direct sums of tensor products and non commutative independence abstract we point out the connection between the three known fundamental notions of non commutative independence in quantum probability and the three basic choices for forming an associative direct sum of alternating tensor products of two vector spaces

positive definite functions on coxeter groups with applications to operator spaces and noncommutative probability a new class of positive definite functions related to colour length function on arbitrary coxeter group is introduced extensions of positive definite functions called the riesz coxeter product from the riesz product on the rademacher abelian coxeter group to arbitrary coxeter group is obtained applications to harmonic analysis operator spaces and noncommutative probability are presented characterization of radial and colour radial functions on dihedral groups and infinite permutation group are shown

legendre polynomials legendre stirling numbers and the left definite spectral analysis of the legendre differential expression in this paper we develop the left definite spectral theory associated with the self adjoint operator a k in l2 1 1 generated from the classical second order legendre differential equation ll k y t 1 t2 y ky y t 1 1 that has the legendre polynomials pm t m 0 as eigenfunctions here k is a fixed nonnegative constant more specifically for k 0 we explicitly determine the unique left definite hilbert sobolev space wn k and its associated inner product n k for each n n moreover for each n n we determine the corresponding unique left definite self adjoint operator an k in wn k and characterize its domain in terms of another left definite space the key to determining these spaces and inner products is in finding the explicit lagrangian symmetric form of the integral composite powers of ll k in turn the key to determining these powers is a remarkable new identity involving a double sequence of numbers which we call legendre stirling numbers

score tests in generalized linear measurement error models summary hypothesis tests in generalized linear models are studied under the condition that a surrogate w is observed in place of the true predictor x the efficient score test for the hypothesis of no association depends on the conditional expectation e xi w which is generally unknown the usual test substitutes w for e xi w and is asymptotically valid but not efficient we investigate two new test statistics appropriate when w x z where z is an independent measurement error the first is a wald test based on estimators corrected for measurement error despite the correction for attenuation in the estimator this test has the same local power as the usual test the second test employs an estimator of e x i w and is both asymptotically efficient for normal errors and approximately efficient when the measurement error variance is small

on errors in variables for binary regression models summary we consider binary regression models when some of the predictors are measured with error for normal measurement errors structural maximum likelihood estimates are considered we show that if the measurement error is large the usual estimate of the probability of the event in question can be substantially in error especially for high risk groups in the situation of large measurement error we investigate a conditional maximum likelihood estimator and its properties

remarks on the r and δ convolutions in this paper we study the properties of the r deformation introduced in b1 we observe that the associated convolution coming from the conditionally free convolution is associative only for r 1 and r 0 we give the realization of some r gaussian random variables and obtain haagerup pisier buchholz type inequalities we also study another convolution defined with the use of the r deformation through a moment cumulant formula ky1 and show that it is associative and in general not positive

multiplicative functions on the lattice of non crossing partitions and free convolution the lattice of non crossing partitions was introduced by kreweras kre in 1972 and examined further by poupard pou and edelman edel ede2 in the last time there has been an increased interest in this lattice on one hand from the purely combinatorial point of view siu eds sire bss and on the other hand from a quantum probabilistic point of view gss spel spe2 spe3 we became interested in this lattice when we noticed that it is connected with some new form of convolution for probability measures the free convolution introduced by voiculescu voil voi2 voi3 this connection between the free convolution and the lattice of non crossing partitions is exactly analogous to the connection between the usual convolution and the lattice of all partitions we shall work out this connection in the last part of this work the intention of this article is to show that both the purely combinatorial and the probabilistic point of view can benefit from each other motivated by voiculescu s formula for the r series of free convolution we examine multiplicative functions on the lattice of non crossing partitions and prove our main theorem which is much in the spirit of voiculescu s formula we shall then use this main theorem for deriving some known results on non crossing partitions in a unified way this part of our work is purely combinatorial the free convolution serves only as a motivation for our theorem in the last section we shall switch to the probabilistie side and show how the lattice of non crossing partitions determines the structure of the free convolution voiculescu s formula will then follow as an easy corollary of our main theorem in this respect we shall get a purely combinatorial proof of this formula without any operator algebraic or functional analytic tools

relations between cumulants in noncommutative probability abstract we express classical free boolean and monotone cumulants in terms of each other using combinatorics of heaps pyramids tutte polynomials and permutations we completely determine the coefficients of these formulas with the exception of the formula for classical cumulants in terms of monotone cumulants whose coefficients are only partially computed

effects of mismodeling on tests of association based on logistic regression models we consider the effects of misspecifying the covariate vector on tests of association between a particular covariate and the response variable in logistic regression models misspecification can include random measurement error discretizing a continuous explanatory variable and mismodeling the functional form of an explanatory variable the effects of misspecification are examined by deriving the asymptotic distribution of the test statistic under a sequence of models applications of this result are discussed

double sampling for exact values in some multivariate measurement error problems abstract increasing attention is being given to measurement error models in which the dimension of the proxy or surrogate values is different from that of the missing true values even if they are of the same dimension the error model may not be the simple additive one of observed true error where the error has mean 0 the use of broader models relating true and observed values requires the use of external or internal data containing some true values to calibrate validate the measurement error model this article considers a multivariate normal framework in which measurement error is allowed in any subset of the variables with a broad class of multivariate regression models relating true and observed values the classical additive model is a special case multiple regression with random regressors the structural case is treated within this framework correcting for measurement error is made possible through double sampling in which true values are obtained for a randomly chosen subset of the main

approximate quasi likelihood estimation in models with surrogate predictors abstract we consider quasi likelihood estimation with estimated parameters in the variance function when some of the predictors are measured with error we review and extend four approaches to estimation in this problem all of them based on small measurement error approximations a taxonomy of the data sets likely to be available in measurement error studies is developed an asymptotic theory based on this taxonomy is obtained and includes measurement error and berkson error models as special cases

superconvergence to the central limit and failure of the cramer theorem for free random variables we show that convergence of the semicircle law in the free central limit theorem for bounded random variables is much better than expected thus the distributions which tend to the semicircle become absolutely continuous in finite time and the densities converge in a very strong sense we also show that the semicircle law is the free convolution of laws which are not semicircular thus proving that cramer s classical result for the normal distribution does not have a free counterpart

power and sample size calculations for generalized regression models with covariate measurement error covariate measurement error is often a feature of scientific data used for regression modelling the consequences of such errors include a loss of power of tests of significance for the regression parameters corresponding to the true covariates power and sample size calculations that ignore covariate measurement error tend to overestimate power and underestimate the actual sample size required to achieve a desired power in this paper we derive a novel measurement error corrected power function for generalized linear models using a generalized score test based on quasi likelihood methods our power function is flexible in that it is adaptable to designs with a discrete or continuous scalar covariate exposure that can be measured with or without error allows for additional confounding variables and applies to a broad class of generalized regression and measurement error models a program is described that provides sample size or power for a continuous exposure with a normal measurement error model and a single normal confounder variable in logistic regression we demonstrate the improved properties of our power calculations with simulations and numerical studies an example is given from an ongoing study of cancer and exposure to arsenic as measured by toenail concentrations and tap water samples

cumulants in noncommutative probability theory iv noncrossing cumulants de finetti s theorem and lp inequalities abstract de finetti s theorem states that any exchangeable sequence of classical random variables is conditionally i i d with respect to some algebra in this paper we prove a free noncommutative analog of this theorem namely we show that any noncrossing exchangeability system with a faithful state which satisfies a so called weak singleton condition can be embedded into an free product with amalgamation over a certain subalgebra such that the interchangeable algebras remain interchangeable with respect to the operator valued expectation vanishing of crossing cumulants can be verified by checking a certain weak freeness condition and the weak singleton condition is satisfied e g when the state is tracial the proof follows the classical proof of de finetti s theorem the main technical tool being a noncommutative l p inequality for i i d sums of centered noncommutative random variables in noncrossing exchangeability systems

noncommutative levy processes for generalized particularly anyon statistics let t mathbb r d let a function qt 2 to mathbb c satisfy q s t overline q t s and q s t 1 a generalized statistics is described by creation operators partial t dagger and annihilation operators t t in t which satisfy the q commutation relations partial s partial dagger t q s t partial dagger t partial s delta s t partial s partial t q t s partial t partial s partial dagger s partial dagger t q t s partial dagger t partial dagger s from the point of view of physics the most important case of a generalized statistics is the anyon statistics for which q s t is equal to q if s t here q in mathbb c q 1 we start the paper with a detailed discussion of a q fock space and operators partial t dagger partial t t in t in it which satisfy the q commutation relations next we consider a noncommutative stochastic process white noise omega t partial t dagger partial t lambda partial t dagger partial t t in t here lambda in mathbb r is a fixed parameter the case 0 corresponds to a q analog of brownian motion while 0 corresponds to a centered q poisson process we study q hermite q charlier respectively polynomials of infinitely many noncommutatative variables omega t t in t the main aim of the paper is to explain the notion of independence for a generalized statistics and to derive corresponding levy processes to this end we recursively define q cumulants of a field xi t t in t this allows us to define a q levy process as a field xi t t in t whose values at different points of t are q independent and which possesses a stationarity of increments in a certain sense we present an explicit construction of a q levy process and derive a nualart schoutens type chaotic decomposition for such a process

gaussian fluctuations of characters of symmetric groups and of young diagrams we study asymptotics of reducible representations of the symmetric groups s q for large q we decompose such a representation as a sum of irreducible components or alternatively young diagrams and we ask what is the character of a randomly chosen component or what is the shape of a randomly chosen young diagram our main result is that for a large class of representations the fluctuations of characters and fluctuations of the shape of the young diagrams are asymptotically gaussian in this way we generalize kerov s central limit theorem the considered class consists of representations for which the characters almost factorize and this class includes for example the left regular representation plancherel measure irreducible representations and tensor representations this class is also closed under induction restriction outer product and tensor product of representations our main tool in the proof is the method of genus expansion well known from the random matrix theory

the left definite spectral theory for the classical hermite differential equation abstract in this paper we develop the left definite spectral theory associated with the self adjoint operator a in l2 exp t2 generated from the classic second order hermite differential equation l h y t y 2ty ky y t that has the hermite polynomials hm t m 0 as eigenfunctions more specifically for each n n we explicitly determine the unique left definite hilbert sobolev space wn and associated inner product n which is generated from the nth integral power lhn of lh moreover for each n n we determine the corresponding unique left definite self adjoint operator an in wn and characterize its domain in terms of another left definite space as a consequence of this we explicitly determine the domain of each integral power of a and in particular we obtain a new characterization of the domain of the classical right definite operator a

covariate measurement errors and parameter estimation in a failure time regression model summary estimation in cox s failure time regression model is considered when the regression vector is subject to measurement error a hazard function model is induced for the failure rate given the measured covariate and a partial likelihood function is derived for the relative risk parameters this partial likelihood function may involve the baseline hazard function as well as the regression parameter but useful inference techniques arise for testing whether the regression parameter equals zero and for more general inferences in important special cases explicit consideration is given to testing equality of survival curves when group membership is subject to misclassification and to relative risk estimation with normally distributed covariates approaches to regression estimation using the overall likelihood function and a marginal likelihood function based on failure time ranks are also indicated illustration of the possible effect of covariate errors on relative risk estimation is provided

freeness of linear and quadratic forms in von neumann algebras we characterize the semicircular distribution by freeness of linear and quadratic forms in noncommutative random variables from tracial w probability spaces with relaxed moment conditions

effects of mismodelling and mismeasuring explanatory variables on tests of their association with a response variable we consider three commonly used statistical tests for assessing the association between an explanatory variable and a measured binary or survival time response variable and investigate the loss in efficiency from mismodelling or mismeasuring the explanatory variable with respect to mismodelling we examine the consequences of using an incorrect dose metameter in a test for trend of mismodelling a continuous explanatory variable and of discretizing a continuous explanatory variable we also examine the consequences of classification errors for a discrete explanatory variable and of measurement errors for a continuous explanatory variable for all three statistical tests the asymptotic relative efficiency are corresponding to each type of mis specification equals the square of the correlation between the correct and fitted form of the explanatory variable this result is evaluated numerically for the different types of mis specification to provide insight into the selection of tests the interpretation of results and the design of studies where the correct explanatory variable cannot be measured exactly

cumulants for random matrices as convolutions on the symmetric group in this paper we show that free cumulants can be naturally seen as the limiting value of cumulants of matrices we define these objects as functions on the symmetric group by some convolution relations involving the generalized moments we state that some characteristic properties of the free cumulants already hold for these cumulants

efficacy of repeated measures in regression models with measurement error ignoring measurement error may cause bias in the estimation of regression parameters when the true covariates are unobservable multiple imprecise measurements can be used in the analysis to correct for the associated bias we suggest a simple estimating procedure that gives consistent estimates of regression parameters by using the repeated measurements with error the relative pitman efficiency of our estimator based on models with and without measurement error has been found to be a simple function of the number of replicates and the ratio of intra to inter variance of the true covariate the procedure thus provides a guide for deciding the number of repeated measurements in the design stage an example from a survey study is presented

noncommutative analogs of probabilistic notions and results abstract considering a random variable as a multiplication operator by a measurable function a natural generalization consists in allowing noncommuting and unbounded operators defined on a common invariant domain with cyclic vector by multiplication and addition these operators generate a algebra which in turn can be considered as a representation of an abstract tensor algebra moments are replaced by m a 1 a n a 1 a n in analogy to the classical case the notions of cumulants addition of independent random variables and infinite divisibility are introduced as well as gaussianness as a generalization of normal random variables previous results are briefly reviewed including a characterization of infinite divisibility among the new results are noncommutative analogs of versions of the central limit theorem and of cramer s theorem all results have direct applications to representations of lie algebras to quantum field theory and to statistical mechanics

r transforms of free joint distributions and non crossing partitions abstract we introduce an extension of the notion of r transform defined by d voiculescu to free joint distributions i e normalized linear functionals on algebras of non commutative polynomials in several indeterminates we point out that the r transform has good behavior with respect to the operations of free product and free convolution of joint distributions we prove that the explicit computation of the inverse r transform of a joint distribution is done via a formula of summation over the lattice of non crossing partitions which shows that the r transform is the operator theoretic counterpart of the free cumulants considered in the combinatorial approach of r speicher moreover in connection to the equivariance of the multidimensional r transform under rotations we show that a natural free analogue of a classical result about rotations of independent random variables is holding

general de finetti type theorems in noncommutative probability we prove general de finetti type theorems for classical and free independence the de finetti type theorems work for all non easy quantum groups which generalize a recent work of banica curran and speicher we determine maximal distributional symmetries which means the corresponding de finetti type theorem fails if a sequence of random variables satisfy more symmetry relations other than the maximal one in addition we define boolean quantum semigroups in analogous to the easy quantum groups by universal conditions on matrix coordinate generators and an orthogonal projection then we show a general de finetti type theorem for boolean independence

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

the asymptotic relative efficiency of score tests in a generalized linear model with surrogate covariates summary in many epidemiological studies the exposure variable of interest cannot be measured directly the classical approaches to errors in variables in regression do not extend easily to the nonlinear models commonly used in epidemiologic al research furthermore the traditional additive measurement error model cannot adequately represent many surrogate relationships by considering the effect of using surrogate independent variables on the efficient score statistic some of the difficulties inherent in the estimation problem may be avoided for the null hypothesis of no association a simple and flexible procedure can be used to calculate the optimal score test the asymptotic relative efficiency of this test to the test based upon the true exposures is derived the optimal test is also compared to the naive procedure of substituting the surrogate into the score test for the true exposure

free products of c algebras small spatial free products of c algebras are constructed under certain conditions they have properties similar to those proved by paschke and salinas for the algebras c gi g2 where g1 g2 are discrete groups the freeproduct analogs of noncommutative bernoulli shifts are discussed 0 introduction let k be a field consider the category of unital algebras over k it is well known that this category admits coproducts free products of algebras 2 heuristically the free product of algebras is the algebra generated by them with no relations except for the identification of unit elements if k c the complex numbers and we consider unital algebras we can easily define a operation on the free products let a b be unital c algebras and a b their free product which is a unital algebra the question arises in what ways may one define a pre c norm on a b that extends the norms on a and b guided by analogy with tensor products we expect to have a choice among many pre c norms giving rise to many c free products of a and b one natural norm is 1c i i sup 1it c ih st representation of a b the representations of a b are in 1 1 correspondence with pairs of representations of a and b which act on the same hilbert space let a b be the completion of a b in this norm it is easy to see that this construction defines a coproduct in the category of c algebras and that a b is the biggest free product of a and b analogous to the biggest tensor product a 0 b if g 1 g2 are discrete groups we obtain c g1 c g2 c gi g2 where g1 g2 is the free product group and this is analogous to the relation c g1 0 c g2 c g1 x g2 this paper is motivated by the question is there a smallest c product a b in analogy to the smallest tensor product a b satisfying a relation cr gi g2 cr gi cr g2 received by the editors january 28 1981 and in revised form may 19 1981 1980 mathematics subject classification primary 46l05 secondary 46l55 1 982 american mathematical society 0002 9947 82 00001022 04 25

operator khintchine inequality in non commutative probability abstract for the operator t sum n k 1 a k otimes r k where a k belongs to the schatten class s 2n and where r k are non commutative random variables with mixed moments satisfying a specific condition we prove the following khintchine inequality vert t vert 2n leq d 2n left left vert left sum n k001 a ka k right frac 1 2 right vert s 2n quad quad left vert left sum n k 1 a k a k right frac 1 2 right vert s 2n right we find the optimal constants d 2n in the case when r k are the q gaussian and circular random variables moreover we show that the moments of any probability symmetric measure appear as the optimal constants for some random variables

second order freeness and fluctuations of random matrices ii unitary random matrices abstract we extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations we show how the concept of second order freeness which was introduced in part i allows one to understand global fluctuations of haar distributed unitary random matrices in particular independence between the unitary ensemble and another ensemble goes in the large n limit over into asymptotic second order freeness two important consequences of our general theory are i we obtain a natural generalization of a theorem of diaconis and shahshahani to the case of several independent unitary matrices ii we can show that global fluctuations in unitarily invariant multi matrix models are not universal

covariance analysis in generalized linear measurement error models we summarize some of the recent work on the errors in variables problem in generalized linear models the focus is on covariance analysis and in particular testing for and estimation of treatment effects there is a considerable difference between the randomized and non randomized models when testing for an effect in randomized studies simple techniques exist for testing for a treatment effect in some instances such as linear and multiplicative regression simple methods exist for estimating the treatment effect in other examples such as logistic regression estimating a treatment effect requires careful attention to measurement error in non randomized studies there is no recourse to understanding and modelling measurement error in particular ignoring measurement error can lead to the wrong conclusions for example the true but unobserved data may indicate a positive effect for treatment while the observed data indicate the opposite some of the possible methods are outlined and compared

the left definite spectral theory for the dunkl hermite differential difference equation abstract in this paper we develop the left definite spectral theory associated with the self adjoint operator a in l 2 r t 2 exp t 2 generated from the dunkl second order hermite differential equation l y t t 2 y t 2tt y t 2 y t y t ky t y t r that has the generalized hermite polynomials hm m 0 as eigenfunctions and where t is a differential difference operator called the dunkl operator on r of index more specifically for each n n we explicitly determine the unique left definite hilbert space wn and associated inner product n which is generated from the nth integral power l n of l moreover for each n n we determine the corresponding unique left definite self adjoint operator a n in wn and characterize its domain in terms of another left definite space as a consequence we explicitly determine the domain of each integral power of a and in particular we obtain a new characterization of the domain of the dunkl right definite operator a

an obstruction for q deformation of the convolution product we consider two independent q gaussian random variables xo and x and a function y chosen in such a way that y xo and xo have the same distribution for q e 0 1 we find that at least the fourth moments of x 4 xi and y x o xi are different we conclude that no deformed convolution product parallelling the known cases q 0 and q 1 can exist

covariate measurement error adjustment for matched case control studies summary we propose a conditional scores procedure for obtaining bias corrected estimates of log odds ratios from matched case control data in which one or more covariates are subject to measurement error the approach involves conditioning on sufficient statistics for the unobservable true covariates that are treated as fixed unknown parameters for the case of gaussian nondifferential measurement error we derive a set of unbiased score equations that can then be solved to estimate the log odds ratio parameters of interest the procedure successfully removes the bias in naive estimates and standard error estimates are obtained by resampling methods we present an example of the procedure applied to data from a matched case control study of prostate cancer and serum hormone levels and we compare its performance to that of regression calibration procedures

deconvolution based score tests in measurement error models consider a generalized linear model with response y and scalar predictor x instead of observing x a surrogate w x z is observed where z represents measurement error and is independent of x and y the efficient score test for the absence of association depends on m w e x mid w w which is generally unknown assuming that the distribution of z is known asymptotically efficient tests are constructed using nonparametric estimators of m w rates of convergence for the estimator of m w are established in the course of proving efficiency of the proposed test

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

loss in efficiency caused by omitting covariates and misspecifying exposure in logistic regression models we examine the effects of omitting covariates and misspecifying the explanatory variable of interest on tests of association between the explanatory variable and response based on logistic regression models we assume throughout that the covariates are statistically independent of the explanatory variable as under simple randomization a general expression for the asymptotic loss in efficiency from omitting covariates and misspecifying the explanatory variable is derived the expression for the asymptotic efficiency of the misspecified test statistic relative to the correctly specified test can be factored into two parts each of which is less than or equal to 1 the first part reflects the consequences of misspecifying exposure whereas the other part captures the effect of omitting needed covariates this result permits numerical evaluation of the approximate loss in efficiency and can be useful in developing guidelines for study design and model fitting procedures we discuss applications of the resul

jacobi stirling numbers jacobi polynomials and the left definite analysis of the classical jacobi differential expression we develop the left definite analysis associated with the self adjoint jacobi operator generated from the classical second order jacobi differential expression in the hilbert space where w t 1 t 1 t that has the jacobi polynomials as eigenfunctions here 1 and k is a fixed non negative constant more specifically for each we explicitly determine the unique left definite hilbert sobolev space and the corresponding unique left definite self adjoint operator in associated with the pair the jacobi polynomials form a complete orthogonal set in each left definite space and are the eigenfunctions of each moreover in this paper we explicitly determine the domain of each as well as each integral power of the key to determining these spaces and operators is in finding the explicit lagrangian symmetric form of the integral composite powers of l k in turn the key to determining these powers is a double sequence of numbers which we introduce in this paper as the jacobi stirling numbers some properties of these numbers which in some ways behave like the classical stirling numbers of the second kind are established including a remarkable and yet somewhat mysterious identity involving these numbers and the eigenvalues of

non crossing cumulants of type b we establish connections between the lattices of non crossing partitions of type b introduced by v reiner and the framework of the free probability theory of d voiculescu lattices of non crossing partitions of type a up to now have played an important role in the combinatorics of free probability primarily via the non crossing cumulants of r speicher here we introduce the concept of non crossing cumulant of type b the inspiration for its definition is found by looking at an operation of restricted convolution of multiplicative functions studied in parallel for functions on symmetric groups in type a and on hyperoctahedral groups in type b the non crossing cumulants of type b live in an appropriate framework of non commutative probability space of type b and are closely related to a type b analogue for the r transform of voiculescu which is the free probabilistic counterpart of the fourier transform by starting from a condition of vanishing of mixed cumulants of type b we obtain an analogue of type b for the concept of free independence for random variables in a non commutative probability space

the invariance of some score tests in the linear model with classical measurement error the linear model with classical measurement error is an alternative to the standard regression model in which it is assumed that the independent variables are subject to error this assumption can cause statistical inferences and parameter estimators to differ dramatically from those obtained by the standard regression model however in some cases inferences remain unchanged even though the independent variables are assumed to be subject to error this article investigates the invariance property of score tests for assessing heteroscedasticity first order autoregressive disturbance and the need for a box cox power transformation under specific constraints we show that the score tests for measurement error models are identical to the corresponding well established tests derived from standard regression models hence practitioners can assess assumptions of constant variance and independent errors as well as the need for a box cox transformation irrespective of whether or not the variables are measu

left definite theory with applications to orthogonal polynomials in the past several years there has been considerable progress made on a general left definite theory associated with a self adjoint operator a that is bounded below in a hilbert space h the term left definite has its origins in differential equations but littlejohn and wellman l l littlejohn r wellman a general left definite theory for certain self adjoint operators with applications to differential equations j differential equations 181 2 2002 280 339 generalized the main ideas to a general abstract setting in particular it is known that such an operator a generates a continuum h r r 0 of hilbert spaces and a continuum of a r r 0 of self adjoint operators in this paper we review the main theoretical results in l l littlejohn r wellman a general left definite theory for certain self adjoint operators with applications to differential equations j differential equations 181 2 2002 280 339 moreover we apply these results to several specific examples including the classical orthogonal polynomials of laguerre hermite and jacobi

free boolean independence for pairs of algebras abstract we construct pairs of algebras with mixed independence relations by using truncations of reduced free products of algebras for example we construct free boolean pairs of algebras and free monotone pairs of algebras we also introduce free boolean cumulants and show that free boolean independence is equivalent to the vanishing of mixed cumulants in addition we provide a moments condition for free boolean independence a central limit law for free boolean independence is studied

sample variance in free probability abstract let x 1 x 2 x n denote i i d centered standard normal random variables then the law of the sample variance q n i 1 n x i x 2 is the 2 distribution with n 1 degrees of freedom it is an open problem in classical probability to characterize all distributions with this property and in particular whether it characterizes the normal law in this paper we present a solution of the free analogue of this question and show that the only distributions whose free sample variance is distributed according to a free 2 distribution are the semicircle law and more generally so called odd laws by which we mean laws with vanishing higher order even cumulants in the way of proof we derive an explicit formula for the free cumulants of q n which shows that indeed the odd cumulants do not contribute and which exhibits an interesting connection to the concept of r cyclicity

measurement error regression with unknown link dimension reduction and data visualization abstract a general nonlinear regression problem is considered with measurement error in the predictors we assume that the response is related to an unknown linear combination of a multidimensional predictor through an unknown link function instead of observing the predictor we instead observe a surrogate with the property that its expectation is linearly related to the true predictor with constant variance we identify an important transformation of the surrogate variable using this transformed variable we show that if one proceeds with the usual analysis ignoring measurement error then both ordinary least squares and sliced inverse regression yield estimates which consistently estimate the true regression parameter up to a constant of proportionality we derive the asymptotic distribution of the estimates a simulation study is conducted applying sliced inverse regression in this context

adjusting for covariate errors with nonparametric assessment of the true covariate distribution a well known and useful method for generalised regression analysis when a linear covariate x is available only through some approximation z is to carry out more or less the usual analysis with e x z substituted for x sometimes but not always the quantity var x z should be used to allow for overdispersion introduced by this substitution these quantities involve the distribution of true covariates x and with some exceptions this requires assessment of that distribution through the distribution of observed values z it is often desirable to take a nonparametric approach to this which inherently involves a deconvolution that is difficult to carry our directly however if covariate errors are assumed to be multiplicative and log normal simple but accurate approximations are available for the quantities e x k z k 1 2 in particular the approximations depend only on the first two derivatives of the logarithm of the density of z at the point under consideration and the coefficient of variation of z x the methods will thus be most useful in large scale observational studies where the distribution of z can be assessed well enough in an essentially nonparametric manner to approximate adequately those derivatives we consider both the classical and berkson error models this approach is applied to radiation dose estimates for atomic bomb survivors

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

r cyclic families of matrices in free probability we introduce the concept of r cyclic family of matrices with entries in a noncommutative probability space the definition consists in asking that only the cyclic noncrossing cumulants of the entries of the matrices are allowed to be nonzero let a1 as be an r cyclic family of d d matrices over a noncommutative probability space a we prove a convolution type formula for the explicit computation of the joint distribution of a1 as considered in md a with the natural state in terms of the joint distribution considered in the original space a of the entries of the s matrices several important situations of families of matrices with tractable joint distributions arise by application of this formula moreover let a1 as be a family of d d matrices over a noncommutative probability space a let d md a denote the algebra of scalar diagonal matrices and let c be the subalgebra of md a generated by a1 as d we prove that the r cyclicity of a1 as is equivalent to a property of c namely that c is free from md c with amalgamation over d

improved score tests for generalized linear models summary we present in matrix notation an asymptotic expansion for the null distribution of the score statistic in generalized linear models corrected up to order n 1 where n is the sample size the expansion has advantages for numerical purposes because it requires only simple operations on matrices it is also sufficiently simple to be used analytically to obtain several closed form expressions analogous to bartlett corrections in a variety of important score tests improved score tests for these models are discussed and various applications to some special tests are given we show by simulation that two asymptotically equivalent adjusted score tests seem to improve at least for continuous data on the usual score test