a zero one law approach to the central limit theorem for the weighted bootstrap mean we develop a new technique to prove the conditional clt for the weighted bootstrap mean through 0 1 laws we show that this conditional clt can be derived from an unconditional one which easily arises conditioning with respect to the weights from the standard lindeberg clt

analogy between approximate reasoning and the method of interpolation abstract in this paper we present an analogy between approximate reasoning and the method of interpolation for a large class of problems we first describe the analogy and then illustrate it through several simple yet concrete examples the new results obtained through the method of interpolation are compared with those obtained by the application of the existing method of approximate reasoning thus we show that approximate reasoning can also be realized effectively by the method of interpolation

an upper bound of the nullity of a graph in terms of order and maximum degree abstract let g be a finite undirected graph without loops and multiple edges by and n we respectively denote the nullity the maximum vertex degree and the order of g in 10 it was proved that n 2 n 1 when g is a tree this result was generalized to a bipartite graph by 25 for a reduced bipartite graph g the above inequality was improved to n 2 2 ln 2 by 27 however the problem of bounding the nullity of an arbitrary graph g in terms of n and is left open for more than ten years in this article we aim to solve such a left problem we prove that 1 n for an arbitrary graph g with order n and maximum degree and the equality holds if and only if g is the disjoint union of some copies of k

a least squares model for fuzzy rules of inference abstract in this paper the method of least squares which uses the power function is applied to the fuzzy inference rules these rules in fuzzy logic are known as the generalized modus ponens and the generalized modus tollens the proposed technique gives reasonable evaluations for fuzzy consequences and is simple for computer applications

investigations on the applicability of fuzzy interference abstract this paper investigates the applicability of some fuzzy implication operators which were evaluated in the literature and are considered to be very good operators we show that iin 1 1 a a b b has the best performance in the cases examined some new fuzzy implication operators are also investigated and found comparable in all cases to the operator given above

on the nullity of the line graph of unicyclic graph with depth one abstract a connected graph with a unique cycle is called a unicyclic graph a unicyclic graph with depth one may be thought of as being obtained from a cycle by appending n i pendent edges on each vertex v i in the cycle c t for some integer t 3 denoted by c n 1 n 2 n t in this paper we give a complete characterization on the nullity of the line graph g l c n 1 n 2 n t as follows let m i n i 0 then i g 2 if and only if m 0 and t 0 mod 4 ii g 1 if and only if m 1 and either 1 n i 0 1 for i 1 t the length of any zero chain of n 1 n 2 n t is even and t m 0 mod 4 or 2 t 0 mod 4 and one of n 1 n 3 n t 1 0 and n 2 n 4 n t 0 must hold iii g 0 otherwise

a class of k sample tests for comparing the cumulative incidence of a competing risk in this paper for right censored competing risks data a class of tests developed for comparing the cumulative incidence of a particular type of failure among different groups the tests are based on comparing weighted averages of the hazards of the subdistribution for the failure type of interest asymptotic results are derived by expressing the statistics in terms of counting processes and using martingale central limit theory it is proposed that weight functions very similar to those for the g p tests from ordinary survival analysis be used simulation results indicate that the asymptotic distributions provide adequate approximations in moderate sized samples

on general resampling algorithms and their performance in distribution estimation recent work of several authors has focussed on first order properties e g consistency of general bootstrap algorithms where the numbers of times that data values are resampled form an exchangeable sequence in the present paper we develop second order properties of such algorithms in a very general setting performance is discussed in the context of distribution estimation and formulae for higher order moments and cumulants are developed arguing thus necessary and sufficient conditions are given for general resampling algorithms to correctly capture second order properties

a characterization of graphs g with nullity v g 2m g 2c g abstract by v g e g g and m g we denote respectively the order the number of edges the nullity and the matching number of a simple graph g recently wang and wong have proved that for every graph g g v g 2 m g 2 c g where c g e g v g g g being the number of connected components of g in this paper graphs g that satisfy the equality g v g 2 m g 2 c g are characterized

proof of a conjecture on the nullity of a connected graph in terms of order and maximum degree abstract let g be a connected graph with n vertices maximum vertex degree 2 and nullity in q zhou et al 2018 22 it was conjectured that 2 n 2 1 and the upper bound is attained if and only if g is a cycle c n with n divisible by 4 or a complete bipartite graph k n 2 n 2 in the present paper this conjecture is proved

relationship between the rank and the matching number of a graph abstract given a simple graph g let a g be its adjacency matrix and g be its matching number the rank of g written as r g refers to the rank of a g in this paper some relations between the rank and the matching number of a graph are studied firstly it is proved that 2 d g r g 2 g n o where d g and no are respectively the dimension of cycle space and the number of odd cycles of g secondly sharp lower bounds on both r g g and r g g are determined all the corresponding extremal graphs are characterized respectively

a characterization of graphs with rank 5 abstract the rank of a graph g is defined to be the rank of its adjacency matrix in this paper we consider the following problem what is the structure of a connected graph g with rank 5 or equivalently what is the structure of a connected n vertex graph g whose adjacency matrix has nullity n 5 in this paper we completely characterize connected graphs g whose adjacency matrix has rank 5

the multiplicity of an arbitrary eigenvalue of a graph in terms of cyclomatic number and number of pendant vertices abstract let g be a graph with adjacency matrix a g the nullity g of g is the multiplicity of zero as an eigenvalue of a g which has received a lot of attention because of its chemical importance the multiplicity of an arbitrary eigenvalue of a g is denoted as m g in 25 the authors proved that g 2 g p g for a connected graph g with equality if and only if g is a cycle of order divisible by 4 where g e g v g 1 is the cyclomatic number of g and p g is the number of pendant vertices of g in the present paper we intend to extend this result from the nullity of g to the multiplicity of an arbitrary eigenvalue of g differing from the method in 25 by mainly applying algebraic method we prove the following result for a connected graph g m g 2 g p g for an arbitrary eigenvalue of g the equality holds if and only if g is a cycle c n and 2 cos 2 k n with k 1 2 n 2 1

trees with maximum nullity abstract the nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum among n vertex trees the star has greatest nullity equal to n 2 we generalize this by showing that among n vertex trees whose vertex degrees do not exceed a certain value d the greatest nullity is n 2 n 1 d methods for constructing such maximum nullity trees are described

nullities of graphs with given order matching number and cyclomatic number revisited abstract for a simple graph g we denote by v g e g g and m g respectively the order the number of edges the nullity and the matching number of g it was shown by wang and wong 2014 that for every graph g v g 2 m g c g g v g 2 m g 2 c g where c g e g v g g is the cyclomatic number of g g being the number of components of g graphs with the maximal nullity condition have been characterized by song et al 2015 and graphs with the minimal nullity condition have also been characterized independently by rula et al 2016 and wang 2016 earlier guo et al 2009 had also shown that for a unicyclic graph g g v g 2 m g can take only one of the values 1 0 or 2 and they characterized these three types of unicyclic graphs in this paper exploiting the concepts of canonical star associated with a rooted tree the canonical unicyclic graph associated with a unicyclic graph and a crucial subgraph of a graph we correct complete and extend the work of previous authors on this topic we give a nontrivial proof for the fact that the crucial subgraphs of a graph as introduced by wang 2016 are unique up to isomorphism more complete lists of characterizations for the three types of unicyclic graphs for nonsingular unicyclic graphs and for graphs with the minimal or maximal nullity conditions are found it is proved that if c n are given positive integers with n 6 c 2 then for any integer k c k 2 c k 2 c 1 there exists a connected graph g of order n that satisfies c g c and g v g 2 m g k however if k is replaced by 2 c 1 then there is no graph g of any order that satisfies the corresponding conditions

characterization of graphs whose signature equals the number of odd cycles abstract let g be a simple graph with vertex set v g and edge set e g the signature s g of g is the difference between the number of positive eigenvalues and the number of negative eigenvalues of the adjacency matrix a g in 20 it was proved that c 1 g s g c 1 g where c 1 g denotes the number of odd cycles in g a problem arises naturally what graphs have signature attaining the upper bound c 1 g resp the lower bound c 1 g in this paper we focus our attention on this problem characterizing graphs g whose signature equals c 1 g resp c 1 g

bootstrapping the kaplan meier estimator abstract randomly censored data consist of iid pairs of observations xi i i 1 n if i 0 xi denotes a censored observation and if i 1 xi denotes an exact survival time which is the variable of interest for estimating the distribution f of the survival times the product limit estimator proposed by kaplan and meier 1958 has been studied extensively and it has been shown to enjoy a number of optimality properties wellner 1982 see gill 1980 chaps 1 4 1983 for a modern treatment and for references with censored data bootstrapping can be carried out using two different resampling plans introduced by efron 1981 and reid 1981 respectively with efron s plan one takes a random sample with replacement from x 1 xn n whereas with reid s plan one takes a random sample from the kaplan meier estimator the purpose of this article is to study the asymptotic behavior of the bootstrapped kaplan meier estimator with both resampling plans the approach adopted uses the

complex unit gain graphs of rank 2 abstract let t be the group of all complex numbers z with z 1 a complex unit gain graph or simply a t gain graph is a triple g t consisting of a graph g v e the circle group t and a gain function e t such that v i v j v j v i 1 v j v i for any adjacent vertices v i and v j the adjacency matrix a of the t gain graph g t of order n is an n n complex matrix a i j where a i j a j i v i v j if v i is adjacent to v j and a i j 0 if otherwise the rank of a t gain graph denoted by r is the rank of the adjacency matrix of yu et al 24 obtained some properties of inertia indexes of a t gain graph and they characterized the t gain unicyclic graphs with small positive or negative index lu 8 characterized the t gain bicyclic graphs with rank 2 3 or 4 motivated by above we determine the rank of a t gain unicyclic graph and classify the t gain graphs with rank 2

approximate reasoning algorithm of interval valued fuzzy sets based on least square method in this paper we develop a new method for bidirectional approximate reasoning of interval valued fuzzy sets by employing least squares method and investigate some properties for our proposed approximate reasoning method furthermore we use four numerical examples to illustrate our proposed method reasonable

some considerations about a least squares model for fuzzy rules of inference abstract in this paper the method of least squares is applied to the fuzzy inference rules we begin studying the conditions in which from a fuzzy set we can build another through the method of least squares then we apply this technique in order to evaluate the conclusions of the generalized modus ponens and the generalized modus tollens we present different theorems and examples that demonstrate the fundamental advantages of the method studied we conclude presenting a method for obtaining the conclusion when working with fuzzy conditional propositions that contain two fuzzy propositions joined by the connective and

on the multiplicity of an arbitrary aα eigenvalue of a connected graph abstract the a matrix of a graph g is defined by nikiforov as a g d g 1 a g where 0 1 a g and d g respectively denotes the adjacency matrix and the degree diagonal matrix of g the eigenvalues of a g are called the a eigenvalues of g in this paper we prove that for a connected graph g of order n and with maximum degree 2 and for any 0 1 the multiplicity of an arbitrary a eigenvalue of g which is denoted as m g is bounded by m g 2 n 2 1 the equality holds if and only if g and satisfy one of the following conditions i g k n with n 3 and n 1 ii g c n with even order and 2 2 1 cos 2 j n j 1 2 n 2 2 iii g c n with odd order and 2 2 1 cos 2 j n j 1 2 n 1 2 iv g k n 2 n 2 with n 4 and n 2 applying this result we solve an open conjecture posed by zhou et al in 32 one year ago

non strange weird resampling for complex survival data this paper introduces a new data dependent multiplier bootstrap for nonparametric analysis of survival data possibly subject to competing risks the nw procedure includes the general wild bootstrap and the weird bootstrap as special cases the data may be subject to independent right censoring and left truncation the asymptotic correctness of the proposed resampling procedure is proven under standard assumptions simulation results on time simultaneous inference suggest that the weird bootstrap performs better than the standard normal multiplier approach

a study of a class of weighted bootstraps for censored data edgeworth expansions are derived for a class of weighted bootstrap methods for the kaplan meier and nelson aalen estimates using the methods contained in the monograph by barbe and bertail von mises representations up to the third order are established for the weighted bootstrap versions of these estimators it is shown that there exists weights which outperform efron s bootstrap method in terms of coverage accuracy moreover it is shown that this holds for a particular choice of gamma weights which are very easy to use in practice the general weighting schemes are also useful in approximating the posterior distribution of survival function with respect to mixtures of beta neutral process priors

the effects of membership function on fuzzy reasoning abstract fuzzy modus ponens has been considered a generalization of the classic modus ponens logic inference rule it has been expected to provide a more powerful and useful inference capability in the process of fuzzy reasoning the definitions of membership functions of fuzzy sets used in the reasoning process may however adversely affect the reasoning results if the insignificant parts of the membership functions are incompatible this paper explains these undesirable effects and proposes a method to resolve this problem in addition an algorithm for the compositional rule of inference is developed to perform the fuzzy modus ponens syllogism test results shown in this paper indicate that the algorithm and the underlying proposed methods are reasonable approaches to a robust fuzzy logic reasoning system

the inertia of weighted unicyclic graphs let gw be a weighted graph the inertia of gw is the triple in gw i gw i gw i0 gw where i gw i gw i0 gw are the numbers of the positive negative and zero eigenvalues of the adjacency matrix a gw of gw including their multiplicities respectively i gw i gw are called the positive negative indices of inertia of gw respectively in this paper we present a lower bound for the positive negative indices of weighted unicyclic graphs of order n with fixed girth and characterize all weighted unicyclic graphs attaining this lower bound moreover we characterize the weighted unicyclic graphs of order n with two positive two negative and at least n 6 zero eigenvalues respectively

non parametric inference for cumulative incidence functions in competing risks studies in the competing risks problem a useful quantity is the cumulative incidence function which is the probability of occurrence by time t for a particular type of failure in the presence of other risks the estimator of this function as given by kalbfleisch and prentice is consistent and properly normalized converges weakly to a zero mean gaussian process with a covariance function for which a consistent estimator is provided a resampling technique is developed to approximate the distribution of this process which enables one to construct confidence bands for the cumulative incidence curve over the entire time span of interest and to perform kolmogorov smirnov type tests for comparing two such curves an aids example is provided

on the nullity and the matching number of unicyclic graphs abstract let g be a graph with n vertices and g be the matching number of g let g denote the nullity of g the multiplicity of the eigenvalue zero of g it is well known that if g is a tree then g n 2 g tan and liu x tan b liu on the nullity of unicyclic graphs linear alg appl 408 2005 212 220 proved that the nullity set of all unicyclic graphs with n vertices is 0 1 n 4 and characterized the unicyclic graphs with g n 4 in this paper we characterize the unicyclic graphs with g n 5 and we prove that if g is a unicyclic graph then g equals n 2 g 1 n 2 g or n 2 g 2 we also give a characterization of these three types of graphs furthermore we determine the unicyclic graphs g with g 0 which answers affirmatively an open problem by tan and liu

a scalable bootstrap for massive data summary the bootstrap provides a simple and powerful means of assessing the quality of estimators however in settings involving large data sets which are increasingly prevalent the calculation of bootstrap based quantities can be prohibitively demanding computationally although variants such as subsampling and the m out of n bootstrap can be used in principle to reduce the cost of bootstrap computations these methods are generally not robust to specification of tuning parameters such as the number of subsampled data points and they often require knowledge of the estimator s convergence rate in contrast with the bootstrap as an alternative we introduce the bag of little bootstraps blb which is a new procedure which incorporates features of both the bootstrap and subsampling to yield a robust computationally efficient means of assessing the quality of estimators the blb is well suited to modern parallel and distributed computing architectures and furthermore retains the generic applicability and statistical efficiency of the bootstrap we demonstrate the blb s favourable statistical performance via a theoretical analysis elucidating the procedure s properties as well as a simulation study comparing the blb with the bootstrap the m out of n bootstrap and subsampling in addition we present results from a large scale distributed implementation of the blb demonstrating its computational superiority on massive data a method for adaptively selecting the blb s tuning parameters an empirical study applying the blb to several real data sets and an extension of the blb to time series data

some nonasymptotic results on resampling in high dimension i confidence regions we study generalized bootstrap confidence regions for the mean of a random vector whose coordinates have an unknown dependency structure the random vector is supposed to be either gaussian or to have a symmetric and bounded distribution the dimensionality of the vector can possibly be much larger than the number of observations and we focus on a non asymptotic control of the confidence level following ideas inspired by recent results in learning theory we consider two approaches the first based on a concentration principle valid for a large class of resampling weights and the second on a direct resampled quantile specifically using rademacher weights several intermediate results established in the approach based on concentration principles are of self interest we also discuss the question of accuracy when using monte carlo approximations of the resampled quantities we present an application of these results to the one sided and two sided multiple testing problem in which we derive several resampling based step down procedures providing a non asymptotic fwer control we compare our different procedures in a simulation study and we show that they can outperform bonferroni s or holm s procedures as soon as the observed vector has sufficiently correlated coordinates

intentionally biased bootstrap methods summary a class of weighted bootstrap techniques called biased bootstrap or b bootstrap methods is introduced it is motivated by the need to adjust empirical methods such as the uniform bootstrap in a surgical way to alter some of their features while leaving others unchanged depending on the nature of the adjustment the b bootstrap can be used to reduce bias or to reduce variance or to render some characteristic equal to a predetermined quantity examples of the last application include a b bootstrap approach to hypothesis testing in nonparametric contexts where the b bootstrap enables simulation under the null hypothesis even when the hypothesis is false and a b bootstrap competitor to tibshirani s variance stabilization method an example of the bias reduction application is adjustment of nadaraya watson kernel estimators to make them competitive with local linear smoothing other applications include density estimation under constraints outlier trimming sensitivity analysis skewness or kurtosis reduction and shrinkage

a characterization of graphs with rank 4 the rank of a graph g is defined to be the rank of its adjacency matrix in this paper we consider the following problem what is the structure of a connected graph with rank 4 this question has not yet been fully answered in the literature and only some partial results are known in this paper we resolve this question by completely characterizing graphs g whose adjacency matrix has rank 4

the dependent wild bootstrap we propose a new resampling procedure the dependent wild bootstrap for stationary time series as a natural extension of the traditional wild bootstrap to time series setting the dependent wild bootstrap offers a viable alternative to the existing block based bootstrap methods whose properties have been extensively studied over the last two decades unlike all of the block based bootstrap methods the dependent wild bootstrap can be easily extended to irregularly spaced time series with no implementational difficulty furthermore it preserves the favorable bias and mean squared error property of the tapered block bootstrap which is the state of the art block based method in terms of asymptotic accuracy of variance estimation and distribution approximation the consistency of the dependent wild bootstrap in distribution approximation is established under the framework of the smooth function model in addition we obtain the bias and variance expansions of the dependent wild bootstrap variance estimat

on the nullity of bicyclic graphs the nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum in this paper we obtain the nullity set of n vertex unicyclic graphs and characterize the unicyclic graphs with extremal nullity

empirical likelihood block bootstrapping monte carlo evidence has made it clear that asymptotic tests based on generalized method of moments gmm estimation have disappointing size the problem is exacerbated when the moment conditions are serially correlated several block bootstrap techniques have been proposed to correct the problem including hall and horowitz 1996 and inoue and shintani 2006 we propose an empirical likelihood block bootstrap procedure to improve inference where models are characterized by nonlinear moment conditions that are serially correlated of possibly infinite order combining the ideas of kitamura 1997 and brown and newey 2002 the parameters of a model are initially estimated by gmm which are then used to compute the empirical likelihood probability weights of the blocks of moment conditions the probability weights serve as the multinomial distribution used in resampling the first order asymptotic validity of the proposed procedure is proven and a series of monte carlo experiments show it may improve test sizes over conventional block bootstrapping

a characterization of long graphs of arbitrary rank abstract the rank of a graph g is defined to be the rank of its adjacency matrix a connected graph g of rank n is called long if it has maximum diameter among all connected graphs with rank n in this paper we characterize the long graphs of rank n for n an arbitrary positive integer

second order correctness of the poisson bootstrap abstract rao pathak and koltchinskii 1997 have recently studied a sequential approach to resampling in which resampling is carried out sequentially one by one with replacement each time until the bootstrap sample contains m approximately 1 e 1 n approximately 632n distinct observations from the original sample in our previous work we have established that the main empirical characteristics of the sequential bootstrap go through in the sense of being within a distance of order o n 3 4 from those of the usual bootstrap however the theoretical justification of the second order correctness of the sequential bootstrap is somewhat difficult it is the main topic of this investigation among other things we accomplish it by approximating our sequential scheme by a resampling scheme based on the poisson distribution with mean u 1 and censored at x 0

the inertia of unicyclic graphs and the implications for closed shells the inertia of a graph is an integer triple specifying the number of negative zero and positive eigenvalues of the adjacency matrix of the graph a unicyclic graph is a simple connected graph with an equal number of vertices and edges this paper characterizes the inertia of a unicyclic graph in terms of maximum matchings and gives a linear time algorithm for computing it chemists are interested in whether the molecular graph of an unsaturated hydrocarbon is properly closed shell having exactly half of its eigenvalues greater than zero because this designates a stable electron configuration the inertia determines whether a graph is closed shell and hence the reported result gives a linear time algorithm for determining this for unicyclic graphs

estimating the confidence interval for prediction errors of support vector machine classifiers support vector machine svm is one of the most popular and promising classification algorithms after a classification rule is constructed via the svm it is essential to evaluate its prediction accuracy in this paper we develop procedures for obtaining both point and interval estimators for the prediction error under mild regularity conditions we derive the consistency and asymptotic normality of the prediction error estimators for svm with finite dimensional kernels a perturbation resampling procedure is proposed to obtain interval estimates for the prediction error in practice with numerical studies on simulated data and a benchmark repository we recommend the use of interval estimates centered at the cross validated point estimates for the prediction error further applications of the proposed procedure in model evaluation and feature selection are illustrated with two examples

the product limit estimator and the bootstrap some asymptotic representations the product limit estimator and its quantile process are represented as i i d mean processes with a remainder of ordern 3 4 logn 3 4 a s corresponding bootstrap versions of these representations are given which can help one visualize how the bootstrap procedure operates in this set up

comparison of fuzzy reasoning methods l a zadeh e h mamdani and m mizumoto et al have proposed methods for fuzzy reasoning in which the antecedent involves a fuzzy conditional proposition if x is a then y is b with a and b being fuzzy concepts mizumoto et al have investigated the properties of their methods in the case of generalized modus ponens this paper deals with the properties of their methods in the case of generalized modus tollens and investigates the other new fuzzy reasoning methods obtained by introducing the implication rules of many valued logic systems finally the properties of syllogism and contrapositive are investigated under each fuzzy reasoning method

the positive and the negative inertia index of line graphs of trees abstract the positive resp negative inertia index of g denoted by p g resp n g is defined to be the number of positive resp negative eigenvalues of its adjacency matrix the nullity of g denoted by g is defined to be the multiplicity of the eigenvalue zero in the adjacency spectrum of g in 2001 gutman and sciriha 11 proved that the nullity of the line graph of a tree is either 0 or 1 in this paper we consider the positive and the negative inertia index for the line graph l t of a tree t showing that e t 1 2 p l t e t 1 where e t denotes the number of internal non pendant edges contained in t the extremal trees for which p l t attains the upper bound and the lower bound are respectively characterized it is shown that l t is nonsingular if p l t attains the upper bound and l t is singular if p l t attains the lower bound