use of the mann whitney u test for clustered data the mann whitney u test is ubiquitous in statistical practice for the comparison of measures of location for two samples where the assumption of normality is questionable frequently one has replicate data for each individual in a group and would like to compare measures of central tendency between groups without assuming normality for this purpose we present a generalization of the mann whitney u test for clustered data the test is performed by computing zc wc mu c sigma c approximately n 0 1 under h0 where wc mu c are the observed and expected mann whitney u statistic based on a comparison of all pairs of replicates in the two groups and sigma c is the standard deviation of wc that is modified to account for clustering effects within a cluster we obtain an explicit variance formula that is a function of four clustering parameters we validate the properties of the test procedure in a simulation study we illustrate the methods with an example comparing the baseline humphrey visual field between two treatment groups in a randomized clinical trial of patients with retinitis pigmentosa rp

multivariate location estimation using extension of r estimates through u statistics type approach we consider a class of u statistics type estimates for multivariate location the estimates extend some r estimates to multivariate data in particular the class of estimates includes the multivariate median considered by gini and galvani 1929 and haldane 1948 and a multivariate extension of the well known hodges lehmann 1963 estimate we explore large sample behavior of these estimates by deriving a bahadur type representation for them in the process of developing these asymptotic results we observe some interesting phenomena that closely resemble the famous shrinkage phenomenon observed by stein 1956 in high dimensions interestingly the phenomena that we observe here occur even in dimension d 2

a weighted multivariate sign test for cluster correlated data we consider the multivariate location problem with cluster correlated data a family of multivariate weighted sign tests is introduced for which observations from different clusters can receive different weights under weak assumptions the test statistic is asymptotically distributed as a chi squared random variable as the number of clusters goes to infinity the asymptotic distribution of the test statistic is also given for a local alternative model under multivariate normality optimal weights maximizing pitman asymptotic efficiency are provided these weights depend on the cluster sizes and on the intracluster correlation several approaches for estimating these weights are presented using pitman asymptotic efficiency we show that appropriate weighting can increase substantially the efficiency compared to a test that gives the same weight to each cluster a multivariate weighted t test is also introduced the finite sample performance of the weighted sign test is explored through a simulation study which shows that the proposed approach is very competitive a real data example illustrates the practical application of the methodology

on an adaptive transformation retransformation estimate of multivariate location an affine equivariant estimate of multivariate location based on an adaptive transformation and retransformation approach is studied the work is primarily motivated by earlier work on different versions of the multivariate median and their properties we explore an issue related to efficiency and equivariance that was originally raised by bickel and subsequently investigated by brown and hettmansperger our estimate has better asymptotic performance than the vector of co ordinatewise medians when the variables are substantially correlated the finite sample performance of the estimate is investigated by using monte carlo simulations some examples are presented to demonstrate the effect of the adaptive transformation retransformation strategy in the construction of multivariate location estimates for real data

rank based estimation and associated inferences for linear models with cluster correlated errors r estimators based on the joint ranks jr of all the residuals have been developed over the last 20 years for fitting linear models with independently distributed errors in this article we extend these estimators to estimating the fixed effects in a linear model with cluster correlated continuous error distributions for general score functions we discuss the asymptotic theory of the estimators and standard errors of the estimators for the related mixed model with a single random effect we discuss robust estimators of the variance components these are used to obtain studentized residuals for the jr fit a real example is discussed which illustrates the efficiency of the jr analysis over the traditional analysis and the efficiency of a prudent choice of a score function simulation studies over situations similar to the example confirm the validity and efficiency of the analysis

duality and approximation of bergman spaces abstract expected duality and approximation properties are shown to fail on bergman spaces of domains in c n via examples when the domain admits an operator satisfying certain mapping properties positive duality and approximation results are proved such operators are constructed on generalized hartogs triangles on a general bounded reinhardt domain norm convergence of laurent series of bergman functions is shown this extends a classical result on hardy spaces of the unit disc

acyclic automata and small expressions using multi tilde bar operators a regular expression with n occurrences of symbol can be converted into an equivalent automaton with n 1 states the so called glushkov automaton of the expression conversely it is possible to decide whether a given n 1 state automaton is a glushkov one and if so to convert it back to an equivalent regular expression of width n our goal is to extend the class of automata for which such a linear retranslation is possible we define new regular operators called multi tilde bars allowing us to simultaneously apply a multi tilde operator and a multi bar one to a list of expressions the main results are that a multi tilde bar expression of width n can be converted into an n 1 state position like automaton and that any acyclic n state automaton can be turned into an extended expression of width o n

the wilcoxon signed rank test for paired comparisons of clustered data the wilcoxon signed rank test is a frequently used nonparametric test for paired data e g consisting of pre and posttreatment measurements based on independent units of analysis this test cannot be used for paired comparisons arising from clustered data e g if paired comparisons are available for each of two eyes of an individual to incorporate clustering a generalization of the randomization test formulation for the signed rank test is proposed where the unit of randomization is at the cluster level e g person while the individual paired units of analysis are at the subunit within cluster level e g eye within person an adjusted variance estimate of the signed rank test statistic is then derived which can be used for either balanced same number of subunits per cluster or unbalanced different number of subunits per cluster data with an exchangeable correlation structure with or without tied values the resulting test statistic is shown to be asymptotically normal as the number of clusters becomes large if the cluster size is bounded simulation studies are performed based on simulating correlated ranked data from a signed log normal distribution these studies indicate appropriate type i error for data sets with or 20 clusters and a superior power profile compared with either the ordinary signed rank test based on the average cluster difference score or the multivariate signed rank test of puri and sen finally the methods are illustrated with two data sets i an ophthalmologic data set involving a comparison of electroretinogram erg data in retinitis pigmentosa rp patients before and after undergoing an experimental surgical procedure and ii a nutritional data set based on a randomized prospective study of nutritional supplements in rp patients where vitamin e intake outside of study capsules is compared before and after randomization to monitor compliance with nutritional protocols

marginal analyses of clustered data when cluster size is informative we propose a new approach to fitting marginal models to clustered data when cluster size is informative this approach uses a generalized estimating equation gee that is weighted inversely with the cluster size we show that our approach is asymptotically equivalent to within cluster resampling hoffman sen and weinberg 2001 biometrika 73 13 22 a computationally intensive approach in which replicate data sets containing a randomly selected observation from each cluster are analyzed and the resulting estimates averaged using simulated data and an example involving dental health we show the superior performance of our approach compared to unweighted gee the equivalence of our approach with wcr for large sample sizes and the superior performance of our approach compared with wcr when sample sizes are small

a signed rank test for clustered data we consider the problem of comparing two outcome measures when the pairs are clustered using the general principle of within cluster resampling we obtain a novel signed rank test for clustered paired data we show by a simple informative cluster size simulation model that only our test maintains the correct size under a null hypothesis of marginal symmetry compared to four other existing signed rank tests further our test has adequate power when cluster size is noninformative in general cluster size is informative if the distribution of pair wise differences within a cluster depends on the cluster size an application of our method to testing radiation toxicity trend is presented

rank sum tests for clustered data the wilcoxon rank sum test is widely used to test the equality of two populations because it makes fewer distributional assumptions than parametric procedures such as the t test however the wilcoxon rank sum test can be used only if data are independent when data are clustered tests based on generalized estimating equations gees that generalize the t test have been proposed here we develop a rank sum test that can be used when data are clustered as an application we use our rank sum test to develop a nonparametric test of association between a genetic marker and a quantitative trait locus we also give a rank sum test for equivalence of three or more populations that generalizes the kruskal wallis test to situations with clustered data unlike previous rank tests for clustered data our proposal is valid when members of the same cluster belong to different groups or when the correlation between cluster members differs across groups

language operations with regular expressions of polynomial size this work deals with questions regarding to what extent regularity preserving language operations affect the descriptional complexity of regular expressions some language operations are identified which are feasible for regular expressions in the sense that the result of the operation can be represented as a regular expression of size polynomial in that of the operands we prove that taking language quotients in particular the prefix and suffix closures of a regular set can incur at most a quadratic blow up on the required expression size the circular shift operation can cause only a cubic increase in size and at least a quadratic bloat can be necessary in the worst case

characterization of glushkov automata abstract glushkov s algorithm computes a nondeterministic finite automaton without e transitions and with n 1 states from a regular expression having n occurrences of letters the aim of this paper is to give a set of necessary and sufficient conditions characterizing this automaton our characterization theorem is formulated in terms of directed graphs moreover these conditions allow us to produce an algorithm of conversion of a glushkov automaton into a regular expression of small size

the complex green operator on cr submanifolds of ℂⁿ of hypersurface type compactness we establish compactness estimates for b on a compact pseudoconvex cr submanifold of c of hypersurface type that satisfies property p when the submanifold is orientable these estimates were proved by a raich in 32 using microlocal methods our proof deduces the estimates from a slight extension when q 1 of those known on hypersurfaces via the fact that locally cr submanifolds of hypersurface type are cr equivalent to a hypersurface the relationship between two potential theoretic conditions is also clarified

obtaining shorter regular expressions from finite state automata we consider the use of state elimination to construct shorter regular expressions from finite state automata fas although state elimination is an intuitive method for computing regular expressions from fas the resulting regular expressions are often very long and complicated we examine the minimization of fas to obtain shorter expressions first then we introduce vertical chopping based on bridge states and horizontal chopping based on the structural properties of given fas we prove that we should not eliminate bridge states until we eliminate all non bridge states to obtain shorter regular expressions in addition we suggest heuristics for state elimination that leads to shorter regular expressions based on vertical chopping and horizontal chopping

boundary invariants of pseudoconvex domains the purpose of this paper is to describe an invariant ax z which is associated with each point z in the boundary of a smoothly bounded pseudoconvex domain in cn the motivation for defining the invariant ax z referred to as the multitype of z comes from the study of the boundary regularity properties of solutions of the d neumann problem on domains of finite type by a domain of finite type we mean one that satisfies the definition given by d angelo 5 namely a domain such that at any boundary point the maximum order of contact of one dimensional complex analytic varieties with the boundary is bounded in 4 the author showed how one can give a proof of global regularity of the a neumann problem for pseudoconvex domains of finite type if there is an invariant such as the one described in this paper in a forthcoming article this invariant will be used to obtain a proof of subelliptic estimates for the same class of domains before giving the definition of the multitype ax z we must introduce some

nonparametric analysis of clustered multivariate data there is wide interest in extending univariate and multivariate nonparametric procedures to clustered and hierarchical data traditionally parametric mixed models have been used to account for the correlation structures among the dependent observational units in this work we extend multivariate nonparametric procedures for one sample and several sample location problems to clustered data settings the results are given for a general score function but with an emphasis on spatial sign and rank methods mixed models notation involving design matrices for fixed and random effects is used throughout the asymptotic variance formulas and limiting distributions of the test statistics under the null hypothesis and under a sequence of alternatives are derived as are the limiting distributions for the corresponding estimates the approach based on a general score function also shows for example how m estimates behave with clustered data efficiency studies demonstrate the practical advantages and disadvantag

real hypersurfaces orders of contact and applications a fundamental problem in the modem theory of several complex variables concerns the boundary behavior of the cauchy riemann equations suppose that d is an open domain in cn and that its boundary m is a smooth real submanifold of cn how does the geometry of m influence the function theory on d one approach to this question is the a neumann problem 6 there one studies the inhomogeneous cauchy riemann equations au a where a is a 0 q form with distribution coefficients on the closed domain d u m satisfying the necessary compatibility condition a a 0 the a neumann problem constructs a particular solution u which is orthogonal to the null space of the complex laplacian kohn 10a has solved this problem and has shown that good local regularity properties for u in terms of a follow from so called subelliptic estimates it is natural to seek geometric conditions on m that are necessary and sufficient for these estimates in this paper we will be concerned with a certain geometric condition on m that we feel is the right one for subellipticity although the motivation for the questions discussed here comes from partial differential equations the techniques come from algebraic geometry to see why we state theorems of kohn greiner and catlin concerning subellipticity kohn 1oa has discovered a sufficient condition for a subelliptic estimate near a fixed point p in m in case d is pseudoconvex and is contained in c2 this condition is equivalent to the finiteness of order of tangency of every complex analytic one dimensional manifold with m at p greiner 13 established the necessity of this condition in

weighted bergman kernels and quantization let be a bounded pseudoconvex domain in c n two positive functions on such that log log are plurisubharmonic and z a point at which log is smooth and strictly plurisubharmonic we show that as k the bergman kernels with respect to the weights k have an asymptotic expansion for x y near z where x y is an almost analytic extension of phi x x x and similarly for further if in addition is of finite type behave reasonably at the boundary and log log are strictly plurisubharmonic on we obtain also an analogous asymptotic expansion for the berezin transform and give applications to the berezin quantization finally for smoothly bounded and strictly pseudoconvex and a smooth strictly plurisubharmonic defining function for we also obtain results on the berezin toeplitz quantization

bootstrapping clustered data summary various bootstraps have been proposed for bootstrapping clustered data from one way arrays the simulation results in the literature suggest that some of these methods work quite well in practice the theoretical results are limited and more mixed in their conclusions for example mccullagh reached negative conclusions about the use of non parametric bootstraps for one way arrays the purpose of this paper is to extend our understanding of the issues by discussing the effect of different ways of modelling clustered data the criteria for successful bootstraps used in the literature and extending the theory from functions of the sample mean to include functions of the between and within sums of squares and non parametric bootstraps to include model based bootstraps we determine that the consistency of variance estimates for a bootstrap method depends on the choice of model with the residual bootstrap giving consistency under the transformation model whereas the cluster bootstrap gives consistent estimates under both the transformation and the random effect model in addition we note that the criteria based on the distribution of the bootstrap observations are not really useful in assessing consistency

analytic discs plurisubharmonic hulls and non compactness of the neumann operator we show that a complex manifold m in the boundary of a smooth bounded pseudoconvex domain in open image in new window is an obstruction to compactness of the open image in new window neumann operator on provided that at some point of m the levi form of b has the maximal possible rank n 1 dim m i e the boundary is strictly pseudoconvex in the directions transverse to m in particular an analytic disc is an obstruction provided that at some point of the disc the levi form has only one zero eigenvalue i e the eigenvalue zero has multiplicity one we also show that a boundary point where the levi form has only one zero eigenvalue can be picked up by the plurisubharmonic hull of a set only via an analytic disc in the boundary

weakly pseudoconvex domains with non compact automorphism groups the purpose of this paper is to prove a uniform extendibility result for sequences of biholomorphic mappings between weakly pseudoconvex domains suppose that f21 and 02 are bounded weakly pseudoconvex domains in e with real analytic boundaries and suppose that fi is a sequence of biholomorphic mappings j i2t o2 which converges to a holomorphic mapping f of 2 into the closure of 122 we remark that a strictly pseudoconvex domain is not a weakly pseudoconvex domain cartan s theorem states that either f is a biholomorphic map of f2 onto 02 or f is a mapping of f21 into the boundary of f22 we shall prove that in either case there exists a domain d which contains 121 together with all the strictly pseudoconvex boundary points of f2x such that each of the mappings f extend to be hoiomorphic on d furthermore the limit map fextends holomorphically to d and the sequence fi converges uniformly on compact subsets of d to f it is interesting to note that the corresponding statement for a sequence of maps between strictly pseudoconvex domains with real analytic boundaries is not true indeed if i21 is strictly pseudoconvex and the limit map f maps into bf22 then both o t and i22 are biholomorphic to the ball by wong s theorem in this situation if the inverse mappings fi f 1 also converge then there is an open set d which contains the closure of o1 minus a single point in bf21 to which all the f extend and the sequence f converges uniformly on compact subsets of d to f if no assumption is made about the convergence of the inverses then the maps need not converge uniformly up to the boundary near any boundary point of o1 when f21 and 02 are assumed to be weakly pseudoconvex and therefore not biholomorphic to the ball the assumption about the convergence of the inverse maps is not needed

on the efficiency of multivariate spatial sign and rank tests conditionally and asymptotically distribution free rotation but not scale invariant competitors of the well known hotelling s t 2 tests in this paper efficiency results are found in the one sample location case but they natu rally hold in the multivariate two sample multisample and general linear model case the lack of means of fully ordering or ranking observations in the multivariate setting seems at first sight to be an obstacle to generalizing the concepts of sign and rank and signed rank however these concepts can be defined also by using l criterion functions as follows consider k 1 variate observations y y utilizing the euclidean distance y s 1 n 2 2 1r2 y q qy the three objective functions 1 k t y s s y y i i i i i

regular expression for a language without empty word in 1 a bruggemann klein asks the following question is there a linear time algorithm transforming a regular expression e into an expression e with oliver in this paper we give a recursive definition of e which enables us to provide such an algorithm furthermore we show that e e 12 log e 1 e 12 where e is the size of e and e is the size of e

sampling different kinds of acyclic automata using markov chains we propose algorithms that use markov chain techniques to generate acyclic automata uniformly at random we first consider deterministic accessible and acyclic automata then focus on the class of minimal acyclic automata in each case we explain how to define random local transformations that describe an ergodic and symmetric markov chain the distribution of the automaton obtained after t random steps in this markov chain tends to the uniform distribution as t tends to infinity

incorporation of clustering effects for the wilcoxon rank sum test a large sample approach the wilcoxon rank sum test is frequently used in statistical practice for the comparison of measures of location when the underlying distributions are far from normal or not known in advance an assumption of the ordinary rank sum test is that individual sampling units are independent in many ophthalmologic clinical trials the early treatment for diabetic retinopathy scale etdrs is a principal endpoint used for measuring the level of diabetic retinopathy this is an ordinal scale and it is natural to consider the wilcoxon rank sum test for the comparison of the level of diabetic retinopathy between treatment groups however under this design unlike the usual wilcoxon rank sum test the subject is the unit of randomization but the eye is the unit of analysis furthermore a person will tend to have different but correlated etdrs scores for fellow eyes thus we propose a correction to the variance of the wilcoxon rank sum statistic that accounts for clustering effects and that can be used for both balanced same number of subunits per cluster or unbalanced different number of subunits per cluster data both in the presence or absence of ties with p value adjusted accordingly in this article we present large sample theory and simulation results for this test procedure and apply it to diabetic retinopathy data from type i diabetics in the sorbinil retinopathy trial