quotient mi groups a many identities group mi group for short is a special algebraic structure in which identity like elements called pseudoidentities are specified and collected into a monoidal substructure in this way many algebraic structures such as monoids of fuzzy intervals numbers or convex bodies possessing behavior very similar to that of a group structure may be well described and investigated using a new approach which seems to be superfluous for the classical structures the concept of mi groups was recently introduced by holcapek and st pnicka in the paper mi algebras a new framework for arithmetics of extensional fuzzy numbers to demonstrate how a standard structure can be generalized in terms of mi algebras this paper is a continuation of the development of mi group theory and is focused on the construction of quotient mi groups and a specification of the conditions under which the isomorphism theorems for groups are fulfilled for mi groups

a pocket calculator algorithm for the shapiro francia test for non normality an application to medicine an easy to calculate approximation to the shapiro francia w test and its p value is proposed its accuracy is sufficient for practical application in samples of size 5 or larger

the use of cusums and other techniques in modelling continuous covariates in logistic regression the assessment of continuous covariates singly as possible predictors in a multivariable logistic regression model is an important first step in the analysis an approach to plotting which uses a cusum cumulative sum of the binary response variable is described extreme deviation statistics associated with the cusum may be used to detect monotonic and non monotonic trends probability plots of the covariate in the two groups defined by the response variable may help to determine the appropriate scale transformation of the covariate and to anticipate possible problems with the logistic fit the ratio of the variances in the response non response groups is informative about the need for a quadratic term in the logistic model smoothed scatterplots of the response are valuable in displaying the observed and fitted values the techniques are illustrated with two data sets

an aggregate subgradient method for nonsmooth convex minimization a class of implementable algorithms is described for minimizing any convex not necessarily differentiable functionf of several variables the methods require only the calculation off and one subgradient off at designated points they generalize lemarechal s bundle method more specifically instead of using all previously computed subgradients in search direction finding subproblems that are quadratic programming problems the methods use an aggregate subgradient which is recursively updated as the algorithms proceed each algorithm yields a minimizing sequence of points and iff has any minimizers then this sequence converges to a solution of the problem particular members of this algorithm class terminate whenf is piecewise linear the methods are easy to implement and have flexible storage requirements and computational effort per iteration that can be controlled by a user

on generalized quotient mi groups abstract many identities group mi group for short is an algebraic structure generalizing the group structure where an involutive anti automorphism satisfying certain properties is used instead of the standard group inversion the concept of mi group in a more general form than in this article has been introduced by holcapek and st pnicka in the paper mi algebras a new frame work for arithmetics of extensional fuzzy numbers to describe properties of different approaches to arithmetics of vaguely specified quantities e g stochastic or fuzzy quantities in a unified way this article is a continuation of the effort to develop the theory of mi groups and is focused on a generalization of the construction of quotient mi groups induced by so called normal full mi subgroups which has been introduced by holcapek et al recently in the paper quotient mi groups besides a more general definition of quotient mi groups we prove three isomorphism theorems for mi groups in this new framework

minimizing piecewise concave functions over polyhedra we introduce an iterative method for solving linearly constrained optimization problems whose nonsmooth nonconvex objective function is defined as the pointwise maximum of finitely many concave functions such problems often arise from reformulations of certain constraint structures e g binary constraints finite max min constraints in diverse areas of optimization we state a local optimization strategy which exploits piecewise concavity of the objective function giving rise to a linearized model corrected by a proximity term in addition we introduce an approximate line search strategy based on a curvilinear model which similarly to bundle methods can return either a satisfactory descent or a null step declaration termination at a point satisfying an approximate stationarity condition is proved we embed the local minimization algorithm into a variable neighborhood search scheme and a coordinate direction search heuristic whose aim is to improve the current estimate of the global minimizer

goodness of fit tests for ordinal response regression models it is well documented that the commonly used pearson chi square and deviance statistics are not adequate for assessing goodness of fit in logistic regression models when continuous covariates are modelled in recent years several methods have been proposed which address this shortcoming in the binary logistic regression setting or assess model fit differently however these techniques have typically not been extended to the ordinal response setting and few techniques exist to assess model fit in that case we present the modified pearson chi square and deviance tests that are appropriate for assessing goodness of fit in ordinal response models when both categorical and continuous covariates are present the methods have good power to detect omitted interaction terms and reasonable power to detect failure of the proportional odds assumption or modelling the wrong functional form of a continuous covariate these tests also provide immediate information as to where a model may not fit well in addition the methods are simple to understand and implement and are non specific that is they do not require prespecification of a type of lack of fit to detect

fuzzy arithmetic with requisite constraints abstract the purpose of this paper is to show that the standard fuzzy arithmetic does not take into account known constraints when applied to states of linguistic variables these constraints referred to as requisite constraints represent additional information when they are ignored as in the standard fuzzy arithmetic the obtained results exhibit in general information deficiency it is thus important as argued in the paper to revise fuzzy arithmetic to take relevant requisite constraints into account some typical requisite constraints are overviewed with a more extensive examination of the requisite equality constraint crisp or fuzzy

transformations of the explanatory variables in the logistic regression model for binary data summary some results are presented on improving the fit of the logistic regression model for binary data by transforming the vector of explanatory variables the methods are based on consideration of the distributions of these variables conditional on outcome group the transformations required are the functions of the explanatory variables which appear in the log density ratio of the conditional distributions

from fuzzy set theory to fuzzy set theory some critical remarks on existing concepts abstract our aim in this paper is to give some foundational remarks on the nature of fuzziness and their modeling by fuzzy sets we will reveal some fundamental misunderstandings which have pervaded fuzzy set theory just from its very beginnings at the end of the paper we give three examples demonstrating the differences and advantages of our new conception our approach in heavily based on an operational and set theoretical background contrasting hisdal s tee model

proximal level bundle methods for convex nondifferentiable optimization saddle point problems and variational inequalities we study proximal level methods for convex optimization that use projections onto successive approximations of level sets of the objective corresponding to estimates of the optimal value we show that they enjoy almost optimal efficiency estimates we give extensions for solving convex constrained problems convex concave saddle point problems and variational inequalities with monotone operators we present several variants establish their efficiency estimates and discuss possible implementations in particular our methods require bounded storage in contrast to the original level methods of lemarechal nemirovskii and nesterov

estimating departure from normality departure of a sample from a normal distribution should be assessed by a quantity that is meaningful in terms of the data rather than merely by the p value from a test statistic indices of departure based on the shapiro francia w and the shapiro wilk w statistics are derived and shown to have a natural interpretation in relation to the normal probability plot a new diagnostic plot is proposed an example is given which shows the relationship between one of the new indices and errors in calculated reference ranges due to non normality of the data

new variants of bundle methods in this paper we describe a number of new variants of bundle methods for nonsmooth unconstrained and constrained convex optimization convex concave games and variational inequalities we outline the ideas underlying these methods and present rate of convergence estimates

fuzzy weighted averages and implementation of the extension principle abstract this paper addresses the computational aspect of the extension principle when the principle is applied to algebraic mappings and in particular to weighted average operations in risk and decision analysis a computational algorithm based on the cut representation of fuzzy sets and interval analysis is described the method provides a discrete but exact solution to extended algebraic operations in a very efficient and simple manner examples are given to illustrate the method and its relation to other discrete methods and the exact approach by non linear programming the algorithm has been implemented in a computer program which can handle very general extended algebraic operations on fuzzy numbers

overweight tails are inefficient test statistics which are almost determined by o n tail order statistics are shown to provide tests of asymptotic relative efficiency 0 against the usual type of contiguous alternative the result is applied to several goodness of fit tests the variance weighted kolmogorov smirnov statistic the kolmogorov smirnov statistic in the stabilized probability plot and the correlation coefficient in a q q plot for a variety of distributions with exponential tails

estimating fatigue curves with the random fatigue limit model in a fatigue limit model units tested below the fatigue limit also known as the threshold stress theoretically will never fail this article uses a random fatigue limit model to describe a the dependence of fatigue life on the stress level b the variation in fatigue life and c the unit tounit variation in the fatigue limit we fit the model to actual fatigue datasets by maximum likelihood methods and study the fits under different distributional assumptions small quantiles of the life distribution are often of interest to designers lower confidence bounds based on likelihood ratio methods are obtained for such quantiles to assess the fits of the model we construct diagnostic plots and perform goodness of fit tests and residual analyses

the stabilized probability plot summary the stabilized probability plot is introduced an attractive feature of the plot that enhances its interpretability is that the variances of the plotted points are approximately equal this prompts the definition of a new and powerful goodness of fit statistic dsp which analogous to the standard kolmogorov smirnov statistic d is defined to be the maximum deviation of the plotted points from their theoretical values using either d or dsp it is shown how to construct acceptance regions for qq pp and the new plots acceptance regions can help remove much of the subjectivity from the interpretation of these probability plots

algebraic properties and topological properties of the quotient space of fuzzy numbers based on mares equivalence relation abstract in this paper we obtain some algebraic properties and topological properties of the quotient space of fuzzy numbers with respect to the equivalence relation defined by mares every fuzzy number has only one mares core and equivalent fuzzy numbers have the same mares core in addition equivalence classes of fuzzy numbers can be only expressed as the sum of its mares core and the set of symmetric fuzzy numbers which shows the notable difference between the equivalence classes of fuzzy numbers and the cosets of the normal subgroup in a group based on these results we introduce a new concept of convergence under which the quotient space is complete as an application of the main results it is shown that if we identify every fuzzy number with the corresponding equivalence class there would be more differentiable fuzzy functions than what is found in the literature

goodness of fit processes for logistic regression simulation results in this paper we use simulations to compare the performance of new goodness of fit tests based on weighted statistical processes to three currently available tests the hosmer lemeshow decile of risk test the pearson chi square and the unweighted sum of squares tests the simulations demonstrate that all tests have the correct size the power for all tests to detect lack of fit due to an omitted quadratic term with a sample of size 100 is close to or exceeds 50 per cent to detect moderate departures from linearity and is over 90 per cent for these same alternatives for sample size 500 all tests have low power with sample size 100 to detect lack of fit due to an omitted interaction between a dichotomous and continuous covariate while the power exceeds 80 per cent to detect extreme interaction with a sample size of 500 the power is low to detect any alternative link function with sample size 100 and for most alternative links for sample size 500 only in the case of sample size 500 and an extremely asymmetric link function is the power over 80 per cent the results from these simulations show that no single test new or current performs best in detecting lack of fit due to an omitted covariate or incorrect link function however one of the new weighted tests has power comparable to other tests in all settings simulated and had the highest power in the difficult case of an omitted interaction term we illustrate the tests within the context of a model for factors associated with abstinence from drug use in a randomized trial of residential treatment programmes we conclude the paper with a summary and specific recommendations for practice

a representation of fuzzy numbers abstract each fuzzy number can be written in a unique way as the sum of a symmetric and a skew fuzzy number these two kinds of fuzzy numbers correspond bijectively to certain functions on the unit interval that are of bounded variation these bijections give rise to a representation of fuzzy numbers by pairs of functions of bounded variation as application mathematical structures on the set of functions of bounded variation are transported to that of fuzzy numbers making the set of fuzzy numbers into a complete metric space a commutative semiring and a commutative residuated lattice

a version of the bundle idea for minimizing a nonsmooth function conceptual idea convergence analysis numerical results during recent years various proposals for the minimization of a nonsmooth functional have been made amongst these the bundle concept turned out to be an especially fruitful idea based on this concept a number of authors have developed codes that can successfully deal with nonsmooth problems the aim of the paper is to show that by adding some features of the trust region philosophy to the bundle concept the end result is a distinguished member of the bundle family with a more stable behaviour than some other bundle versions the reliability and efficiency of this code is demonstrated on the standard academic test examples and on some real life problems

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

quotients with respect to similarity relations abstract the problem of the quotient w r t a similarity relation is closely related to poincare s paradox and central problems in cluster analysis amazingly this problem remains open for more than 30 years as a generalization of strong m categories we introduce the concept of m valued sets which leads to the definition of the category m set in m set we give a complete solution of the quotient problem

tables and large sample distribution theory for censored data correlation statistics for testing normality abstract plotting order statistics versus some variant of the normal scores is a standard graphical technique for assessing the assumption of normality to obtain an objective evaluation of the normal assumption it is customary to calculate the correlation coefficient associated with this plot the shapiro francia statistic is the square of the correlation between the observed order statistics and the expected values of standard normal order statistics whereas the shapiro wilk statistic also involves the covariances of the standard normal order statistics in a wide variety of applications an investigation of the plausibility of the normal or lognormal model is needed when the observations on strength or life length are right censored the plotting procedure still applies if the observations are censored at a fixed order statistic or a fixed time here the corresponding distribution theory for some modified versions of the shapiro wilk correlation statistic is investigated because the asymptotic th

fuzzy quantities in networks one of possible applications of the fuzzy sets theory in network analysis is described and discussed it is shown that the uncertainty connected with the length of activities forming the network can be represented by fuzzy quantities summing these fuzzy quantities it is possible to derive the uncertainty connected with more complex paths and reserves an illustrative numerical example is presented at the end of the paper

weak arithmetics of fuzzy numbers abstract computation with fuzzy numbers appears to be a perspective branch of fuzzy set theory namely regarding the data processing applications a very compact survey of the actual state of art is given in dubois and prade 1988 this paper aims to offer in a brief and concentrated form a survey of some results completing the classical ones of dubois and prade most of its content summarizes the approach submitted and discussed by mares 1994

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

nonsmooth optimization techniques for semisupervised classification we apply nonsmooth optimization techniques to classification problems with particular reference to the transductive support vector machine tsvm approach where the considered decision function is nonconvex and nondifferentiable hence difficult to minimize we present some numerical results obtained by running the proposed method on some standard test problems drawn from the binary classification literature

on quasilinear spaces of convex bodies and intervals algebraic systems abstracting properties of convex bodies and intervals with respect to addition and multiplication by scalars known as quasilinear spaces are studied axiomatically we discuss special quasilinear spaces with group structure called quasivector spaces we show that every quasivector space is a direct sum of a vector space and a symmetric quasivector space a complete characterization of symmetric quasivector spaces in the finite dimensional case is given which permits to reduce computation in quasilinear spaces to computation in familiar vector spaces

goodness of fit tests based on p p probability plots percentage percentage p p probability plots constructed from standardized observations possess some attractive features that are not shared by more commonly used quantile quantile q q plots in particular the identification of viable alternatives to a proposed probability model can be greatly facilitated by displaying curves on p p plots to represent families of alternative models a single curve can represent an entire family of alternatives indexed by both location and scale parameters two goodness of fit statistics based on measures of linearity for standardized p p plots are proposed and simple approximations for percentage points of these statistics are presented for testing the fit of exponential gumbel weibull and normal lognormal probability models with unknown parameters results of extensive monte carlo power comparisons with other goodness of fit tests are summarized the proposed tests are shown to have superior power for detecting light tailed and moderate tailed alternatives to

piecewise quadratic approximations in convex numerical optimization we present a bundle method for convex nondifferentiable minimization where the model is a piecewise quadratic convex approximation of the objective function unlike standard bundle approaches the model only needs to support the objective function from below at a properly chosen small subset of points as opposed to everywhere we provide the convergence analysis for the algorithm with a general form of master problem which combines features of trust region stabilization and proximal stabilization taking care of all the important practical aspects such as proper handling of the proximity parameters and the bundle of information numerical results are also reported

a robust gradient sampling algorithm for nonsmooth nonconvex optimization let f be a continuous function on rl n and suppose f is continuously differentiable on an open dense subset such functions arise in many applications and very often minimizers are points at which f is not differentiable of particular interest is the case where f is not convex and perhaps not even locally lipschitz but is a function whose gradient is easily computed where it is defined we present a practical robust algorithm to locally minimize such functions based on gradient sampling no subgradient information is required by the algorithm when f is locally lipschitz and has bounded level sets and the sampling radius eps is fixed we show that with probability 1 the algorithm generates a sequence with a cluster point that is clarke eps stationary furthermore we show that if f has a unique clarke stationary point bar x then the set of all cluster points generated by the algorithm converges to bar x as eps is reduced to zero numerical results are presented demonstrating the robustness of the algorithm and its applicability in a wide variety of contexts including cases where f is not locally lipschitz at minimizers we report approximate local minimizers for functions in the applications literature which have not to our knowledge been obtained previously when the termination criteria of the algorithm are satisfied a precise statement about nearness to clarke eps stationarity is available a matlab implementation of the algorithm is posted at http www cs nyu edu overton papers gradsamp alg

tests for the exponential weibull and gumbel distributions based on the stabilized probability plot goodness of fit procedures for composite testing of the exponential weibull and gumbel distributions are investigated these procedures are such that they can be used graphically with and to aid interpretation of probability plots in the situations of interest here the new procedure is generally more powerful than the kolmogorovsmirnov test which appears to be the only other readily available goodness of fit procedure that can be used graphically with probability plots

on the mathematical foundations of nondifferentiable optimization in engineering design it is shown by example that a large class of engineering design problems can be transcribed into the form of a canonical optimization problem with inequality constraints involving mar functions such problems are commonly referred to as semi infinite optimization problems the bulk of this paper is devoted to the development of a mathematical theory for the construction of first order nondifferentiable optimization algorithms related to phase i phase ii methods of feasible directions which solve these semi infinite optimization problems the applicability of the theory is illustrated with examples that are relevant to engineering design

additive decomposition of fuzzy quantities abstract the object of this paper is to define a more refined equivalent relation than mares 3 and to improve mares result about additive decomposition of fuzzy quantities concerning an equivalent relation this result proves one of his open questions

solving nonlinear multicommodity flow problems by the analytic center cutting plane method the paper deals with nonlinear multicommodity flow problems with convex costs a decomposition method is proposed to solve them the approach applies a potential reduction algorithm to solve the master problem approximately and a column generation technique to define a sequence of primal linear programming problems each subproblem consists of finding a minimum cost flow between an origin and a destination node in an uncapacited network it is thus formulated as a shortest path problem and solved with dijkstra s d heap algorithm an implementation is described that takes full advantage of the supersparsity of the network in the linear algebra operations computational results show the efficiency of this approach on well known nondifferentiable problems and also large scale randomly generated problems up to 1000 arcs and 5000 commodities

piecewise linear approximations in nonconvex nonsmooth optimization we present a bundle type method for minimizing nonconvex nondifferentiable functions of several variables the algorithm is based on the construction of both a lower and an upper polyhedral approximation of the objective function in particular at each iteration a search direction is computed by solving a quadratic program aiming at maximizing the difference between the lower and the upper model a proximal approach is used to guarantee convergence to a stationary point under the hypothesis of weak semismoothness

an incremental method for solving convex finite min max problems we introduce a new approach to minimizing a function defined as the pointwise maximum over finitely many convex real functions next referred to as the component functions with the aim of working on the basis of incomplete knowledge of the objective function a descent algorithm is proposed which need not require at the current point the evaluation of the actual value of the objective function namely of all the component functions thus extending to min max problems the philosophy of the incremental approaches widely adopted in the nonlinear least squares literature given the nonsmooth nature of the problem we resort to the well established machinery of bundle methods we provide global convergence analysis of our method and in addition we study a subgradient aggregation scheme aimed at simplifying the problem of finding a tentative step this paper is completed by the numerical results obtained on a set of standard test problems

minimizing nonconvex nonsmooth functions via cutting planes and proximity control we describe an extension of the classical cutting plane algorithm to tackle the unconstrained minimization of a nonconvex not necessarily differentiable function of several variables the method is based on the construction of both a lower and an upper polyhedral approximation to the objective function and is related to the use of the concept of proximal trajectory convergence to a stationary point is proved for weakly semismooth functions

on goodness of fit tests for the two parameter weibull distribution derived from the stabilized probability plot summary michael s statistic dsp based on the stabilized probability plot is considered for the assessment of goodness of fit of a two parameter weibull distribution this statistic is shown to have high power when the parameters are estimated using a procedure due to blom 1958 the correlation coefficient of the stabilized plot rsp is also proposed as a test statistic for this situation finally a family of transformations is obtained to reduce the null distribution of rsp to standard normality

a numerically stable dual method for solving strictly convex quadratic programs an efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage of the fact that the unconstrained minimum of the objective function can be used as a starting point its implementation utilizes the cholesky and qr factorizations and procedures for updating them the performance of the dual algorithm is compared against that of primal algorithms when used to solve randomly generated test problems and quadratic programs generated in the course of solving nonlinear programming problems by a successive quadratic programming code the principal motivation for the development of the algorithm these computational results indicate that the dual algorithm is superior to primal algorithms when a primal feasible point is not readily available the algorithm is also compared theoretically to the modified simplex type dual methods of lemke and van de panne and whinston and it is illustrated by a numerical example

transformations and influential observations in minimum sum of absolute errors regression the box cox power transformation model is considered for minimum sum of absolute errors msae regression a long tailed error distribution specifically the laplace distribution is included and could accommodate observations that in the normal case are outlying or unduly influencing a choice of transformation the log likelihood procedure of box and cox 1964 for obtaining the optimal transformation parameter is adapted for laplace errors and msae regression graphical methods for detecting the influence of individual observations on the choice of transformation are described application to examples illustrates that this approach can provide valuable additional information to the data analyst

a comparison of goodness of fit tests for the logistic regression model recent work has shown that there may be disadvantages in the use of the chi square like goodness of fit tests for the logistic regression model proposed by hosmer and lemeshow that use fixed groups of the estimated probabilities a particular concern with these grouping strategies based on estimated probabilities fitted values is that groups may contain subjects with widely different values of the covariates it is possible to demonstrate situations where one set of fixed groups shows the model fits while the test rejects fit using a different set of fixed groups we compare the performance by simulation of these tests to tests based on smoothed residuals proposed by le cessie and van houwelingen and royston a score test for an extended logistic regression model proposed by stukel the pearson chi square and the unweighted residual sum of squares these simulations demonstrate that all but one of royston s tests have the correct size an examination of the performance of the tests when the correct model has a quadratic term but a model containing only the linear term has been fit shows that the pearson chi square the unweighted sum of squares the hosmer lemeshow decile of risk the smoothed residual sum of squares and stukel s score test have power exceeding 50 per cent to detect moderate departures from linearity when the sample size is 100 and have power over 90 per cent for these same alternatives for samples of size 500 all tests had no power when the correct model had an interaction between a dichotomous and continuous covariate but only the continuous covariate model was fit power to detect an incorrectly specified link was poor for samples of size 100 for samples of size 500 stukel s score test had the best power but it only exceeded 50 per cent to detect an asymmetric link function the power of the unweighted sum of squares test to detect an incorrectly specified link function was slightly less than stukel s score test we illustrate the tests within the context of a model for factors associated with low birth weight

additions of interactive fuzzy numbers this paper provides an account of an approach to modeling unknown data by means of fuzzy set theory and addresses the problem of deriving the uncertainty on a sum of variables whose values lie within fuzzy intervals the first part is an extensive presentation of the theoretical background of the approach the extension principle is stated in terms of possibility of an event the concept of variable interaction is investigated at length section ii gives new results regarding the effective practical computation of additions of fuzzy numbers its originality lies in the introduction of interaction which enables to control the growth of uncertainty in calculations moreover the problem of computing mathematical expectations with fuzzy probabilities is solved the results derived in this paper can easily be used in decision problems where values of parameters or decision variables are not yet precisely fixed or assessed typical applications could be multicriteria optimization and decision making under uncertainty where fuzzy expected utilities can be obtained out of uncompletely assessed probabilities more generally fuzzy arithmetic can be an important tool for sophisticated computationally tractable sensitivity analysis in systems modeling computer aided design and operations research

two goodness of fit tests for logistic regression models with continuous covariates assessing goodness of fit in logistic regression models can be problematic in that commonly used deviance or pearson chi square statistics do not have approximate chi square distributions under the null hypothesis of no lack of fit when continuous covariates are modelled we present two easy to implement test statistics similar to the deviance and pearson chi square tests that are appropriate when continuous covariates are present the methodology uses an approach similar to that incorporated by the hosmer and lemeshow goodness of fit test in that observations are classified into distinct groups according to fitted probabilities allowing sufficient cell sizes for chi square testing the major difference is that the proposed tests perform this grouping within the cross classification of all categorical covariates in the model and in some situations allow for a more powerful assessment of where model predicted and observed counts may differ a variety of simulations are performed comparing the proposed tests to the hosmer lemeshow test

confidence bands for survival functions with censored data a comparative study consider the problem of obtaining simultaneous confidence bands for the survival function s x when the data are arbitrarily right censored the usual pointwise confidence intervals based on greenwood s variance formula can be adapted to yield a large sample confidence band this band has in a certain sense equal precision at each point of s x it is compared with the censored versions of the kolmogorov band and the renyi band the comparisons are made in terms of the widths and the adequacy of large sample approximations and are carried out under various censoring models and degrees of censoring the bands are illustrated by applying them to data from a mechanical switch life test

additive decomposition of fuzzy quantities with finite supports abstract the results on group properties of the addition operation over fuzzy quantities presented in a previous paper by the author 5 imply some interesting consequences concerning the equivalence relation between fuzzy quantities the equivalence is necessary for the validity of the group axioms for the class of fuzzy quantities more detailed analysis of the equivalence properties implies some conclusions regarding the interpretation of different sources of fuzziness entering the fuzzy quantities model the method used to prove the main theorem of this paper demands to limit the investigation to fuzzy quantities with finite support this limitation however unpleasant does not influence the applicability of the presented results in a wide class of practical situations

mi algebras the existing arithmetics of fuzzy numbers usually stemming from the cut arithmetic do not preserve some of the important properties of the standard arithmetics of real numbers although one cannot expect a generalization of standard arithmetic to perfectly preserve all its properties the most important properties should be preserved at least in a weakened form we present a novel framework for arithmetics of extensional fuzzy numbers that more or less preserves all the important algebraic properties of the arithmetic of real numbers and thus seems to be an important seed for further investigations on this topic the suggested approach to arithmetics of extensional fuzzy numbers is demonstrated by many examples and besides its algebraic properties it is also shown to carry some desirable practical properties the investigation leads to novel algebraic structures mi algebras mi monoids mi groups mi fields that abstract the discussed properties the main idea of these structures is based on a set of pseudoidentities that complements the only commonly used identity element in a monoid group structure these pseudoidentities are elements that generalize the identity like behavior and allow us to weaken the standard forms of algebraic properties that are highly desirable to preserve at least in a weakened form appropriate properties of suggested mi structures are introduced and demonstrated by a number of practical examples

the concept of residual confounding in regression models and some applications in this paper the concept of residual confounding is generalized to various types of regression models such as logistic regression or cox regression residual confounding and a newly suggested parameter the relative residual confounding are defined on the regression parameters of the models the estimator gives the proportion of confounding which has been removed by incomplete adjustment the concept quantifies the effects of categorizing continuous covariables and of model misspecification these are investigated by a simulation study and with data from an epidemiological investigation a case control study of laryngeal cancer is used to illustrate the residual confounding effect of arbitrary transformation of a continuous confounder smoking on the effect of alcohol consumption on laryngeal cancer risk the data also showed that categorization into two levels can yield high residual confounding the parameters described in this paper are of some use in quantifying the effect of inadequate adjustment for confounding variables

solving fuzzy equations we start in the next section 10 2 with looking at possible solutions to the simple fuzzy linear equation overline a cdot overline x overline b overline c we discuss three different types of solution which we have studied before in solving fuzzy equations then we present a fourth type of solution based on our fuzzy monte carlo method in section 10 2 2 this new solution is based on random fuzzy numbers in section 10 3 we look at only classical solutions to the fuzzy quadratic equation and apply our fuzzy monte carlo method to obtain new solutions then in section 10 4 we consider the fuzzy matrix equation overline a cdot overline x overline b and a number of solution types for overline x and then another solution based on fuzzy monte carlo techniques the last section contains a brief summary and our conclusions

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12

representation theorems for l fuzzy quantities abstract in this paper representation theorems are given for l fuzzy quantities which permit a better understanding of fuzziness in particular representation theorems due to negoita and ralescu 17 and to sherwood and taylor 23 are extended to the scope of complete and brouwerian lattices

on fuzzy differential equations in the quotient space of fuzzy numbers since addition possesses an inverse in the quotient space of fuzzy numbers based on mares equivalence relation in this paper we introduce a metric on the quotient space of fuzzy numbers and then deal with fuzzy mappings of a real variable whose values are equivalence classes of fuzzy numbers we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation finally some examples are given to illustrate the main theorems