ordering and selecting extreme populations by means of entropies and divergences this paper studies the simultaneous selection of extreme populations from a set of independent populations two types of subset selection rules for k populations are proposed and studied the first type selects one subset of populations that should contain the population with the smallest and another subset of populations that should contain the population with the largest f entropy the second type selects analogously but in terms of the extreme f divergences with respect a known control population properties of the proposed procedures are stated and studied examples are presented in order to illustrate the results

an empirical non parametric likelihood family of data based benford like distributions abstract a mathematical expression known as benford s law provides an example of an unexpected relationship among randomly selected sequences of first significant digits fsds newcomb note on the frequency of use of the different digits in natural numbers am j math 4 1881 39 40 and later benford the law of anomalous numbers proc am philos soc 78 4 1938 551 572 conjectured that fsds would exhibit a weakly monotonic decreasing distribution and proposed a frequency proportional to the logarithmic rule unfortunately the benford fsd function does not hold for a wide range of scale invariant multiplicative data to confront this problem we use information theoretic methods to develop a data based family of alternative benford like exponential distributions that provide null hypotheses for testing purposes two data sets are used to illustrate the performance of generalized benford like distributions

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

robust fitting of the binomial model we consider the problem of robust inference for the binomial m pi model the discreteness of the data and the fact that the parameter and sample spaces are bounded mean that standard robustness theory gives surprising results for example the maximum likelihood estimator mle is quite robust it cannot be improved on for m 1 but can be for m 1 we discuss four other classes of estimators m estimators minimum disparity estimators optimal mgp estimators and a new class of estimators which we call e estimators we show that e estimators have a non standard asymptotic theory which challenges the accepted relationship between robustness concepts and thereby provided new perspectives on these concepts

the hellinger correlation in this paper the defining properties of any valid measure of the dependence between two continuous random variables are revisited and complemented with two original ones shown to imply other usu

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

the mm me ml el ef and gmm approaches to estimation a synthesis abstract the 20th century began on an auspicious statistical note with the publication of karl pearson s philos mag ser 50 1900 157 goodness of fit test which is regarded as one of the most important scientific breakthroughs the basic motivation behind this test was to see whether an assumed probability model adequately described the data at hand pearson philos trans roy soc london ser a 185 1894 71 also introduced a formal approach to statistical estimation through his method of moments mm estimation ronald a fisher while he was a third year undergraduate at the gonville and caius college cambridge suggested the maximum likelihood estimation mle procedure as an alternative to pearson s mm approach in 1922 fisher published a monumental paper that introduced such basic concepts as consistency efficiency sufficiency and even the term parameter with its present meaning fisher philos trans roy soc london ser a 222 1922 309 provided the analytical foundation of mle and studied its efficiency relative to the mm estimator fisher j roy statist soc 87 1924a 442 established the asymptotic equivalence of minimum 2 and ml estimators and wrote in favor of using minimum 2 method rather than pearson s mm approach recently econometricians have found working under assumed likelihood functions restrictive and have suggested using a generalized version of pearson s mm approach commonly known as the gmm estimation procedure as advocated in hansen econometrica 50 1982 1029 earlier godambe ann math statist 31 1960 1208 and durbin j roy statist soc ser b 22 1960 139 developed the estimating function ef approach to estimation that has been proven very useful for many statistical models a fundamental result is that score is the optimum ef ferguson ann math statist 29 1958 1046 considered an approach very similar to gmm and showed that estimation based on the pearson 2 statistic is equivalent to efficient gmm golan et al maximum entropy econometrics robust estimation with limited data wiley new york 1996 developed entropy based formulation that allowed them to solve a wide range of estimation and inference problems in econometrics more recently imbens et al econometrica 66 1998 333 kitamura and stutzer econometrica 65 1997 861 and mittelhammer et al econometric foundations cambridge university press cambridge 2000 put gmm within the framework of empirical likelihood el and maximum entropy me estimation it can be shown that many of these estimation techniques can be obtained as special cases of minimizing cressie and read j roy statist soc ser b 46 1984 440 power divergence criterion that comes directly from the pearson 1900 2 statistic in this way we are able to assimilate a number of seemingly unrelated estimation techniques into a unified framework

confidence intervals in generalized method of moments models we consider the construction of confidence intervals for parameters characterized by moment restrictions in the standard approach to generalized method of moments gmm estimation confidence intervals are based on the normal approximation to the sampling distribution of the parameters there is often considerable disagreement between the nominal and actual coverage rates of these intervals especially in cases with a large degree of overidentification we consider alternative confidence intervals based on empirical likelihood methods which exploit the normal approximation to the lagrange multipliers calculated as a byproduct in empirical likelihood estimation in large samples such confidence intervals are identical to the standard gmm ones but in finite samples their properties can be substantially different in some of the examples we consider the proposed confidence intervals have coverage rates much closer to the nominal coverage rates than the corresponding gmm intervals

majorization and divergence abstract a majorization ordering of the distance or divergence of a positive function from a fixed positive function q on a measure space is studied if a function p1 is smaller than a function p2 relative to the ordering then p1 is closer to q than p2 applications are given to measures of divergence markov chains and qualitative conditions for probability densities closest to q subject to first and second moment constraints in addition a generalization of the path result of r f muirhead 1903 proc edinburgh math soc 21 144 157 and g h hardy j e littlewood and g polya inequalities 2nd ed cambridge univ press london 1952 is given

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

jackknifing and bootstrapping goodness of fit statistics in sparse multinomials abstract many goodness of fit problems can be put in the following form n n 1 nk is a multinomial n vector and it is desired to test the composite null hypothesis h 0 p against all possible alternatives the usual tests used are pearson s 1900 statistic or the likelihood ratio statistic neyman and pearson 1928 cressie and read 1984 pointed out that both of these statistics are in the power family of statistics with 1 and 0 respectively they suggested an alternative statistic with although all of these statistics are asymptotically 2 in the usual situation of k fixed and n this is not the case if the multinomial is sparse specifically morris 1975 showed that under certain regularity conditions with k and n both x 2 and g 2 are asymptotically normal with different mean and variance under a simple null hypothesis cressie and read 1984 extended these results to the general 2ni family although these results have not been proven for composite nul

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

on goodness of fit tests for multiple recursive random number generators abstract this paper considers the problem of reducing the computational time in testing uniformity for a full period multiple recursive generator mrg if a sequence of random numbers generated by a mrg is divided into even number of segments say 2 s then the multinomial goodness of fit tests and the empirical distribution function goodness of fit tests calculated from the i th segment are the same as those of the s i th segment the equivalence properties of the goodness of fit test statistics for a mrg and its associated reverse and additive inverse mrgs are also discussed

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

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

power approximations to multinomial tests of fit abstract multinomial tests for the fit of iid observations x 1 xn to a specified distribution f are based on the counts ni of observations falling in k cells e 1 ek that partition the range of the x j the earliest such test is based on the pearson 1900 chi squared statistic x 2 k i 1 ni npi 2 npi where pi pf xj in ei are the cell probabilities under the null hypothesis a common competing test is the likelihood ratio test based on lr 2 k i 1 ni log ni npi cressie and read 1984 introduced a class of multinomial goodness of fit statistics r based on measures of the divergence between discrete distributions this class includes both x 2 when 1 and lr when 0 all of the r have the same chi squared limiting null distribution the power of the commonly used members of the class is usually approximated from a noncentral chi squared distribution that is also the same for all we propose new approximations to the power that vary with the statistic chosen bot

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

about distances of discrete distributions satisfying the data processing theorem of information theory the distances of discrete probability distributions are considered necessary and sufficient conditions for validity of the data processing theorem of information theory are established these conditions are applied to the burbea rao 1982 divergences and bregman 1967 distances

testing goodness of fit with multinomial data abstract several new test procedures are proposed for assessing the goodness of fit of a postulated multinomial distribution the new tests are neyman smooth type tests with orders selected adaptively from the data they are shown through fourier and large sample analyses to provide potential improvements over classical methods in terms of their ability to detect certain types of alternatives simulation results and a real example illustrate the finite sample validity of the large sample theory and the practical utility of the proposed methods

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

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

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

on testing independence in multidimensional contingency tables with stratified random sampling abstract h divergence statistics quantify the divergence between a joint probability and the product of its marginal probabilities on the basis of multidimensional contingency tables asymptotic distributions of these statistics are investigated in a stratified random sampling the five hypotheses that may be tested in connection with mutidimensional contingency tables are considered as well as their corresponding asymptotic powers the problem of optimum allocation is studied and the relative precision of stratified and simple random sampling is analyzed

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

testing a point null hypothesis the irreconcilability of p values and evidence abstract the problem of testing a point null hypothesis or a small interval null hypothesis is considered of interest is the relationship between the p value or observed significance level and conditional and bayesian measures of evidence against the null hypothesis although one might presume that a small p value indicates the presence of strong evidence against the null such is not necessarily the case expanding on earlier work especially edwards lindman and savage 1963 and dickey 1977 it is shown that actual evidence against a null as measured say by posterior probability or comparative likelihood can differ by an order of magnitude from the p value for instance data that yield a p value of 05 when testing a normal mean result in a posterior probability of the null of at least 30 for any objective prior distribution objective here means that equal prior weight is given the two hypotheses and that the prior is symmetric and nonincreasing away from the null other definiti

sample selection and information theoretic alternatives to gmm information theoretic alternatives to general method of moments gmm use over identifying moments to estimate the data generating distribution jointly with the parameters of interest this paper demonstrates how these estimates can be interpreted when the sample is not a random draw from the population of interest i make explicit the selection probability implied by the empirical likelihood and exponential tilting estimators two commonly used estimators in this class in addition i propose an alternative estimator that corresponds to a logisitic selection model the small sample properties of the estimators are demonstrated with a monte carlo experiment

goodness of fit tests for large sparse multinomial distributions abstract a goodness of fit statistic d 2 is introduced for use in multinomial distributions pearson s x2 and d 2 are both approximately normally distributed when the sample size n is not large relative to the number of multinomial categories k under sequences of local alternative hypotheses the test based on d 2 exhibits moderate power when the x2 test is biased application is made to the analysis of large sparse contingency tables a theorem is presented that describes the likelihood ratio for testing a simple multinomial distribution against a mixture of multinomial distributions a wide variety of mixing distributions is considered and the d 2 statistic is a special case of log when testing for dirichlet mixtures of multinomial distributions in the case where n is large relative to k x2 and d 2 k behave approximately as chi squared random variables and differ by a very small amount under the null hypothesis in this situation x2 and d 2 k will generally yield the same inference with a l

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 number of classes in chi squared goodness of fit tests abstract the power of pearson chi squared and likelihood ratio goodness of fit tests based on different partitions is studied by considering families of densities between the null density and fixed alternative densities for sample sizes n local asymptotic theory with respect to the number of classes k is developed for such families simple sufficient and almost necessary conditions are derived under which it is asymptotically optimal to let k tend to infinity with n a numerical study shows that the results of the asymptotic local theory for contamination families agree well with the actual power performance of the tests for heavy tailed alternatives the tests have the best power when k is relatively large unbalanced partitions with some small classes in the tails perform surprisingly well in particular when the alternatives have fairly heavy tails

robust estimation for grouped data here we investigate the robustness properties of the class of minimum power divergence estimators for grouped data this class contains the classical maximum likelihood estimators for grouped data we find that the bias of these estimators due to deviations from the assumed underlying model can be large therefore we propose a more general class of estimators that allows us to construct robust procedures by analogy with hampel s theorem we define optimal bounded influence function estimators and by a simulation study we show that under small model contaminations these estimators are more stable than the classical estimators for grouped data finally we apply our results to a particular real example

generalized empirical likelihood non nested tests this paper examines non nested tests for competing moment condition models using a semi parametric generalized empirical likelihood gel framework the resultant gel estimators are first order asymptotically equivalent to those based on generalized method of moments gmm cox type moment encompassing and parametric encompassing non nested tests for competing moment condition models are proposed simulation experiments are conducted to examine the efficacy of the proposed gel statistics in terms of their size and power properties and to compare their properties with those of corresponding non nested test statistics based on gmm estimation

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

online markov decision processes we consider a markov decision process mdp setting in which the reward function is allowed to change after each time step possibly in an adversarial manner yet the dynamics remain fixed similar to the experts setting we address the question of how well an agent can do when compared to the reward achieved under the best stationary policy over time we provide efficient algorithms which have regret bounds with no dependence on the size of state space instead these bounds depend only on a certain horizon time of the process and logarithmically on the number of actions

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

fast calculation of scattering by 3 d inhomogeneities in uniaxial anisotropic multilayers a fast algorithm to calculate the response of uniaxial layered media on a rectilinear mesh is proposed in order to handle scattering phenomena due to inhomogeneities within them the uniaxial layered media are with their optical axes parallel to the interfaces of the layered structure the generalized poisson summation formula is used to efficiently calculate the discrete fourier transform of spatial responses of the layeredmedia from its continuous fourier spectrum using the windowing technique the integral involved in constructing the discrete fourier spectrum can be performed in a small domain without compromising the accuracy of the result furthermore a fast numerical integration method based on the recently proposed padua points is used which together with the windowing technique yields a fast solution to such scattering problems numerical simulations illustrate the accuracy and efficiency of the proposed algorithm

rendering parametric procedures more robust by empirically tilting the model summary we suggest methods for tilting a likelihood so as to enhance the robustness of maximum likelihood procedures from the viewpoint of computation tilting amounts to choosing unequal weights for the score function in such a way as to maximise likelihood subject to moving a given distance from equally weighted scores empirical methods based on standard parametric q q plots are used to determine the appropriate amount of tilting distance may be measured in a variety of ways and we devote particular attention to power divergence approaches in this context one of the two kullback leibler distance measures is shown to be advantageous

simulation study of the tests of uniform association based on the power divergence in this paper a simulation study is presented to analyze the behavior of the family of test statistics 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 using the phi divergence measures that include as special case the power divergence n cressie t r c read multinomial goodness of fit tests journal of the royal statistic society series b 46 1984 440 464 for the analysis of uniform association between two classification processes based on the local odd ratios for the above test statistics the significance level and its power are evaluated for different sample sizes when we consider a 3 x 2 contingency table

assessing misspecified asset pricing models with empirical likelihood estimators abstract hansen and jagannathan 1997 compare misspecified asset pricing models based on least square projections on a family of admissible stochastic discount factors we extend their fundamental contribution by considering minimum discrepancy projections where misspecification is measured by a family of convex functions that take into account higher moments of asset returns the minimum discrepancy problems are solved on dual spaces producing a family of estimators that captures the least square problem as a particular case we derive the asymptotic distributions of the estimators for the cressie read family of discrepancies and illustrate their use with an assessment of the consumption asset pricing model

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

a new class of asymptotically efficient estimators for moment condition models in this paper we propose a new class of asymptotically efficient estimators for moment condition models these estimators share the same higher order bias properties as the generalized empirical likelihood estimators and once bias corrected have the same higher order efficiency properties as the bias corrected generalized empirical likelihood estimators unlike the generalized empirical likelihood estimators our new estimators are much easier to compute a simulation study finds that our estimators have better finite sample performance than the two step gmm and compare well to several potential alternatives in terms of both computational stability and overall performance

small sample comparisons for the power divergence goodness of fit statistics abstract the small sample properties of the power divergence family of goodness of fit statistics cressie and read in press are examined the asymptotic 2 and three alternative approximations are used to obtain appropriate test levels for comparison with the exact level it is shown that if the 2 approximation is to be used then there are many preferable statistics to the log likelihood ratio g 2 including pearson s 2 furthermore exact power comparisons under specific alternatives to the symmetric null hypothesis show that the power of the well known tests based on g 2 and 2 can be improved by choosing other statistics from the family an explanation is proposed in terms of the effect of relative deviations between observed and expected counts

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