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

2 matchings the traveling salesman problem and the subtour lp a proof of the boyd carr conjecture determining the precise integrality gap for the subtour linear programming lp relaxation of the traveling salesman problem is a significant open question with little progress made in thirty years in the general case of symmetric costs that obey triangle inequality boyd and carr boyd s carr r 2011 finding low cost tsp and 2 matching solutions using certain half integer subtour vertices discrete optim 8 525 539 prior version accessed june 27 2011 http www site uottawa ca sylvia recentpapers halftri pdf observe that we do not even know the worst case upper bound on the ratio of the optimal 2 matching to the subtour lp they conjecture the ratio is at most 10 9 in this paper we prove the boyd carr conjecture in the case that the support of a fractional 2 matching has no cut edge we can further prove that an optimal 2 matching has cost at most 10 9 times the cost of the fractional 2 matching

point estimation with exponentially tilted empirical likelihood parameters defined via general estimating equations gee can be estimated by maximizing the empirical likelihood el newey and smith econometrica 72 2004 219 255 have recently shown that this el estimator exhibits desirable higher order asymptotic properties namely that its o n 1 bias is small and that bias corrected el is higher order efficient although el possesses these properties when the model is correctly specified this paper shows that in the presence of model misspecification el may cease to be root n convergent when the functions defining the moment conditions are unbounded even when their expectations are bounded in contrast the related exponential tilting et estimator avoids this problem this paper shows that the et and el estimators can be naturally combined to yield an estimator called exponentially tilted empirical likelihood etel exhibiting the same o n 1 bias and the same o n 2 variance as el while maintaining root n convergence under model misspecification

higher order properties of gmm and generalized empirical likelihood estimators in an effort to improve the small sample properties of generalized method of moments gmm estimators a number of alternative estimators have been suggested these include empirical likelihood el continuous updating and exponential tilting estimators we show that these estimators share a common structure being members of a class of generalized empirical likelihood gel estimators we use this structure to compare their higher order asymptotic properties we find that gel has no asymptotic bias due to correlation of the moment functions with their jacobian eliminating an important source of bias for gmm in models with endogeneity we also find that el has no asymptotic bias from estimating the optimal weight matrix eliminating a further important source of bias for gmm in panel data models we give bias corrected gmm and gel estimators we also show that bias corrected el inherits the higher order property of maximum likelihood that it is higher order asymptotically efficient relative to the other bias corrected estimators

inequalities in entanglement percolation we present proofs of two results concerning entanglement in three dimensional bond percolation firstly the critical probability for entanglement with free boundary conditions is strictly less than the critical probability for connectivity percolation the proof presented here is a detailed justification of the ideas sketched in aizenman and grimmett secondly under the hypothesis that the critical probabilities for entanglement with free and wired boundary conditions are different for p between the two critical probabilities the size of the entangled cluster at the origin with free boundary conditions does not have exponentially decaying tails

empirical likelihood and general estimating equations for some time so called empirical likelihoods have been used heuristically for purposes of nonparametric estimation owen showed that empirical likelihood ratio statistics for various parameters theta f of an unknown distribution f have limiting chi square distributions and may be used to obtain tests or confidence intervals in a way that is completely analogous to that used with parametric likelihoods our objective in this paper is twofold first to link estimating functions or equations and empirical likelihood second to develop methods of combining information about parameters we do this by assuming that information about f and theta is available in the form of unbiased estimating functions empirical likelihoods for parameters are developed and shown to have properties similar to those for parametric likelihood efficiency results for estimates of both theta and f are obtained the methods are illustrated on several problems and areas for future investigation are noted

empirical likelihood ratio confidence intervals for a single functional summary the empirical distribution function based on a sample is well known to be the maximum likelihood estimate of the distribution from which the sample was taken in this paper the likelihood function for distributions is used to define a likelihood ratio function for distributions it is shown that this empirical likelihood ratio function can be used to construct confidence intervals for the sample mean for a class of m estimates that includes quantiles and for differentiable statistical functionals the results are nonparametric extensions of wilks s 1938 theorem for parametric likelihood ratios the intervals are illustrated on some real data and compared in a simulation to some bootstrap confidence intervals and to intervals based on student s t statistic a hybrid method that uses the bootstrap to determine critical values of the likelihood ratio is introduced

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

asymptotic optimality of empirical likelihood for testing moment restrictions we show by example that empirical likelihood and other commonly used tests for moment restrictions are unable to control the exponential rate at which the probability of a type i error tends to zero unless the possible distributions for the observed data are restricted appropriately from this it follows that for the optimality claim for empirical likelihood in kitamura 2001 to hold additional assumptions and qualifications are required under stronger assumptions than those in kitamura 2001 we establish the following optimality result i empirical likelihood controls the rate at which the probability of a type i error tends to zero and ii among all procedures for which the probability of a type i error tends to zero at least as fast empirical likelihood maximizes the rate at which the probability of a type ii error tends to zero for most alternatives this result further implies that empirical likelihood maximizes the rate at which the probability of a type ii error tends to zero for all alternatives among a class of tests that satisfy a weaker criterion for their type i error probabilities

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

hodges lehmann optimality for testing moment conditions this paper studies the hodges and lehmann 1956 optimality of tests in a general setup the tests are compared by the exponential rates of growth to one of the power functions evaluated at a fixed alternative while keeping the asymptotic sizes bounded by some constant we present two sets of sufficient conditions for a test to be hodges lehmann optimal these new conditions extend the scope of the hodges lehmann optimality analysis to setups that cannot be covered by other conditions in the literature the general result is illustrated by our applications of interest testing for moment conditions and overidentifying restrictions in particular we show that i the empirical likelihood test does not necessarily satisfy existing conditions for optimality but does satisfy our new conditions and ii the generalized method of moments gmm test and the generalized empirical likelihood gel tests are hodges lehmann optimal under mild primitive conditions these results support the belief that the hodges lehmann optimality is a weak asymptotic requirement

entanglement in percolation we study the existence of finite and infinite entangled graphs in the bond percolation process in three dimensions with density p after a discussion of the relevant definitions the entanglement critical probabilities are defined the size of the maximal entangled graph at the origin is studied for small p and it is shown that this graph has radius whose tail decays at least as fast as exp n log n indeed the logarithm may be replaced by any iterate of logarithm for an appropriate positive constant we explore the question of almost sure uniqueness of the infinite maximal open entangled graph when p is large and we establish two relevant theorems we make several conjectures concerning the properties of entangled graphs in percolation

strict monotonicity for critical points in percolation and ferromagnetic models when is the numerical value of the critical point changed by an enhancement of the process or of the interaction ferromagnetic spin models independent percolation and the contact process are known to be endowed with monotonicity properties in that certain enhancements are capable of shifting the corresponding phase transition in only an obvious direction e g the addition of ferromagnetic couplings can only increase the transition temperature the question explored here is whether enhancements do indeed change the value of the critical point we present a generally applicable approach to this issue for ferromagnetic ising spin systems with pair interactions of finite range ind 2 dimensions it is shown that the critical temperaturetc is strictly monotone increasing in each coupling with the first order derivatives bounded by positive functions which are continuous on the set of fullyd dimensional interactions for independent percolation with 0 pc 1 we prove that any essential enhancement of the process has an effect on the critical probability a result with applications to the question of the existence of entanglements and to invasion percolation with trapping

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

bootstrap critical values for tests based on generalized method of moments estimators monte carlo experiments have shown that tests based on generalized method ofmoments estimators often have true levels that differ greatly from their nominal levels when asymptotic critical values are used this paper gives conditions under which the bootstrap provides asymptotic refinements to the critical values of t tests and the test of overidentifying restrictions particular attention is given to the case of dependent data it is shown that with such data the bootstrap must sample blocks of data and that the formulae for the bootstrap versions of test statistics differ from the formulae that apply with the original data the results of monte carlo experiments on the numerical performance of the bootstrap show that it usually reduces the errors in level that occur when critical values based on first order asymptotic theory are used the bootstrap also provides an indication of the accuracy of critical values obtained from first order asymptotic theory

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

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

minimum weight two connected spanning networks we consider the problem of constructing a minimum weight two connected network spanning all the points in a setv we assume a symmetric nonnegative distance functiond defined onv v which satisfies the triangle inequality we obtain a structural characterization of optimal solutions specifically there exists an optimal two connected solution whose vertices all have degree 2 or 3 and such that the removal of any edge or pair of edges leaves a bridge in the resulting connected components these are the strongest possible conditions on the structure of an optimal solution since we also show thatany two connected graph satisfying these conditions is theunique optimal solution for a particular choice of canonical distances satisfying the triangle inequality we use these properties to show that the weight of an optimal traveling salesman cycle is at most 4 3 times the weight of an optimal two connected solution examples are provided which approach this bound arbitrarily closely in addition we obtain similar results for the variation of this problem where the network need only span a prespecified subset of the points

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

worst case comparison of valid inequalities for the tsp we consider most of the known classes of valid inequalities for the graphical travelling salesman polyhedron and compute the worst case improvement resulting from their addition to the subtour polyhedron for example we show that the comb inequalities cannot improve the subtour bound by a factor greater than 10 9 the corresponding factor for the class of clique tree inequalities is 8 7 while it is 4 3 for the path configuration inequalities

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

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

an information theoretic alternative to generalized method of moments estimation while optimally weighted gmm estimation has desirable large sample properties its small sample performance is poor in some applications we propose a computationally simple alternative for weakly dependent data generating mechanisms based on minimization of the kullback leibler information criterion conditions are derived under which the large sample properties of this estimator are similar to gmm i e the estimator will be consistent and asymptotically normal with the same asymptotic covariance matrix as gmm in addition we propose overidentifying and parametric restrictions tests as alternatives to analogous gmm procedures

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

uniqueness of the infinite entangled component in three dimensional bond percolation we prove uniqueness of the infinite entangled component for bond percolation on the three dimensional cubic lattice above the entanglement critical probability this improves earlier results by grimmett and holroyd

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

on the number of entangled clusters we prove that the number of entangled clusters with n edges in the simple cubic lattice grows exponentially in n this answers an open question posed by grimmett and holroyd proc lond math soc 81 485 512 2000 our result has immediate implications for entanglement percolation we obtain an improved rigorous lower bound on the critical probability and we prove that the radius of the entangled component of the origin has exponentially decaying tail when p is small

estimation and confidence regions for parameter sets in econometric models this paper develops a framework for performing estimation and inference in econometric models with partial identification focusing particularly on models characterized by moment inequalities and equalities applications of this framework include the analysis of game theoretic models revealed preference restrictions regressions with missing and corrupted data auction models structural quantile regressions and asset pricing models specifically we provide estimators and confidence regions for the set of minimizers i of an econometric criterion function q in applications the criterion function embodies testable restrictions on economic models a parameter value that describes an economic model satisfies these restrictions if q attains its minimum at this value interest therefore focuses on the set of minimizers called the identified set we use the inversion of the sample analog q n of the population criterion q to construct estimators and confidence regions for the identified set and develop consistency rates of convergence and inference results for these estimators and regions to derive these results we develop methods for analyzing the asymptotic properties of sample criterion functions under set identification

on bahadur efficiency of empirical likelihood this paper studies the bahadur efficiency of empirical likelihood for testing moment condition models it is shown that under mild regularity conditions the empirical likelihood overidentifying restriction test is bahadur efficient i e its p value attains the fastest convergence rate under each fixed alternative hypothesis analogous results are derived for parameter hypothesis testing and set inference problems

information theoretic approaches to inference in moment condition models one step efficient gmm estimation has been developed in the recent papers of back and brown 1990 imbens 1993 and qin and lawless 1994 these papers emphasized methods that correspond to using owen s 1988 method of empirical likelihood to reweight the data so that the reweighted sample obeys all the moment restrictions at the parameter estimates in this paper we consider an alternative klic motivated weighting and show how it and similar discrete reweightings define a class of unconstrained optimization problems which includes gmm as a special case such klic motivated reweightings introduce m auxiliary tilting parameters where m is the number of moments parameter and overidentification hypotheses can be recast in terms of these tilting parameters such tests when appropriately conditioned on the estimates of the original parameters are often startlingly more effective than their conventional counterparts this is apparently due to the local ancillarity of the original parameters for the tilting parameters

gel estimation and tests of spatial autoregressive models abstract this paper considers the generalized empirical likelihood gel estimation and tests of high order spatial autoregressive sar models by exploring an inherent martingale structure the gel estimator has the same asymptotic distribution as the generalized method of moments estimator explored with same moment conditions for estimation but circumvents a first step estimation of the optimal weighting matrix with a preliminary estimator and thus can be robust to unknown heteroskedasticity and non normality while the gel removes the asymptotic bias from the preliminary estimator and partially removes the bias due to the correlation between the moment conditions and their jacobian the empirical likelihood as a special member of gels further partially removes the bias from estimating the second moment matrix we also formulate the gel overidentification test moran s i test and gel ratio tests for parameter restrictions and non nested hypotheses while some of the conventional tests might not be robust to non normality and or unknown heteroskedasticity the corresponding gel tests can be robust

on universal hypotheses testing via large deviations a prototype problem in hypotheses testing is discussed the problem of deciding whether an i i d sequence of random variables has originated from a known source p sub 1 or an unknown source p sub 2 is considered the exponential rate of decrease in type ii probability of error under a constraint on the minimal rate of decrease in type i probability of error is chosen for a criterion of optimality using large deviations estimates a decision rule that is based on the relative entropy of the empirical measure with respect to p sub 1 is proposed in the case of discrete random variables this approach yields weaker results than the combinatorial approach used by hoeffding 1965 however it enables the analysis to be extended to the general case of r sup n valued random variables finally the results are extended to the case where p sub 1 is an unknown parameter dependent distribution that is known to belong to a set of distributions p sup 0 sub 1 theta in theta

testing conditional moment restrictions let x z be a pair of observable random vectors we construct a new smoothed empirical likelihood based test for the hypothesis e g z x 0 w p 1 where g is a vector of known functions and an unknown finite dimensional parameter we show that the test statistic is asymptotically normal under the null hypothesis and derive its asymptotic distribution under a sequence of local alternatives furthermore the test is shown to possess an optimality property in large samples simulation evidence suggests that it also behaves well in small samples

el inference for partially identified models large deviations optimality and bootstrap validity this paper addresses the issue of optimal inference for parameters that are partially identied in models with moment inequalities there currently exists a variety of inferential methods for use in this setting however the question of choosing optimally among contending procedures is unresolved in this paper i rst consider a canonical large deviations criterion for optimality and show that inference based on the empirical likelihood ratio statistic is optimal this nding is a direct analog to that in kitamura 2001 for moment equality models second i introduce a new empirical likelihood bootstrap that provides a valid resampling method for moment inequality models and overcomes the implementation challenges that arise as a result of non pivotal limit distributions lastly i analyze the nite sample properties of the proposed framework using monte carlo simulations the

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 bound for the ratio between the 2 matching problem and its linear programming relaxation consider the 2 matching problem defined on the complete graph with edge costs which satisfy the triangle inequality we prove that the value of a minimum cost 2 matching is bounded above by 4 3 times the value of its linear programming relaxation the fractional 2 matching problem this lends credibility to a long standing conjecture that the optimal value for the traveling salesman problem is bounded above by 4 3 times the value of its linear programming relaxation the subtour elimination problem

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

moment matching and best entropy estimation abstract given the first n moments of an unknown function x on the unit interval a common estimate of x is n where n is a polynomial of degree n taking values in a prescribed interval is a given monotone function and n is chosen so that the moments of n equal those of x this moment matching procedure is closely related to best entropy estimation of x two classical cases arise when is the exponential function corresponding to the boltzmann shannon entropy and the reciprocal function corresponding to the burg entropy general conditions ensuring the existence and uniqueness of n are given using convex programming duality techniques and it is shown that the estimate n converges uniformly to x providing x is sufficiently smooth