combining information from independent sources through confidence distributions this paper develops new methodology together with related theories for combining information from independent studies through confidence distributions a formal definition of a confidence distribution and its asymptotic counterpart i e asymptotic confidence distribution are given and illustrated in the context of combining information two general combination methods are developed the first along the lines of combining p values with some notable differences in regard to optimality of bahadur type efficiency the second by multiplying and normalizing confidence densities the latter approach is inspired by the common approach of multiplying likelihood functions for combining parametric information the paper also develops adaptive combining methods with supporting asymptotic theory which should be of practical interest the key point of the adaptive development is that the methods attempt to combine only the correct information downweighting or excluding studies containing little or wrong information about the true parameter of interest the combination methodologies are illustrated in simulated and real data examples with a variety of applications

absorbing boundary in one dimensional anomalous transport in this paper we study the space time probability distribution q x t of a random walk subject to an absorbing boundary at the origin x 0 for motion controlled by l evy flights and l evy walks characterized by the exponent ensuremath gamma we find that the method of images usually applicable to brownian motion may break down for l evy processes we calculate the distribution q x t to be at xg0 after time tg0 having started at the origin assuming that the boundary is effective at time tg0 we show that q x t depends on the details of the underlying process q x t ensuremath sim mathit x ensuremath gamma 2 mathit t 1 1 ensuremath gamma 1 ensuremath le ensuremath gamma ensuremath le 2 for small x while total survival is independent of the spatial realization of motion and displays a universal behavior we also discuss the related smoluchowski boundary condition problem

modeling record breaking stock prices abstract we study the statistics of record breaking events in daily stock prices of 366 stocks from the standard and poor s 500 stock index both the record events in the daily stock prices themselves and the records in the daily returns are discussed in both cases we try to describe the record statistics of the stock data with simple theoretical models the daily returns are compared to i i d rvs and the stock prices are modeled using a biased random walk for which the record statistics are known these models agree partly with the behavior of the stock data but we also identify several interesting deviations most importantly the number of records in the stocks appears to be systematically decreased in comparison with the random walk model considering the autoregressive ar 1 process we can predict the record statistics of the daily stock prices more accurately we also compare the stock data with simulations of the record statistics of the more complicated garch 1 1 model which in combination with the ar 1 model gives the best agreement with the observational data to better understand our findings we discuss the survival and first passage times of stock prices on certain intervals and analyze the correlations between the individual record events after recapitulating some recent results for the record statistics of ensembles of n stocks we also present some new observations for the weekly distributions of record events

on the removal of skewness by transformation summary we suggest empirical transformations for removing most of the skewness of an asymmetric statistic these transformations may be used as the basis for accurate confidence procedures or hypothesis tests and can be employed in conjunction with either a normal approximation or the bootstrap our approach differs from those of other researchers in that the transformation is monotone and invertible in this paper we suggest elementary but general transformations for eliminating skewness from the distribution of a studentized statistic the transformations are monotone and have simple explicit inversion formulae and so are readily applied to confidence interval problems when the transformations are used in conjunction with the bootstrap they can produce confidence intervals with particularly low levels of coverage error that confidence intervals for symmetric statistics have lower coverage error than their counterparts in the case of asymmetry may be deduced from edgeworth expansions the first term in an expansion of size n 1 2 where n is the sample size describes the error in the usual normal approximation and is due entirely to skewness if it can be eliminated for example by transformation then the normal approxima tion will be in error by only o n 1 thus our method involves transforming one statistic into another where the distribution is virtually symmetric applying the normal approximation or the bootstrap to the new statistic and then regaining the asymmetry of the original problem by inverse transformation at the same time retain ing the high coverage accuracy conferred by applying confidence interval procedures to a symmetric statistic

score confidence intervals and sample sizes for stratified comparisons of binomial proportions in a series of articles gart and nam construct the efficient score tests and confidence intervals with or without skewness correction for stratified comparisons of binomial proportions on the risk difference relative risk and odds ratio effect metrics however the stratified score methods and their properties are not well understood we rederive the efficient score tests which reveals their theoretical relationship with the contrast based score tests and provides a basis for adapting the method by using other weighting schemes the inverse variance weight is optimal for a common treatment effect in large samples we explore the behavior of the score approach in the presence of extreme outcomes when either no or all subjects in some strata are responders and provide guidance on the choice of weights in the analysis of rare events the score method is recommended for studies with a small number of moderate or large sized strata a general framework is proposed to calculate the asymptotic power and sample size for the score test in superiority noninferiority and equivalence clinical trials or case control studies we also describe a nearly exact procedure that underestimates the exact power but the degree of underestimation can be controlled to a negligible level the proposed methods are illustrated by numerical examples

a relaxed extragradient like method for a generalized mixed equilibrium problem a general system of generalized equilibria and a fixed point problem abstract very recently takahashi and takahashi s takahashi w takahashi strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a hilbert space nonlinear anal 69 2008 1025 1033 suggested and analyzed an iterative method for finding a common solution of a generalized equilibrium problem and a fixed point problem of a nonexpansive mapping in a hilbert space in this paper based on takahashi takahashi s iterative method and well known extragradient method we introduce a relaxed extragradient like method for finding a common solution of a generalized mixed equilibrium problem a general system of generalized equilibria and a fixed point problem of a strictly pseudocontractive mapping in a hilbert space and then obtain a strong convergence theorem utilizing this theorem we establish some new strong convergence results in fixed point problems variational inequalities mixed equilibrium problems and systems of generalized equilibria

inferential methods for elasticity estimates abstract elasticities are often estimated from the results of demand analysis however drawing inferences from them may involve assumptions that could influence the outcome in this paper we investigate one of the most common forms of elasticity which is defined as a ratio of estimated relationships and demonstrate how the fieller method for the construction of confidence intervals can be used to draw inferences we estimate the elasticities of expenditure from engel curves using a variety of estimation models parametric engel curves are modelled using ols mm robust regression and tobit semiparametric engel curves are estimated using a penalized spline regression we demonstrate the construction of confidence intervals of the expenditure elasticities for a series of expenditure levels as well as the estimated cumulative density function for the elasticity evaluated for a particular household

record statistics for biased random walks with an application to financial data we consider the occurrence of record breaking events in random walks with asymmetric jump distributions the statistics of records in symmetric random walks was previously analyzed by majumdar and ziff phys rev lett 101 050601 2008 and is well understood unlike the case of symmetric jump distributions in the asymmetric case the statistics of records depends on the choice of the jump distribution we compute the record rate p n c defined as the probability for the nth value to be larger than all previous values for a gaussian jump distribution with standard deviation that is shifted by a constant drift c for small drift in the sense of c n 1 2 the correction to p n c grows proportional to arctan n and saturates at the value c 2 for large n the record rate approaches a constant which is approximately given by 1 2 c exp c 2 2 2 for c 1 these asymptotic results carry over to other continuous jump distributions with finite variance as an application we compare our analytical results to the record statistics of 366 daily stock prices from the standard poor s 500 index the biased random walk accounts quantitatively for the increase in the number of upper records due to the overall trend in the stock prices and after detrending the number of upper records is in good agreement with the symmetric random walk however the number of lower records in the detrended data is significantly reduced by a mechanism that remains to be identified

breaking records in the evolutionary race abstract we explore some aspects of the relationship between biological evolution processes and the mathematical theory of records for eigen s quasispecies model with an uncorrelated fitness landscape we show that the evolutionary trajectories traced out by a population initially localized at a randomly chosen point in sequence space can be described in close analogy to record dynamics with two complications first the increasing number of genotypes that become available with increasing distance from the starting point implies that fitness records are more frequent than for the standard case of independent identically distributed random variables second fitness records can be bypassed which strongly reduces the number of genotypes that take part in an evolutionary trajectory for exponential and gaussian fitness distributions this number scales with sequence length n as n and it is of order unity for distributions with a power law tail this is in strong contrast to the number of records which is of order n for any fitness distribution

nonlinear proximal point algorithms using bregman functions with applications to convex programming a bregman function is a strictly convex differentiable function that induces a well behaved distance measure or d function on euclidean space this paper shows that for every bregman function there exists a nonlinear version of the proximal point algorithm and presents an accompanying convergence theory applying this generalization of the proximal point algorithm to convex programming one obtains the d function proximal minimization algorithm of censor and zenios and a wide variety of new multiplier methods these multiplier methods are different from those studied by kort and bertsekas and include nonquadratic variations on the proximal method of multipliers

correlations of record events as a test for heavy tailed distributions a record is an entry in a time series that is larger or smaller than all previous entries if the time series consists of independent identically distributed random variables with a superimposed linear trend record events are positively negatively correlated when the tail of the distribution is heavier lighter than exponential here we use these correlations to detect heavy tailed behavior in small sets of independent random variables the method consists of converting random subsets of the data into time series with a tunable linear drift and computing the resulting record correlations

correlations between record events in sequences of random variables with a linear trend the statistics of records in sequences of independent identically distributed random variables is a classic subject of study one of the earliest results concerns the stochastic independence of record events recently records statistics beyond the case of i i d random variables have received much attention but the question of independence of record events has not been addressed systematically in this paper we study this question in detail for the case of independent non identically distributed random variables specifically for random variables with a linearly moving mean we find a rich pattern of positive and negative correlations and show how their asymptotics is determined by the universality classes of extreme value statistics

record statistics and persistence for a random walk with a drift we study the statistics of records of a one dimensional random walk of n steps starting from the origin and in presence of a constant bias c at each time step the walker makes a random jump of length eta drawn from a continuous distribution f eta which is symmetric around a constant drift c we focus in particular on the case were f eta is a symmetric stable law with a levy index 0 mu leq 2 the record statistics depends crucially on the persistence probability which as we show here exhibits different behaviors depending on the sign of c and the value of the parameter mu hence in the limit of a large number of steps n the record statistics is sensitive to these parameters c and mu of the jump distribution we compute the asymptotic mean record number after n steps as well as its full distribution p r n we also compute the statistics of the ages of the longest and the shortest lasting record our exact computations show the existence of five distinct regions in the c 0 mu leq 2 strip where these quantities display qualitatively different behaviors we also present numerical simulation results that verify our analytical predictions

levy flights on the half line we study the probability distribution function pdf of the position of a levy flight of index 0 2 in the presence of an absorbing wall at the origin the solution of the associated fractional fokker planck equation can be constructed using a perturbation scheme around the brownian solution corresponding to 2 as an expansion in e 2 we obtain an explicit analytical solution exact at the first order in e which allows us to conjecture the precise asymptotic behavior of this pdf including the first subleading corrections for any careful numerical simulations as well as an exact computation for 1 confirm our conjecture

constructing confidence intervals for cost effectiveness ratios an evaluation of parametric and non parametric techniques using monte carlo simulation the statistic of interest in most health economic evaluations is the incremental cost effectiveness ratio since the variance of a ratio estimator is intractable the health economics literature has suggested a number of alternative approaches to estimating confidence intervals for the cost effectiveness ratio in this paper monte carlo simulation techniques are employed to address the question of which of the proposed methods is most appropriate by repeatedly sampling from a known distribution and applying the different methods of confidence interval estimation it is possible to calculate the coverage properties of each method to see if these correspond to the chosen confidence level as the results of a single monte carlo experiment would be valid only for that particular set of circumstances a series of experiments was conducted in order to examine the performance of the different methods under a variety of conditions relating to the sample size the coefficient of variation of the numerator and denominator of the ratio and the covariance between costs and effects in the underlying data response surface analysis was used to analyse the results and substantial differences between the different methods of confidence interval estimation were identified the methods both parametric and non parametric which assume a normal sampling distribution performed poorly as did the approach based on simply combining the separate intervals on costs and effects the choice of method for confidence interval estimation can lead to large differences in the estimated confidence limits for cost effectiveness ratios the importance of such differences is an empirical question and will depend to a large extent on the role of hypothesis testing in economic appraisal however where it is suspected that the sampling distribution is skewed normal approximation methods produce particularly poor results and should be avoided

approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method in this paper we introduce an iterative method based on the extragradient method for finding a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping in a real hilbert space furthermore we prove that the studied iterative method strongly converges to a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping under some mild conditions imposed on algorithm parameters

lagrangian duality and related multiplier methods for variational inequality problems we consider a new class of multiplier interior point methods for solving variational inequality problems with maximal monotone operators and explicit convex constraint inequalities developing a simple lagrangian duality scheme which is combined with the recent logarithmic quadratic proximal lqp theory introduced by the authors we derive three algorithms for solving the variational inequality vi problem this provides a natural extension of the methods of multipliers used in convex optimization and leads to smooth interior point multiplier algorithms we prove primal dual and primal dual convergence under very mild assumptions eliminating all the usual assumptions used until now in the literature for related algorithms applications to complementarity problems are also discussed

better bootstrap confidence intervals abstract we consider the problem of setting approximate confidence intervals for a single parameter in a multiparameter family the standard approximate intervals based on maximum likelihood theory can be quite misleading in practice tricks based on transformations bias corrections and so forth are often used to improve their accuracy the bootstrap confidence intervals discussed in this article automatically incorporate such tricks without requiring the statistician to think them through for each new application at the price of a considerable increase in computational effort the new intervals incorporate an improvement over previously suggested methods which results in second order correctness in a wide variety of problems in addition to parametric families bootstrap intervals are also developed for nonparametric situations

universal first passage properties of discrete time random walks and levy flights on a line statistics of the global maximum and records abstract in these lecture notes i will discuss the universal first passage properties of a simple correlated discrete time sequence x 0 0 x 1 x 2 x n up to n steps where x i represents the position at step i of a random walker hopping on a continuous line by drawing independently at each time step a random jump length from an arbitrary symmetric and continuous distribution it includes e g the levy flights i will focus on the statistics of two extreme observables associated with the sequence i its global maximum and the time step at which the maximum occurs and ii the number of records in the sequence and their ages i will demonstrate how the universal statistics of these observables emerge as a consequence of pollaczek spitzer formula and the associated sparre andersen theorem

on borwein wiersma decompositions of monotone linear relations monotone operators are of basic importance in optimization as they generalize simultaneously subdifferential operators of convex functions and positive semidefinite not necessarily symmetric matrices in 1970 asplund a monotone convergence theorem for sequences of nonlinear mapping in nonlinear functional analysis american math society providence ri 1970 pp 1 9 studied the additive decomposition of a maximal monotone operator as the sum of a subdifferential operator and an irreducible monotone operator in 2007 borwein and wiersma siam j optim 18 2007 pp 946 960 introduced another additive decomposition where the maximal monotone operator is written as the sum of a subdifferential operator and a skew monotone operator both decompositions are variants of the well known additive decomposition of a matrix via its symmetric and skew part this paper presents a detailed study of the borwein wiersma decomposition of a maximal monotone linear relation we give sufficient conditions and characterizations for a maximal monotone linear relation to be borwein wiersma decomposable and show that borwein wiersma decomposability implies asplund decomposability we exhibit irreducible linear maximal monotone operators without full domain thus answering one of the questions raised by borwein and wiersma the borwein wiersma decomposition of any maximal monotone linear relation is made quite explicit in hilbert space

calibrating confidence coefficients abstract two approaches for dealing with the problem of poor coverage probabilities of certain standard confidence intervals are proposed the first is a recommendation that the actual coverage be estimated directly from the data and its value reported in addition to the nominal level this is achieved through a combination of computer simulation and density estimation the asymptotic validity of the procedure is proved for a number of common situations a classical example is the nonparametric estimation of the variance of a population using the normal theory interval here it is shown that the estimated coverage probability consistently estimates the true coverage probability if the population distribution possesses a finite sixth moment the second approach is more traditional it is a procedure for modifying an interval to yield improved coverage properties given a confidence interval its estimated coverage probability obtained in the first approach is used to alter the nominal level of the interval

time since maximum of brownian motion and asymmetric levy processes motivated by recent studies of record statistics in relation to strongly correlated time series we consider explicitly the drawdown time of a levy process which is defined as the time since it last achieved its running maximum when observed over a fixed time period we show that the density function of this drawdown time in the case of a completely asymmetric jump process may be factored as a function of t multiplied by a function of t t this extends a known result for the case of pure brownian motion we state the factors explicitly for the cases of exponential down jumps with drift and for the downward inverse gaussian levy process with drift

universal and nonuniversal properties of cross correlations in financial time series we use methods of random matrix theory to analyze the cross correlation matrix c of stock price changes of the largest 1000 u s companies for the 2 year period 1994 1995 we find that the statistics of most of the eigenvalues in the spectrum of c agree with the predictions of random matrix theory but there are deviations for a few of the largest eigenvalues we find that c has the universal properties of the gaussian orthogonal ensemble of random matrices furthermore we analyze the eigenvectors of c through their inverse participation ratio and find eigenvectors with large ratios at both edges of the eigenvalue spectrum a situation reminiscent of localization theory results

a geometric confidence ellipse approach to the estimation of the ratio of two variables one is often interested in the ratio of two variables for example in genetics assessing drug effectiveness and in health economics in this paper we derive an explicit geometric solution to the general problem of identifying the two tangents from an arbitrary external point to an ellipse this solution permits numerical integration of a bivariate normal distribution over a wedge shaped region bounded by the tangents which yields an evaluation of the tangent slopes as confidence limits on the ratio of the component variables after suitable adjustment of the confidence coverage of the ellipse these confidence limits are shown to be equivalent to those from fieller s method however the geometric approach allows additional interpretation of the data through identification of the points of tangency the ellipse itself and expressions for the coverage probability of the confidence interval numerical evaluations using the theoretical expressions for the geometric confidence intervals but ignoring sample variation in the underlying parameters suggested that they perform well overall and are slightly conservative simulations that do take account of sample variation in the underlying parameters again suggested that the intervals perform well overall although here they are slightly anti conservative coverage probabilities for the confidence intervals were only weakly dependent on the distance and correlation of the ellipse but there were asymmetries in the failure rates of the upper and lower confidence limits in some configurations the probability of no real solution existing was also evaluated these ideas are illustrated by a practical example

an interior point iterative maximum likelihood reconstruction algorithm incorporating upper and lower bounds with application to spect transmission imaging the algorithm we consider here is a block iterative or ordered subset version of the inferior point algorithm for transmission reconstruction our algorithm is an interior point method because each vector of the iterative sequence x sup k k 0 1 2 satisfies the constraints a sub j x sub j sup k b sub j j 1 j because it is a block iterative algorithm that reconstructs the transmission attenuation map and places constraints above and below the pixel values of the reconstructed image we call it the bitab method computer simulations using the three dimensional mathematical cardiac and torso phantom reveal that the bitab algorithm in conjunction with reasonably selected prior upper and lower bounds has the potential to improve the accuracy of the reconstructed attenuation coefficients from truncated fan beam transmission projections by suitably selecting the bounds it is possible to restrict the over estimation of coefficients outside the fully sampled region that results from reconstructing truncated fan beam projections with iterative transmission algorithms such as the maximum likelihood gradient type algorithm

smooth methods of multipliers for complementarity problems this paper describes several methods for solving nonlinear complementarity problems a general duality framework for pairs of monotone operators is developed and then applied to the monotone complementarity problem obtaining primal dual and primal dual formulations we derive bregman function based generalized proximal algorithms for each of these formulations generating three classes of complementarity algorithms the primal class is well known the dual class is new and constitutes a general collection of methods of multipliers or augmented lagrangian methods for complementarity problems in a special case it corresponds to a class of variational inequality algorithms proposed by gabay by appropriate choice of bregman function the augmented lagrangian subproblem in these methods can be made continuously differentiable the primal dual class of methods is entirely new and combines the best theoretical features of the primal and dual methods some preliminary computation shows that this class of algorithms is effective at solving many of the standard complementarity test problems

multiple testing in clinical trials the basic ideas of multiple testing are outlined and the problem of how to control the probability of erroneous decisions is discussed the main emphasis is on the concept of the multiple level of significance controlling the experiment or family error in the strong sense which can be achieved by applying the principle of closed tests various practical situations encountered in multiple testing in clinical trials are considered more than one end point more than two treatments such as comparisons with a single control comparisons using ordered alternatives all pairwise comparisons and contrast methods and more than one trial tests based on global statistics the union intersection principle and other criteria are discussed the application of the multiple test concept in sequential sampling is investigated finally some comments are made on multiple power multiple confidence intervals and directed decisions

universal statistics of longest lasting records of random walks and levy flights we study the record statistics of random walks after n steps x0 x1 xn with arbitrary symmetric and continuous distribution p of the jumps i xi xi 1 we consider the age of the records i e the time up to which a record survives depending on how the age of the current last record is defined we propose three distinct sequences of ages indexed by i ii iii associated to a given sequence of records we then focus on the longest lasting record which is the longest element among this sequence of ages to characterize the statistics of these longest lasting records we compute i the probability that the record of the longest age is broken at step n denoted by q n which we call the probability of record breaking and ii the duration of the longest lasting record we show that both q n and the full statistics of are universal i e independent of the jump distribution p we compute exactly the large n asymptotic behaviors of q n as well as when it exists and show that each case gives rise to a different universal constant associated to random walks including levy flights while two of them appeared before in the excursion theory of brownian motion for which we provide here a simpler derivation the third case gives rise to a non trivial new constant ciii 0 241 749 associated to the records of random walks other observables characterizing the ages of the records exhibiting an interesting universal behavior are also discussed

a proximal point method for the variational inequality problem in banach spaces in this paper we prove well definedness and weak convergence of the generalized proximal point method when applied to the variational inequality problem in reflexive banach spaces the proximal version we consider makes use of bregman functions whose original definition for finite dimensional spaces has here been properly extended to our more general framework

convergence analysis of a proximal like minimization algorithm using bregman functions an alternative convergence proof of a proximal like minimization algorithm using bregman functions recently proposed by censor and zenios is presented the analysis allows the establishment of a global convergence rate of the algorithm expressed in terms of function values

iterative algorithms for a general system of generalized nonlinear mixed composite type equilibria in this paper we consider and study a general system of generalized nonlinear mixed composite type equilibria in hilbert spaces first we prove the existence and uniqueness of the solution for this system of generalized nonlinear mixed composite type equilibria second the mann iterative method with errors is extended to develop some new iterative algorithms for finding approximate solutions for this system of generalized nonlinear mixed composite type equilibria we also derive the strong convergence of the sequences generated by these iterative algorithms in hilbert spaces

the primal dual algorithm as a constraint set manipulation device a general primal dual algorithm for linearly constrained optimization problems is formulated in which the dual variables are updated by a dual algorithmic operator convergence is proved under the assumption that the dual algorithmic operator implies asymptotic feasibility of the primal iterates with respect to the linear constraints a general result relating the minimal values of an infinite sequence of constrained problems to the minimal value of a limiting problem constrained by the limit of the sequence of constraints sets is established and invoked the applicability of the general theory is demonstrated by analyzing a specific dual algorithmic operator this leads to the mart algorithm for constrained entropy maximization used in image reconstruction from projections

on the bootstrap and confidence intervals we derive an explicit formula for the first term in an unconditional edgeworth type expansion of coverage probability for the nonparametric bootstrap technique applied to a very broad class of studentized statistics the class includes sample mean k sample mean sample correlation coefficient maximum likelihood estimators expressible as functions of vector means etc we suggest that the bootstrap is really an empiric one term edgeworth inversion with the bootstrap simulations implicitly estimating the first term in an edgeworth expansion this view of the bootstrap is reinforced by our discussion of the iterated bootstrap which inverts an edgeworth expansion to arbitrary order by simulating simulations

multiple tests of cost effectiveness angles cost effectiveness angles are an attractive measure of performance when comparing effects and costs of health care therapies because they have a clear interpretation and are well suited for statistical inference in clinical trials a common setup is the comparison of multiple new therapies with a single control if cost effectiveness angles are calculated for each comparison multiplicity issues should be taken into account when quantifying uncertainty of the point estimates therefore this paper proposes a parametric test for multiple cost effectiveness angles that guarantees strong family wise error rate control the idea is to replace the test of m cost effectiveness angles as a union intersection test of 3m linear hypotheses considering the correlation structure of the individual test statistics for the linear hypotheses leads to a maximum type test for the intersection hypothesis inverting these test decisions then gives simultaneous cis of cost effectiveness angles with the appropriate coverage probabilities

central paths generalized proximal point methods and cauchy trajectories in riemannian manifolds we study the relationships between three concepts which arise in connection with variational inequality problems central paths defined by arbitrary barriers generalized proximal point methods where a bregman distance substitutes for the euclidean one and cauchy trajectory in riemannian manifolds first we prove that under rather general hypotheses the central path defined by a general barrier for a monotone variational inequality problem is well defined bounded and continuous and converges to the analytic center of the solution set with respect to the given barrier thus generalizing results which deal only with complementarity problems and with the logarithmic barrier next we prove that a sequence generated by the proximal point method with the bregman distance naturally induced by the barrier function converges precisely to the same point furthermore for a certain class of problems including linear programming such a sequence is contained in the central path making the concepts of central path and generalized proximal point sequence virtually equivalent finally we prove that for this class of problems the central path also coincides with the cauchy trajectory in the riemannian manifold defined on the positive orthant by a metric given by the hessian of the barrier i e a curve whose direction at each point is the negative gradient of the objective function at that point in the riemannian metric

on bootstrap resampling and iteration summary we propose a single unifying approach to bootstrap resampling applicable to a very wide range of statistical problems it enables attention to be focused sharply on one or more characteristics which are of major importance in any particular problem such as coverage error or length for confidence intervals or bias for point estimation our approach leads easily and directly to a very general form of bootstrap iteration unifying and generalizing present disparate accounts of this subject it also provides simple solutions to relatively complex problems such as a suggestion by lehmann 1986 for conditionally short confidence intervals we set out a single unifying principle guiding the operation of bootstrap resampling applicable to a very wide range of statistical problems including bias reduction shrinkage hypothesis testing and confidence interval construction our principle differs from other approaches in that it focuses attention directly on a measure of quality or accuracy expressed in the form of an equation whose solution is sought a very general form of bootstrap iteration is an immediate consequence of iterating the empirical solution to this equation so as to improve accuracy when employed for bias reduction iteration of the resampling principle yields a competitor to the generalized jackknife enabling bias to be reduced to arbitrarily low levels when applied to confidence intervals it produces the techniques of hall 1986 and beran 1987 the resampling principle leads easily to solutions of new complex problems such as empirical versions of confidence intervals proposed by lehmann 1986 lehmann argued that an ideal confidence interval is one which is short when it covers the true parameter value but not necessarily otherwise the resampling principle suggests a simple empirical means of constructing such intervals section 2 describes the general principle and 3 shows how it leads naturally to bootstrap iteration there we show that in many problems of practical interest such as bias reduction and coverage error reduction in two sided confidence intervals each iteration reduces error by the factor n 1 where n is sample size in the case of confidence intervals our result sharpens one of beran 1987 who showed that coverage error is reduced by the factor n 2 in two sided intervals the main exception to our n 1 rule is coverage error of one sided intervals where error is reduced by the factor n a at each iteration our approach to bootstrap iteration serves to unify not just the philosophy of iteration for different statistical problems but also different techniques of iteration for the same

saddlepoint approximations of the distribution of the person parameter in the two parameter logistic model large sample theory states the asymptotic normality of the maximum likelihood estimator of the person parameter in the two parameter logistic 2pl model in short tests however the assumption of normality can be grossly wrong as a consequence intended coverage rates may be exceeded and confidence intervals are revealed to be overly conservative methods belonging to the higher order theory more specifically saddlepoint approximations are a convenient way to deal with small sample problems confidence bounds obtained by these means hold the approximate confidence level for a broad range of the person parameter moreover an approximation to the exact distribution permits to compute median unbiased estimates mue that are as likely to overestimate as to underestimate the true person parameter additionally in small samples these mue are less mean biased than the often used maximum likelihood estimator

nonparametric statistical methods for cost effectiveness analyses two measures often used in a cost effectiveness analysis are the incremental cost effectiveness ratio icer and the net health benefit nhb inferences on these two quantities are often hindered by highly skewed cost data in this article we derive the edgeworth expansions for the studentized t statistics for the two measures and show how they could be used to guide inferences in particular we use the expansions to study the theoretical performance of existing confidence intervals based on normal theory and to derive new confidence intervals for the icer and the nhb we conduct a simulation study to compare our new intervals with several existing methods the methods evaluated include taylor s interval fieller s interval the bootstrap percentile interval and the bootstrap bias corrected acceleration interval we found that our new intervals give good coverage accuracy and are narrower compared to the current recommended intervals

prepivoting to reduce level error of confidence sets summary approximate confidence sets for a parameter 0 may be obtained by referring a function of 0 and of the sample to an estimated quantile of that function s sampling distribution we call this function the root of the confidence set either asymptotic theory or bootstrap methods can be used to estimate the desired quantile when the root is not a pivot in the sense of classical statistics the actual level of the approximate confidence set may differ substantially from the intended level prepivoting is the transformation of a confidence set root by its estimated bootstrap cumulative distribution function prepivoting can be iterated bootstrap confidence sets generated from a root prepivoted one or more times have smaller error in level than do confidence sets based on the original root the first prepivoting is nearly equivalent to studentizing when that operation is appropriate further iterations of prepivoting make higher order corrections automatically

universal properties of a run and tumble particle in arbitrary dimension we consider an active run and tumble particle rtp in d dimensions starting from the origin and evolving over a time interval 0 t we examine three different models for the dynamics of the rtp the standard rtp model with instantaneous tumblings a variant with instantaneous runs and a general model in which both the tumblings and the runs are noninstantaneous for each of these models we use the sparre andersen theorem for discrete time random walks to compute exactly the probability that the x component does not change sign up to time t showing that it does not depend on d as a consequence of this result we compute exactly other x component properties namely the distribution of the time of the maximum and the record statistics showing that they are universal i e they do not depend on d moreover we show that these universal results hold also if the speed v of the particle after each tumbling is random drawn from a generic probability distribution our findings are confirmed by numerical simulations some of these results have been announced in a recent letter phys rev lett 124 090603 2020 10 1103 physrevlett 124 090603

records for the number of distinct sites visited by a random walk on the fully connected lattice we consider a random walk on the fully connected lattice with n sites and study the time evolution of the number of distinct sites s visited by the walker on a subset with n sites a record value v is obtained for s at a record time t when the walker visits a site of the subset for the first time the record time t is a partial covering time when v n and a total covering time when v n the probability distributions for the number of records s the record value v and the record covering time t involving r stirling numbers are obtained using generating function techniques the mean values variances and skewnesses are deduced from the generating functions in the scaling limit the probability distributions for s and v lead to the same gaussian density the fluctuations of the record time t are also gaussian at partial covering when n v mathrm o n they are distributed according to the type i gumbel extreme value distribution at total covering when v n a discrete sequence of generalized gumbel distributions indexed by n v is obtained at almost total covering when n v mathrm o 1 these generalized gumbel distributions are crossing over to the gaussian distribution when n v increases

on dual convergence and the rate of primal convergence of bregman s convex programming method bregman s method is an iterative algorithm for solving optimization problems with convex objective and linear inequality constraints it generates two sequences one the primal one is known to converge to the solution of the problem under the assumption of smoothness of the objective function at the solution it is proved that the other sequence the dual one converges to a solution of the dual problem and that the rate of convergence of the primal sequence is at least linear

inexact proximal point methods for variational inequality problems we present a new family of proximal point methods for solving monotone variational inequalities our algorithm has a relative error tolerance criterion in solving the proximal subproblems our convergence analysis covers a wide family of regularization functions including double regularizations recently introduced by silva eckstein and humes jr siam j optim 12 2001 pp 238 261 and the bregman distance induced by h x sum i 1 n x i log x i we do not use in our analysis the assumption of paramonotonicity which is standard in proving convergence of bregman based proximal methods

order statistics of 1 fα signals order statistics of periodic gaussian noise with 1 f power spectrum is investigated using simulations and phenomenological arguments we find three scaling regimes for the average gap d k x k x k 1 between the kth and k 1 st largest values of the signal the result d k k 1 known for independent identically distributed variables remains valid for 0 5 the spectra of average ordered values e k x 1 x k k is also examined the exponent is derived from the gap scaling as well as by relating e k to the density of near extreme states known results for the density of near extreme states combined with scaling suggest that 2 1 2 4 3 2 and 2 are exact values we also show that parallels can be drawn between e k and the quantum mechanical spectra of a particle in power law potentials

the two parameter poisson dirichlet distribution derived from a stable subordinator the two parameter poisson dirichlet distribution denoted mathsf pd alpha theta is a probability distribution on the set of decreasing positive sequences with sum 1 the usual poisson dirichlet distribution with a single parameter theta introduced by kingman is mathsf pd 0 theta known properties of mathsf pd 0 theta including the markov chain description due to vershik shmidt and ignatov are generalized to the two parameter case the size biased random permutation of mathsf pd alpha theta is a simple residual allocation model proposed by engen in the context of species diversity and rediscovered by perman and the authors in the study of excursions of brownian motion and bessel processes for 0 alpha 1 mathsf pd alpha 0 is the asymptotic distribution of ranked lengths of excursions of a markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index alpha formulae in this case trace back to work of darling lamperti and wendel in the 1950s and 1960s the distribution of ranked lengths of excursions of a one dimensional brownian motion is mathsf pd 1 2 0 and the corresponding distribution for a brownian bredge is mathsf pd 1 2 1 2 the mathsf pd alpha 0 and mathsf pd alpha alpha distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semistable markov process whose zero set is the range of a stable subordinator of index alpha

an outer approximation method for the variational inequality problem we study two outer approximation schemes applied to the variational inequality problem in reflexive banach spaces first we propose a generic outer approximation scheme and its convergence analysis unifies a wide class of outer approximation methods applied to the constrained optimization problem as is standard in this setting boundedness and optimality of weak limit points are proved to hold under two alternative conditions i boundedness of the feasible set or ii coerciveness of the operator to develop a convergence analysis where i and ii do not hold we consider a second scheme in which the approximated subproblems use a coercive approximation of the original operator under conditions alternative to both i and ii we obtain standard convergence results furthermore when the space is uniformly convex we establish full strong convergence of the second scheme to a solution

principal component models for correlation matrices summary distributional theory regarding principal components is less well developed for correlation matrices than it is for covariance matrices the intent of this paper is to reduce this disparity methods are proposed that enable investigators to fit and to make inferences about flexible principal components models for correlation matrices the models allow arbitrary eigenvalue multiplicities and allow the distinct eigenvalues to be modelled parametrically or nonparametrically local parameterisations and implicit functions are used to construct full rank unconstrained parameterisations first order asymptotic distributions are obtained directly from the theory of estimating functions second order accurate distributions for making inferences under normality are obtained directly from likelihood theory simulation studies show that the bartlett correction is effective in controlling the size of the tests and that first order approximations to nonnull distributions are reasonably accurate the methods are illustrated on a dataset

probabilistic measures of cost effectiveness several probability based measures are introduced in order to assess the cost effectiveness of a treatment the basic measure consists of the probability that one treatment is less costly and more effective compared with another several variants of this measure are suggested as flexible options for cost effectiveness analysis the proposed measures are invariant under monotone transformations of the cost and effectiveness measures interval estimation of the proposed measures are investigated under a parametric model assuming bivariate normality and also non parametrically the delta method and a generalized pivotal quantity approach are both investigated under the bivariate normal model a non parametric u statistics based approach is also investigated for computing confidence intervals numerical results show that under bivariate normality the solution based on generalized pivotal quantities exhibits accurate performance in terms of maintaining the coverage probability of the confidence interval the non parametric u statistics based solution is accurate for sample sizes that are at least moderately large the results are illustrated using data from a clinical trial for prostate cancer therapy copyright 2016 john wiley sons ltd

universal survival probability for a correlated random walk and applications to records we consider a model of space continuous one dimensional random walk with simple correlation between the steps the probability that two consecutive steps have same sign is q with 0 leq q leq 1 the parameter q allows thus to control the persistence of the random walk we compute analytically the survival probability of a walk of n steps showing that it is independent of the jump distribution for any finite n this universality is a consequence of the sparre andersen theorem for random walks with uncorrelated and symmetric steps we then apply this result to derive the distribution of the step at which the random walk reaches its maximum and the record statistics of the walk which show the same universality in particular we show that the distribution of the number of records for a walk of n gg 1 steps is the same as for a random walk with n rm eff q n 2 1 q uncorrelated and symmetrically distributed steps we also show that in the regime where n to infty and q to 1 with y n 1 q this model converges to the run and tumble particle a persistent random walk often used to model the motion of bacteria our theoretical results are confirmed by numerical simulations

a generalized proximal point algorithm for the variational inequality problem in a hilbert space we consider a generalized proximal point method for solving variational inequality problems with monotone operators in a hilbert space it differs from the classical proximal point method as discussed by rockafellar for the problem of finding zeroes of monotone operators in the use of generalized distances called bregman distances instead of the euclidean one these distances play not only a regularization role but also a penalization one forcing the sequence generated by the method to remain in the interior of the feasible set so that the method becomes an interior point one under appropriate assumptions on the bregman distance and the monotone operator we prove that the sequence converges weakly if and only if the problem has solutions in which case the weak limit is a solution if the problem does not have solutions then the sequence is unbounded we extend similar previous results for the proximal point method with bregman distances which dealt only with the finite dimensional case and which applied only to convex optimization problems or to finding zeroes of monotone operators which are particular cases of variational inequality problems

on symmetric bootstrap confidence intervals on montre que dans des circonstances generales les intervalles de confiance du bootstrap de percentile t symetrique ont une erreur de recouvrement o n 2 ou n denote la taille d echantillon

universal survival probability for a d dimensional run and tumble particle we consider an active run and tumble particle rtp in d dimensions and compute exactly the probability s t that the x component of the position of the rtp does not change sign up to time t when the tumblings occur at a constant rate we show that s t is independent of d for any finite time t and not just for large t as a consequence of the celebrated sparre andersen theorem for discrete time random walks in one dimension moreover we show that this universal result holds for a much wider class of rtp models in which the speed v of the particle after each tumbling is random drawn from an arbitrary probability distribution we further demonstrate as a consequence the universality of the record statistics in the rtp problem

on variable step relaxed projection algorithm for variational inequalities projection algorithms are practically useful for solving variational inequalities vi however some among them require the knowledge related to vi in advance such as lipschitz constant usually it is impossible in practice this paper studies the variable step basic projection algorithm and its relaxed version under weakly co coercive condition the algorithms discussed need not know constant function associated with the co coercivity or weak co coercivity and the step size is varied from one iteration to the next under certain conditions the convergence of the variable step basic projection algorithm is established for the practical consideration we also give the relaxed version of this algorithm in which the projection onto a closed convex set is replaced by another projection at each iteration and latter is easy to calculate the convergence of relaxed scheme is also obtained under certain assumptions finally we apply these two algorithms to the split feasibility problem sfp

unusual properties of bootstrap confidence intervals in regression problems we show that the percentile t method and one of the two percentile methods have unusually good performance when employed to construct bootstrap confidence intervals in a regression setting in the case of slope parameters percentile t produces two sided intervals with coverage errorn 2 and one sided intervals with coverage errorn 3 2 wheren is sample size the errors are onlyn 1 in most other problems one of the percentile methods produces critical points which are third order correct for efron s 11 relatively complex accelerated bias corrected points

rescaling and stepsize selection in proximal methods using separable generalized distances this paper presents a convergence proof technique for a broad class of proximal algorithms in which the perturbation term is separable and may contain barriers enforcing interval constraints there are two key ingredients in the analysis a mild regularity condition on the differential behavior of the barrier as one approaches an interval boundary and a lower stepsize limit that takes into account the curvature of the proximal term we give two applications of our approach first we prove subsequential convergence of a very broad class of proximal minimization algorithms for convex optimization where different stepsizes can be used for each coordinate applying these methods to the dual of a convex program we obtain a wide class of multiplier methods with subsequential convergence of both primal and dual iterates and independent adjustment of the penalty parameter for each constraint the adjustment rules for the penalty parameters generalize a well established scheme for the exponential method of multipliers the results may also be viewed as a generalization of recent work by ben tal and zibulevsky siam j optim 7 1997 pp 347 366 and auslender teboulle and ben tiba comput optim appl 12 1999 pp 31 40 math oper res 24 1999 pp 645 668 on methods derived from varphi divergences the second application established full convergence under a novel stepsize condition of bregman function based proximal methods for general monotone operator problems over a box prior results in this area required strong restrictive assumptions on the monotone operator

record statistics of a strongly correlated time series random walks and levy flights we review recent advances on the record statistics of strongly correlated time series whose entries denote the positions of a random walk or a l evy flight on a line after a brief survey of the theory of records for independent and identically distributed random variables we focus on random walks during the last few years it was indeed realized that random walks are a very useful laboratory to test the effects of correlations on the record statistics we start with the simple one dimensional random walk with symmetric jumps both continuous and discrete and discuss in detail the statistics of the number of records as well as of the ages of the records i e the lapses of time between two successive record breaking events then we review the results that were obtained for a wide variety of random walk models including random walks with a linear drift continuous time random walks constrained random walks like the random walk bridge and the case of multiple independent random walkers finally we discuss further observables related to records like the record increments as well as some questions raised by physical applications of record statistics like the effects of measurement error and noise

record statistics for multiple random walks we study the statistics of the number of records r n n for n identical and independent symmetric discrete time random walks of n steps in one dimension all starting at the origin at step 0 at each time step each walker jumps by a random length drawn independently from a symmetric and continuous distribution we consider two cases i when the variance ensuremath sigma 2 of the jump distribution is finite and ii when ensuremath sigma 2 is divergent as in the case of l evy flights with index 0l ensuremath mu l2 in both cases we find that the mean record number ensuremath langle r n n ensuremath rangle grows universally as ensuremath sim ensuremath alpha n sqrt n for large n but with a very different behavior of the amplitude ensuremath alpha n for ng1 in the two cases we find that for large n ensuremath alpha n ensuremath approx 2 sqrt mathrm ln n independently of ensuremath sigma 2 in case i in contrast in case ii the amplitude approaches to an n independent constant for large n ensuremath alpha n ensuremath approx 4 sqrt ensuremath pi independently of 0l ensuremath mu l2 for finite ensuremath sigma 2 we argue and this is confirmed by our numerical simulations that the full distribution of r n n sqrt n ensuremath 2 sqrt mathrm ln n sqrt mathrm ln n converges to a gumbel law as n ensuremath rightarrow ensuremath infty and n ensuremath rightarrow ensuremath infty in case ii our numerical simulations indicate that the distribution of r n n sqrt n converges for n ensuremath rightarrow ensuremath infty and n ensuremath rightarrow ensuremath infty to a universal nontrivial distribution independently of ensuremath mu we discuss the applications of our results to the study of the record statistics of 366 daily stock prices from the standard poor s 500 index

a new hybrid iterative algorithm for variational inequalities let h be a real hilbert space let f h h be a strongly monotone and lipschitzian mapping let t n n 1 h h be an infinite family of non expansive mappings with common fixed points set n 1 fix t n a we devise an iterative algorithmy n x email protected nf x n x n 1 email protected n y email protected nw ny n n 0 where l n is a sequence in 0 a n is a sequence in 0 1 and w n is the w mapping we prove that the sequence x n converges in norm to x email protected n 1 fix t n which is the unique solution of the following variational inequality 0 email protected email protected n 1 fix t n

an interior point method with bregman functions for the variational inequality problem with paramonotone operators we present an algorithm for the variational inequality problem on convex sets with nonempty interior the use of bregman functions whose zone is the convex set allows for the generation of a sequence contained in the interior without taking explicitly into account the constraints which define the convex set we establish full convergence to a solution with minimal conditions upon the monotone operatorf weaker than strong monotonicity or lipschitz continuity for instance and including cases where the solution needs not be unique we apply our algorithm to several relevant classes of convex sets including orthants boxes polyhedra and balls for which bregman functions are presented which give rise to explicit iteration formulae up to the determination of two scalar stepsizes which can be found through finite search procedures 1998 the mathematical programming society inc published by elsevier science b v