ma estimation in polynomial time the parameter estimation of moving average ma signals from second order statistics was deemed for a long time to be a difficult nonlinear problem for which no computationally convenient and reliable solution was possible we show how the problem of ma parameter estimation from sample covariances can be formulated as a semidefinite program that can be solved in a time that is a polynomial function of the ma order two methods are proposed that rely on two specific over parametrizations of the ma covariance sequence whose use makes the minimization of a covariance fitting criterion a convex problem the ma estimation algorithms proposed here are computationally fast statistically accurate and reliable none of the previously available algorithms for ma estimation methods based on higher order statistics included shares all these desirable properties our methods can also be used to obtain the optimal least squares approximant of an invalid estimated ma spectrum that takes on negative values at some frequencies which was another long standing problem in the signal processing literature awaiting a satisfactory solution

strong consistency of maximum quasi likelihood estimators in generalized linear models with fixed and adaptive designs strong consistency for maximum quasi likelihood estimators of regression parameters in generalized linear regression models is studied results parallel to the elegant work of lai robbins and wei and lai and wei on least squares estimation under both fixed and adaptive designs are obtained let y 1 dots y n and x 1 dots x n be the observed responses and their corresponding design points p times 1 vectors respectively for fixed designs it is shown that if the minimum eigenvalue of sigma x i x prime i goes to infinity then the maximum quasi likelihood estimator for the regression parameter vector is strongly consistent for adaptive designs it is shown that a sufficient condition for strong consistency to hold is that the ratio of the minimum eigenvalue of sigma x i x prime i to the logarithm of the maximum eigenvalues goes to infinity use of the results for the adaptive design case in quantal response experiments is also discussed

computing stochastic continuous time models from arma models some algorithms for computing the underlying continuous time stochastic model from a sampled arma model are presented three algorithms are given all having a modest computational complexity the properties of the algorithms are analysed and also illustrated by means of numerical examples

penalized generalized estimating equations for high dimensional longitudinal data analysis we consider the penalized generalized estimating equations gees for analyzing longitudinal data with high dimensional covariates which often arise in microarray experiments and large scale health studies existing high dimensional regression procedures often assume independent data and rely on the likelihood function construction of a feasible joint likelihood function for high dimensional longitudinal data is challenging particularly for correlated discrete outcome data the penalized gee procedure only requires specifying the first two marginal moments and a working correlation structure we establish the asymptotic theory in a high dimensional framework where the number of covariates p n increases as the number of clusters n increases and p n can reach the same order as n one important feature of the new procedure is that the consistency of model selection holds even if the working correlation structure is misspecified we evaluate the performance of the proposed method using monte carlo simulations and demonstrate its application using a yeast cell cycle gene expression data set

the effects of rapid sampling in system identification this paper deals with the effects of rapid sampling in system identification in continuous time noise models of non zero relative degree imply that one has to differentiate the measured data to find the prediction error parameter estimate this problem can be avoided by using noise models of relative degree zero or by introducing prefilters the corresponding difficulty for the discrete time case is less obvious however by linking the discrete time and continuous time parameter estimation problem we show that the same problem arises for rapid sampled systems

asymptotic theory in generalized ilinear models with nuisance scale parameters this paper establishes the asymptotic normality and the consistencyrobustness of the weighted least squares estimator wlse in the generalized linear models with multiple nuisance scale parameters in addition noting that the asymptotic robust statistical inference in presence of nuisance scale parameters requires a consistency robust estimator of the asymptotic covariance matrix of the wlse this paper derives a class of covariance estimators and proves their consistency robustness

dependent generalized functional linear models this paper considers testing for no effect of functional covariates on response variables in multivariate regression we use generalized estimating equations to determine the underlying parameters and establish their joint asymptotic normality this is then used to test the significance of the effect of predictors on the vector of response variables simulations demonstrate the importance of considering existing correlation structures in the data to explore the effect of treating genetic data as a function we perform a simulation study using gene sequencing data and find that the performance of our test is comparable to that of another popular method used in sequencing studies we present simulations to explore the behaviour of our test under varying sample size cluster size and dimension of the parameter to be estimated and an application where we are able to confirm known associations between nicotine dependence and neuronal nicotinic acetylcholine receptor subunit genes

asymptotics for generalized estimating equations with large cluster sizes generalized estimating equations are used in regression analysis of longitudinal data where observations on each subject are correlated statistical analysis using such methods is based on the asymptotic properties of regression parameter estimators this paper presents asymptotic results when either the number of independent subjects or the cluster sizes the number of observations on each subject or both go to infinity a set of information matrix based general conditions is developed which leads to the weak and strong consistency as well as the asymptotic normality of the estimators most of the results are parallel to the elegant work of fahrmeir and kaufmann on maximum likelihood estimators related to the generalized linear models the conditions for weak consistency and asymptotic normality are verified for several examples of general interest

gee analysis of clustered binary data with diverging number of covariates clustered binary data with a large number of covariates have become increasingly common in many scientific disciplines this paper develops an asymptotic theory for generalized estimating equations gee analysis of clustered binary data when the number of covariates grows to infinity with the number of clusters in this large n diverging p framework we provide appropriate regularity conditions and establish the existence consistency and asymptotic normality of the gee estimator furthermore we prove that the sandwich variance formula remains valid even when the working correlation matrix is misspecified the use of the sandwich variance formula leads to an asymptotically valid confidence interval and wald test for an estimable linear combination of the unknown parameters the accuracy of the asymptotic approximation is examined via numerical simulations we also discuss the diverging p asymptotic theory for general gee the results in this paper extend the recent elegant work of xie and yang ann statist 31 2003 310 347 and balan and schiopu kratina ann statist 32 2005 522 541 in the fixed p setting

modeling continuous time processes via input to state filters a direct algorithm to estimate continuous time arma carma models is proposed in this paper in this approach we first pass the observed data through an input to state filter and compute the state covariance matrix the properties of the state covariance matrix are then exploited to estimate the half spectrum of the observed data at a set of user defined points on the right half plane finally the continuous time parameters are obtained from the half spectrum estimates by solving an analytic interpolation problem with a positive real constraint as shown by simulations the proposed algorithm delivers much more reliable estimates than indirect modeling approaches which rely on estimating an intermediate discrete time model

brief performance evaluation of methods for identifying continuous time autoregressive processes identification of continuous time autoregressive processes from discrete time data by replacing the differentiation operator by an approximation is considered a linear regression model can then be formulated the least squares method and the instrumental variables method must be used with some care to get parameter estimates of good quality the bias is studied explicitly in the paper together with the asymptotic distribution and expressions are presented for the covariance matrix of the estimated parameters it turns out that there are small differences in the dominating bias term for the different methods whereas the statistical properties are comparable overall the performance is similar to that of a prediction error method for short sampling intervals

a bias correction for covariance estimators to improve inference with generalized estimating equations that use an unstructured correlation matrix generalized estimating equations gees are routinely used for the marginal analysis of correlated data the efficiency of gee depends on how closely the working covariance structure resembles the true structure and therefore accurate modeling of the working correlation of the data is important a popular approach is the use of an unstructured working correlation matrix as it is not as restrictive as simpler structures such as exchangeable and ar 1 and thus can theoretically improve efficiency however because of the potential for having to estimate a large number of correlation parameters variances of regression parameter estimates can be larger than theoretically expected when utilizing the unstructured working correlation matrix therefore standard error estimates can be negatively biased to account for this additional finite sample variability we derive a bias correction that can be applied to typical estimators of the covariance matrix of parameter estimates via simulation and in application to a longitudinal study we show that our proposed correction improves standard error estimation and statistical inference

brief identification of continuous time ar processes from unevenly sampled data when identifying a continuous time ar process from discrete time data an obvious approach is to replace the derivative operator in the continuous time model by an approximation in some cases a linear regression model can then be formulated the well known least squares method would be very desirable to apply since it enjoy good numerical properties and low computational complexity in particular for fast or nonuniform sampling the focus of this paper is the latter i e nonuniform sampling two consistent least squares schemes for the case of unevenly sampled data are presented the precise choice of derivative approximation turns out to be crucial the obtained results are compared to a prediction error method

a generalized entropy criterion for nevanlinna pick interpolation with degree constraint we present a generalized entropy criterion for solving the rational nevanlinna pick problem for n 1 interpolating conditions and the degree of interpolants bounded by n the primal problem of maximizing this entropy gain has a very well behaved dual problem this dual is a convex optimization problem in a finite dimensional space and gives rise to an algorithm for finding all interpolants which are positive real and rational of degree at most n the criterion requires a selection of a monic schur polynomial of degree n it follows that this class of monic polynomials completely parameterizes all such rational interpolants and it therefore provides a set of design parameters for specifying such interpolants the algorithm is implemented in a state space form and applied to several illustrative problems in systems and control namely sensitivity minimization maximal power transfer and spectral estimation

spectral estimation via selective harmonic amplification the state covariance of a linear filter is characterized by a certain algebraic commutativity property with the state matrix of the filter and also imposes a generalized interpolation constraint on the power spectrum of the input process this algebraic property and the relationship between state covariance and the power spectrum of the input allow the use of matrix pencils and analytic interpolation theory for spectral analysis several algorithms for spectral estimation are developed with resolution higher than state of the art

consistency and asymptotic normality of the maximum likelihood estimator in generalized linear models generalized linear models are used for regression analysis in a number of cases including categorical responses where the classical assumptions are violated the statistical analysis of such models is based on the asymptotic properties of the maximum likelihood estimator we present mild general conditions which respectively assure weak or strong consistency or asymptotic normality most of the previous work has been concerned with natural link functions in this case our normality condition though obtained by a different approach is closely related to a condition of haberman 1977a examples show how the general conditions reduce to weak requirements for special exponential families further for regressors with a compact range sufficient conditions are given which do not involve the unknown parameter and are therefore easy to check in practice responses with a bounded range e g categorical responses and stochastic regressors also are treated

a singular boundary value problem for uniformly elliptic operators abstract strong solvability in the sobolev space w 2 p is proved for the oblique derivative problem j i 1 n a ij x d ij u f x almost every where in u l x u x in the trace sense on in the case when the vector field l x has a contact of infinite order with at the points of some non empty subset e

the interpolation problem with a degree constraint the author previously 1983 1987 showed that there is a correspondence between nonnegative hermitian trigonometric polynomials of degree spl les n and solutions to the standard nevanlinna pick caratheodory interpolation problem with n 1 constraints which are rational and also of degree spl les n it was conjectured that the correspondence under suitable normalization is bijective and thereby that it results in a complete parametrization of rational solutions of degree spl les n the conjecture was proven by byrnes et al 1995 along with a detailed study of this parametrization however byrnes et al used a slightly restrictive assumption that the trigonometric polynomials are positive and accordingly the corresponding solutions have positive real part the purpose of the present note is to extend the result to the case of nonnegative trigonometric polynomials as well we present the arguments in the context of the general nevanlinna pick caratheodory fejer interpolation

consistent model selection and data driven smooth tests for longitudinal data in the estimating equations approach summary model selection for marginal regression analysis of longitudinal data is challenging owing to the presence of correlation and the difficulty of specifying the full likelihood particularly for correlated categorical data the paper introduces a novel bayesian information criterion type model selection procedure based on the quadratic inference function which does not require the full likelihood or quasi likelihood with probability approaching 1 the criterion selects the most parsimonious correct model although a working correlation matrix is assumed there is no need to estimate the nuisance parameters in the working correlation matrix moreover the model selection procedure is robust against the misspecification of the working correlation matrix the criterion proposed can also be used to construct a data driven neyman smooth test for checking the goodness of fit of a postulated model this test is especially useful and often yields much higher power in situations where the classical directional test behaves poorly the finite sample performance of the model selection and model checking procedures is demonstrated through monte carlo studies and analysis of a clinical trial data set

on the expectation of the product of four matrix valued gaussian random variables the formula for the expectation of the product of four scalar real gaussian random variables is generalized to matrix valued real or complex gaussian random variables as an application of the extended formula a simple derivation is presented of the covariance matrix of instrumental variable estimates of parameters in multivariable regression models

estimation of continuous time ar process parameters from discrete time data the problem of estimating continuous time autoregressive process parameters from discrete time data is considered the basic approach used here is based on replacing the derivatives in the model by discrete time differences forming a linear regression and using the least squares method such a procedure is simple to apply computationally flexible and efficient and may have good numerical properties it is known however that all standard approximations of the highest order derivative such as repeated use of the delta operator gives a biased least squares estimate even as the sampling interval tends to zero some of our previous approaches to overcome this problem are reviewed then two new methods which avoid the shift in our previous results are presented one of them which is termed bias compensation is computationally very efficient finally the relationship of the above least squares approaches with an instrumental variable method is investigated comparative simulation results are also presented

asymptotic results with generalized estimating equations for longitudinal data we consider the marginal models of liang and zeger biometrika 73 1986 13 22 for the analysis of longitudinal data and we develop a theory of statistical inference for such models we prove the existence weak consistency and asymptotic normality of a sequence of estimators defined as roots of pseudo likelihood equations

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures