proportional likelihood ratio models for mean regression the proportional likelihood ratio model introduced in luo tsai 2012 is adapted to explicitly model the means of observations this is useful for the estimation of and inference on treatment effects particularly in designed experiments and allows the data analyst greater control over model specification and parameter interpretation

fitting regression models to case control data by maximum likelihood summary we consider fitting categorical regression models to data obtained by either stratified or nonstratified case control or response selective sampling from a finite population with known population totals in each response category with certain models such as the logistic with appropriate constant terms a method variously known as conditional maximum likelihood breslow cain 1988 or pseudo conditional likelihood wild 1991 which involves the prospective fitting of a pseudo model results in maximum likelihood estimates of case control data we extend these results by showing the maximum likelihood estimates for any model can be found by iterating this process with a simple updating of offset parameters attention is also paid to estimation of the asymptotic covariance matrix one benefit of the results of this paper is the ability to obtain maximum likelihood estimates of the parameters of logistic models for stratified case control studies compare breslow cain 1988 scott wild 1991 using an ordinary logistic regression program even when the stratum constants are modelled

a note on efficient estimation from case control data summary it is well known that it is valid to treat case control data as if they had been obtained by random sampling when estimating the odds ratio parameters of logistic regression models the result has been generalised to estimation of the regression parameters in multiplicative intercept models and it has further been pointed out that the approach may be applied to non multiplicative intercept models if the parameter space is first enlarged to include a multiplicative intercept term this note is concerned with the efficiency of the approach in epidemiological case control studies it is common to model the influence of predictors on the risk of disease through a finite dimensional regression parameter while treating the marginal distribution of the predictors as an infinite dimensional nuisance parameter see for example breslow day 1980 it is well known that case control data may be treated as if they had been obtained by random sampling when estimating the regression parameters in a logistic regression model in a multiplicative intercept model the log odds ratio of the risk of disease is modelled as a known function of the regression parameters and the predictors plus a variation independent intercept parameter it is also known that the result in breslow day 1980 holds more generally for estimating other than the intercept parameter in any multiplicative intercept model see for example prentice pyke 1979 hsieh manski mcfadden 1985 anderson 1972 wacholder weinberg 1993 robins rotnitzky zhao 1994 or scott wild 1989 with non multiplicative intercept models the data may be treated as if they had been obtained by random sampling if the model for the risk of disease is first replaced in the likelihood by a larger model that has the multiplicative intercept property the larger model is defined implicitly by treating the logit of the risk of disease as if it were equal to the logit of the risk of disease in the original model plus an unspecified intercept term see for example scott wild 1986 robins blevins 1987 breslow storer 1985 or storer wacholder breslow 1983 the results of cosslett 1981 imply that joint maximisation of the likelihood for the marginal distribution of the predictors and the regression parameters results in efficient estimates of the regression parameters in the asymptotic setting where the numbers of cases and controls tend to infinity together scott wild 1986 note that with randomly sampled data in multiplicative intercept models the case control indicators are ancillary for the regression coefficients thus joint maximisation of the semiparametric likelihood for the case control data and maximisation of the likelihood obtained by treating the data as if they were randomly sampled both result in the same estimate of the regression parameters j robins personal communication has pointed out that the argument may be applied to show efficiency of the approach in non multiplicative intercept models as well because the case control likelihood in the larger model and the case control likelihood in the smaller model are equivalent this note presents a direct calculation that demonstrates the efficiency result both for multiplicative and non multiplicative intercept models

semiparametric maximum likelihood estimation exploiting gene environment independence in case control studies we consider the problem of maximum likelihood estimation in case control studies of gene environment associations with disease when genetic and environmental exposures can be assumed to be independent in the underlying population traditional logistic regression analysis may not be efficient in this setting we study the semiparametric maximum likelihood estimates of logistic regression parameters that exploit the gene environment independence assumption and leave the distribution of the environmental exposures to be nonparametric we use a profile likelihood technique to derive a simple algorithm for obtaining the estimator and we study the asymptotic theory the results are extended to situations where genetic and environmental factors are independent conditional on some other factors simulation studies investigate small sample properties the method is illustrated using data from a case control study designed to investigate the interplay of brca1 2 mutations and oral contraceptive use in the aetiology of ovarian cancer

total positivity and its applications preface i spline functions total positivity of the spline kernel and its applications b bojanov total positivity properties of lb splines t lyche l l schumaker total positivity and splines k morken totally positive systems and interpolation by subspaces of generalized splines m sommer h strauss ii matrix theory preservation of total nonnegativity under the hadamard product and related topics j garloff d g wagner vertex implications for totally nonnegative under the hadamard product and related topics j garloff on factorizations of totally positive matrices j garloff on factorizations of totally positive matrices m gasca j m pena iii geometric modeling total positivity and optimal bases j m carnicer j m pena total positivity and the shape of curves t n t goodman tchebycheff curves m l mazure h pottmann iv probability and mathematical biology analytical and probabilistic methods for totally positive matrices g s goodman some inequalities of total positivity in pure and applied mathematics s karlin z ziegler new directions and problems of multivariate total positivity motivated by genetic recombination probabilities s karlin v approximation theory tchebycheff spaces and total positivity j m carnicer j m pena cebysev and weak cebysev systems r a zalik vi complex analysis zero sets of entire fourier transforms of multiply positive density functions o m katkova vii statistics total nonnegativity of moment matrices and its application to e optimal designs in polynomial regression b heiligers analysis of regression dependence in 2 xn bivariate distributions and some applications in contingency tables s kasala m b rao viii real analysis total positivity and nonlinear analysis r q jia solutions of certain functional equations s karlin alternance and snakes in the nonlinear finite moment problem k tatalian ix combinatorics the applications of total positivity to combinatorics and conversely f brenti x integral equations spectral properties of totally positive kernels and matrices a pinkus index

a conjecture concerning the hadamard product of inverses of m matrices abstract we conjecture that for an n n matrix a which is an inverse of an m matrix the hadamard product a a is also an inverse of an m matrix we have checked this conjecture without failure on many many examples but here we show that for quite a few well known classes of inverses of m matrices the conjecture is true it is known that the more general conjecture that when a and b are n n inverses of m matrices then a b is also an inverse of an m matrix is false however here too we are able to display some classes of inverses of m matrices which are closed under taking hadamard products

testing the order of a model using locally conic parametrization population mixtures and stationary arma processes in this paper we address the problem of testing hypotheses using the likelihood ratio test statistic in nonidentifiable models with application to model selection in situations where the parametrization for the larger model leads to nonidentifiability in the smaller model we give two major applications the case where the number of populations has to be tested in a mixture and the case of stationary arma p q processes where the order p q has to be tested we give the asymptotic distribution for the likelihood ratio test statistic when testing the order of the model in the case of order selection for armas the asymptotic distribution is invariant with respect to the parameters generating the process a locally conic parametrization is a key tool in deriving the limiting distributions it allows one to discover the deep similarity between the two problems

hadamard products of stable polynomials are stable a real polynomial is asymptotically stable when all of its zeros lie in the open left half of the complex plane we show that the hadamard coefficient wise product of two stable polynomials is again stable improving upon some known results via the associated hurwitz matrices we find another example of a class of totally nonnegative matrices which is closed under hadamard multiplication

the multivariate skew normal distribution summary the paper extends earlier work on the so called skew normal distribution a family of distributions including the normal but with an extra parameter to regulate skewness the present work introduces a multivariate parametric family such that the marginal densities are scalar skew normal and studies its properties with special emphasis on the bivariate case

closure properties of certain positivity classes of matrices under various algebraic operations abstract we present a table indicating whether or not each of five positivity classes of matrices positive definite hermitian matrices m matrices inverse m matrices totally positive matrices and inverse totally positive matrices is closed under each of seven algebraic operations conventional multiplication addition powers extraction of roots hadamard multiplication the hadamard product of one element and the inverse of another and lu factorization three of these 35 facts seem not to have been previously known and key examples and references are given

total nonnegativity of gcd matrices and kernels abstract let x x 1 x n be a vector of distinct positive integers the n n matrix s s x gcd x i x j i j 1 n where gcd x i x j denotes the greatest common divisor of x i and x j is called the greatest common divisor gcd matrix on x by a surprising result of beslin and ligh 1989 7 all gcd matrices are positive definite in this paper we completely characterize the gcd matrices satisfying the stronger property of being totally nonnegative tn or totally positive tp as we show a gcd matrix is never tp when n 3 and is tn if and only if it is t n 2 i e all its 2 2 minors are nonnegative we next demonstrate that a gcd matrix is tn 2 if and only if the exponents of each prime divisor in the prime factorization of the x i s form a monotonic sequence reformulated in the language of kernels our results characterize the subsets of integers over which the kernel k x y gcd x y is totally nonnegative the proofs of our characterizations depend on gantmacher and krein s notion of a green s matrix we conclude by showing that a gcd matrix is tn if and only if it is a green s matrix as a consequence we obtain explicit formulas for all the minors and for the inverse of totally nonnegative gcd matrices

on the extension of some total positivity inequalities abstract some inequalities recently obtained for totally positive matrices are extended to more general classes of matrices among these larger classes of matrices sign consistent sign regular and matrices with signed bidiagonal decomposition are considered the class of matrices with a signed bidiagonal decomposition contains the nonsingular totally positive matrices and their inverses

inverse m matrices ii this is an update of the 1981 survey by the first author in the meantime a considerable amount has been learned about the very special structure of the important class of inverse m matrices developments since the earlier survey are emphasized but we have tried to be somewhat complete and some results have not previously been published some proofs are given where appropriate and references are given for others after some elementary preliminaries results are grouped by certain natural categories

nonparametric and semiparametric models for missing covariates in parametric regression robustness of covariate modeling for the missing covariate problem in parametric regression is studied under the missing at random assumption for a simple missing covariate pattern nonparametric covariate model is proposed and is shown to yield a consistent and semiparametrically efficient estimator for the regression parameter total robustness is achieved in this situation for more general missingcovariate patterns a novel semiparametric modeling approach is proposed for the covariates in this approach the covariate distribution is first decomposed into the product of a series of conditional distributions according to the overall missing data patterns and the conditional distributions are then represented in the general odds ratio form the general odds ratios are modeled parametrically and the other components of the covariate distribution are modeled nonparametrically maximum semiparametric likelihood is used to find the parameter estimates the proposed method yields a consistent estimator f

on the solvability of derived matrix problems including completions and duals abstract a class of m by n real complex matrices is semi algebraic sa if membership in it may be checked via a finite list of polynomial inequalities in the entries the real and imaginary parts of the entries real algebraic geometry is used to show that 1 very many familiar classes of matrices some perhaps surprising are sa and 2 that two important derived problems matrix completions and hadamard duals lead to an sa solution when they come from an sa class this means that at least there exists a finite solution to these problems though it may be difficult to find the methodology likely extends to other problems of interest

regression analysis under non standard situations a pairwise pseudolikelihood approach regression analysis is one of the most used statistical methods for data analysis there are however many situations in which one cannot base inference solely on f y x the conditional probability density function for the response variable y given x the covariates examples include missing data where the missingness is non ignorable sampling surveys in which subjects are selected on the basis of the y values and meta analysis where published studies are subject to selection bias the conventional approaches require the correct specification of the missingness mechanism sampling probability and probability for being published respectively in this paper we propose an alternative estimating procedure for based on an idea originated by kalbfleisch the novelty of this method is that no assumption on the missingness probability mechanisms etc mentioned above is required to be specified asymptotic efficiency calculations and simulation studies were conducted to compare the method proposed with the two existing methods the conditional likelihood and the weighted estimating function approaches

hadamard powers and totally positive matrices considered are continuous positive hadamard powers of entry wise positive nonnegative matrices those that are eventually in the sense of all hadamard powers beyond some point totally positive totally nonnegative doubly nonnegative and doubly positive are characterized for example for matrices with at least four rows and four columns hadamard powers greater than one of totally positive matrices need not be totally positive but they are eventually totally positive

on asymptotically optimal tests under loss of identifiability in semiparametric models we consider tests of hypotheses when the parameters are not identifiable under the null in semiparametric models where regularity conditions for profile likelihood theory fail exponential average tests based on integrated profile likelihood are constructed and shown to be asymptotically optimal under a weighted average power criterion with respect to a prior on the nonidentifiable aspect of the model these results extend existing results for parametric models which involve more restrictive assumptions on the form of the alternative than do our results moreover the proposed tests accommodate models with infinite dimensional nuisance parameters which either may not be identifiable or may not be estimable at the usual parametric rate examples include tests of the presence of a change point in the cox model with current status data and tests of regression parameters in odds rate models with right censored data optimal tests have not previously been studied for these scenarios we study the asymptotic distribution of the proposed tests under the null fixed contiguous alternatives and random contiguous alternatives we also propose a weighted bootstrap procedure for computing the critical values of the test statistics the optimal tests perform well in simulation studies where they may exhibit improved power over alternative tests

asymptotics for likelihood ratio tests under loss of identifiability this paper describes the large sample properties of the likelihood ratio test statistic lrts when the parameters characterizing the true null distribution are not unique it is well known that the classical asymptotic theory for the likelihood ratio test does not apply to such problems and the lrts may not have the typical chi squared type limiting distribution this paper establishes a general quadratic approximation of the log likelihood ratio function in a hellinger neighborhood of the true density which is valid with or without loss of identifiability of the true distribution under suitable conditions the asymptotic null distribution of the lrts under loss of identifiability can be obtained by maximizing the quadratic form these results extend the work of chernoff and le cam in particular applications to testing the number of mixture components in finite mixture models are discussed

the hadamard core of the totally nonnegative matrices abstract an m by n matrix a is called totally nonnegative if every minor of a is nonnegative the hadamard product of two matrices is simply their entry wise product this paper introduces the subclass of totally nonnegative matrices whose hadamard product with any totally nonnegative matrix is again totally nonnegative many properties concerning this class are discussed including a complete characterization for min m n a characterization of the zero nonzero patterns for which all totally nonnegative matrices lie in this class and connections to oppenheim s inequality

inferences for case control and semiparametric two sample density ratio models summary we consider inference in general binary response regression models under retrospective sampling plans prentice pyke 1979 discovered that inference for the odds ratio parameter in a logistic model can be based on a prospective likelihood even though the sampling scheme is retrospective we show that the estimating function obtained from the prospective likelihood is optimal in a class of unbiased estimating functions also we link casecontrol sampling with a two sample biased sampling problem where the ratio of two densities is assumed to take a known parametric form connections between this model and the cox proportional hazards model are pointed out large and small sample size behaviour of the proposed estimators is studied

an operator like description of love affairs we adopt the so called occupation number representation originally used in quantum mechanics and recently considered in the description of stock markets in the analysis of the dynamics of love relations we start with a simple model involving two actors alice and bob in the linear case we obtain periodic dynamics whereas in the nonlinear regime either periodic or quasi periodic solutions are found then we extend the model to a love triangle involving alice bob and a third actress carla interesting features appear and in particular we find analytical conditions for the linear model of the love triangle to have periodic or quasi periodic solutions numerical solutions are exhibited in the nonlinear case

the minimal length uncertainty and the quantum model for the stock market abstract we generalize the recently proposed quantum model for the stock market by zhang and huang to make it consistent with the discrete nature of the stock price in this formalism the price of the stock and its trend satisfy the generalized uncertainty relation and the corresponding generalized hamiltonian contains an additional term proportional to the fourth power of the trend we study a driven infinite quantum well where information as the external field periodically fluctuates and show that the presence of the minimal trading value of stocks results in a positive shift in the characteristic frequencies of the quantum system the connection between the information frequency and the transition probabilities is discussed finally

a proportional likelihood ratio model we propose a semiparametric proportional likelihood ratio model which is particularly suitable for modelling a nonlinear monotonic relationship between the outcome variable and a covariate this model extends the generalized linear model by leaving the distribution unspecified and has a strong connection with semiparametric models such as the selection bias model gilbert et al 1999 the density ratio model qin 1998 fokianos kaimi 2006 the single index model ichimura 1993 and the exponential tilt regression model rathouz gao 2009 a maximum likelihood estimator is obtained for the new model and its asymptotic properties are derived an example and simulation study illustrate the use of the model copyright 2012 oxford university press

on the statistical analysis of dirty pictures may 7th 1986 professor a f m smith in the chair summary a continuous two dimensional region is partitioned into a fine rectangular array of sites or pixels each pixel having a particular colour belonging to a prescribed finite set the true colouring of the region is unknown but associated with each pixel there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model the aim is to reconstruct the true scene with the additional knowledge that pixels close together tend to have the same or similar colours in this paper it is assumed that the local characteristics of the true scene can be represented by a nondegenerate markov random field such information can be combined with the records by bayes theorem and the true scene can be estimated according to standard criteria however the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field thus a simple iterative method of reconstruction is proposed which does not depend on these large scale characteristics the method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field some complications including parameter estimation are discussed potential applications are mentioned briefly

total positivity of sums hadamard products and hadamard powers results and counterexamples abstract we show that for hankel matrices total nonnegativity resp total positivity of order r is preserved by sum hadamard product and hadamard power with real exponent t r 2 we give examples to show that our results are sharp relative to matrix size and structure general symmetric or hankel some of these examples also resolve the hadamard critical exponent problem for totally positive and totally nonnegative matrices

a semiparametric odds ratio model for measuring association we propose a semiparametric odds ratio model to measure the association between two variables taking discrete values continuous values or a mixture of both methods for estimation and inference with varying degrees of robustness to model assumptions are studied semiparametric efficient estimation and inference procedures are also considered the estimation methods are compared in a simulation study and applied to the study of associations among genital tract bacterial counts in hiv infected women

bidiagonal factorizations and quasi oscillatory rectangular matrices abstract a real matrix a of size m n is called totally nonnegative totally positive if all its minors are nonnegative positive a variant of the neville elimination process is studied in relation to the existence of a totally nonnegative elementary bidiagonal factorization of a the class of quasi oscillatory rectangular matrices which in the square case contains the oscillatory matrices is introduced and a characterization of this class of matrices by incorporating bidiagonal factorization is showed

maximum likelihood estimation in semiparametric selection bias models with application to aids vaccine trials summary the following problem is treated given s possibly selection biased samples from an unknown distribution function and assuming that the sampling rule weight functions for each of the samples are mathematically specified up to a common unknown finite dimensional parameter how can we use the data to estimate the unknown parameters we propose a simple maximum partial likelihood method for deriving the semiparametric maximum likelihood estimator a discussion of assumptions under which the selection bias model is identifiable and uniquely estimable is presented we motivate the need for the methodology by discussing the generalised logistic regression model gilbert self ashby 1998 a semiparametric selection bias model which is useful for assessing from vaccine trial data how the efficacy of an hiv vaccine varies with characteristics of the exposing virus we show through simulations and an example that the maximum likelihood estimator in the generalised logistic regression model has satisfactory finite sample properties

quantum spatial periodic harmonic model for daily price limited stock markets abstract we investigate the behaviors of stocks in daily price limited stock markets by purposing a quantum spatial periodic harmonic model the stock price is considered to be oscillating and damping in a quantum spatial periodic harmonic oscillator potential well a complicated non linear relation including inter band positive correlation and intra band negative correlation between the volatility and trading volume of a stock is numerically derived with the energy band structure of the model concerned the effectiveness of price limit is re examined with some observed characteristics of price limited stock markets in china studied by applying our quantum model

a quantum model for the stock market abstract beginning with several basic hypotheses of quantum mechanics we give a new quantum model in econophysics in this model we define wave functions and operators of the stock market to establish the schrodinger equation for stock price based on this theoretical framework an example of a driven infinite quantum well is considered in which we use a cosine distribution to simulate the state of stock price in equilibrium after adding an external field into the hamiltonian to analytically calculate the wave function the distribution and the average value of the rate of return are shown

total positivity of hadamard products abstract the hadamard product of two totally positive toeplitz matrices m and n need not be totally positive when only finitely many diagonals of m and of n are non zero preservation of total positivity by hadamard product is essentially a theorem of malo here we establish another sufficient condition for the preservation of total positivity if both m and n are totally positive lower triangular toeplitz matrices such that the value on the nth diagonal is a polynomial function of n then the hadamard product m n is totally positive we use the characterization of the generating functions of polya frequency sequences given by aissen edrei schoenberg and whitney and in the course of the proof we extend the concept of sturm sequences and develop some other results on polynomials with only real roots which are of independent interest

a unified framework for studying parameter identifiability and estimation in biased sampling designs summary based on the odds ratio representation of a joint density we propose a unified framework to study parameter identifiability in biased sampling designs it is shown that most of these designs encountered in practice can be reformulated within the proposed framework and as a result the question of parameter identifiability can be largely clarified estimation of the identifiable parameters is considered and traditional results on the equivalence of the prospective and retrospective likelihoods are extended information contained in data on certain identifiable parameters is often very limited such parameters can be poorly estimated by the likelihood approach with practically attainable sample sizes which can substantially affect the estimates of parameters of primary interest a partially penalized likelihood approach is proposed to address this simulation results suggest that the proposed approach has good performance

a note on the prospective analysis of outcome dependent samples summary two likelihood representations corresponding to the prospective and retrospective analyses of the case control design are derived for general outcome dependent samples with arbitrary discrete or continuous outcomes and possibly non multiplicative models parameter identification in the general outcome dependent design is reduced to the simple problem of parameter identification in the general odds ratio function both likelihoods are shown to generate the same profile likelihood for the common parameter of interest maximum like lihood estimators based on either likelihood are semiparametric efficient for the identifiable parameters

asymptotic inference for semiparametric association models association models for a pair of random elements x and y e g vectors are considered which specify the odds ratio function up to an unknown parameter 0 these models are shown to be semiparametric in the sense that they do not restrict the marginal distributions of x and y inference for the odds ratio parameter may be obtained from sampling either y conditionally on x or vice versa generalizing results from prentice and pyke weinberg and wacholder and scott and wild we show that asymptotic inference for 0 under sampling conditional on y is the same as if sampling had been conditional on x common regression models for example generalized linear models with canonical link or multivariate linear respectively logistic models are association models where the regression parameter is closely related to the odds ratio parameter 0 hence inference for may be drawn from samples conditional on y using an association model

a quantum statistical approach to simplified stock markets we use standard perturbation techniques originally formulated in quantum statistical mechanics in the analysis of a toy model of a stock market which is given in terms of bosonic operators in particular we discuss the probability of transition from a given value of the portfolio of a certain trader to a different one this computation can also be carried out using some kind of feynman graphs adapted to the present context

quantum brownian motion model for the stock market abstract it is believed by the majority today that the efficient market hypothesis is imperfect because of market irrationality using the physical concepts and mathematical structures of quantum mechanics we construct an econophysical framework for the stock market based on which we analogously map massive numbers of single stocks into a reservoir consisting of many quantum harmonic oscillators and their stock index into a typical quantum open system a quantum brownian particle in particular the irrationality of stock transactions is quantitatively considered as the planck constant within heisenberg s uncertainty relationship of quantum mechanics in an analogous manner we analyze real stock data of shanghai stock exchange of china and investigate fat tail phenomena and non markovian behaviors of the stock index with the assistance of the quantum brownian motion model thereby interpreting and studying the limitations of the classical brownian motion model for the efficient market hypothesis from a new perspective of quantum open system dynamics

stock markets and quantum dynamics a second quantized description abstract in this paper we continue our description of stock markets in terms of some non abelian operators which are used to describe the portfolio of the various traders and other observable quantities after a first prototype model with only two traders we discuss a more realistic model of market involving an arbitrary number of traders for both models we find approximated solutions for the time evolution of the portfolio of each trader in particular for the more realistic model we use the stochastic limit approach and a fixed point like approximation

likelihood inferences on semiparametric odds ratio model a flexible semiparametric odds ratio model has been proposed to unify and to extend both the log linear model and the joint normal model for data with a mix of discrete and continuous variables the semiparametric odds ratio model is particularly useful for analyzing biased sampling designs however statistical inference of the model has not been systematically studied when more than one nonparametric component is involved in the model in this article we study the maximum semiparametric likelihood approach to estimation and inference of the semiparametric odds ratio model we show that the maximum semiparametric likelihood estimator of the odds ratio parameter is consistent and asymptotically normally distributed we also establish statistical inference under a misspecified semiparametric odds ratio model which is important when handling weak identifiability in conditionally specified models under biased sampling designs we use simulation studies to demonstrate that the proposed approaches have satis