new construction of 3nj symbols polynomials reducing the wavefunction of coupled momenta to products of single ones are introduced and discussed composition of n such polynomials serves as a generating function for 3nj coefficients of vector addition for su 1 1 the polynomials are calculated explicitly and a compact integral representation for 9j coefficients is obtained as an example the method is rather general and may be applied to other groups and to higher vector addition coefficients

hopf bifurcation in a leaky faucet experiment in a leaky faucet experiment an inverse hopf bifurcation was observed as one increases the water flux before the occurrence of the continuous flow for values of the drop rate smaller than the critical drop rate the movement is periodic or quasiperiodic with a finite amplitude of the time series at the critical point the amplitude of the time series vanishes and it suggests the bifurcation point as the threshold of the continuous flow c 1995 the american physical society

the multivariate krawtchouk polynomials as matrix elements of the rotation group representations on oscillator states an algebraic interpretation of the bivariate krawtchouk polynomials is provided in the framework of the three dimensional isotropic harmonic oscillator model these polynomials in two discrete variables are shown to arise as matrix elements of unitary reducible representations of the rotation group in three dimensions many of their properties are derived by exploiting the group theoretic setting the bivariate tratnik polynomials of krawtchouk type are seen to be special cases of the general polynomials that correspond to particular rotations involving only two parameters it is explained how the approach generalizes naturally to d 1 dimensions and allows us to interpret multivariate krawtchouk polynomials as matrix elements of so d 1 unitary representations indications are given on the connection with other algebraic models for these polynomials

empirical bayes methods for combining likelihoods comment suppose that several independent experiments are observed each one yielding a likelihood l k k for a real valued parameter of interest k for example k might be the log odds ratio for a 2 x 2 table relating to the kth population in a series of medical experiments this article concerns the following empirical bayes question how can we combine all of the likelihoods l k to get an interval estimate for any one of the k s say 1 the results are presented in the form of a realistic computational scheme that allows model building and model checking in the spirit of a regression analysis no special mathematical forms are required for he priors or the likelihoods this scheme is designed to take advantage of recent methods that produce approximate numerical likelihoods l k k even in very complicated situations with all nuisance parameters eliminated the empirical bayes likelihood theory is extended to situations where the k s have a regression structure as well as an empirical bayes relationship most of the discussion is presented in terms of a hierarchical bayes model and concerns how such a model can be implemented without requiring large amounts of bayesian input frequentist approaches such as bias correction and robustness play a central role in the methodology

generalized linear mixed models for meta analysis we examine two strategies for meta analysis of a series of 2 x 2 tables with the odds ratio modelled as a linear combination of study level covariates and random effects representing between study variation penalized quasi likelihood pql an approximate inference technique for generalized linear mixed models and a linear model fitted by weighted least squares to the observed log odds ratios are used to estimate regression coefficients and dispersion parameters simulation results demonstrate that both methods perform adequate approximate inference under many conditions but that neither method works well in the presence of highly sparse data under certain conditions with small cell frequencies the pql method provides better inference

the multivariate meixner polynomials as matrix elements of so d 1 representations on oscillator states the multivariate meixner polynomials are shown to arise as matrix elements of unitary representations of the so d 1 group on oscillator states these polynomials depend on d discrete variables and are orthogonal with respect to the negative multinomial distribution the emphasis is put on the bivariate case for which the so 2 1 connection is used to derive the main properties of the polynomials orthogonality relation raising lowering relations generating function recurrence relations and difference equations as well as explicit expressions in terms of standard univariate krawtchouk and meixner polynomials it is explained how these results generalize directly to d variables

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

connection coefficients for classical orthogonal polynomials of several variables abstract connection coefficients between different orthonormal bases satisfy two discrete orthogonal relations themselves for classical orthogonal polynomials whose weights are invariant under the action of the symmetric group connection coefficients between a basis consisting of products of hypergeometric functions and another basis obtained from the first one by applying a permutation are studied for the jacobi polynomials on the simplex it is shown that the connection coefficients can be expressed in terms of tratnik s multivariable racah polynomials and their weights this gives in particular a new interpretation of the hidden duality between the variables and the degree indices of the racah polynomials which lies at the heart of their bispectral properties these techniques also lead to explicit formulas for connection coefficients of hahn and krawtchouk polynomials of several variables as well as for orthogonal polynomials on balls and spheres

interbasis expansions for the isotropic 3d harmonic oscillator and bivariate krawtchouk polynomials an explicit expression for the general bivariate krawtchouk polynomials is obtained in terms of the standard krawtchouk and dual hahn polynomials the bivariate krawtchouk polynomials occur as matrix elements of the unitary reducible representations of so 3 on the energy eigenspaces of the three dimensional isotropic harmonic oscillator and the explicit formula is obtained from the decomposition of these representations into their irreducible components the decomposition entails expanding the cartesian basis states in the spherical bases that span irreducible so 3 representations the overlap coefficients are obtained from the clebsch gordan problem for the lie algebra

natural exponential families with quadratic variance functions statistical theory the normal poisson gamma binomial negative binomial and nefghs distributions are the six univariate natural exponential families nef with quadratic variance functions qvf this sequel to morris 1982 treats certain statistical topics that can be handled within this unified nef qvf formulation including unbiased estimation bhattacharyya and cramer rao lower bounds conditional distributions and moments quadratic regression conjugate prior distributions moments of conjugate priors and posterior distributions empirical bayes and g 2 minimax marginal distributions and their moments parametric empirical bayes and characterizations

long range anticorrelations and non gaussian behavior of the heartbeat we find that the successive increments in the cardiac beat to beat interals of healthy subjects display scale invariant long range anticorrelations up to 10 4 heart beats furthermore we find that the histogram for the heartbeat intervals increments is well described by a l evy stable distribution for a group of subjects with severe heart disease we find that the distribution is unchanged but the long range correlations vanish therefore the different scaling behavior in health and disease must relate to the underlying dynamics of the heartbeat

discrete orthogonal polynomials and difference equations of several variables the goal of this work is to characterize all second order difference operators of several variables that have discrete orthogonal polynomials as eigenfunctions under some mild assumptions we give a complete solution of the problem

some multivariable orthogonal polynomials of the askey tableau discrete families a multivariable generalization is presented for all the discrete families of the askey tableau this significantly extends the multivariable hahn polynomials introduced by karlin and mcgregor the latter are recovered as a limit case from a family of multivariable racah polynomials

symbolic dynamics analysis in the dripping faucet experiment abstract we applied symbolic dynamics analysis to data from the dripping faucet experiment three chaotic attractors at different drop rates were analyzed as well as an intermittent behavior by the graphs construction which are approximation to minimal machines we also propose a new graph representation of symbolic sequences whose pattern reflects the set complexity associated to the experimental attractors

inferring statistical complexity in the dripping faucet experiment abstract we have used topological e machine reconstruction to calculate statistical complexity associated to period five orbits in the dripping faucet experiment basic structures related to the periodic movement and to peripheral tracks in the corresponding directed graphs were identified

long range anticorrelations and non gaussian behavior of a leaky faucet we find that intervals between successive drops from a leaky faucet display scale invariant long range anticorrelations characterized by the same exponents of heart beat to beat intervals of healthy subjects this behavior is also confirmed by numerical simulations on lattice and it is faucet width and flow rate independent the histogram for the drop intervals is also well described by a levy distribution with the same index for both histograms of healthy and diseased subjects this additional result corroborates the evidence for similarities between leaky faucets and healthy hearts underlying dynamics

a random effects regression model for meta analysis many meta analyses use a random effects model to account for heterogeneity among study results beyond the variation associated with fixed effects a random effects regression approach for the synthesis of 2 x 2 tables allows the inclusion of covariates that may explain heterogeneity a simulation study found that the random effects regression method performs well in the context of a meta analysis of the efficacy of a vaccine for the prevention of tuberculosis where certain factors are thought to modify vaccine efficacy a smoothed estimator of the within study variances produced less bias in the estimated regression coefficients the method provided very good power for detecting a non zero intercept term representing overall treatment efficacy but low power for detecting a weak covariate in a meta analysis of 10 studies we illustrate the model by exploring the relationship between vaccine efficacy and one factor thought to modify efficacy the model also applies to the meta analysis of continuous outcomes when covariates are present

the multivariate hahn polynomials and the singular oscillator karlin and mcgregor s d variable hahn polynomials are shown to arise in the dimensional singular oscillator model as the overlap coefficients between bases associated with the separation of variables in cartesian and hyperspherical coordinates these polynomials in d discrete variables depend on real parameters and are orthogonal with respect to the multidimensional hypergeometric distribution the focus is put on the d 2 case for which the connection with the three dimensional singular oscillator is used to derive the main properties of the polynomials forward backward shift operators orthogonality relation generating function recurrence relations bispectrality difference equations and explicit expression in terms of the univariate hahn polynomials the extension of these results to an arbitrary number of variables is presented at the end of the paper

symbol statistics and spatio temporal systems abstract we consider the problem of estimating parameters from time series observations of spatio temporal systems two types of models are considered a a one dimensional coupled map lattice with nearest neighbor diffusive coupling and b the complex ginzburg landau equation in one and two spatial dimensions model parameters are to be estimated using time series observations from only a few sites a symbolic partition of the time series is introduced and the probabilities of observing various symbol sequences in the data are measured the parameter fitting is accomplished by adjusting parameters of the model until it produces time series whose symbol sequences have the same probabilities as the data we show that it is possible to reliably estimate the parameters from a single time series when the spatio temporal dynamics is turbulent i e it displays a wide range of space and time scales with no discernible patterns

parametric empirical bayes inference theory and applications abstract this article reviews the state of multiparameter shrinkage estimators with emphasis on the empirical bayes viewpoint particularly in the case of parametric prior distributions some successful applications of major importance are considered recent results concerning estimates of error and confidence intervals are described and illustrated with data

continued fraction representation for expected cell counts of a 2 x 2 table a rapid and exact method for conditional maximum likelihood estimation the expected cell count for a 2 x 2 contingency table governed by the noncentral extended hypergeometric distribution is expressed as a terminating continued fraction the coefficients in the continued fraction are better behaved than the multinomial coefficients required for the usual moment calculation the expected cell count must be calculated repeatedly in a conditional maximum likelihood analysis of k2 x 2 contingency tables since the continued fraction can be easily evaluated a rapid and numerically stable computational algorithm results once this first moment is known higher moments can be obtained as shown by harkness 1965 annals of mathematical statistics 36 938 945 a basic program to implement the continued fraction algorithm is given in an appendix

meta analysis for 2 2 tables a bayesian approach this paper develops and implements a fully bayesian approach to meta analysis in which uncertainty about effects in distinct but comparable studies is represented by an exchangeable prior distribution specifically hierarchical normal models are used along with a parametrization that allows a unified approach to deal easily with both clinical trial and case control study data monte carlo methods are used to obtain posterior distributions for parameters of interest integrating out the unknown parameters of the exchangeable prior or random effects distribution the approach is illustrated with two examples the first involving a data set on the effect of beta blockers after myocardial infarction and the second based on a classic data set comprising 14 case control studies on the effects of smoking on lung cancer in both examples rather different conclusions from those previously published are obtained in particular it is claimed that widely used methods for meta analysis which involve complete pooling of o e values lead to understatement of uncertainty in the estimation of overall or typical effect size

approximate inference in generalized linear mixed models statistical approaches to overdispersion correlated errors shrinkage estimation and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model glmm given an unobserved vector of random effects observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function known prior weights and a scale factor the random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components for problems involving time series spatial aggregation and smoothing the dispersion may be specified in terms of a rank deficient inverse covariance matrix approximation of the marginal quasi likelihood using laplace s method leads eventually to estimating equations based on penalized quasilikelihood or pql for the mean parameters and pseudo likelihood for the variances im

methods of estimation in log odds ratio regression models mccullagh s 1984 journal of the royal statistical society series b 46 250 256 approximation to the conditional maximum likelihood estimator in log odds ratio regression models is shown to have negligible asymptotic bias unless the odds ratios are large and the sample sizes in individual 2 x 2 tables are very small in application to two sets of case control data it yields results virtually indistinguishable from those of the conditional analysis a generalization of the mantel haenszel estimator proposed by davis 1985 biometrics 41 487 495 does not approximate the conditional results nearly as well

a comparison of heterogeneity variance estimators in combining results of studies for random effects meta analysis seven different estimators of the heterogeneity variance are compared and assessed using a simulation study the seven estimators are the variance component type estimator vc the method of moments estimator mm the maximum likelihood estimator ml the restricted maximum likelihood estimator reml the empirical bayes estimator eb the model error variance type estimator mv and a variation of the mv estimator mvvc the performance of the estimators is compared in terms of both bias and mean squared error using monte carlo simulation the results show that the reml and especially the ml and mm estimators are not accurate having large biases unless the true heterogeneity variance is small the vc estimator tends to overestimate the heterogeneity variance in general but is quite accurate when the number of studies is large the mv estimator is not a good estimator when the heterogeneity variance is small to moderate but it is reasonably accurate when the heterogeneity variance is large the mvvc estimator is an improved estimator compared to the mv estimator especially for small to moderate values of the heterogeneity variance the two estimators mvvc and eb are found to be the most accurate in general particularly when the heterogeneity variance is moderate to large

the multivariate charlier polynomials as matrix elements of the euclidean group representation on oscillator states a family of multivariate orthogonal polynomials generalizing the standard univariate charlier polynomials is shown to arise in the matrix elements of the unitary representation of the euclidean group e d on oscillator states these polynomials in d discrete variables are orthogonal on the product of d poisson distributions the accent is put on the d 2 case and the group theoretical setting is used to obtain the main properties of the polynomials orthogonality and recurrence relations difference equation raising lowering relations generating function hypergeometric and integral representations and explicit expression in terms of standard charlier and krawtchouk polynomials the approach is seen to extend straightforwardly to an arbitrary number of variables the contraction of so 3 to e 2 is used to show that the bivariate charlier polynomials correspond to a limit of the bivariate krawtchouk polynomials

wilson polynomials and the generic superintegrable system on the 2 sphere for complex two dimensional riemannian spaces every classical or quantum second order superintegrable system can be obtained from a single generic 3 parameter potential on the complex 2 sphere by delicate limit operations and through stackel transforms between manifolds here we derive families of finite and infinite dimensional irreducible representations of the corresponding quadratic quantum algebra for the 2 sphere and point out their role in explaining the degeneracy of the energy eigenspaces corresponding to bound state and continuous spectra of quantum and wave equation analogs of this system the algebra is exactly the one that describes the wilson and racah polynomials in their full generality

an algorithm for the mean and variance of the noncentral hypergeometric distribution the mean and variance of the noncentral hypergeometric distribution are essential for obtaining maximum likelihood estimates for conditional logistic models for 2 x 2 tables the numerical difficulty in directly computing them using the definition is discussed in detail by satten and kupper 1990 they also give an algorithm based on a continued fraction representation for the mean we present here another algorithm based on two equations in harkness 1965

analysis of binary data from a multicentre clinical trial summary we develop several methods for estimating the treatment effect difference defined as the overall log odds ratio of favourable response in a multicentre clinical trial comparing two treatments with binary response a simulation study compares the bias and mean squared error of the point estimates and the exact coverage probabilities of confidence intervals obtained distributions multicentre randomized clinical trials are frequently used to test the efficacy of new medical regimens these trials are attractive due to the ease of getting the desired sample size in a short period of time and the broad coverage of the patient population and medical practitioners the analysis of data from such trials requires combining information from the centres in a way that properly accounts for the variation due to the centres and the differential efficacy of treatments between the centres in this paper we investigate several approaches to pooling the data across centres and for handling the effects associated with the centres chakravarthi grizzle 1975 and beitler landis 1985 argue for treating these effects as random and fleiss 1986 argues against because the centres are seldom chosen at random from a well defined population of clinics and therefore the effects cannot be associated with any population we prefer the random effect model approach for the following reasons the goal of a trial is to extend the results of the study to the general population duly accounting for variation in the patient population and the medical skills of practitioners hence if we believe that the variability between centres as estimated by the data is typical of the variability in the population of centres then the inference based on the random effects model is more appropriate than the one based on the fixed effects model furthermore the variation among the centres may be more than that predicted from samples from a binomial distribution treating the centre effects as random introduces this intra class correlation into our model assumptions let the outcome variable from each centre be a pair of binomial random variables xaj

bayesian approaches to random effects meta analysis a comparative study current methods for meta analysis still leave a number of unresolved issues such as the choice between fixed and random effects models the choice of population distribution in a random effects analysis the treatment of small studies and extreme results and incorporation of study specific covariates we describe how a full bayesian analysis can deal with these and other issues in a natural way illustrated by a recent published example that displays a number of problems such analyses are now generally available using the bugs implementation of markov chain monte carlo numerical integration techniques appropriate proper prior distributions are derived and sensitivity analysis to a variety of prior assumptions carried out current methods are briefly summarized and compared to the full bayes analysis

bispectral commuting difference operators for multivariable askey wilson polynomials we construct a commutative algebra a z generated by d algebraically independent q difference operators acting on variables z 1 z 2 z d which is diagonalized by the multivariable askey wilson polynomials p n z considered by gasper and rahman 2005 iterating sears 4 3 transformation formula we show that the polynomials p n z possess a certain duality between z and n analytic continuation allows us to obtain another commutative algebra a n generated by d algebraically independent difference operators acting on the discrete variables n 1 n 2 n d which is also diagonalized by p n z this leads to a multivariable q askey scheme of bispectral orthogonal polynomials which parallels the theory of symmetric functions

empirical bayes methods for combining likelihoods abstract suppose that several independent experiments are observed each one yielding a likelihood l k k for a real valued parameter of interest k for example k might be the log odds ratio for a 2 2 table relating to the kth population in a series of medical experiments this article concerns the following empirical bayes question how can we combine all of the likelihoods l k to get an interval estimate for any one of the k s say 1 the results are presented in the form of a realistic computational scheme that allows model building and model checking in the spirit of a regression analysis no special mathematical forms are required for the priors or the likelihoods this scheme is designed to take advantage of recent methods that produce approximate numerical likelihoods l k k even in very complicated situations with all nuisance parameters eliminated the empirical bayes likelihood theory is extended to situations where the k s have a regression structure as well as an empiri

coupling coefficients for lie algebra representations and addition formulas for special functions representations of the lie algebra su 1 1 and of a generalization of the oscillator algebra b 1 are considered the paper then introduces polynomials which are related by the coupling or clebsch gordan coefficients of the lie algebra in question by making a proper choice these polynomials themselves are related to known special functions the coupling of two or three representations of the lie algebra then leads to interesting addition formulas for these special functions the polynomials appearing here are generalized laguerre and jacobi polynomials for the su 1 1 case and hermite polynomials for the b 1 algebra

structure relations for monic orthogonal polynomials in two discrete variables in this paper extensions of several relations linking differences of bivariate discrete orthogonal polynomials and polynomials themselves are given by using an appropriate vector matrix notation three term recurrence relations are presented for the partial differences of the monic polynomial solutions of admissible second order partial difference equation of hypergeometric type structure relations difference representations as well as lowering and raising operators are obtained finally expressions for all matrix coefficients appearing in these finite type relations are explicitly presented for a finite set of hahn and kravchuk orthogonal polynomials