character degrees and nilpotence class of finite p groups an approach via pro p groups let s be a finite set of powers of p containing 1 it is known that for some choices of s if p is a finite p group whose set of character degrees is s then the nilpotence class of p is bounded by some integer that depends on s while for some other choices of s such an integer does not exist the sets of the first type are called class bounding sets the problem of determining the class bounding sets has been studied in several papers whose results made it tempting to conjecture that a set s is class bounding if and only if p s in this article we provide a new approach to this problem our main result shows the relevance of certain p adic space groups in this problem with its help we are able to prove some results that provide new class bounding sets we also show that there exist non class bounding sets s such that p s

efficient estimation for the proportional hazards model with interval censoring the maximum likelihood estimator mle for the proportional hazards model with case 1 interval censored data is studied it is shown that the mle for the regression parameter is asymptotically normal with n convergence rate and achieves the information bound even though the mle for the baseline cumulative hazard function only converges at n 1 3 rate estimation of the asymptotic variance matrix for the mle of the regression parameter is also considered to prove our main results we also establish a general theorem showing that the mle of the finite dimensional parameter in a class of semiparametric models is asymptotically efficient even though the mle of the infinite dimensional parameter converges at a rate slower than n the results are iliustrated by applying them to a data set from a tumorigenicity study

cox regression in cohort studies with validation sampling summary an estimation procedure is proposed for the cox model in cohort studies with validation sampling where crude covariate information is observed for the full cohort and true covariate information is collected on a validation set sampkled randomly from the full cohort the method proposed makes use of the partial information from data that are available on the entire cohort by fitting a working cox model relating crude covariates to the failure time the resulting estimator is consistent regardless of the specification of the working model and is asymptotically more efficient than the validation set only estimator approximate asymptotic relative efficiencies with respect ot some alternative methods are derived under a simple scenaria and further studied numerically the finite sample performance is investigated and compared with alternative methods via simulation studies a similar procedure also works for the case where the validation set is a stratified random sample from the cohort

a semiparametric proportional odds regression model for the analysis of current status data abstract an approach to covariate analysis of current status data is described the method is applicable when the logit of the conditional probability of survival given the covariates is some increasing function of time plus a linear combination of the covariates the approach is based on approximating the infinite dimensional nuisance parameter the baseline log odds of failure with a step function and carrying out a maximum likelihood procedure the resulting finite dimensional parameter estimates for the regression parameters are shown to be asymptotically normal and semiparametric efficient numerical studies with small and moderate samples are discussed

a unified approach to regression analysis under double sampling designs we propose a unified approach to the estimation of regression parameters under double sampling designs in which a primary sample consisting of data on the rough or proxy measures for the response and or explanatory variables as well as a validation subsample consisting of data on the exact measurements are available we assume that the validation sample is a simple random subsample from the primary sample our proposal utilizes a specific parametric model to extract the partial information contained in the primary sample the resulting estimator is consistent even if such a model is misspecified and it achieves higher asymptotic efficiency than the estimator based only on the validation data specific cases are discussed to illustrate the application of the estimator proposed

a model for the measurement of membership and the consequences of its empirical implementation representation and uniqueness theorems are presented for the fundamental measurement of the membership of a fuzzy set when the domain of discourse is order dense the conclusion that membership is on an interval scale is justified by the inapplicability of extensive measurement to fuzziness and the lack of a natural origin for membership some descriptive statistics are also presented from an empirical study currently in progress to construct membership functions they suggest that subjects differ in their perception of fuzziness and that the models in the literature for certain linguistic operators such as not very and antonym are not sufficiently general to account for the range of individual perceptions of vague attributes a more general mathematical model is proposed the question of the meaningfulness of statements about membership functions is also discussed in the light of the assertion that membership can be fundamentally measured at best on an interval scale some of the difficulties which arise in fuzzy set theory due to this property are illustrated and a solution is suggested involving the use of a function which is derived from membership but is on an absolute scale

assessing age at onset risk factors with incomplete covariate current status data under proportional odds models motivated by an epidemiological survey of fracture in elderly women we develop a semiparametric regression analysis of current status data with incompletely observed covariate under the proportional odds model to accommodate both the interval censored nature of current status failure time data and the incompletely observed covariate data we propose an analysis based on the validation likelihood vl which is derived from likelihood pertaining to the validation sample namely the subset of the sample where the data are completely observed the missing data mechanism is assumed to be missing at random and is explicitly modeled and estimated in the vl approach we propose implementing the vl method by integrating self consistency and newton raphson algorithms asymptotic normality and standard error estimation for the proposed estimator of the regression parameter are guaranteed simulation results reveal good performance of the vl estimator the vl method has some gain in efficiency compared with the naive complete case method but the vl method leads to unbiased estimators whereas the complete case method does not when missing covariates are not missing completely at random application of the vl approach to the fracture data confirms that osteoporosis low bone density is a strong risk factor for the age at onset of fracture in elderly women copyright 2012 john wiley sons ltd

bayesian estimation in forest surveys when samples or prior information are fuzzy abstract fuzzy phenomena are frequently encountered in forestry due to the complexity of the systems under investigation in this paper existing inference methods are presented to accommodate situations in which random events take place in fuzzy circumstances fuzzy concepts are incorporated into these commonly accepted approaches to account for uncertainties used by factors other than randomness which traditional statistical approaches deal primarily with based on two extensions of likelihood functions for fuzzy samples bayesian estimates are generalized for use when sample information and prior distribution of parameters are fuzzy the estimators are developed for some typical membership functions for fuzzy samples and fuzzy priors applications are illustrated using examples for forestry

fuzzy logic of lukasiewicz logic a clarification pavelka 10 had shown in 1979 that the only natural way of formalizing fuzzy logic for truth value in the unit interval 0 1 is by using lukasiewicz s implication operator a b min 1 1 a b or some isomorphic form of it a considerable number of other papers around the same time had attempted to formulate alternative definitions for a b by giving intuitive justifications for them there continues to be some confusion however even today about the right notion of fuzzy logic much of this has its origin in the use of improper and or and the not operations and a misunderstanding of some of the key differences between proofs or inferencing in fuzzy logic and those in lukasiewicz s logic we point out the need for defining the strong conjunction operator in connection with fuzzy modus ponens rule and why we do not need the fuzzy syllogism rule we also point out the shortcomings of many of the alternative definitions of a b which indicate further support for pavelka s result we hope that these discussions help to clarify the misconceptions about fuzzy logic

constructing membership functions using statistical data abstract determination of the membership functions is vital in practical applications of the fuzzy set theory this paper presents a guideline to construct the membership functions for fuzzy sets whose elements have a defining feature with a known probability density function pdf in the universe of discourse the method finds the smallest fuzzy set which assigns high average membership values to those objects with the defining features distributed according to the given pdf it is show that for any pdf the method is capable of generating membership functions in accordance with the possibility probability consistency principle membership functions derived from some of the well known pdfs and an application in solving noise contaminated linear system of equations are presented

analysis of incomplete repeated measurements with dependent censoring times summary a frequently encountered complication in repeated measurements data analysis is that a substantial number of subjects drop out of the study in this paper several conceptually simple methods are proposed for analysing incomplete repeated measures when the measurement times vary substantially from subject to subject and the subject s drop out time may depend on the observed and unobserved outcome variables numerical studies are conducted to examine the adequacy of our procedures under practical settings a detailed illustration with an aids clinical trial is also provided

contribution to application of energy measure of fuzzy sets abstract the works of de luca termini continued by for example knopfmacher loo and gottwald are the most important on the topic of determination of measures of fuzzy sets the matter is to evaluate how fuzzy a fuzzy set is there are two general concepts of measures of fuzzy set i e entropy and energy measures we show that the special kind of energy measure is better suited than the entropy kind of measure in many practical situations applications of the use of energy measure discussed in detail include decision making fuzzy process control and prediction in fuzzy systems

semiparametric analysis of incomplete current status outcome data under transformation models summary this work motivated by an osteoporosis survey study considers regression analysis with incompletely observed current status data here the current status data including an examination time and an indicator for whether or not the event of interest has occurred by the examination time is not observed for all subjects instead a surrogate outcome subject to misclassification of the current status is available for all subjects we focus on semiparametric regression under transformation models including the proportional hazards and proportional odds models as special cases under the missing at random mechanism where the missingness of the current status outcome can depend only on the observed surrogate outcome and covariates we propose an approach of validation likelihood based on the likelihood from the validation subsample where the data are fully observed with adjustments of the probability of observing the current status outcome as well as the distribution of the surrogate outcome in the validation subsample we propose an efficient computation algorithm for implementation and derive consistency and asymptotic normality for inference with the proposed estimator the application to the osteoporosis survey data and simulations reveal that the validation likelihood performs well it removes the bias from the complete case analysis discarding subjects with missing data and achieves higher efficiency than the inverse probability weighting analysis

the generalized truth degree of quantitative logic in the logic system l n n valued nm logic system the concept of the truth degree of a formula is the crucial tool and the building block in quantitative logic from which the concept of logic metric in quantitative logic is derived logic metric takes an important role in quantitative logic related to which are other concepts in quantitative logic such as divergence consistency etc in the present paper having combined the theory of generalized tautologies with the theory of truth degrees in quantitative logic we have proposed the theory of s c truth degrees of formulas related to theory c in the logic system l n n valued nm logic system and discussed some of its properties with the help of the properties of s c truth degrees t c a email protected c a b email protected c b we have obtained the c logic metric on the set f s of formulas in the propositional logic system l n n valued nm logic system by the work of this paper we can generalize the theory of quantitative logic in all round way and establish an approximate reasoning s framework related to theory c in the logic system l n n valued nm logic system

weighted breslow type and maximum likelihood estimation in semiparametric transformation models a semiparametric transformation model comprises a parametric component for covariate effects and a nonparametric component for the baseline hazard intensity the breslow type estimator has been proposed for estimating the nonparametric component in some inefficient estimation procedures we show that introducing weights into this estimator leads to nonparametric maximum likelihood estimation with the weights depending on the martingale residuals the weighted breslow type estimator suggests an iterative reweighting algorithm for nonparametric maximum likelihood estimation which can be implemented by a weighted variant of the existing algorithms for inefficient estimation and can be computationally more efficient than an em type algorithm the weighting idea is further extended to semiparametric transformation models with mismeasured covariates

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

an algebraic approach to linguistic hedges in zadeh s fuzzy logic the paper addresses the mathematical modelling of domains of linguistic variables i e term sets of linguistic variables in order to obtain a suitable algebraic structure for the set of truth values of zadeh s fuzzy logic we shall give a unified algebraic approach to the natural structure of domains of linguistic variables which was proposed by ho and wechler fuzzy sets and systems 35 1990 281 and then by ho and nam proc ncst vietnam 9 1 1997 15 logic algebra and computer science vol 46 banach center publications pwn warsaw 1999 p 63 in this approach every linguistic domain can be considered as an algebraic structure called hedge algebra because properties of its unary operations reflect semantic characteristics of linguistic hedges many fundamental properties of refined hedge algebras rh algebras are examined especially it is shown that every rh algebra of a linguistic variable with a chain of the primary terms is a distributive lattice rh algebras with exactly two distinct primary terms one being an antonym of the other will also be investigated and they will be called symmetrical rh algebras it is shown that a class of finite symmetrical rh algebras has a rich enough algebraic structure

induction of fuzzy decision trees abstract most decision tree induction methods used for extracting knowledge in classification problems do not deal with cognitive uncertainties such as vagueness and ambiguity associated with human thinking and perception in this paper cognitive uncertainties involved in classification problems are explicitly represented measured and incorporated into the knowledge induction process a fuzzy decision tree induction method which is based on the reduction of classification ambiguity with fuzzy evidence is developed fuzzy decision trees represent classification knowledge more naturally to the way of human thinking and are more robust in tolerating imprecise conflict and missing information

finite solvable groups with at most two nonlinear irreducible characters of each degree abstract the aim of this paper is to investigate the finite solvable groups with at most two nonlinear irreducible characters of each degree

fuzzy control from the logical point of view abstract this paper is a discussion of the logical aspect in fuzzy control which is an application of the theory of approximate reasoning first we analyze approximate reasoning and show that it is a concise logical theory based on many valued logic integrated with some parts of linguistics then we demonstrate that fuzzy control based on equality relations can transparently be formulated as a result of the logical theory finally we demonstrate that fuzzy control can be seen as a repeated derivation in a continuously modified logical theory and propose a hypothesis on the role of linguistics in the controllability problem

fitting heights of odd order groups with few character degrees abstract if g is any finite solvable group having a normal sylow 2 subgroup in particular if g is odd and satisfying cd g 5 where cd g is the set of ordinary irreducible character degrees of g we show that the fitting height of g does not exceed cd g 2 in case cd g 5 this upper bound on the fitting height of g is best possible

solvability of fuzzy relational equations and manipulation of fuzzy data abstract the problem of characterization of fuzzy relational equations with respect to their solvability property is studied some aspects of manipulation of fuzzy data with the aid of fuzzy relational equations are considered two stages of manipulation process are indicated combining pieces of evidence and inferring their mutual correspondence both of them are formulated and solved by the use of fuzzy relational equations the role of a solvability index is extensively studied numerical considerations give an illustration of the approach we propose and deal with some real data used for fuzzy controllers

equality relations as a basis for fuzzy control abstract the aim of this paper is to introduce a fuzzy control model with well founded semantics in order to explain the concepts applied in fuzzy control assuming that the domains of the input and output variables for the process are endowed with equality relations that reflect the indistinguishability of values lying closely together the use of triangular and trapezoidal membership functions can be justified and max inference where is a t norm turns out to be a consequence of our model distinguishing between a functional and a relational view of the control rules it is possible to explain when defuzzification strategies like mom or coa are appropriate or lead to undesired results

pseudo and conditional score approach to joint analysis of current count and current status data we develop a joint analysis approach for recurrent and nonrecurrent event processes subject to case i interval censorship which are also known in literature as current count and current status data respectively we use a shared frailty to link the recurrent and nonrecurrent event processes while leaving the distribution of the frailty fully unspecified conditional on the frailty the recurrent event is assumed to follow a nonhomogeneous poisson process and the mean function of the recurrent event and the survival function of the nonrecurrent event are assumed to follow some general form of semiparametric transformation models estimation of the models is based on the pseudo likelihood and the conditional score techniques the resulting estimators for the regression parameters and the unspecified baseline functions are shown to be consistent with rates of square and cubic roots of the sample size respectively asymptotic normality with closed form asymptotic variance is derived for the estimator of the regression parameters we apply the proposed method to a fracture osteoporosis survey data to identify risk factors jointly for fracture and osteoporosis in elders while accounting for association between the two events within a subject

auxiliary covariate data in failure time regression summary we consider the problem of missing covariate data in the context of censored failure time relative risk regression auxiliary covariate data which are considered informative about the missing data but which are not explicitly part of the relative risk regression model may be available full covariate information is available for a validation set an estimated partial likelihood method is proposed for estimating relative risk parameters this method is an extension of the estimated likelihood regression analysis method for uncensored data pepe 1992 pepe fleming 1991 a key feature of the method is that it is nonparametric with respect to the association between the missing and observed including auxiliary covariate components asymptotic distribution theory is derived for the proposed estimated partial likelihood estimator in the case where the auxiliary or mismeasured covariates are categorical asymptotic efficiencies are calculated for exponential failure times using an exponential relative risk model the estimated partial likelihood estimator compares favourably with a fully parametric maximum likelihood analysis comparisons are also made with a standard partial likelihood analysis which ignores the incomplete observations important efficiency gains can be made with the estimated partial likelihood method small sample properties are investigated through simulation studies

logistic regression for two stage case control data summary samples of diseased cases and nondiseased controls are drawn at random from the population at risk after classification according to the exposure of interest subsamples of cases and controls are selected for purposes of covariable ascertainment a modification of the usual logistic regression analysis yields consistent estimates of covariable adjusted relative risks and their standard errors by balancing the numbers of exposed and nonexposed for whom covariable inforniation is ascertained within case and control samples some efficiency may be gained over the usual single stage design particularly when the exposure is rare and the relative risks associated with the covariables are large the procedure may be useful also when covariable information is missing for a large part of the sample

the relation between inference and interpolation in the framework of fuzzy systems abstract this papers aims at clarifying the meaning of different interpretations of the max min or more generally the max t norm rule in fuzzy systems it turns out that basically two distinct approaches play an important role in fuzzy logic and its applications fuzzy interpolation on the basis of an imprecisely known function and logical inference in the presence of fuzzy information

maximum likelihood estimation in semiparametric regression models with censored data summary semiparametric regression models play a central role in formulating the effects of covariates on potentially censored failure times and in the joint modelling of incomplete repeated measures and failure times in longitudinal studies the presence of infinite dimensional parameters poses considerable theoretical and computational challenges in the statistical analysis of such models we present several classes of semiparametric regression models which extend the existing models in important directions we construct appropriate likelihood functions involving both finite dimensional and infinite dimensional parameters the maximum likelihood estimators are consistent and asymptotically normal with efficient variances we develop simple and stable numerical techniques to implement the corresponding inference procedures extensive simulation experiments demonstrate that the inferential and computational methods proposed perform well in practical settings applications to three medical studies yield important new insights we conclude that there is no reason theoretical or numerical not to use maximum likelihood estimation for semiparametric regression models we discuss several areas that need further research