differentiable interpolants for high order runge kutta methods for a particular family of pairs of explicit runge kutta methods of orders p 1 and p sets of efficient continuously differentiable interpolants of several orders up to p are characters zed algorithmically in terms of several arbitrary parameters the approach can be applied in an obvious way to yield interpolants for other types of explicit runge kutta methods as well as families of other types of explicit and implicit methods derivative evaluations required for each pair of methods are reused and additional derivative evaluations are selected in an attempt to minimize the total number of stages required the analysis provides a lower bound on the number of stages required and indicates for example why twelve stages are required to provide interpolants for eight stage pairs of orders 5 and 6 in contrast to the eighteen stages used to obtain a known interpolant of order 7 for a pair of methods of orders 6 and 7 only sixteen stages are required by the proposed derivation details for interpola

asymptotic properties of the product limit estimate under random truncation on donne une description precise du comportement asymptotique de l estimateur limite produit incluant une forme explicite simple de la structure de covariance asymptotique

piecewise linearized methods for initial value problems piecewise linearized methods for the solution of initial value problems in ordinary differential equations are developed by approximating the right hand sides of the equations by means of a taylor polynomial of degree one the resulting approximation can be integrated analytically to obtain the solution in each interval and yields the exact solution for linear problems three adaptive methods based on the norm of the jacobian matrix maintaining constant the value of the approximation errors incurred by the linearization of the right hand sides of the ordinary differential equations and richardson s extrapolation are developed numerical experiments with some nonstiff first and second order ordinary differential equations indicate that the accuracy of piecewise linearized methods is in general superior to those of the explicit modified second order accurate euler method and the implicit trapezoidal rule but lower than that of the explicit fourth order accurate runge kutta technique it is also shown that piecewise linearized methods do not exhibit computational i e spurious modes for the relaxation oscillations of the van der pol oscillator and for those systems of equations which satisfy certain conservation principles conserve more accurately the invariants than the trapezoidal rule an error bound for piecewise linearized methods is provided for ordinary differential equations whose right hand sides satisfy certain lipschitz conditions

asymptotic properties of censored linear rank tests a conjecture of prentice is established which states that for censored linear rank test exact scores based on conditional expectations can be replaced by approximate scores obtained by evaluating the score function at an estimate of the survival function we show that under minimal conditions asymptotically equivalent tests are obtained when either the kaplan meier altshuler or moment estimator of the survival function is used asymptotic normality is also established for a general random censorship model under the null hypothesis and for contiguous alternatives this is used to calculate efficacies and when the censoring times are i i d an expression for the asymptotic relative efficiency is given which is a natural generalization of the one for classical uncensored linear rank tests 1 introduction prentice 1978 has constructed a general class of linear rank tests for a regression model with censored data his approach was to specify a score function for a classical linear rank test in the absence of censoring chernoff and savage 1958 hajek and sidak 1967 to construct all possible rankings of the unobserved uncensored values which were consistent with observed censored sample and then to assign to each observation the average of all possible scores it could have received in the absence of censoring giving equal weighting to each possible uncensored ranking except for a few special scoring functions notably logarithmic or power law scores these averages are unwieldy when there is appreciable censoring and prentice suggested that asymptotically equivalent tests might arise if the score function was evaluated at some suitable estimate of the survivor function this is well known to be true for uncensored data but requires further justification when censoring is present because the estimate of the survivor function is more complicated in this work we show that prentice s conjecture is true under minimal conditions and also we show that it remains true if prentice s moment estimator of the survivor function is replaced by either the kaplan meier estimator or altshuler s estimator asymptotic normality of these tests is established under a random censorship model for the null hypothesis and contiguous alternatives by martingale methods this is used to compute efficacies for a general random censorship model and asymptotic relative efficiencies for i i d potential censoring times where fully efficient linear rank tests exist asymptotic normality for general alternatives has been established in cuzick 1982 by more complicated methods

traveling wave solutions of the camassa holm equation all weak traveling wave solutions of the camassa holm equation are classified we show that in addition to smooth solutions there are a multitude of traveling waves with singularities peakons cuspons stumpons and composite waves

a lil type result for the product limit estimator let x1 x2 xn be i i d r v s with p x u f u and y1 y2 yn be i i d p y u g u where both f and g are unknown continuous survival functions for i 1 2 n set i 1 if xi yi and 0 if xi yi and zi min itxi yi one way to estimate f from the observations zi i i l n is by means of the product limit p l estimator fn kaplan meier 1958 6

strong approximations of some biometric estimates under random censorship the empirical distribution function has been widely used as an estimator for the distribution function of the elements of a random sample it is not however appropriate when the observations are incomplete this is the case for example when the observations have been randomly censored on the right in the latter case the so called product limit estimate of kaplan and meier 16 has been generally accepted as a substitute for the empirical distribution function and it possesses many of the same properties cf meier 19 developing the corresponding statement of efron 8 weak convergence over a finite interval of the kaplan meier product limit estimate to a gaussian process when there is censorship on the right has been obtained by breslow and crowley 3 cf also meier 19 breslow and crowley obtained their result via the limiting distribution of the closely related cumulative hazard process under random censorship a generalisation of the efron breslow crowley theorem was formulated by yang 24 for the case when censorship on the right is performed by more competing risks these results have numerous statistical applications in areas such as medical follow up studies life testing actuarial sciences and demography koziol and green 18 have computed tables for the limiting distribution of the cram6r von mises statistic based on the kaplan meier product limit estimate under a specific random censorship model a point will be stressed in this paper however that by a certain transformation which seems to have escaped many researchers attention and is due to efron 8 it is always possible to obtain the standard wiener process in the limit the distributions of many functionals of which are well known this transformation was only recently used in a stimulating note by gillespie and fisher 14 to construct asymptotic confidence bands for the unknown distribution function in the kaplan meier model a general scheme of random censorship can be defined in the following way let x be a real random variable with distribution function f t

large sample behaviour of the product limit estimator on the whole line weak convergence results are proved for the product limit estimator on the whole line applications are given to confidence band construction estimation of mean lifetime and to the theory of q functions the results are obtained using stochastic calculus and in probability linear bounds for empirical processes

single peak solitary wave solutions for the ch kp 2 1 equation under boundary condition abstract the qualitative theory of differential equations is applied to the ch kp 2 1 equation u t 2 u x u 2 x x t u u x x u y y 0 our procedure shows that the ch kp 2 1 equation either has the regular peakon soliton cuspon soliton and smooth soliton solutions when sitting on the non zero constant pedestal lim x u a 0 or possesses compacton solutions only when lim x u a 0 in particular mathematical analysis and numerical simulations of the ch kp 2 1 equation are provided for those peakon cuspon compacton loop soliton and smooth soliton solutions

estimating time dependent roc curves using data under prevalent sampling prevalent sampling is frequently a convenient and economical sampling technique for the collection of time to event data and thus is commonly used in studies of the natural history of a disease however it is biased by design because it tends to recruit individuals with longer survival times this paper considers estimation of time dependent receiver operating characteristic curves when data are collected under prevalent sampling to correct the sampling bias we develop both nonparametric and semiparametric estimators using extended risk sets and the inverse probability weighting techniques the proposed estimators are consistent and converge to gaussian processes while substantial bias may arise if standard estimators for right censored data are used to illustrate our method we analyze data from an ovarian cancer study and estimate receiver operating characteristic curves that assess the accuracy of the composite markers in distinguishing subjects who died within 3 5 years from subjects who remained alive copyright 2016 john wiley sons ltd

the rate of strong uniform consistency for the product limit estimator the maximal deviation of the product limit estimate from the estimated distribution function is investigated as a consequence of a functional law of the iterated logarithm the log log law is proved for this deviation on appropriate half lines with the precise constants this result implies that the log log law need not hold in general for the maximal deviation on the whole line then a general asymptotic order of magnitude is determined for the latter deviation this order depends on the joint behaviour of the censoring and censored distributions in a well defined way corresponding to specific joint behaviours several limsup results are deduced as consequence including all the previously known log log type laws in improved form the results are also extended to the variable censoring model

functional limit laws for the increments of kaplan meier product limit processes and applications we prove functional limit laws for the increment functions of empirical processes based upon randomly right censored data the increment sizes we consider are classified into different classes covering the whole possible spectrum we apply these results to obtain a description of the strong limiting behavior of a series of estimators of local functionals of lifetime distributions in particular we treat the case of kernel density and hazard rate estimators

order barriers for continuous explicit runge kutta methods in this paper we deal with continuous numerical methods for solving initial value problems for ordinary differential equations the need for which occurs frequently in applications whereas most of the commonly used multistep methods provide continuous extensions by means of an interpolant which is available without making extra function evaluations this is not always the case for one step methods we consider the class of explicit runge kutta methods and provide theorems used to obtain lower bounds for the number of stages required to construct methods of a given unifonn order p these bounds are similar to the butcher barriers known for the discrete case and are derived up to order p 5 as far as we know the examples we present of 8 stage continuous runge kutta methods of uniform order 5 are the first of their kind

central limit theorems for l_p distances of kernel estimators of densities under random censorship a sequence of independent nonnegative random variables with common distribution function f is censored on the right by another sequence of independent identically distributed random variables these two sequences are also assumed to be independent we estimate the density function f of f by a sequence of kernel estimators f n t int infty infty k t x h n d hat f n x h n where h n is a sequence of numbers k is kernel density function and hat f n is the product limit estimator of f we prove central limit theorems for int t 0 f n t f t p d mu t 1 leq p infty 0 t leq infty where mu is a measure on the borel sets of the real line the result is tested in monte carlo trials and applied for goodness of fit

semiparametric analysis of survival data with left truncation and dependent right censoring summary studies of chronic life threatening diseases often involve both mortality and morbidity in observational studies the data may also be subject to administrative left truncation and right censoring because mortality and morbidity may be correlated and mortality may censor morbidity the lynden bell estimator for left truncated and right censored data may be biased for estimating the marginal survival function of the nonterminal event we propose a semiparametric estimator for this survival function based on a joint model for the two time to event variables which utilizes the gamma frailty specification in the region of the observable data first we develop a novel estimator for the gamma frailty parameter under left truncation using this estimator we then derive a closed form estimator for the marginal distribution of the nonterminal event the large sample properties of the estimators are established via asymptotic theory the methodology performs well with moderate sample sizes both in simulations and in an analysis of data from a diabetes registry

asymptotics for a censored generalized linear model with unknown link function for censored response variable against projected co variable a generalized linear model with an unknown link function can cover almost all existing models under censorship its special cases include the accelerated failure time model with censored data such a model in the uncensored case is called the single index model in econometrics in this paper we systematically study the asymptotic properties we derive the central limit theorem and the law of the iterated logarithm for an estimator of the direction parameter we also obtain the optimal convergence rate of an estimator of the unknown link function in the model

special exact soliton solutions for the k 2 2 equation with non zero constant pedestal abstract special exact solutions of the k 2 2 equation u t u 2 x u 2 xxx 0 are investigated by employing the qualitative theory of differential equations our procedure shows that the k 2 2 equation either has loop soliton cusped soliton and smooth soliton solutions when sitting on the non zero constant pedestal lim x u a 0 or possesses compacton solutions only when lim x u 0 mathematical analysis and numerical simulations are provided for these soliton solutions of the k 2 2 equation

estimating a distribution function with truncated and censored data a minor modification of the product limit estimator is proposed for estimating a distribution function not necessarily continuous when the data are subject to either truncation or censoring or to both by independent but not necessarily identically distributed truncation censoring variables making use of martingale integral representations and empirical process theory uniform strong consistency of the estimator is established and weak convergence results are proved for the entire observable range of the function numerical results are also given to illustrate the usefulness of the modification particularly in the context of truncated data

continuous explicit runge kutta methods of order 5 a continuous explicit runge kutta cerk method provides a continuous approximation to an initial value problem such a method may be obtained by appending additional stages to a discrete method or alternatively by solving the appropriate order conditions directly owren and zennaro have shown for order 5 that the latter approach yields some cerk methods that require fewer derivative evaluations than methods obtained by appending stages in contrast continuous methods of order 6 that require the minimum number of stages can be obtained by appending additional stages to certain discrete methods this article begins a study to understand why this occurs by making no assumptions to simplify solution of the order conditions the existence of other types of cerk methods of order 5 is established while methods of the new families may not be as good for implementation as the owren zennaro methods the structure is expected to lead to a better understanding of how to construct families of methods of higher order

repeated significance testing for a general class of statistics used in censored survival analysis abstract the asymptotic joint distribution of a class of sequentially computed statistics used in survival analysis is derived when the individuals under study enter serially and are subject to random loss to follow up these results are used for constructing repeated significance tests over real time

estimation of the truncation probability in the random truncation model under random truncation a pair of independent random variables x and y is observable only if x is larger than y the resulting model is the conditional probability distribution h x y p x x y y for the truncation probability p x y a proper estimate is not the sample proportion but n g n s df n s where f n and g n are product limit estimates of the distribution functions f and g of x and y respectivly we obtain a much simpler representation n for n with this the strong consistency an iid representation and hence asymptotic normality and a lil for the estimate are established the results are true for arbitrary f and g the continuity restriction on f and g often imposed in the literature is not necessary furthermore the representation n of n facilitates the establishment of the strong law for the product limit estimates f n and g n

a reliable treatment of the physical structure for the nonlinear equation k m n in this work the nonlinear equation k m n is studied for all possible values of m and n we show that this equation may exhibit compactons solitons or periodic solutions the analysis reveals the change of the physical structure of the solutions as a result of the change of m and n

b spline linear multistep methods and their continuous extensions in this paper starting from a sequence of results which can be traced back to i j schoenberg we analyze a class of spline collocation methods for the numerical solution of ordinary differential equations odes with collocation points coinciding with the knots such collocation methods are naturally associated to a special class of linear multistep methods here called b spline bs methods which are able to generate the spline values at the knots we prove that provided the additional conditions are appropriately chosen such methods are all convergent and a stable the convergence property of the bs methods is naturally inherited by the related spline extensions which by the way are easily and safely computable using their b spline representation

solutions of compact and noncompact structures for nonlinear klein gordon type equation the focusing and the defocusing branches of nonlinear klein gordon type equation kg n n are considered a framework is implemented to show that the first model exhibits compactons solitons that do not have exponential tails whereas the second model demonstrates solitary patterns solutions the two variants of kg n n equation will be examined in one and higher dimensions general formulas will be developed to present a fairly complete understanding of the solutions of compact and noncompact structures

on the rate of uniform convergence of the product limit estimator strong and weak laws by approximating the classical product limit estimator of a distribution function with an average of iid random variables we derive sufficient and necessary conditions for the rate of both strong and weak uniform convergence of the product limit estimator over the whole line these findings somehow fill a longstanding gap in the asymptotic theory of survival analysis the result suggests a natural way of estimating the rate of convergent we also prove a related conjecture raised by gill and discuss its application to the construction of a confidence interval for a survival function near the endpoint

the camassa holm kp equations with compact and noncompact travelling wave solutions abstract in this paper two variants of camassa holm kp equations are investigated compactons solitons with the absence of infinite tails solitons nonlinear localized waves of infinite support solitary patterns having infinite slopes or cusps and plane periodic solutions are formally derived the work highlights the qualitative change in the physical structures of the obtained solutions

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

high order embedded runge kutta formulae abstract the criteria to be satisfied by embedded runge kutta pairs of formulae are reviewed two new formulae of orders 6 and 8 are presented together with tests on their efficiency relative to other high order formulae in current use

estimating a distribution function with truncated data let mathscr p be a finite population with n geq 1 elements for each e in mathscr p let x e and y e be independent positive random variables with unknown distribution functions f and g and suppose that the pairs x e y e are i i d we consider the problem of estimating f g and n when the data consist of those pairs x e y e for which e in mathscr p and y e leq x e the nonparametric maximum likelihood estimators mles of f and g are described and their asymptotic properties as n rightarrow infty are derived it is shown that the mles are consistent against pairs f g for which f and g are continuous g 1 0 leq f 1 0 and g 1 1 leq f 1 1 sqrt n times estimation error for f converges in distribution to a gaussian process if int infty 0 1 g df infty but may fail to converge if this integral is infinite

nonparametric estimation with left truncated semicompeting risks data summary nonparametric estimators for competing risks data can be applied to semicompeting risks data a type of multi state data where a terminating event may censor a non terminating event after forcing the data into the competing risks format complications may arise with left truncation of the terminating event where the competing risks analysis naively truncates the nonterminating event using the left truncation time for the terminating event which may lead to large efficiency losses we propose nonparametric estimators which use all semicompeting risks information and do not require artificial truncation the uniform consistency and weak convergence of the estimators are established and variance estimators are provided simulation studies and an analysis of a diabetes registry demonstrate large efficiency gains over the naive estimators

functional laws of the iterated logarithm for the product limit estimator of a distribution function under random censorship or truncation functional laws of the iterated logarithm are established for a modified version of the classical product limit estimator of a distribution function when the data are subject to random censorship or truncation making use of stochastic integral representations and empirical process theory strong approximations involving i i d continuous parameter martingales are developed for the product limit estimator and these strong approximations are then applied to derive the functional laws

a class of nonlinear fourth order variant of a generalized camassa holm equation with compact and noncompact solutions abstract in this paper we study a class of nonlinear fourth order analogue of a generalized camassa holm equation it is shown that this class gives compactons conventional solitons solitary patterns and periodic solutions it is also found that the qualitative change in the physical structure of solutions depends mainly on the exponent of the wave function u x t positive or negative and on the coefficient of u n as well

travelling wave solutions of the fornberg whitham equation zhou and tian j b zhou l x tian a type of bounded travelling wave solutions for the fornberg whitham equation j math anal appl 346 2008 255 261 successfully found a type of bounded travelling wave solutions of the fornberg whitham equation in this paper we improve the previous result by using the phase portrait analytical technology moreover some smooth periodic wave smooth solitary wave periodic cusp wave and loop soliton solutions are given and the numerical simulation is made the results show that our theoretical analysis agrees with the numerical simulation