distribution estimation consistent in total variation and in two types of information divergence the problem of the nonparametric estimation of a probability distribution is considered from three viewpoints the consistency in total variation the consistency in information divergence and consistency in reversed order information divergence these types of consistencies are relatively strong criteria of convergence and a probability distribution cannot be consistently estimated in either type of convergence without any restrictions on the class of probability distributions allowed histogram based estimators of distribution are presented which under certain conditions converge in total variation in information divergence and in reversed order information divergence to the unknown probability distribution some a priori information about the true probability distribution is assumed in each case as the concept of consistency in information divergence is stronger than that of convergence in total variation additional assumptions are imposed in the cases of informational divergences

on the estimation of a probability density function by the maximum penalized likelihood method abstract a class of probability density estimates can be obtained by penalizing the likelihood by a functional which depends on the roughness of the logarithm of the density the limiting case of the estimates as the amount of smoothing increasing has a natural form which makes the method attractive for data analysis and which provides a rationale for a particular choice of roughness penalty the estimates are shown to be the solution of an unconstrained convex optimization problem and mild natural conditions are given for them to exist rates of consistency in various norms and conditions for asymptotic normality and approximation by a gaussian process are given thus breaking new ground in the theory of maximum penalized likelihood density estimation author

maximum entropy density estimation with generalized regularization and an application to species distribution modeling we present a unified and complete account of maximum entropy density estimation subject to constraints represented by convex potential functions or alternatively by convex regularization we provide fully general performance guarantees and an algorithm with a complete convergence proof as special cases we easily derive performance guarantees for many known regularization types including l1 l2 l22 and l2 l22 style regularization we propose an algorithm solving a large and general subclass of generalized maximum entropy problems including all discussed in the paper and prove its convergence our approach generalizes and unifies techniques based on information geometry and bregman divergences as well as those based more directly on compactness our work is motivated by a novel application of maximum entropy to species distribution modeling an important problem in conservation biology and ecology in a set of experiments on real world data we demonstrate the utility of maximum entropy in this setting we explore effects of different feature types sample sizes and regularization levels on the performance of maxent and discuss interpretability of the resulting models

asymptotic normality of the posterior in relative entropy we show that the relative entropy between a posterior density formed from a smooth likelihood and prior and a limiting normal form tends to zero in the independent and identically distributed case the mode of convergence is in probability and in mean applications to code lengths in stochastic complexity and to sample size selection are discussed

homogeneity test with a generalized mantel haenszel estimator for l2 k contingency tables abstract a conditional test for the homogeneity of the odds ratios in l2 k k 2 tables is developed based on a multiple hypergeometric distribution the test is a generalization of the breslow day test 1980 and is useful in sparse tables for computational feasibility the conditional maximum likelihood estimator is replaced by a consistent but inefficient estimator in the test statistic if this is the case it is shown that the asymptotic distribution is not chi squared as has been noticed by tarone 1985 in the case of 2 2 tables following tarone we develop a correction method so that the statistic follows an asymptotic chi squared distribution an algorithm is developed for calculating the homogeneity test with a generalized mantel haenszel estimator

a universal prior for integers and estimation by minimum description length of the number of bits required to write down the observed data has been reformulated to extend the classical maximum likelihood principle the principle permits estimation of the number of the parameters in statistical models in addition to their values and even of the way the parameters appear in the models i e of the model structures the principle rests on a new way to interpret and construct a universal prior distribution for the integers which makes sense even when the parameter is an individual object truncated realvalued parameters are converted to integers by dividing them by their precision and their prior is determined from the universal prior for the integers by optimizing the precision 1 introduction in this paper we study estimation based upon the principle of minimizing the total number of binary digits required to rewrite the observed data when each observation is given with some precision instead of attempting at an absolutely shortest description which would be futile we look for the optimum relative to a class of parametrically given distributions this minimum description length mdl principle which we introduced in a less comprehensive form in 25 turns out to degenerate to the more familiar maximum likelihood ml principle in case the number of parameters in the models is fixed so that the description length of the parameters themselves can be ignored in another extreme case where the parameters determine the data it similarly degenerates to jaynes s principle of maximum entropy 14 but the main power of the new criterion is that it permits estimates of the entire model its parameters their number and even the way the parameters appear in the model i e the model structure hence there will be no need to supplement the estimated parameters with a separate hypothesis test to decide whether a model is adequately parameterized or perhaps over parameterized

approximation of probability density functions by the multilevel monte carlo maximum entropy method we develop a complete convergence theory for the maximum entropy method based on moment matching for a sequence of approximate statistical moments estimated by the multilevel monte carlo method under appropriate regularity assumptions on the target probability density function the proposed method is superior to the maximum entropy method with moments estimated by the monte carlo method new theoretical results are illustrated in numerical examples

the leavitt path algebra of a graph abstract for any row finite graph e and any field k we construct the leavitt path algebra l e having coefficients in k when k is the field of complex numbers then l e is the algebraic analog of the cuntz krieger algebra c e described in i raeburn graph algebras in cbms reg conf ser math vol 103 amer math soc 2005 the matrix rings m n k and the leavitt algebras l 1 n appear as algebras of the form l e for various graphs e in our main result we give necessary and sufficient conditions on e which imply that l e is simple

maximum entropy and conditional probability it is well known that maximum entropy distributions subject to appropriate moment constraints arise in physics and mathematics in an attempt to find a physical reason for the appearance of maximum entropy distributions the following theorem is offered the conditional distribution of x l given the empirical observation 1 n sum n i l h x i alpha where x 1 x 2 cdots are independent identically distributed random variables with common density g converges to f lambda x e lambda t h x g x suitably normalized where lambda is chosen to satisfy int f lambda x h x dx alpha thus the conditional distribution of a given random variable x is the normalized product of the maximum entropy distribution and the initial distribution this distribution is the maximum entropy distribution when g is uniform the proof of this and related results relies heavily on the work of zabell and lanford

exponential series estimator of multivariate densities abstract we present an exponential series estimator ese of multivariate densities which has an appealing information theoretic interpretation for a d dimensional random variable x with density p 0 the ese takes the form p x exp i 1 0 m 1 i d 0 m d i i x where i are some real valued linearly independent functions defined on the support of p 0 we derive the convergence rate of the ese in terms of the kullback leibler information criterion the integrated squared error and some other metrics we also derive its almost sure uniform convergence rate we then establish the asymptotic normality of p we undertake two sets of monte carlo experiments the first experiment examines the ese performance using mixtures of multivariate normal densities the second estimates copula density functions the results demonstrate the efficacy of the ese an empirical application on the joint distributions of stock returns is presented

generalized mantel haenszel estimators for k 2 x j tables mickey and elashoff 1985 biometrics 41 623 635 gave an extension of mantel haenszel estimation to log linear models for 2 x j x k tables their extension yields two generalizations of the mantel haenszel odds ratio estimator to k 2 x j tables this paper provides variance and covariance estimators for these generalized mantel haenszel estimators that are dually consistent i e consistent in both large strata and sparse data and presents comparisons of the efficiency of the generalized mantel haenszel estimators

joint fixed rate universal lossy coding and identification of continuous alphabetmemoryless sources the problem of joint universal source coding and density estimation is considered in the setting of fixed rate lossy coding of continuous alphabet memoryless sources for a wide class of bounded distortion measures it is shown that any compactly parametrized family of rd valued independent and identically distributed i i d sources with absolutely continuous distributions satisfying appropriate smoothness and vapnik chervonenkis vc learnability conditions admits a joint scheme for universal lossy block coding and parameter estimation such that when the block length n tends to infinity the overhead per letter rate and the distortion redundancies converge to zero as o n 1 log n and o radicn 1 log n respectively moreover the active source can be determined at the decoder up to a ball of radius o radicn 1 log n in variational distance asymptotically almost surely the system has finite memory length equal to the block length and can be thought of as blockwise application of a time invariant nonlinear filter with initial conditions determined from the previous block comparisons are presented with several existing schemes for universal vector quantization which do not include parameter estimation explicitly and an extension to unbounded distortion measures is outlined finally finite mixture classes and exponential families are given as explicit examples of parametric sources admitting joint universal compression and modeling schemes of the kind studied here

online density estimation of nonstationary sources using exponential family of distributions we investigate online probability density estimation or learning of nonstationary and memoryless sources using exponential family of distributions to this end we introduce a truly sequential algorithm that achieves hannan consistent log loss regret performance against true probability distribution without requiring any information about the observation sequence e g the time horizon t and the drift of the underlying distribution c to optimize its parameters our results are guaranteed to hold in an individual sequence manner our log loss performance with respect to the true probability density has regret bounds of o ct 1 2 where c is the total change drift in the natural parameters of the underlying distribution to achieve this we design a variety of probability density estimators with exponentially quantized learning rates and merge them with a mixture of experts notion hence we achieve this square root regret with computational complexity only logarithmic in the time horizon thus our algorithm can be efficiently used in big data applications apart from the regret bounds through synthetic and real life experiments we demonstrate substantial performance gains with respect to the state of the art probability density estimation algorithms in the literature

a generalization of the mantel haenszel estimator of partial association for 2 x j x k tables summary a generalization of the mantel haenszel estimator of a common odds ratio for a series of 2 x 2 tables is proposed for the estimation of log linear partial association parameters for a series of 2 x j j 2 tables the assumption of no three way interaction among row column and stratifying variables is made two approximate covariance estimators are presented small sample properties of these estimators are investigated by simulation studies the results indicate that use of the proposed partial association technique produces estimates with negligible bias both covariance estimation techniques provide confidence intervals for functions of partial association parameters for example odds ratios or adjusted rates a test of no three way interaction results from these procedures

gmm estimation of a maximum entropy distribution with interval data we develop a generalized method of moments gmm estimator for the distribution of a variable where summary statistics are available only for intervals of the random variable without individual data one cannot calculate the weighting matrix for the gmm estimator instead we propose a simulated weighting matrix based on a first step consistent estimate when the functional form of the underlying distribution is unknown we estimate it using a simple yet flexible maximum entropy density our monte carlo simulations show that the proposed maximum entropy density is able to approximate various distributions extremely well the two step gmm estimator with a simulated weighting matrix improves the efficiency of the one step gmm considerably we use this method to estimate the u s income distribution and compare these results with those based on the underlying raw income data

estimators of the variance of the mantel haenszel log odds ratio estimate summary a new approach to large sample studies of the variance of the log odds ratio estimate of mantel and haenszel is developed and used to derive formulas that lead to useful standard error formulas for the mantel haenszel estimate the variance estimator of robins breslow and greenland 1986 biometrics 42 311 324 is derived in a new way and a new variance estimator valid only for the case of 1 m matching is also derived using the new approach

crossed products by twisted partial actions and graded algebras abstract for a twisted partial action of a group g on an associative non necessarily unital algebra a over a commutative unital ring k the crossed product a g is proved to be associative given a g graded k algebra b g g b g with the mild restriction of homogeneous non degeneracy a criteria is established for b to be isomorphic to the crossed product b 1 g for some twisted partial action of g on b 1 the equality b g b g 1 b g b g g g is one of the ingredients of the criteria and if it holds and moreover b has enough local units then it is shown that b is stably isomorphic to a crossed product by a twisted partial action of g

nonparametric likelihood ratio model selection tests between parametric likelihood and moment condition models we propose a nonparametric likelihood ratio testing procedure for choosing between a parametric likelihood model and a moment condition model when both models could be misspecified our procedure is based on comparing the kullback leibler information criterion klic between the parametric model and moment condition model we construct the klic for the parametric model using the difference between the parametric log likelihood and a sieve nonparametric estimate of population entropy and obtain the klic for the moment model using the empirical likelihood statistic we also consider multiple 2 model comparison tests when all the competing models could be misspecified and some models are parametric while others are moment based we evaluate the performance of our tests in a monte carlo study and apply the tests to an example from industrial organization

convergence rates for bayesian density estimation of infinite dimensional exponential families we study the rate of convergence of posterior distributions in density estimation problems for log densities in periodic sobolev classes characterized by a smoothness parameter p the posterior expected density provides a nonparametric estimation procedure attaining the optimal minimax rate of convergence under hellinger loss if the posterior distribution achieves the optimal rate over certain uniformity classes a prior on the density class of interest is induced by a prior on the coefficients of the trigonometric series expansion of the log density we show that when p is known the posterior distribution of a gaussian prior achieves the optimal rate provided the prior variances die off sufficiently rapidly for a mixture of normal distributions the mixing weights on the dimension of the exponential family are assumed to be bounded below by an exponentially decreasing sequence to avoid the use of infinite bases we develop priors that cut off the series at a sample size dependent truncation point when the degree of smoothness is unknown a finite mixture of normal priors indexed by the smoothness parameter which is also assigned a prior produces the best rate a rate adaptive estimator is derived

adaptive estimation of the intensity of inhomogeneous poisson processes via concentration inequalities in this paper we establish oracle inequalities for penalized projection estimators of the intensity of an inhomogeneous poisson process we study consequently the adaptive properties of penalized projection estimators at first we provide lower bounds for the minimax risk over various sets of smoothness for the intensity and then we prove that our estimators achieve these lower bounds up to some constants the crucial tools to obtain the oracle inequalities are new concentration inequalities for suprema of integral functionals of poisson processes which are analogous to talagrand s inequalities for empirical processes

asymptotic behavior of likelihood methods for exponential families when the number of parameters tends to infinity consider a sample of size n from a regular exponential family in p n dimensions let hat theta n denote the maximum likelihood estimator and consider the case where p n tends to infinity with n and where theta n is a sequence of parameter values in r p n moment conditions are provided under which hat theta n theta n o p sqrt p n n and hat theta n theta n overline x n o p p n n where overline x n is the sample mean the latter result provides normal approximation results when p 2 n n rightarrow 0 it is shown by example that even for a single coordinate of hat theta n theta n p 2 n n rightarrow 0 may be needed for normal approximation however if p 3 2 n n rightarrow 0 the likelihood ratio test statistic lambda for a simple hypothesis has a chi square approximation in the sense that 2 log lambda p n sqrt 2p n rightarrow d mathscr n 0 1

odds ratio inference with dependent data summary when data can be presented as a series of k 2 x 2 tables with cell counts xi ni xi yi mi yi it is often assumed that xi and yi are binomially distributed this paper deals with inference for the common odds ratio i l when the binomial assumption is invalid when k increases the consistency and asymptotic normality of the mantel haenszel 1959 estimator is derived the conditional maximum likelihood estimator is shown to be inconsistent and the asymptotic bias is computed when either the first order markov chain or beta binomial model is assumed the mantel haenszel test for testing f 1 is also shown to be inappropriate through some simulation studies two consistent test statistics are proposed and shown to be comparable to each other in terms of size and efficiency some possible further work is described

nonparametric density estimation the l 1 view differentiation of integrals consistency lower bounds for rates of convergence rates of convergence in l1 the automatic kernel estimate l1 and pointwise convergence estimates related to the kernel estimate and the histogram estimate simulation inequalities and random variate generation the transformed kernel estimate applications in discrimination operations on density estimates estimators based on orthogonal series index

on the recovery of joint distributions from limited information the paper considers the role of entropy and other information theoretic concepts in the description and formation of joint distributions for m random variables the discussion includes a review of methods for constructing discrete and continuous joint distributions from the component marginal distributions we then propose a minimum cross entropy approach that recovers continuous joint distributions from the joint and marginal moments and the marginal densities the large sample properties of the associated estimator are outlined and a simple demonstration problem that highlights the advantages and drawbacks of the proposed method is presented

multiscale poisson intensity and density estimation the nonparametric poisson intensity and density estimation methods studied in this paper offer near minimax convergence rates for broad classes of densities and intensities with arbitrary levels of smoothness the methods and theory presented here share many of the desirable features associated with wavelet based estimators computational speed spatial adaptivity and the capability of detecting discontinuities and singularities with high resolution unlike traditional wavelet based approaches which impose an upper bound on the degree of smoothness to which they can adapt the estimators studied here guarantee nonnegativity and do not require any a priori knowledge of the underlying signal s smoothness to guarantee near optimal performance at the heart of these methods lie multiscale decompositions based on free knot free degree piecewise polynomial functions and penalized likelihood estimation the degrees as well as the locations of the polynomial pieces can be adapted to the observed data resulting in near minimax optimal convergence rates for piecewise analytic signals in particular the error of this estimator converges at nearly the parametric rate these methods can be further refined in two dimensions and it is demonstrated that platelet based estimators in two dimensions exhibit similar near optimal error convergence rates for images consisting of smooth surfaces separated by smooth boundaries

an estimating equation approach for the analysis of case control studies with exposure measured at several levels liang gave an extension of the mantel haenszel estimating procedure for a common odds ratio to logistic regression models it is applicable to case control studies with multiple exposure levels which yield k 2 j tables this paper provides variance and covariance estimators which are consistent in both sparse data and large strata for liang s estimating functions in the k 2 j tables case and proposes an approximate confidence interval method for the common odds ratios

sieve maximum likelihood estimation of the spatial autoregressive tobit model abstract this paper extends the ml estimation of a spatial autoregressive tobit model under normal disturbances in xu and lee 2015b journal of econometrics to distribution free estimation we examine the sieve mle of the model where the disturbances are i i d with an unknown distribution we show that the spatial autoregressive process with tobit censoring and related variables are spatial near epoch dependent ned a related contribution is that we develop some exponential inequalities for spatial ned random fields with these inequalities we establish the consistency of the estimator asymptotic distributions of structural parameters of the model are derived from a functional central limit theorem and projection simulations show that the sieve mle can improve the finite sample performance upon misspecified normal mles as an empirical application we examine the school district income surtax rates in iowa our results show that the spatial spillover effects are significant but they may be overestimated if disturbances are restricted to be normally distributed

approximation theoretic derivation of logarithmic entropy principles for inverse problems and unique extension of the maximum entropy method to incorporate prior knowledge the minimum distance approach for reconstructing a positive function based on knowledge of finitely many linear functional values is examined two important classes of directed distances for signal processing and statistical inference are discussed by imposing conditions analogous to those satisfied by linear projections in hilbert space two logarithmic entropy principles are derived one of these involves the itakura saito distortion measure of communication theory and uniquely extends burg s maximum entropy method to incorporate prior knowledge the other uses the kullback leibler distance of statistics

associativity of crossed products by partial actions enveloping actions and partial representations given a partial action a of a group g on an associative algebra a we consider the crossed product a g using the algebras of multipliers we generalize a result of exel 1997 on the associativity of a g obtained in the context of c algebras in particular we prove that a g is associative provided that a is semiprime we also give a criterion for the existence of a global extension of a given partial action on an algebra and use crossed products to study relations between partial actions of groups on algebras and partial representations as an application we endow partial group algebras with a crossed product structure

catching up faster by switching sooner a predictive approach to adaptive estimation with an application to the aic bic dilemma summary prediction and estimation based on bayesian model selection and model averaging and derived methods such as the bayesian information criterion bic do not always converge at the fastest possible rate we identify the catch up phenomenon as a novel explanation for the slow convergence of bayesian methods which inspires a modification of the bayesian predictive distribution called the switch distribution when used as an adaptive estimator the switch distribution does achieve optimal cumulative risk convergence rates in non parametric density estimation and gaussian regression problems we show that the minimax cumulative risk is obtained under very weak conditions and without knowledge of the underlying degree of smoothness unlike other adaptive model selection procedures such as the akaike information criterion aic and leave one out cross validation bic and bayes factor model selection are typically statistically consistent we show that this property is retained by the switch distribution which thus solves the aic bic dilemma for cumulative risk the switch distribution has an efficient implementation we compare its performance with aic bic and bayesian model selection and averaging on a regression problem with simulated data

globalization of group cohomology in the sense of alvares alves redondo abstract recently e r alvares m m alves and m j redondo introduced a cohomology for a group g with values in a module over the partial group algebra k par g which is different from the partial group cohomology defined earlier by the first two named authors of the present paper given a unital partial action of g on a unital algebra a we consider a as a k par g module in a natural way and study the globalization problem for the cohomology in the sense of alvares alves redondo with values in a the problem is reduced to an extendibility property of cocycles furthermore assuming that a is a product of blocks we prove that any cocycle is globalizable and globalizations of cohomologous cocycles are also cohomologous as a consequence we obtain that the alvares alves redondo cohomology group h p a r n g a is isomorphic to the usual cohomology group h n g m b where m b is the multiplier algebra of b and b is the algebra under the enveloping action of

approximation of density functions by sequences of exponential families 1 introduction consider the estimation of a probability density func tion p x defined on a bounded interval we approximate the logarithm of the density by a basis function expansion consisting of polynomials splines or trigonometric series the expansion yields a regular exponential family within which we estimate the density by the method of maximum likelihood this method of density estimation arises by application of the principle of maxi mum entropy or minimum relative entropy subject to empirical constraints we show that if the logarithm of the density has r square integrable deriva tives f idr log p12 ox then the sequence of density estimators p converges to p in the sense of relative entropy kullback leibler fp log p pn at rate opr 1 m2r m n as m x c and m2 n 0 in the spline and trigonometric cases and m3 n o 0 in the polynomial case where m is the dimension of the family and n is the sample size boundary conditions are assumed for the density in the trigonometric case this convergence rate specializes to op r n 2r 2r 1 by setting m nl 2r 1 when the log density is known to have degree of smoothness at least r analogous convergence results for the relative entropy are shown to hold in general for any class of log density functions and sequence of finite dimensional linear spaces having l2 and l approximation properties the approximation of log densities using polynomials has previously been considered by neyman 1937 to define alternatives for goodness of fit tests by good 1963 as an application of the method of maximum entropy or minimum relative entropy by crain 1974 1976a b 1977 who demonstrates existence and consistency of the maximum likelihood estimator and by mead and

on the choice of a model to fit data from an exponential family let x 1 cdots x n be iid observations coming from an exponential family the problem of interest is this given a finite number of models m j smoothly curved manifolds in mathbb r k choose the best model to fit the observations with some penalty for choosing models with dimensions which are too large a result of schwarz is made more specific and is extended to the case where the models are curved manifolds if s y n j is up to a constant c n independent of the model the log of the posterior probability of the j th model where the sample mean y n 1 n sum n i 1 x i has been replaced by y schwarz suggested an asymptotic expansion of s y n j whose leading terms are gamma y n j n sup psi in m j cap theta y psi b psi frac 1 2 k j log n in the case where the models are affine subspaces of mathbb r k we establish a similar asymptotic expansion including the next term with uniform bounds for y in a compact neighborhood of nabla b theta where theta is the true value of the parameter we suggest a criterion for the choice of the best model that consists of maximizing the three leading terms in the expansion s y n j we show that the criterion gives the correct model with probabilities p n theta rightarrow 1 as n rightarrow infty

approximation of least squares regression on nested subspaces on presente une methode generale pour approximer le biais et la variance dans une echelle de normes hilbertiennes

projection method mantel haenszel estimator for k 2 j tables abstract a projection method is introduced to generalize the mantel haenszel estimator for estimating the common odds ratios in k 2 j j 2 tables the method provides estimators in an explicit form and needs no iterative computation it is shown that the generalized mantel haenszel estimator proposed by greenland and yanagawa and fujii is obtained by this method as a special case furthermore selecting appropriate weight it is shown that the method leads to an estimator that is asymptotically equivalent to the generalized mantel haenszel estimator proposed by liang a dually consistent estimator of the variance of the estimator is proposed the characteristics of these estimators have been evaluated by monte carlo means

maximum entropy in the problem of moments the maximum entropy approach to the solution of underdetermined inverse problems is studied in detail in the context of the classical moment problem in important special cases such as the hausdorff moment problem we establish necessary and sufficient conditions for the existence of a maximum entropy solution and examine the convergence of the resulting sequence of approximations a number of explicit illustrations are presented in addition to some elementary examples we analyze the maximum entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems we also briefly indicate possible applications to the lee yang theory of ising models to the summation of divergent series and so on the general conclusion is that maximum entropy provides a valuable approximation scheme a serious competitor of traditional pade like procedures

minimum complexity density estimation the authors introduce an index of resolvability that is proved to bound the rate of convergence of minimum complexity density estimators as well as the information theoretic redundancy of the corresponding total description length the results on the index of resolvability demonstrate the statistical effectiveness of the minimum description length principle as a method of inference the minimum complexity estimator converges to true density nearly as fast as an estimator based on prior knowledge of the true subclass of densities interpretations and basic properties of minimum complexity estimators are discussed some regression and classification problems that can be examined from the minimum description length framework are considered

bounds on approximate steepest descent for likelihood maximization in exponential families an approximate steepest descent strategy is described converging in families of regular exponential densities to maximum likelihood estimates of density functions these density estimates are also obtained by an application of the principle of minimum relative entropy subject to empirical constraints we prove tight bounds on the increase of the log likelihood at each iteration of our strategy for families of exponential densities whose log densities are spanned by a set of bounded basis functions

discretizing distributions with exact moments error estimate and convergence analysis the maximum entropy principle is a powerful tool for solving underdetermined inverse problems this paper considers the problem of discretizing a continuous distribution which arises in various applied fields we obtain the approximating distribution by minimizing the kullback leibler information relative entropy of the unknown discrete distribution relative to an initial discretization based on a quadrature formula subject to some moment constraints we study the theoretical error bound and the convergence of this approximation method as the number of discrete points increases we prove that i the theoretical error bound of the approximate expectation of any bounded continuous function has at most the same order as the quadrature formula we start with and ii the approximate discrete distribution weakly converges to the given continuous distribution moreover we present some numerical examples that show the advantage of the method and apply to numerically solving an optimal portfolio problem

large sample inference for log spline models let f be a continuous and positive unknown density on a known compact interval mathscr y let f denote the distribution function of f and let q f 1 denote its quantile function a finite parameter exponential family model based on b splines is constructed maximum likelihood estimation of the parameters of the model based on a random sample of size n from f yields estimates hat f f and hat q of f f and q respectively under mild conditions if the number of parameters tends to infinity in a suitable manner as n rightarrow infty these estimates achieve the optimal rate of convergence the asymptotic behavior of the corresponding confidence bounds is also investigated in particular it is shown that the standard errors of hat f and hat q are asymptotically equal to those of the usual empirical distribution function and empirical quantile function

a bound on the financial value of information it is shown that each bit of information at most doubles the resulting wealth in the general stock market setup this information bound on the growth of wealth is actually attained for certain probability distributions on the market investigated by j kelly 1956 the bound is shown to be a special case of the result that the increase in exponential growth of wealth achieved with true knowledge of the stock market distribution f over that achieved with incorrect knowledge g is bounded above by the entropy of f relative to g

extended mantel haenszel estimating procedure for multivariate logistic regression models a class of estimating functions is proposed for the estimation of multivariate relative risk in stratified case control studies it reduces to the well known mantel haenszel estimator when there is a single binary risk factor large sample properties of the solutions to the proposed estimating equations are established for two distinct situations efficiency calculations suggest that the proposed estimators are nearly fully efficient relative to the conditional maximum likelihood estimator for the parameters considered application of the proposed method to family data and longitudinal data where the conditional likelihood approach fails is discussed two examples from case control studies and one example from a study on familial aggregation are presented

simplicity of partial skew group rings with applications to leavitt path algebras and topological dynamics let a be a commutative and associative ring not necessarily unital g a group and a partial action of g on ideals of a all of which have local units we show that a is maximal commutative in the partial skew group ring a g if and only if a has the ideal intersection property in a g from this we derive a criterion for simplicity of a g in terms of maximal commutativity and g simplicity of a we also provide two applications of our main results first we give a new proof of the simplicity criterion for leavitt path algebras as well as a new proof of the cuntz krieger uniqueness theorem secondly we study topological dynamics arising from partial actions on clopen subsets of a compact set

nonparametric maximum likelihood estimation by the method of sieves maximum likelihood estimation often fails when the parameter takes values in an infinite dimensional space for example the maximum likelihood method cannot be applied to the completely nonparametric estimation of a density function from an operatorname iid sample the maximum of the likelihood is not attained by any density in this example as in many other examples the parameter space positive functions with area one is too big but the likelihood method can often be salvaged if we first maximize over a constrained subspace of the parameter space and then relax the constraint as the sample size grows this is grenander s method of sieves application of the method sometimes leads to new estimators for familiar problems or to a new motivation for an already well studied technique we will establish some general consistency results for the method and then we will focus on three applications

an optimal selection of regression variables summary an asymptotically optimal selection of regression variables is proposed the key assumption is that the number of control variables is infinite or increases with the sample size it is also shown that mallows s qp akaike s fpe and aic methods are all asymptotically equivalent to this method

cohomology of partial smash products abstract we define the partial group cohomology as the right derived functor of the functor of partial invariants we relate this cohomology with partial derivations and with the partial augmentation ideal and we show that there exists a grothendieck spectral sequence relating cohomology of partial smash products with partial group cohomology and algebra cohomology

data driven k sample tests a new solution of the general nonparametric k sample problem for independent continuous random variables is developed our approach to solving the nonparametric problem is via introducing a net of semiparametric models solving the testing problem for members of this approximating net and then combining the resulting statistics via model selection rules which mimic the rule proposed recently in inglot and ledwina 2006 we call such a solution a data driven test since we in fact use a class of model selection criteria we also obtain a class of k sample data driven tests the class is flexible and allows us to construct omnibus as well as more specialized solutions simulations show that the new omnibus test has power comparable to existing k sample tests in case of changes of location or scale and is more powerful for much more complex changes a variant of the new solution focused on detecting high frequency alternations is also briefly discussed some theoretical support is given this article is a