divergence statistics for testing uniform association in cross classifications in this paper we consider the problem of testing uniform association in cross classifications having ordered categories taking as test statistic one in the family proposed by conde and salicru j conde m salicru uniform association in contingency tables associated to csiszar divergence statistics and probability letters 37 1998 149 154 we consider two approximations to the null distribution of the test statistics in this family an estimation of the asymptotic null distribution and a bootstrap estimator we prove that both approximations are asymptotically equivalent to study their finite sample performance we carried out two simulation experiments whose results are presented from the simulations it can be concluded that the bootstrap estimator behaves much better than the estimated asymptotic null distribution

goodness of fit tests based on rao s divergence under sparseness assumptions in many practical situations the classical fixed cells assumptions to test goodness of fit are inappropriate and we consider an alternative set of assumptions which we call sparseness assumptions it is proved that under general conditions the proposed family of statistics based on rao s divergence is asymptotically normal when the sample size n and the number of cells m n tend to infinity so that n m n n 0 this result is extended to contiguous alternatives and subsequently it is possible to find the asymptotically most efficient member of the family

category o gradings and translation functors abstract in this article we consider a graded version of category o we reprove some results of beilinson et al j amer math soc 9 1996 473 527 using a different approach furthermore we define a graded version of translation functors and duality this provides the construction of various graded modules on the other hand we describe how to get modules which are not gradable

categorification of induced cell modules and the rough structure of generalised verma modules this paper presents categorifications of right cell modules and induced cell modules for hecke algebras of finite weyl groups fit type a we show that these categorifications depend only on the isomorphism class of the cell module not on the cell itself our main application is multiplicity formulas for parabolically induced modules over a reductive lie algebra of type a which finally determines the so called rough structure of generalised verma modules on the way we present several categorification results and give a positive answer to kostant s problem from a joseph kostant s problem goldie rank and the gelfand kirillov conjecture invent math 56 3 1980 191 213 in many cases we also present a general setup of decategorification precategorification and categorification

powerful goodness of fit tests based on the likelihood ratio summary a new approach of parameterization is proposed to construct a general goodnessof fit test it can not only generate traditional tests including the kolmogorov smirnov cramer von mises and anderson darling tests but also produce new types of omnibus tests which are generally much more powerful than the old ones

representation type of schur algebras we give a complete classification of the classical schur algebras and the infinitesimal schur algebras which have tame representation type in combination with earlier work of some of the authors on semisimplicity and finiteness this completes the classification of representation type of all classical and infinitesimal schur algebras in all characteristics

confidence limits to the risk ratio for the problem of obtaining confidence limits to the risk ratio or the ratio of two binomial probabilities katz et al 1978 biometrics 34 469 474 proposed a method based on the logarithm of the observed ratio and using fieller s result we propose a method based on a power of the observed ratio the power being chosen to minimize the skewness of the pivotal random variable the method is simple to use and more stable than that of katz et al a continuity correction is suggested if a conservative interval is desired

combining estimators to improve structural model estimation and inference under quadratic loss asymptotically semi parametric estimators of the parameters in linear structural models have the same sampling properties in finite samples the sampling properties of these estimators vary and large biases may result for sample sizes often found in practice with a goal of improving asymptotic risk performance and finite sample efficiency properties we investigate the idea of combining correlated structural equation estimators with different finite and asymptotic sampling characteristics based on a quadratic loss measure we present evidence that the finite sample performance of the resulting combination estimator can be notably superior to that of a leading traditional moment based estimator

tests of independence for bivariate data with random censoring a contingency table approach procedures are proposed for testing the hypothesis of independence between two discrete or discretized random variables one or both of which may be randomly censored the censoring mechanisms can be either independent bivariate censoring or identical univariate censoring the proposed tests reduce to goodness of fit tests for log linear models applied to either complete or incomplete contingency tables three examples are analyzed for illustrative purposes

asymptotic normality for the k π divergence goodness of fit tests in this paper for a wide class of goodness of fit statistics based k divergences the asymptotic normality is established under the assumption n mn a 0 where n denotes sample size and mn the number of cells this result is extended to contiguous alternatives to study asymptotic efficiency

representation type of schur algebras we give a complete classification of the classical schur algebras and the infinitesimal schur algebras which have tame representation type in combination with earlier work of some of the authors on semisimplicity and finiteness this completes the classification of representation type of all classical and infinitesimal schur algebras in all characteristics

projective modules in the category _ self duality given a parabolic subalgebra ps of a complex semisimple lie algebra there is a naturally defined category s of g modules which includes all the g modules induced from finite dimensional ps modules this paper treats the question of whether certain projective modules in s are isomorphic to their dual modules the projectives in question are the projective covers of those simple modules occurring in the socles of generalized verma modules their self duality is proved in a number of cases and additional information is obtained on the generalized verma modules

nonparametric kernel regression subject to monotonicity constraints we suggest a method for monotonizing general kernel type estimators for example local linear estimators and nadaraya watson estimators attributes of our approach include the fact that it produces smooth estimates indeed with the same smoothness as the unconstrained estimate the method is applicable to a particularly wide range of estimator types it can be trivially modified to render an estimator strictly monotone and it can be employed after the smoothing step has been implemented therefore an experimenter may use his or her favorite kernel estimator and their favorite bandwidth selector to construct the basic nonparametric smoother and then use our technique to render it monotone in a smooth way implementation involves only an off the shelf programming routine the method is based on maximizing fidelity to the conventional empirical approach subject to monotonicity we adjust the unconstrained estimator by tilting the empirical distribution so as to make the least possible change in the sense of a distance measure subject to imposing the constraint of monotonicity

representation type of the blocks of category os abstract in this paper the authors investigate the representation type of the blocks of the relative parabolic category o s for complex semisimple lie algebras a complete classification of the blocks corresponding to regular weights is given the main results of the paper provide a classification of the blocks in the mixed case when the simple roots corresponding to the singular set and s do not meet

bootstrap confidence regions in multinomial sampling power divergences can be used to give a measure of distance between two probability vectors in multinomial sampling arguments can be substituted by empirical and theoretical proportions to obtain confidence regions of parameters in this paper the bootstrap versions of these confidence regions are constructed monte carlo simulation experiments are carried out to calculate average coverage probabilities and to compare the behavior of the introduced procedures

efficient derivative pricing by the extended method of moments in this paper we introduce the extended method of moments xmm estimator this estimator accommodates a more general set of moment restrictions than the standard generalized method of moments gmm estimator more specifically the xmm differs from the gmm in that it can handle not only uniform conditional moment restrictions i e valid for any value of the conditioning variable but also local conditional moment restrictions valid for a given fixed value of the conditioning variable the local conditional moment restrictions are of special relevance in derivative pricing to reconstruct the pricing operator on a given day by using the information in a few cross sections of observed traded derivative prices and a time series of underlying asset returns the estimated derivative prices are consistent for a large time series dimension but a fixed number of cross sectionally observed derivative prices the asymptotic properties of the xmm estimator are nonstandard since the combination of uniform and local conditional moment restrictions induces different rates of convergence parametric and nonparametric for the parameters

natural bundles iii resolving the singularities in the space of immersed submanifolds this is the third of a sequence of articles dealing with the geometry and applications to field theories of natural infinite dimensional bundles in this paper a method for resolving the singularities in the space of immersed submanifolds imm m n diff m is described this desingularization is completely natural emerging as it does from a consideration of principal bundles associated with the manifold pair m n geometrical aspects of the bundles involved in our construction are discussed and the potential role of this geometry in physical theories is outlined in the final section a brief description of the action of the codomain diffeomorphism group diff n on the manifold of immersions imm m n is provided

resolution of simple singularities yielding particle symmetries in a space time a finite subgroup of the conformal group sl 2 c can be related to invariant polynomials on a hypersurface in c3 the latter then carries a simple singularity which resolves by a finite iteration of basic cycles of deprojections the homological intersection graph of these cycles is the dynkin graph of an ade lie group i e a lie group from the cartan series a d or e the deformation of the simple singularity corresponds to ade symmetry breaking a 3 1 dimensional topological model of observation is constructed transforming consistently under sl 2 c as an evolving three dimensional system of world tubes which connect possible points of observation the existence of an initial singularity for the four dimensional space time is related to its global topological structure associating the geometry of ade singularities to the vertex structure of the topological model puts forward the conjecture on a likewise relation of inner symmetries of elementary particles to local space time structure

complexity for modules over finite chevalley groups and classical lie algebras 1 1 let g be a connected reductive algebraic group defined over a finite prime field fp of p elements for a given rational g module m one can consider the restrictions of m to the restricted lie algebra lie g and to the finite group g fp of fp rational points even though there is no direct functorial connection between the categories of restricted lie g modules and g fp modules the ambient algebraic group g can serve as a bridge linking these two important module categories early results of curtis in the 1960s provided a one to one correspondence between the simple restricted modules over k fp of lie g and those of g fp this correspondence is given by simply restricting certain classes of simple g modules more importantly these results along with work of steinberg allow one to easily transfer the questions pertaining to the computation of irreducible characters among these three categories for over thirty years this approach has been one of the main ideas used to relate the representation and cohomology theory of algebraic groups lie algebras and finite groups of lie type in the defining characteristic see e g cpsvan jan2 hum3 and jan1 given the previous results in this area a natural course of investigation should be to find relationships between canonical modules like projective modules for lie g and g fp more precisely

approximating the moments and distribution of the likelihood ratio statistic for multinomial goodness of fit abstract approximations were derived for the mean and variance of g 2 the likelihood ratio statistic for testing goodness of fit in a k cell multinomial distribution these approximate moments accurate to o n 3 may be used in fitting the distribution of g 2 extensive numerical studies of the exact and approximate distributions were performed in the special case of equiprobability the asymptotic chi squared distribution was found to fit poorly for k 4 satisfactory results were obtained using a multiplicative correction to the chi squared fit more sophisticated procedures using the beta distribution produced increased accuracy but at the cost of excessive computational labor

information and entropy econometrics editor s view amos golan department of economics american university roper 200 4400 massachusetts ave nw washington dc 20016 usa

distributed estimation and detection for sensor networks using hidden markov random field models we develop a hidden markov random field hmrf framework for distributed signal processing in sensor network environments under this framework spatially distributed observations collected at the sensors form a noisy realization of an underlying random field that has a simple structure with markovian dependence we derive iterated conditional modes icm algorithms for distributed estimation of the hidden random field from the noisy measurements we consider both parametric and nonparametric measurement error models the proposed distributed estimators are computationally simple applicable to a wide range of sensing environments and localized implying that the nodes communicate only with their neighbors to obtain the desired results we also develop a calibration method for estimating markov random field model parameters from training data and discuss initialization of the icm algorithms the hmrf framework and icm algorithms are applied to event region detection numerical simulations demonstrate the performance of the proposed approach

maxallent maximizers of all entropies and uncertainty of uncertainty a b s t r a c t the entropy maximum approach maxent was developed as a minimization of the subjective uncertainty measured by the boltzmann gibbs shannon entropy many new entropies have been invented in the second half of the 20th century now there exists a rich choice of entropies for fitting needs this diversity of entropies gave rise to a maxent anarchism the maxent approach is now the conditional maximization of an appropriate entropy for the evaluation of the probability distribution when our information is partial and incomplete the rich choice of non classical entropies causes a new problem which entropy is better for a given class of applications we understand entropy as a measure of uncertainty which increases in markov processes in this work we describe the most general ordering of the distribution space with respect to which all continuous time markov processes are monotonic the markov order for inference this approach results in a set of conditionally most random distributions each distribution from this set is a maximizer of its own entropy this uncertainty of uncertainty is unavoidable in the analysis of nonequilibrium systems surprisingly the constructive description of this set of maximizers is possible two decomposition theorems for markov processes provide a tool for this description

efficient information theoretic inference for conditional moment restrictions abstract the generalised method of moments estimator may be substantially biased in finite samples especially so when there are large numbers of unconditional moment conditions this paper develops a class of first order equivalent semi parametric efficient estimators and tests for conditional moment restrictions models based on a local or kernel weighted version of the cressie read power divergence family of discrepancies this approach is similar in spirit to the empirical likelihood methods of kitamura et al 2004 empirical likelihood based inference in conditional moment restrictions models econometrica 72 1667 1714 and tripathi and kitamura 2003 testing conditional moment restrictions annals of statistics 31 2059 2095 these efficient local methods avoid the necessity of explicit estimation of the conditional jacobian and variance matrices of the conditional moment restrictions and provide empirical conditional probabilities for the observations

using permutations to detect dependence between time series abstract in this paper we propose an independence test between two time series which is based on permutations the proposed test can be carried out by means of different common statistics such as pearson s chi square or the likelihood ratio we also point out why an exact test is necessary simulated and real data return exchange rates between several currencies reveal the capacity of this test to detect linear and nonlinear dependences

goodness of fit tests in models for life history data based on cumulative hazard rates to check the validity of an assumed parametric model for survival data one may compare t the nonparametric nelson aalen plot of the cumulative hazard rate with a t the estimated parametric cumulative hazard rate being for example the maximum likelihood estimator convergence in distribution of n t a t and more general processes is studied employing the general framework of counting processes which allows for quite general models for life history data and for quite general censoring schemes the results are applied to the construction of 2 type statistics for goodness of fit cramer von mises and kolmogorov smirnov type tests are presented in the case where the unknown parameter is one dimensional

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

neglected heterogeneity in moment condition models abstract the central concern of this paper is parameter heterogeneity in models specified by a number of unconditional or conditional moment conditions and thereby the provision of a framework for the development of apposite optimal m tests against its potential presence we initially consider the unconditional moment restrictions framework optimal m tests against moment condition parameter heterogeneity are derived with the relevant jacobian matrix obtained in terms of the second order own derivatives of the moment indicator in a leading case gmm and gel tests of specification based on generalized information matrix equalities appropriate for moment based models are described and their relation to optimal m tests against moment condition parameter heterogeneity examined a fundamental and important difference is noted between gmm and gel constructions the paper is concluded by a generalization of these tests to the conditional moment context and the provision of a limited set of simulation experiments to illustrate the efficacy of the proposed tests

benford s law and the fsd distribution of economic behavioral micro data abstract in this paper we focus on the first significant digit fsd distribution of european micro income data and use information theoretic entropy based methods to investigate the degree to which benford s fsd law is consistent with the nature of these economic behavioral systems we demonstrate that benford s law is not an empirical phenomenon that occurs only in important distributions in physical statistics but that it also arises in self organizing dynamic economic behavioral systems the empirical likelihood member of the minimum divergence entropy family is used to recover country based income fsd probability density functions and to demonstrate the implications of using a benford prior reference distribution in economic behavioral system information recovery

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

on schur algebras and related algebras ii this paper is a continuation of 7 the purpose of 7 was to set up in the context of reductive algebraic groups a theory of generalized schur algebras a special case of our construction gives the schur algebras s n r studied extensively in lo we concentrate here on the schur algebras themselves and show that our construction yields new results in their representation theory by means of the schur functor see 10 chap 6 we obtain new results in the representation theory of the symmetric groups the main results are as follows the schur algebras s n r defined over a principal ideal domain r have finite global dimension see also 1 i over a field or complete valuation ring r every finitely generated projective s n r module has a weyl module filtration we prove the irreducibility of the socle of certain weyl modules see also 15 we describe a connection between weights of projective indecomposable modules for the schur algebras in terms of the involution on irreducible representations of symmetric groups given by tensoring with the sign representation we show that over a complete valuation ring or a field every young module for a symmetric group has a specht series in particular the projective indecomposables have specht series finally we show that for r n and a complete valuation ring or field r the blocks of the schur algebra s n r are given by the rule embodied in nakayama s conjecture the notations and conventions are as in the first part of this work 7

robust and efficient estimation by minimising a density power divergence a minimum divergence estimation method is developed for robust parameter estimation the proposed approach uses new density based divergences which unlike existing methods of this type such as minimum hellinger distance estimation avoid the use of nonparametric density estimation and associated complications such as bandwidth selection the proposed class of density power divergences is indexed by a single parameter which controls the trade off between robustness and efficiency the methodology affords a robust extension of maximum likelihood estimation for which 0 choices of near zero afford considerable robustness while retaining efficiency close to that of maximum likelihood

the moment corrected phi divergence test statistics for symmetry in this paper we consider the family of f divergence test statistics t n f s for the problem of symmetry in ixi contingency tables whose asymptotic distribution is chi square with i i 1 2 degrees of freedom and we propose a moment corrected f divergence test statistic in order to improve the accuracy of the chi square approximation of the distribution of t n f s under the hypothesis of symmetry

data tilting for time series summary we develop a general methodology for tilting time series data attention is focused on a large class of regression problems where errors are expressed through autoregressive processes the class has a range of important applications and in the context of our work may be used to illustrate the application of tilting methods to interval estimation in regression robust statistical inference and estimation subject to constraints the method can be viewed as empirical likelihood with nuisance parameters

tests of additional conditional moment restrictions abstract the primary focus of this article is the provision of tests for the validity of a set of conditional moment constraints additional to those defining the maintained hypothesis that are relevant for independent cross sectional data contexts the point of departure and principal contribution of the paper is the explicit and full incorporation of the conditional moment information defining the maintained hypothesis in the design of the test statistics thus the approach mirrors that of the classical parametric likelihood setting by defining restricted tests in contradistinction to unrestricted tests that partially or completely fail to incorporate the maintained information in their formulation the framework is quite general allowing the parameters defining the additional and maintained conditional moment restrictions to differ and permitting the conditioning variates to differ likewise gmm and generalised empirical likelihood test statistics are suggested the asymptotic properties of the statistics are described under both null hypothesis and a suitable sequence of local alternatives an extensive set of simulation experiments explores the practical efficacy of the various test statistics in terms of empirical size and size adjusted power confirming the superiority of restricted over unrestricted tests a number of restricted tests possess both sufficiently satisfactory empirical size and power characteristics to allow their recommendation for econometric practice

normal goodness of fit tests for multinomial models with large degrees of freedom abstract goodness of fit tests for independent multinomials with parameters to be estimated are usually based on pearson s x 2 or the likelihood ratio g 2 both are included in the family of power divergence statistics sd for increasing sample sizes each sd has an asymptotic x 2 distribution provided that the number of cells remains fixed we are dealing with an increasing cells approach where the number j of independent multinomials increases while the number of classes for each multinomial and the number of parameters remain fixed extending results on x 2 and g 2 the asymptotic normality of any sd is obtained for increasing cells the corresponding normal goodness of fit tests discussed here apply for models with large degrees of freedom with no restrictions imposed on the sizes n j of each multinomial allowing large as well as small expectations sparse data within each cell the asymptotic expectation and variance of sd are easy to compute for pearson s x 2 and simplify considerably for

natural bundles ii spin and the diffeomorphism group in this the second of a sequence of papers dealing with the geometry and applications of natural infinite dimensional bundles a framework is described in which the interaction between spinors metrics and diffeomorphisms can be discussed in this framework a coupled metric spinor field configuration manifold and a grand spin frame bundle can be constructed geometrically in addition a notion of infinite dimensional spin structure is defined and an example of such a structure is provided applications of these ideas to general relativity and other field theories are suggested

natural bundles i a minimal resolution of superspace in this the first of a sequence of papers applying infinite dimensional natural bundle techniques to problems in general relativity and other field theories we construct a minimal resolution of the singularities in the space of geometries or superspace of a closed manifold m the basic properties of two natural bundles associated with m are also discussed subsequent articles will develop the theory in the context of spin and the diffeomorphism group the yang mills moduli space and nonlinear sigma models

approximate confidence intervals for the parameters of a stationary binary markov chain the power divergence family of test statistics introduced by cressie and read 1984 is used to obtain approximate confidence intervals for the parameters of a stationary first order binary markov chain intervals based on inverting the likelihood ratio pearson and freeman tukey statistics are included in this family small sample comparisons are used to select the best intervals for the different parameters the results show that the best power divergence intervals for the probability of success and in particular the likelihood ratio interval have error rates closer to nominal levels than the sample proportion based intervals suggested by crow 1979 the properties of power divergence joint confidence regions for the transition probabilities are also explored

power divergence family of statistics for person parameters in irt models we generalize the power divergence pd family of statistics to the two parameter logistic irt model for the purpose of constructing hypothesis tests and confidence intervals of the person parameter the well known score test statistic is a special case of the proposed pd family we also prove the proposed pd statistics are asymptotically equivalent and converge in distribution to chi 1 2 in addition a moment matching method is introduced to compare statistics and choose the optimal one within the pd family simulation results suggest that the coverage rate of the associated confidence interval is well controlled even under small sample sizes for some pd statistics compared to some other approaches the associated confidence intervals exhibit smaller lengths while maintaining adequate coverage rates the utilities of the proposed method are demonstrated by analyzing a real data set

efficient robust regression via two stage generalized empirical likelihood large and finite sample efficiency and resistance to outliers are the key goals of robust statistics although often not simultaneously attainable we develop and study a linear regression estimator that comes close efficiency is obtained from the estimator s close connection to generalized empirical likelihood and its favorable robustness properties are obtained by constraining the associated sum of weighted squared residuals we prove maximum attainable finite sample replacement breakdown point and full asymptotic efficiency for normal errors simulation evidence shows that compared to existing robust regression estimators the new estimator has relatively high efficiency for small sample sizes and comparable outlier resistance the estimator is further illustrated and compared to existing methods via application to a real dataset with purported outliers

a tilting approach to ranking influence summary we suggest a new approach which is applicable for general statistics computed from random samples of univariate or vector valued or functional data to assessing the influence that individual data have on the value of a statistic and to ranking the data in terms of that influence our method is based on first perturbing the value of the statistic by tilting or reweighting each data value where the total amount of tilt is constrained to be the least possible subject to achieving a given small perturbation of the statistic and then taking the ranking of the influence of data values to be that which corresponds to ranking the changes in data weights it is shown both theoretically and numerically that this ranking does not depend on the size of the perturbation provided that the perturbation is sufficiently small that simple result leads directly to an elegant geometric interpretation of the ranks they are the ranks of the lengths of projections of the weights onto a line determined by the first empirical principal component function in a generalized measure of covariance to illustrate the generality of the method we introduce and explore it in the case of functional data where for example it leads to generalized boxplots the method has the advantage of providing an interpretable ranking that depends on the statistic under consideration for example the ranking of data in terms of their influence on the value of a statistic is different for a measure of location and for a measure of scale this is as it should be a ranking of data in terms of their influence should depend on the manner in which the data are used additionally the ranking recognizes rather than ignores sign and in particular can identify left and right hand tails of the distribution of a random function or vector

representation type of the blocks of category os in types f4 and g2 abstract in this paper a complete classification of the representation type of the infinitesimal blocks of the parabolic category o s for the complex simple lie algebras of types f 4 and g 2 is given

on the convexity of some divergence measures based on entropy functions three measures of divergence between vectors in a convex set of a n dimensional real vector space are defined in terms of certain types of entropy functions and their convexity property is studied among other results a classification of the entropies of degree alpha is obtained by the convexity of these measures these results have applications in information theory and biological studies

estimating model discrepancy a quantity 2 is defined as the discrepancy between an actual and an assumed statistical model an estimator is developed for the case in which data are grouped the estimator is based on pearson

modelling trends in ordered correspondence analysis using orthogonal polynomials the core of the paper consists of the treatment of two special decompositions for correspondence analysis of two way ordered contingency tables the bivariate moment decomposition and the hybrid decomposition both using orthogonal polynomials rather than the commonly used singular vectors to this end we will detail and explain the basic characteristics of a particular set of orthogonal polynomials called emerson polynomials it is shown that such polynomials when used as bases for the row and or column spaces can enhance the interpretations via linear quadratic and higher order moments of the ordered categories to aid such interpretations we propose a new type of graphical display the polynomial biplot