a large sample study of cox s regression model strong consistency and asymptotic normality are established for the maximum partial likelihood estimate of the regression parameter in cox s regression model estimates are also derived for the underlying cumulative hazard function and survival distribution we establish the asymptotic normality of these estimates and calculate the limiting variances

on inconsistency of breslow s estimator as an estimator of the hazard rate in the cox model in the cox proportional hazards model breslow s estimator of the cumulative hazard function is well known this estimator is sometimes presented in such a way that it appears to offer a method of estimating the hazard rate function as well we state and prove an asymptotic result for this estimator of the hazard rate which shows that the estimator is inconsistent and that it is asymptotically unbiased

an application of rough sets to graph theory this paper studies the testing bipartiteness of simple undirected graphs based on generalized rough sets a new binary relation induced from a simple undirected graph is first defined the concept of quasi outer definable sets is then introduced and some of its important properties are investigated finally based on the quasi outer definable set a new algorithm is designed for testing bipartiteness of a simple undirected graph experiments show that the proposed algorithm is more efficient than the traditional one when the graph is represented by an adjacency matrix

clustering in complex networks i general formalism we develop a full theoretical approach to clustering in complex networks a key concept is introduced the edge multiplicity that measures the number of triangles passing through an edge this quantity extends the clustering coefficient in that it involves the properties of two and not just one vertices the formalism is completed with the definition of a three vertex correlation function which is the fundamental quantity describing the properties of clustered networks the formalism suggests different metrics that are able to thoroughly characterize transitive relations a rigorous analysis of several real networks which makes use of this formalism and the metrics is also provided it is also found that clustered networks can be classified into two main groups the weak and the strong transitivity classes in the first class edge multiplicity is small with triangles being disjoint in the second class edge multiplicity is high and so triangles share many edges as we shall see in the following paper the class a network belongs to has strong implications in its percolation properties

local em estimation of the hazard function for interval censored data summary we propose a smooth hazard estimator for interval censored survival data using the method of local likelihood the model is fit using a local em algorithm the estimator is more descriptive than traditional empirical estimates in regions of concentrated information and takes on a parametric flavor in regions of sparse information we derive two different standard error estimates for the smooth curve one based on asymptotic theory and the other on the bootstrap we illustrate the local em method for times to breast cosmesis deterioration finkelstein 1986 biometrics42 845 854 and for times to hiv 1 infection for individuals with hemophilia kroner et al 1994 journal of aids7 279 286 our hazard estimates for each of these data sets show interesting structures that would not be found using a standard parametric hazard model or empirical survivorship estimates

estimation of the mean function with panel count data using monotone polynomial splines we study nonparametric likelihood based estimators of the mean function of counting processes with panel count data using monotone polynomial splines the generalized rosen algorithm proposed by zhang jamshidian 2004 is used to compute the estimators we show that the proposed spline likelihood based estimators are consistent and that their rate of convergence can be faster than n 1 3 simulation studies with moderate samples show that the estimators have smaller variances and mean squared errors than their alternatives proposed by wellner zhang 2000 a real example from a bladder tumour clinical trial is used to illustrate this method

on computing the edge connectivity of an uncertain graph uncertain graphs are employed to describe graph models with imprecise expert data in uncertain graphs due to the existence of uncertain edges edge connectivity is essentially an uncertain variable different from that in a deterministic graph it is more suitable to investigate the possibility uncertain measure that an uncertain graph is k edge connected which is the main aim of this paper we first deduce a theoretical formula to calculate the uncertain measure and on this basis we then propose an algorithm which is derived from maximum flow algorithms to numerically calculate the uncertain measure the proposed algorithm is also proved to be a polynomial time algorithm and its effectiveness and efficiency are illustrated by numerical examples

efficient pattern matching on big uncertain graphs a significant amount of research has been devoted to seeking efficient solutions to the problem of pattern matching over graphs this interest is largely due to the many applications that require such efficient solutions including protein complex prediction social network analysis and structural pattern recognition however in many real applications the graph data are often noisy incomplete and inaccurate in other words there exist many uncertain graphs therefore in this paper we study pattern matching in the context of large uncertain graphs specifically we want to retrieve all qualified matches of a query pattern in the uncertain graph though pattern matching over uncertain graphs is np hard we employ a filtering and verification framework to speed up the search in the filtering phase we propose a probabilistic matching tree pm tree built from match cuts obtained by a cut selection process based on the pm tree we devise a collective pruning strategy to prune a large number of unqualified matches during the verification phase we develop an efficient sampling algorithm to validate the remaining candidates extensive experimental results demonstrate the effectiveness and efficiency of the proposed algorithms finally we show how our solution can be applied to querying knowledge graphs

rough set theory for the interval valued fuzzy information systems the notion of a rough set was originally proposed by pawlak z pawlak rough sets international journal of computer and information sciences 11 5 1982 341 356 later on dubois and prade d dubois h prade rough fuzzy sets and fuzzy rough sets international journal of general system 17 2 3 1990 191 209 introduced rough fuzzy sets and fuzzy rough sets as a generalization of rough sets this paper deals with an interval valued fuzzy information system by means of integrating the classical pawlak rough set theory with the interval valued fuzzy set theory and discusses the basic rough set theory for the interval valued fuzzy information systems in this paper we firstly define the rough approximation of an interval valued fuzzy set on the universe u in the classical pawlak approximation space and the generalized approximation space respectively i e the space on which the interval valued rough fuzzy set model is built secondly several interesting properties of the approximation operators are examined and the interrelationships of the interval valued rough fuzzy set models in the classical pawlak approximation space and the generalized approximation space are investigated thirdly we discuss the attribute reduction of the interval valued fuzzy information systems finally the methods of the knowledge discovery for the interval valued fuzzy information systems are presented with an example

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

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

uncertain wave equation with infinite half boundary wave equation is a type of second order and hyperbolic partial differential equation it is a commonly used tool to model many kinds of wave propagations such as sound wave electromagnetic wave water wave and string vibration propagations similarly uncertain wave equation is a type of uncertain partial equation driven by liu process which is widely used to model the wave propagation with uncertain noise such as vibrating string in uncertain environment the existing literature has studied uncertain wave equation with infinite boundary since infinite boundary is a much ideal condition this paper aims at studying the uncertain wave equation with infinite half boundary

shortest path problem with uncertain arc lengths uncertainty theory provides a new tool to deal with the shortest path problem with nondeterministic arc lengths with help from the operational law of uncertainty theory this paper gives the uncertainty distribution of the shortest path length also it investigates solutions to the a shortest path and the most shortest path in an uncertain network it points out that there exists an equivalence relation between the a shortest path in an uncertain network and the shortest path in a corresponding deterministic network which leads to an effective algorithm to find the a shortest path and the most shortest path roughly speaking this algorithm can be broken down into two parts constructing a deterministic network and then invoking the dijkstra algorithm

uncertain multi product newsboy problem with chance constraint a multi product newsboy problem with uncertain demand and uncertain storage space is investigated under a warehousing chance constraint it is assumed that indeterminacy may appear in demand and storage space due to rough estimation of their distributions by decision maker uncertainty theory provides a new tool to deal with the human uncertainty we develop an uncertain expected value programming model uevpm based on uncertainty theory uncertain variables functions from uncertainty space to the set of real numbers are used to describe subjective estimation the objective function is to maximize the expected profit of newsboy at a predetermined confidence level furthermore we convert the chance constrained uncertain programming model to its equivalent deterministic form so as to be solved by classic integer programming method finally three numerical examples are given to illustrate the potential usefulness of the study

reduction methods of type 2 uncertain variables and their applications to solid transportation problem uncertainty theory is a branch of mathematics for dealing with realistic uncertainties arising out of complexity changeability and non decidability of practical environments an uncertain variable is defined as a function from the uncertainty space to the set of real numbers and is characterized by an uncertainty distribution this paper proposes the definition of type 2 uncertain variables within the framework of uncertainty theory through introduction of generalized uncertain measures and focuses on more complex twofold uncertainties some uncertainty reduction methods associated with type 2 uncertain variables are also proposed for convenience of applicability including reduction of optimistic value pessimistic value and expected value moreover four classes of type 2 uncertain variables are reduced to type 1 uncertain variables with specific uncertainty distributions type 2 uncertain optimization methods are applied to solving the fixed charge solid transportation problem with the type 2 uncertain parameters where the solution methods are also provided for the proposed models finally numerical experiments are implemented to demonstrate application and sensitivity analysis of the proposed approaches

the maximum flow problem of uncertain network the maximum flow problem is one of the classic combinatorial optimization problems with many applications in electrical power systems communication networks computer networks and logistic networks the goal of the problem is to find the maximum amount of flow from the source to the sink in a network a network is called uncertain if the arc capacities of the network are uncertain variables the main purpose of this paper is to solve the maximum flow in an uncertain network by under the framework of uncertainty theory

on distribution function of the diameter in uncertain graph in uncertain graphs the existence of some edges is not predetermined the diameter of an uncertain graph is essentially an uncertain variable which indicates the suitability for investigation of its distribution function the main focus of this paper is to propose an algorithm to determine the distribution function of the diameter of an uncertain graph we first discuss the characteristics of the uncertain diameter and the distribution function is derived an efficient algorithm is designed based on floyd s algorithm further some numerical examples are illustrated to show the efficiency and application of the algorithm

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

the problem of a covariate time qualitative interaction in a survival study we introduce a test for the equality of two survival distributions against the specific alternative of crossing hazards although this kind of alternative is somewhat rare designing a test specifically aimed at detecting such departures from the null hypothesis in this direction leads to powerful procedures upon which we can call in those few cases where such departures are suspected furthermore the proposed test and an approximate version of the test are seen to suffer only moderate losses in power when compared with their optimal counterparts should the alternative be one of proportional hazards our interest in the problem is motivated by clinical studies on the role of acute graft versus host disease as a risk factor in leukemic children and we discuss the analysis of this study in detail the model we use in this work is a special case of the one introduced by anderson and senthilselvan 1982 applied statistics 31 44 51 we propose overcoming an inferential problem stemming from their model by using the methods of davies 1977 biometrika 64 247 254 1987 biometrika 74 33 43 backed up by resampling techniques we also look at an approach relying directly on resampling techniques the distributional aspects of this approach under the null hypothesis are interesting but practically its behaviour is such that its use cannot be generally recommended outlines of the necessary asymptotic theory are presented and for this we use the tools of martingale theory

single period inventory problem under uncertain environment this paper investigates the single period inventory newsboy problem in a general uncertain environment here it is assumed that the uncertainty appears in the market demand of the product uncertain demand is used to describe a subjective estimate not estimated by random fuzzy demand linguistically expressed by demand is about x using identification functions of the uncertain variable we derive the optimum order quantity and finally the expression of total expected profit for a fixed time period the computational aspects of the model in uncertain environment and their interpretations are illustrated by some examples

a nonhyperbolic cauchy problem for b c and its applications to elasticity theory on etudie l unicite et la stabilite d une fonction u satisfaisant les conditions suivantes b c u a 3 u f sur t t g n j u g j sur t t j 3 ou b u bu tt u a k est un operateur differentiel d ordre k g r n une partie de sa frontiere et n la normale a

uncertainty theory as a basis for belief reliability abstract belief reliability is a newly developed model based reliability metric which considers both what we know expressed as reliability models and what we don t know expressed as epistemic uncertainty in the reliability models about the reliability in this paper we show that due to the explicit representation of epistemic uncertainty belief reliability should not be regarded as a probability measure rather it should be treated as an uncertain measure in uncertainty theory a minimal cut set based method is developed to calculate the belief reliability of coherent systems a numerical algorithm is then presented for belief reliability analysis based on fault tree models the results of application show that the developed methods require less computations than the structure function based method of classical reliability theory

link based models for survival data with interval and continuous time censoring we present a flexible family of link based regression models with time independent baseline covariates particular choices of the link function yield a proportional hazard or a proportional odds model among others the background hazard function in the model is estimated using b splines this yields a full likelihood under both continuous time and random interval censoring residuals for influence and goodness of fit are described two examples are shown one in continuous time and the other under interval censoring

the restricted cubic spline as baseline hazard in the proportional hazards model with step function time dependent covariables we incorporate a cubic spline function where the tails are linearly constrained as the baseline hazard into the proportional hazards model we show estimation of covariable coefficients and survival probabilities with this model to be as efficient statistically as with the cox proportional hazards model when covariables are fixed examples show that the inclusion of time dependent covariables defined as step functions into the restricted cubic spline proportional hazards model reduces computation time by a factor of 213 over the cox model advantages of the spline model also include flexibility of the hazard smooth survival curves and confidence limits for the survival and hazard estimates when there are time dependent covariables present