial likelihood estimates of the linear model parameters in the log linear proportional hazard model and on breslow s estimate of the baseline hazard function the asymptotic properties of these semiparametric estimates are developed and they are compared with the optimal parametric estimates for the weibull regression model for the parameter values considered the more generally valid semiparametric estimate of mean survival loses little efficiency relative to the optimal parametric estimates in this model unless the weibull shape parameter is close to zero the efficiency loss for the semiparametric estimate of median survival is greater but not severe unless censoring is heavy we also compare the optimal parametric estimates of mean and median survival with the weibull shape parameter known and unknown and find that the efficiency loss for the unknown shape estimate is small

nonparametric estimation for the difference or ratio of median failure times a simple and purely nonparametric confidence interval procedure for the difference or ratio of two median failure times is proposed with censored observations the new proposal does not involve unstable and complicated nonparametric density estimates and is valid asymptotically even when the two underlying distribution functions differ in shape extensive numerical studies are carried out to examine the appropriateness of the new procedure for practical sample sizes

bootstrap type confidence intervals for quantiles of the survival distribution in this paper we outline and illustrate an easy to program method for analytically calculating both parametric and non parametric bootstrap type confidence intervals for quantiles of the survival distribution based on right censored data this new approach allows for the incorporation of covariates within the framework of parametric models the procedure is based upon the notion of fractional order statistics and is carried forth using a simple beta transformation of the estimated survival function parametric or non parametric it is the only direct method currently available in the sense that all other methods are based on inverting test statistics or employing confidence intervals for other survival quantities we illustrate that the new method has favourable coverage probabilities for median confidence intervals as compared to six other competing methods

objective priors for discrete parameter spaces this article considers the development of objective prior distributions for discrete parameter spaces formal approaches to such development such as the reference prior approach often result in a constant prior for a discrete parameter which is questionable for problems that exhibit certain types of structure to take advantage of structure this article proposes embedding the original problem in a continuous problem that preserves the structure and then using standard reference prior theory to determine the appropriate objective prior four different possibilities for this embedding are explored and applied to a population size model the hypergeometric distribution the multivariate hypergeometric distribution the binomial beta distribution and the binomial distribution the recommended objective priors for the first third and fourth problems are new

generalized fiducial inference for wavelet regression we apply fisher s fiducial idea to conduct statistical inference for wavelet regression we first develop a general methodology for handling model selection problems within the fiducial framework with this new methodology we then propose fiduical based methods for performing wavelet curve estimation as well as constructing both pointwise and curvewise confidence intervals it is shown that under some mild regularity conditions both the new fiducial based pointwise and curvewise confidence intervals have asymptotically correct coverage furthermore simulation results show that these new fiducial based methods especially for constructing pointwise confidence intervals also possess promising empirical properties to the best of our knowledge this is the first time that the fiducial idea has been applied to a nonparametric estimation problem

two sample inference for median survival times based on one sample procedures for censored survival data abstract confidence intervals for median survival times are derived for censored survival data the intervals are obtained by using the quantiles of the kaplan meier product limit estimator and have the same pitman efficiency as the intervals found by inverting the sign test two sample tests and confidence intervals for the difference in median survival times are then developed based on the comparison of the one sample confidence intervals several methods for choosing the confidence coefficients of the corresponding one sample confidence intervals are developed the pitman efficiencies of these two sample tests are the same as that of the median test proposed by brookmeyer and crowley 1982a the procedures can also be used for the accelerated failure time model proportional hazard model and the behrens fisher problem nonparametric two sample inference procedures are useful in comparing the responses of treatment and control groups in medical follow up studies the data are usually subjected to cens

on quasi orthogonal polynomials their differential equations discriminants and electrostatics abstract in this paper we develop a general theory of quasi orthogonal polynomials we first derive three term recurrence relation and second order differential equations for quasi orthogonal polynomials we also give an expression for their discriminants in terms of the recursion coefficients of the corresponding orthogonal polynomials in addition we investigate an electrostatic equilibrium problem where the equilibrium position of movable charges is attained at the zeros of the quasi orthogonal polynomials the examples of the freud weight w x e x 4 2 t x 2 and the jacobi weight w x 1 x 1 x are discussed in some detail finally we consider the nonlinear orthogonality preserving transformation and related matrix problem

contiguous relations continued fractions and orthogonality an 8 p 7 model abstract a contiguous relation for very well poised 8o7 basic hypergeometric functions is used to derive an explict expression for the associated continued fraction when the continued fraction terminates it has a singularity structure of simple poles which can be associated with a system of biorthogonal rational functions having a discrete weight function however in the nonterminating case the continued fraction does not have explicit singularities and thus we are unable to exhibit the associated orthogonality this is the first example of this type that we have encountered

comparison of median survival times with adjustment for covariates brookmeyer and crowley derived a non parametric confidence interval for the median survival time of a homogeneous population by inverting a generalization of the sign test for censored data the 1 alpha confidence interval for the median is essentially the set of all values t such that the kaplan meier estimate of the survival curve at time t does not differ significantly from one half at the two sided alpha level su and wei extended this approach to the two sample problem and derived a confidence interval for the difference in median survival times based on the kaplan meier estimates of the individual survival curves and a minimum dispersion test statistic here i incorporate covariates into the analysis by assuming a proportional hazards model for the covariate effects while leaving the two underlying survival curves virtually unconstrained i generate a simultaneous confidence region for the two median survival times adjusted to any selected value z of the covariate vector using a test based approach analogous to brookmeyer and crowley s for the one sample case this region is in turn used to derive a confidence interval for the difference in median survival times between the two treatment groups at the selected value of z employment of a procedure suggested by aitchison sets the level of the simultaneous region to a value that should yield at least approximately the desired confidence coefficient for the difference in medians simulation studies indicate that the method provides reasonably accurate coverage probabilities

a correlation goodness of fit test for randomly censored data summary for testing a composite goodness of fit hypothesis with randomly censored data a correlation statistic which is an analogue of the shapiro francia statistic is presented the test for exponentiality in case of light censoring is asymptotically robust against departures from the koziol green model of random censorship the power of the test is compared with the total time on test procedure and the new better than used test using monte carlo simulation as an example the test is applied to some data with prostate cancer