a continuous version of de finetti s theorem a continuous version of de finetti s theorem is proved in which the role of the homogeneous product states is played by the independent increment stationary processes on the real line the proof is based on a conditional finite de finetti s theorem i e a result involving only a finite number of random variables and exchangeable conditional expectations rather than exchangeable probabilities our technique of proof improves and simplifies a result of freedman and includes a generalization of the quantum de finetti s theorem as well as some more recent variants of it the last section of the paper is an attempt to answer a question of diaconis and freedman

advantages to transforming the receiver operating characteristic roc curve into likelihood ratio co ordinates traditionally the receiver operating characteristic roc curve for a diagnostic test plots true positives sensitivity against false positives one minus specificity however this representation brings with it several drawbacks a transformation to positive and negative likelihood ratio co ordinates scaled by base ten logarithms offers several advantages first we motivate the use of positive and negative likelihood ratios emphasizing their relationship to modification of the odds ratio then we highlight properties of likelihood ratios using the traditional roc axes finally we demonstrate roc curves and their properties after conversion to likelihood ratio co ordinates these graphs do not waste space for tests lacking diagnostic power and offer a simple visual assessment of a test s impact on the odds ratio

a distribution free procedure for comparing receiver operating characteristic curves from a paired experiment summary a distribution free permutation test procedure is proposed for comparing receiver operating characteristic curves based on continuous data from a paired design the method tests the hypothesis that the two curves are identical for all operating points unlike previously proposed methods which test the equivalence of the areas under the curves the new test is shown by simulation to have very similar operating characteristics to the standard method based on comparisons of the areas when the curves are parallel but markedly superior power when the curves cross that is when the curves are different but have similar areas the prospects of generalising the approach to unpaired experiments and to comparisons of ordinal rating data are discussed

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

convexity and w compactness in banach spaces suppose l x we construct examples of bounded sets m x such that these examples show that the previous results of the authors on quantitative versions of krein s theorem are optimal

confidence intervals for a binomial proportion thirteen methods for computing binomial confidence intervals are compared based on their coverage properties widths and errors relative to exact limits the use of the standard textbook method x n 1 96 square root of x n 1 x n n or its continuity corrected version is strongly discouraged a commonly cited rule of thumb stating that alternatives to exact methods may be used when the estimated proportion p is such that np and n 1 p both exceed 5 does not ensure adequate accuracy score limits are easily calculated from closed form solutions to quadratic equations and can be used at all times based on coverage functions the continuity corrected score method is recommended over exact methods its conservative nature should be kept in mind as should the wider fluctuation of actual coverage that accompanies omission of the continuity correction

comparing diagnostic tests a simple graphic using likelihood ratios the diagnostic abilities of two or more diagnostic tests are traditionally compared by their respective sensitivities and specificities either separately or using a summary of them such as youden s index several authors have argued that the likelihood ratios provide a more appropriate if in practice a less intuitive comparison we present a simple graphic which incorporates all these measures and admits easily interpreted comparison of two or more diagnostic tests we show using likelihood ratios and this graphic that a test can be superior to a competitor in terms of predictive values while having either sensitivity or specificity smaller a decision theoretic basis for the interpretation of the graph is given by relating it to the tent graph of hilden and glasziou statistics in medicine 1996 finally a brief example comparing two serodiagnostic tests for lyme disease is presented published in 2000 by john wiley sons ltd

baire class 1 selectors for upper semicontinuous set valued maps let t be a metric space and x a banach space let f t x be a set valued map assuming arbitrary values and satisfying the upper semicontinuity condition t t f t c o is closed for each weakly closed set c in x then there is a sequence of norm continuous functions converging pointwise in the norm to a selection for f we prove a statement of similar precision and generality when x is a metric space

quantum symmetry groups of finite spaces we determine the quantum automorphism groups of finite spaces these are compact matrix quantum groups in the sense of woronowicz

σ fragmentability of multivalued maps and selection theorems abstract we answer the question as to when a weak star upper semi continuous map f with arbitrary non empty values from a metric space t to the dual x of an asplund banach space x has a selector of the first baire class to the norm

regret graphs diagnostic uncertainty and youden s index regret is the difference in outcome between the action we took and the best action we could in retrospect have taken tent graphs representing decision problems under uncertainty in terms of expected regret offer an instructive geometric supplement to standard formulae allow instant calculations and suggest as we shall illustrate novel algebraic properties for instance the problem of finding the pre test probability range in which it is worthwhile doing a diagnostic test becomes trivial even when test costs depend on the unknown disease status we take the opportunity to explore the non trivial algebraic properties of this problem the graphs also provide a simple way to illustrate and read off the expected value of information and expected value of perfect information this property is used to derive a clinical interpretation of youden s index sensitivity specificity 1 namely it is the maximum proportional reduction in expected regret achieved by the test finally we examine the relationship between youden s index and the area under an roc curve

a noncommutative de finetti theorem invariance under quantum permutations is equivalent to freeness with amalgamation we show that the classical de finetti theorem has a canonical noncommutative counterpart if we strengthen exchangeability i e invariance of the joint distribution of the random variables under the action of the permutation group to invariance under the action of the quantum permutation group more precisely for an infinite sequence of noncommutative random variables x i i in mathbb n we prove that invariance of the joint distribution of the xi s under quantum permutations is equivalent to the fact that the xi s are identically distributed and free with respect to the conditional expectation onto the tail algebra of the xi s

on ergodic theorems for free group actions on noncommutative spaces we extend in a noncommutative setting the individual ergodic theorem of nevo and stein concerning measure preserving actions of free groups and averages on spheres s 2 n of even radius here we study state preserving actions of free groups on a von neumann algebra a and the behaviour of s 2 n x for x in noncommutative spaces l p a for the cesaro means this problem was solved by walker our approach is based on ideas of bufetov we prove a noncommutative version of rota alternierende verfahren theorem to this end we introduce specific dilations of the powers of some noncommutative markov operators

radon nikodým indexes and measures of weak noncompactness abstract the aim of this paper is to present a quantitative version of the radon nikod m property and some other results related to it this approach gives an extra insight to the classical results we introduce two indexes an index of representability of measures and an index of dentability we review classic results in order to obtain relationships between these and other indexes

a non iterative accurate asymptotic confidence interval for the difference between two proportions i propose a new confidence interval for the difference between two binomial probabilities that requires only the solution of a quadratic equation the procedure is based on estimating the variance of the observed difference at the boundaries of the confidence interval and uses least squares estimation rather than maximum likelihood as previously suggested the proposed procedure is non iterative agrees with the conventional test of equality of two binomial probabilities and even for fairly small sample sizes appears to yields actual 95 per cent confidence intervals with mean or median probabilities of coverage very close to 0 95 the yates continuity correction appears to generate confidence intervals with the conditional probability of coverage at least equal to nominal levels 1997 by john wiley sons ltd

a family of confidence intervals for the ratio of two binomial proportions the power divergence family of statistics introduced by cressie and read 1984 journal of the royal statistical society series b 46 440 464 is used to obtain approximate confidence intervals for the ratio of two proportions intervals based on inverting the pearson likelihood ratio and freeman tukey statistics are included in this family small sample comparisons of the intervals are presented

distances to spaces of baire one functions given a metric space x and a banach space e we use an index of fragmentability for maps f in e x to estimate the distance of f to the space b 1 x e of baire one functions from x into e when x is polish we use our estimations for these distances to give a quantitative version of the well known rosenthal s result stating that in b 1 x mathbb r the pointwise relatively countably compact sets are pointwise relatively compact we also obtain a quantitative version of a srivatsa s result that states that whenever x is metric any weakly continuous function f in e x belongs to b 1 x e our result here says that for an arbitrary f in e x we have d f b 1 x e leq 2 sup x in b e ast rm osc x circ f where osc x circ f stands for the supremum of the oscillations of x circ f at all points x in x as a consequence of the above we prove that for functions in two variables f x times k to mathbb r x complete metric and k compact there exists a g dense set d subset x such that the oscillation of f at each x k in d times k is bounded by the oscillations of the partial functions f x and f k a representative result in this direction that we prove using games is the following if x is a unfavorable space and k is a compact space then there exists a dense g subset d of x such that for each y k in d times k rm osc f y k le 6 sup x in x rm osc f x 8 sup k in k rm osc f k when the right hand side of the above inequality is zero we are dealing with separately continuous functions f x times k to mathbb r and we obtain as a particular case some well known results obtained by the third named author in the mid 1970s

a noncommutative central limit theorem for ccr algebras abstract a noncommutative generalization of the central limit theorem for even completely positive mappings between two ccr algebras is proved quasi free completely positive mappings are found to be the generalizations of the gaussian distributed random variables

cumulants in noncommutative probability theory i noncommutative exchangeability systems cumulants linearize convolution of measures we use a formula of good to define noncommutative cumulants in a very general setting it turns out that the essential property needed is exchangeability of random variables roughly speaking the formula says that cumulants are moments of a certain discrete fourier transform of a random variable this provides a simple unified method to understand the known examples of cumulants like classical free and various q cumulants

some applications of projective resolutions of identity we show that the unit ball k of the bidual of an asplund space is a corson compact or contains 0 1 and that it has the namioka property on separate to joint continuity the same results are shown for k a valdivia compact a by product is that all dyadic compacts have the namioka property some connections with weakly compactly generated dual spaces and renormings are given

comparative analysis of two rates in this paper we examine comparative analysis of rates with a view to each of the usual comparative parameters rate difference rd rate ratio rr and odds ratio or and with particular reference to first principles for rd and rr we show the prevailing statistical practices to be rather poor we stress the need for restricted estimation of variance in the chi square function underlying interval estimation and also point estimation and hypothesis testing for rr analysis we propose a chi square formulation analogous to that for rd and thus one which obviates the present practice of log transformation and its associated use of taylor series approximation of the variance as for or analysis we emphasize that the chi square function introduced by cornfield for unstratified data and extended by gart to the case of stratified analysis is based on the efficient score and thus embodies its optimality properties we provide simulation results to evince the better performance of the proposed parameter constrained procedures over the traditional ones

cumulants in noncommutative probability theory ii we continue the investigation of noncommutative cumulants in this paper various characterizations of generalized gaussian random variables are proved

addition of certain non commuting random variables soit g le groupe libre non commutatif sur deux generateurs et soit u j j 1 2 les unites dans l 2 g correspondant a la translation a gauche par les generateurs et on considere l 2 g la fonction g g e soient x j des operateurs de la forme x j j u j les operateurs x j peuvent etre consideres comme des variables aleatoires de moments ou de facon equivalente avec des distributions donnees par les fonctionnelles analytiques j ou j f avec ces conventions la distribution de x 1 x 2 depend seulement des distributions de x 1 et x 2 on explique cette relation

efficient statistical tests to compare youden index accounting for contingency correlation youden index is widely utilized in studies evaluating accuracy of diagnostic tests and performance of predictive prognostic or risk models however both one and two independent sample tests on youden index have been derived ignoring the dependence association between sensitivity and specificity resulting in potentially misleading findings besides paired sample test on youden index is currently unavailable this article develops efficient statistical inference procedures for one sample independent and paired sample tests on youden index by accounting for contingency correlation namely associations between sensitivity and specificity and paired samples typically represented in contingency tables for one and two independent sample tests the variances are estimated by delta method and the statistical inference is based on the central limit theory which are then verified by bootstrap estimates for paired samples test we show that the estimated covariance of the two sensitivities and specificities can be represented as a function of kappa statistic so the test can be readily carried out we then show the remarkable accuracy of the estimated variance using a constrained optimization approach simulation is performed to evaluate the statistical properties of the derived tests the proposed approaches yield more stable type i errors at the nominal level and substantially higher power efficiency than does the original youden s approach therefore the simple explicit large sample solution performs very well because we can readily implement the asymptotic and exact bootstrap computation with common software like r the method is broadly applicable to the evaluation of diagnostic tests and model performance

tests for equivalence or non inferiority for paired binary data assessment of therapeutic equivalence or non inferiority between two medical diagnostic procedures often involves comparisons of the response rates between paired binary endpoints the commonly used and accepted approach to assessing equivalence is by comparing the asymptotic confidence interval on the difference of two response rates with some clinical meaningful equivalence limits this paper investigates two asymptotic test statistics a wald type sample based test statistic and a restricted maximum likelihood estimation rmle based test statistic to assess equivalence or non inferiority based on paired binary endpoints the sample size and power functions of the two tests are derived the actual type i error and power of the two tests are computed by enumerating the exact probabilities in the rejection region the results show that the rmle based test controls type i error better than the sample based test to establish an equivalence between two treatments with a symmetric equivalence limit of 0 15 a minimal sample size of 120 is needed the rmle based test without the continuity correction performs well at the boundary point 0 a numerical example illustrates the proposed procedures

the quantitative difference between countable compactness and compactness abstract we establish here some inequalities between distances of pointwise bounded subsets h of r x to the space of real valued continuous functions c x that allow us to examine the quantitative difference between pointwise countable compactness and compactness of h relative to c x we prove amongst other things that if x is a countably k determined space the worst distance of the pointwise closure h of h to c x is at most 5 times the worst distance of the sets of cluster points of sequences in h to c x here distance refers to the metric of uniform convergence in r x we study the quantitative behavior of sequences in h approximating points in h as a particular case we obtain the results known about angelicity for these c p x spaces obtained by orihuela we indeed prove our results for spaces c x z hence for banach valued functions and we give examples that show when our estimates are sharp

kms quantum symmetric states let a be a unital c algebra and let be a one parameter automorphism group of a we consider qss a the set of all quantum symmetric states on 1 a that are also kms states for a fixed inverse temperature for specificity taken to be 1 for the free product automorphism group 1 we characterize the elements of qss a we show that qss a is a choquet simplex whenever it is nonempty and we characterize its extreme points

a noncommutative extended de finetti theorem abstract the extended de finetti theorem characterizes exchangeable infinite sequences of random variables as conditionally i i d and shows that the apparently weaker distributional symmetry of spreadability is equivalent to exchangeability our main result is a noncommutative version of this theorem in contrast to the classical result of ryll nardzewski exchangeability turns out to be stronger than spreadability for infinite sequences of noncommutative random variables out of our investigations emerges noncommutative conditional independence in terms of a von neumann algebraic structure closely related to popa s notion of commuting squares and kummerer s generalized bernoulli shifts our main result is applicable to classical probability quantum probability in particular free probability braid group representations and jones subfactors

comparison of the likelihood ratios of two binary diagnostic tests in paired designs the performance of two binary diagnostic tests is traditionally compared by their respective sensitivities and specificities other measures to describe the performance of a binary diagnostic test are likelihood ratios defined as the ratio between the likelihood of a diagnostic test result in a group of diseased patients and the likelihood of a diagnostic test result in a group of non diseased patients in this study we propose a method based on the log transformation of the ratio of the likelihood ratios to compare the likelihood ratios of two binary diagnostic tests in paired designs we have deduced hypothesis tests to compare the likelihood ratios and we have carried out simulation experiments to study the power and the type i error of the hypothesis tests deduced we have also deduced a joint hypothesis test to simultaneously compare the likelihood ratios the procedure used has been extended to the situation in which more than two binary diagnostic tests are applied to the same sample and the situation in which two diagnostic tests with multilevel results are compared