a monte carlo based method for the estimation of lower and upper probabilities of events using infinite random sets of indexable type random set theory is a useful tool to quantify lower and upper bounds on the probability of the occurrence of events given uncertain information represented for example by possibility distributions probability boxes or dempster shafer structures among others in this paper it is shown that the belief and plausibility estimated by dempster shafer evidence theory are basically approximations by riemann stieltjes sums of the integrals of the lower and upper probability employed when using infinite random sets of indexable type in addition it is shown that the evaluation of the lower and upper probability is more efficient if it is done by pseudo monte carlo strategies this discourages the use of dempster shafer evidence theory and suggests the use of infinite random sets of indexable type specially in high dimensions not only because the initial discretization step of the basic variables is not required anymore but also because the evaluation of the lower and upper probability of events is much more efficient using the different techniques for multidimensional integration like monte carlo simulation

local well posedness to the cauchy problem of the 2d compressible navier stokes smoluchowski equations with vacuum abstract this paper concerns the local strong solution to the compressible navier stokes smoluchowski equations with vacuum as far field density on the whole space r 2 without additional cho choe kim type compatibility conditions which provided the initial density and the initial particles density decay not too slow at infinity in particular we extend the results of ding et al 2016 13 and yang 2020 24 to the 2d case

parametric approaches to quality adjusted survival analysis international breast cancer study group we present a parametric methodology for performing quality of life adjusted survival analysis using multivariate censored survival data it represents a generalization of the nonparametric q twist method quality adjusted time without symptoms and toxicity the event times correspond to transitions between states of health that differ in terms of quality of life each transition is governed by a competing risks model where the health states are the competing risks overall survival is the sum of the amount of time spent in each health state the first step of the proposed methodology consists of defining a quality function that assigns a score to a life having given health state transitions it is a composite measure of both quantity and quality of life in general the quality function assigns a small value to a short life with poor quality and a high value to a long life with good quality in the second step parametric survival models are fit to the data this is done by repeatedly modeling the conditional cause specific hazard functions given the previous transitions covariates are incorporated by accelerated failure time regression and the model parameters are estimated by maximum likelihood lastly the modeling results are used to estimate the expectation of quality functions standard errors and confidence intervals are computed using the bootstrap and delta methods the results are useful for simultaneously evaluating treatments in terms of quantity and quality of life to demonstrate the proposed methods we perform an analysis of data from the international breast cancer study group trial v which compared short duration chemotherapy versus long duration chemotherapy in the treatment of node positive breast cancer the events studied are 1 the end of treatment toxicity 2 disease recurrence and 3 overall survival

strongly degenerate parabolic hyperbolic systems modeling polydisperse sedimentation with compression we show how existing models for the sedimentation of monodisperse flocculated suspensions and of polydisperse suspensions of rigid spheres differing in size can be combined to yield a new theory of the sedimentation processes of polydisperse suspensions forming compressible sediments sedimentation with compression or sedimentation consolidation process for n solid particle species this theory reduces in one space dimension to an n times n coupled system of quasi linear degenerate convection diffusion equations analyses of the characteristic polynomials of the jacobian of the convective flux vector and of the diffusion matrix show that this system is of strongly degenerate parabolic hyperbolic type for arbitrary n and particle size distributions bounds for the eigenvalues of both matrices are derived the mathematical model for n 3 is illustrated by a numerical simulation obtained by the kurganov tadmor central difference scheme for convection diffusion problems the numerical scheme exploits

strong solutions of the navier stokes equations for isentropic compressible fluids we study strong solutions of the isentropic compressible navier stokes equations in a domain r3 we first prove the local existence of unique strong solutions provided that the initial data 0 and u0 satisfy a natural compatibility condition the important point in this paper is that we allow the initial vacuum the initial density may vanish in an open subset of we then prove a new uniqueness result and stability result our results are valid for unbounded domains as well as bounded ones

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

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

sklar s theorem in an imprecise setting sklar s theorem is an important tool that connects bidimensional distribution functions with their marginals by means of a copula when there is imprecision about the marginals we can model the available information by means of p boxes that are pairs of ordered distribution functions similarly we can consider a set of copulas instead of a single one we study the extension of sklar s theorem under these conditions and link the obtained results to stochastic ordering with imprecision

empirical likelihood ratio test for median and mean residual lifetime prompted by several recent papers on the inference on median and mean residual life time we note that the testing involving the mean or median residual life function in censored survival data can be obtained by an easy application of the general empirical likelihood ratio test this approach has several advantages 1 there is no need to estimate the variance covariance at all which may become prohibitively complicated for other procedures that require the estimation of such 2 when inverting the tests to obtain confidence regions intervals this procedure inherits all the good properties of a likelihood ratio test 3 free software implementation of the test is readily available

summarising censored survival data using the mean residual life function the mean residual life function provides a clear and simple summary of the effect of a treatment or a risk factor in units of time avoiding hazard ratios or probability scales which require careful interpretation estimation of the mean residual life is complicated by the upper tail of the survival distribution not being observed as for example patients may still be alive at the end of the follow up period various approaches have been developed to estimate the mean residual life in the presence of such right censoring in this work a novel semi parametric method that combines existing non parametric methods and an extreme value tail model is presented where the limited sample information in the tail prior to study termination is used to estimate the upper tail behaviour this approach will be demonstrated with simulated and real life examples copyright 2015 john wiley sons ltd

a random set characterization of possibility measures several authors have pointed out the relationship between consonant random sets and possibility measures however this relationship has only been proven for the finite case where the inverse mobius of the upper probability induced by the random set simplifies the computations to a great extent in this paper we study the connection between both concepts for arbitrary referential spaces we complete existing results about the lack of an implication in general with necessary and sufficient conditions for the most interesting cases

a geometric and game theoretic study of the conjunction of possibility measures we study the conjunction of possibility measures within imprecise probability theory we identify conditions under which the conjunction remains a possibility measure we provide a graphical way to check these conditions using zero sum games we provide a graphical way to adjust possibility measures so their conjunction is a possibility measure we identify conditions under which the conjunction is coherent in this paper we study the conjunction of possibility measures when they are interpreted as coherent upper probabilities that is as upper bounds for some set of probability measures we identify conditions under which the minimum of two possibility measures remains a possibility measure we provide graphical way to check these conditions by means of a zero sum game formulation of the problem this also gives us a nice way to adjust the initial possibility measures so their minimum is guaranteed to be a possibility measure finally we identify conditions under which the minimum of two possibility measures is a coherent upper probability or in other words conditions under which the minimum of two possibility measures is an exact upper bound for the intersection of the credal sets of those two possibility measures

the encounter of interval and probabilistic approaches to structural reliability at the design point abstract the consideration of uncertainties in structural analysis has traditionally been made in the field of structural reliability within the framework of classical probability theory in many cases however there is no sufficient probabilistic information to substantiate such an approach in recent years several non probabilistic methods have been proposed as alternatives one of them is the interval analysis oriented to estimate the range of variation of output structural variables given the ranges of the input ones the method however requires the use of special arithmetics which renders its application unpractical for structural analysis based on finite elements or other numerical techniques in this paper a monte carlo method that can be applied to probabilistic as well as to interval approaches to reliability analysis is proposed it is based on the optimal properties of the classical design point of probabilistic structural reliability in particular it is shown that the order statistics of the output variable is concealed in a plot defined by the design point vector this facilitates the selection of the relevant samples for either interval or probability based analysis the examples show that the desired interval of the response or the failure probability can be accurately estimated on such a basis the paper also discusses the use of monte carlo methods for both reliability and interval analysis from the point of view of the proposed representation

representing parametric probabilistic models tainted with imprecision numerical possibility theory belief functions have been suggested as useful tools to represent imprecise vague or incomplete information they are particularly appropriate in uncertainty analysis where information is typically tainted with imprecision or incompleteness based on their experience or their knowledge about a random phenomenon experts can sometimes provide a class of distributions without being able to precisely specify the parameters of a probability model frequentists use two dimensional monte carlo simulation to account for imprecision associated with the parameters of probability models they hence hope to discover how variability and imprecision interact this paper presents the limitations and disadvantages of this approach and propose a fuzzy random variable approach to treat this kind of knowledge

representation theorem for probabilities on ifs events riecan gave in riecan 2004 9 an axiomatic characterization of a probability on ifs events and proved in riecan 2006 10 a representation theorem for it in rencova 9 rencova introduced the notions of additive f probability and strongly additive f probability and gave a representation theorem for strongly additive f probabilities in this paper we give a generalization of the axiomatic characterization given by riecan in riecan 2006 10 from this the representation theorem for strongly additive f probabilities will follow

random sets and fuzzy interval analysis abstract a general expression of function with random set valued arguments is stated which encompasses zadeh s extension principle as well as functions of random variables and interval analysis a monotonicity property is derived for algebraic operations performed between random set valued variables this property extends the monotonicity of functions with set valued arguments in the sense of inclusion the application of this result to the approximate calculation of functions of random variables is suggested and illustrated by an example

the mean value of a fuzzy number abstract the notion of expectation is defined for fuzzy numbers of intervals viewing them as consonant random sets this expectation is interval valued and remains additive in the sense of fuzzy interval addition as a by product some light is shed on the debate between probability and possibility theory and the problem of how to carry measure theoretic notions into the field of possibility theory

imprecise probabilistic evaluation of sewer flooding in urban drainage systems using random set theory 1 uncertainty analysis is widely applied in water system modeling to quantify prediction uncertainty from models and data conventional methods typically handle various kinds of uncertainty using a single characterizing approach be it probability theory or fuzzy set theory however using a single approach may not be appropriate particularly when uncertainties are of different types for example in sewer flood estimation problems random rainfall variables are used as model inputs and imprecise or subjective information is used to define model parameters this paper presents a general framework for sewer flood estimation that enables simultaneous consideration of two types of uncertainty randomness from rainfall data represented using imprecise probabilities and imprecision from model parameters represented by fuzzy numbers these two types of uncertainties are combined using random set theory and then propagated through a hydrodynamic urban drainage model two propagation methods i e discretization and monte carlo based methods are presented and compared with the latter shown to be much more computationally efficient and hence recommended for high dimensional problems the model output flood depth is generated in the form of lower and upper cumulative probabilities which are best estimates given the various stochastic and epistemic uncertainties considered and which embrace the unknown true cumulative probability the distance between the cumulative probabilities represents the extent of imprecise incomplete or conflicting information and can be reduced only when more knowledge is available the proposed methodology has a more complete and thus more accurate representation of uncertainty in data and models and can effectively handle different uncertainty characterizations in a single integrated framework for sewer flood estimation

joint propagation and exploitation of probabilistic and possibilistic information in risk assessment random variability and imprecision are two distinct facets of the uncertainty affecting parameters that influence the assessment of risk while random variability can be represented by probability distribution functions imprecision or partial ignorance is better accounted for by possibility distributions or families of probability distributions because practical situations of risk computation often involve both types of uncertainty methods are needed to combine these two modes of uncertainty representation in the propagation step a hybrid method is presented here which jointly propagates probabilistic and possibilistic uncertainty it produces results in the form of a random fuzzy interval this paper focuses on how to properly summarize this kind of information and how to address questions pertaining to the potential violation of some tolerance threshold while exploitation procedures proposed previously entertain a confusion between variability and imprecision thus yielding overly conservative results a new approach is proposed based on the theory of evidence and is illustrated using synthetic examples

extremes and local dependence in stationary sequences extensions of classical extreme value theory to apply to stationary sequences generally make use of two types of dependence restriction a a weak mixing condition restricting long range dependence b a local condition restricting the clustering of high level exceedances

a comparison of several methods of estimating the survival function when there is extreme right censoring summary when there is extreme censoring on the right the kaplan meier product limit estimator is known to be a biased estimator of the survival function several modifications of the kaplan meier estimator are examined and compared with respect to bias and mean squared error