higher order properties of the wild bootstrap under misspecification abstract we examine the higher order properties of the wild bootstrap wu 1986 in a linear regression model with stochastic regressors we find that the ability of the wild bootstrap to provide a higher order refinement is contingent upon whether the errors are mean independent of the regressors or merely uncorrelated with them in the latter case the wild bootstrap may fail to match some of the terms in an edgeworth expansion of the full sample test statistic nonetheless we show that the wild bootstrap still has a lower maximal asymptotic risk as an estimator of the true distribution than a normal approximation in shrinking neighborhoods of properly specified models to assess the practical implications of this result we conduct a monte carlo study contrasting the performance of the wild bootstrap with a normal approximation and the traditional nonparametric bootstrap

efficient rank reduction of correlation matrices geometric optimisation algorithms are developed that efficiently find the nearest low rank correlation matrix we show in numerical tests that our methods compare favourably with the existing methods in the literature the connection with the lagrange multiplier method is established along with an identification of whether a local minimum is a global minimum an additional benefit of the geometric approach is that any weighted norm can be applied the problem of finding the nearest low rank correlation matrix occurs as part of the calibration of multi factor interest rate market models to correlation

large scale estimation of variance and covariance components this paper concerns matrix computations within algorithms for variance and covariance component estimation hemmerle and hartley technometrics 15 1973 pp 819 831 showed how to compute the objective function and its derivatives for maximum likelihood estimation of variance components using matrices with dimensions of the order of the number of coefficients rather than that of the number of observations their approach was extended by corbeil and searle technometrics 18 1976 pp 31 38 for restricted maximum likelihood estimation a similar reduction in dimension is possible using expectation maximization em algorithms in most cases variance components are assumed to be strictly positive we advocate the use of a modification that is numerically stable even if variance component estimates are small in magnitude for problems in which the number of coefficients is large fellner proc statistical computing section american statistical association 1984 pp 150 154 comm statist simulat

bootstrapping regression models the regression and correlation models are considered it is shown that the bootstrap approximation to the distribution of the least squares estimates is valid and some error bounds are given

estimation in covariance components models abstract estimation techniques for linear covariance components models are developed and illustrated with special emphasis on explaining computational processes the estimation of fixed and random effects when the variances and covariances are known is presented in bayesian terms point estimates of the unknown variances and covariances are computed using the em algorithm for maximum likelihood estimation from incomplete data the techniques are illustrated with data on law schools field mice and professional football teams

computing the nearest low rank correlation matrix by a simplified sqp algorithm in this paper we propose a numerical method for computing the nearest low rank correlation matrix lrcm motivated by the fact that the nearest lrcm problem can be reformulated as a standard nonlinear equality constrained optimization problem with matrix variables via the gramian representation we propose a new algorithm based on the sequential quadratic programming sqp method on each iteration we do not solve the quadratic program qp corresponding to the exact hessian but a modified qp with a simpler hessian this qp subproblem can be solved efficiently by equivalently transforming it to a sparse linear system global convergence is established and preliminary numerical results are presented to demonstrate the proposed method is potentially useful

nonnegative matrix factorization of a correlation matrix we present a dedicated algorithm for the nonnegative factorization of a correlation matrix from an application in financial engineering we look for a low rank approximation the origin of the problem is discussed in some detail next to the description of the algorithm we prove by means of a counter example that an exact nonnegative decomposition of a general positive semidefinite matrix is not always available

some heteroskedasticity consistent covariance matrix estimators with improved finite sample properties abstract we examine several modified versions of the heteroskedasticity consistent covariance matrix estimator of hinkley 1977 and white 1980 on the basis of sampling experiments which compare the performance of quasi t statistics we find that one estimator based on the jackknife performs better in small samples than the rest we also examine the finite sample properties of using modified critical values based on edgeworth approximations as proposed by rothenberg 1984 in addition we compare the power of several tests for heteroskedasticity and find that it may be wise to employ the jackknife heteroskedasticity consistent covariance matrix even in the absence of detected heteroskedasticity

optimal low rank approximation to a correlation matrix abstract low rank approximation of a correlation matrix is a constrained minimization problem that can be translated into a minimization maximization problem by the method of lagrange multiplier in this paper we solve the inner maximization problems with a single spectral decomposition and the outer minimization problems with gradient based descending methods an in depth analysis is done to characterize the solutions of the inner maximization problem for the case when they are non unique the well posedness of the lagrange multiplier problem and the convergence of the descending methods are rigorously justified numerical results are presented

a majorization algorithm for constrained correlation matrix approximation abstract we desire to find a correlation matrix r of a given rank that is as close as possible to an input matrix r subject to the constraint that specified elements in r must be zero our optimality criterion is the weighted frobenius norm of the approximation error and we use a constrained majorization algorithm to solve the problem although many correlation matrix approximation approaches have been proposed this specific problem with the rank specification and the r ij 0 constraints has not been studied until now we discuss solution feasibility convergence and computational effort we also present several examples

improved heteroscedasticity consistent covariance matrix estimators summary the heteroscedasticity consistent covariance matrix estimator proposed by white 1980 is commonly used in practical applications and is implemented into a number of pieces of statistical software however although consistent it can display substantial bias in small to moderately large samples as shown by monte carlo simulations elsewhere this paper defines modified white estimators which are approximately bias free numerical results show that the modified estimators display much smaller bias than white s estimator in small samples we also show that the bias correction leads to some variance inflation in hypothesis testing based on heteroscedasticity consistent covariance matrix estimators numerical results suggest that tests based on the proposed bias corrected estimators typically display smaller size distortions

unbalanced repeated measures models with structured covariance matrices the question of how to analyze unbalanced or incomplete repeated measures data is a common problem facing analysts we address this problem through maximum likelihood analysis using a general linear model for expected responses and arbitrary structural models for the within subject covariances models that can be fit include standard univariate and multivariate models with incomplete data random effects models and models with time series and factor analytic error structures we describe newton raphson and fisher scoring algorithms for computing maximum likelihood estimates and generalized em algorithms for computing restricted and unrestricted maximum likelihood estimates an example fitting several models to a set of growth data is included

on the generalized low rank approximation of the correlation matrices arising in the asset portfolio abstract in this paper we consider the generalized low rank approximation of the correlation matrices problem which arises in the asset portfolio we first characterize the feasible set by using the gramian representation together with a special trigonometric function transform and then transform the generalized low rank approximation of the correlation matrices problem into an unconstrained optimization problem finally we use the conjugate gradient algorithm with the strong wolfe line search to solve the unconstrained optimization problem numerical examples show that our new method is feasible and effective

heteroscedasticity robustness of jackknife variance estimators in linear models on etudie l absence de biais et la consistance asymptotiques de trois types d estimateurs de variance du jackknife en presence d heteroscedasticite de la variance de l erreur dans les modeles lineaires

minimum rank skew hermitian solutions to the matrix approximation problem in the spectral norm in this paper we discuss the following two minimum rank matrix approximation problems in the spectral norm i for given a a h c m i m b c m i n determining x s 1 such that rank x min y s 1 rank y s 1 y y h c n i n a b y b h 2 min ii for given a a h c m i m b c m i n determining x s 2 such that rank x min y s 2 rank y s 2 y y h c n i n a b y b h 2 min by applying the norm preserving dilation theorem the hermitian type skew hermitian type generalized singular value decomposition hgsvd shgsvd we characterize the expressions of the minimum rank and derive a general form of minimum rank skew hermitian solutions to the matrix approximation problem

low rank approximation of the symmetric positive semidefinite matrix in this paper we consider the low rank approximation of the symmetric positive semidefinite matrix which arises in machine learning quantum chemistry and inverse problem we first characterize the feasible set by x yy t email protected r n x k and then transform low rank approximation into an unconstrained optimization problem finally we use the nonlinear conjugate gradient method with exact line search to compute the optimal low rank symmetric positive semidefinite approximation of the given matrix numerical examples show that the new method is feasible and effective

inference in linear regression models with many covariates and heteroskedasticity the linear regression model is widely used in empirical work in economics statistics and many other disciplines researchers often include many covariates in their linear model specification in a

a sequential semismooth newton method for the nearest low rank correlation matrix problem based on the well known result that the sum of the largest eigenvalues of a symmetric matrix can be represented as a semidefinite programming problem sdp we formulate the nearest low rank correlation matrix problem as a nonconvex sdp and propose a numerical method that solves a sequence of least square problems each of the least square problems can be solved by a specifically designed semismooth newton method which is shown to be quadratically convergent the sequential method is guaranteed to produce a stationary point of the nonconvex sdp our numerical results demonstrate the high efficiency of the proposed method on large scale problems

inference for misspecified models with fixed regressors following the work by eicker huber and white it is common in empirical work to report standard errors that are robust against general misspecification in a regression setting these standard errors are valid for the parameter that minimizes the squared difference between the conditional expectation and a linear approximation averaged over the population distribution of the covariates here we discuss an alternative parameter that corresponds to the approximation to the conditional expectation based on minimization of the squared difference averaged over the sample rather than the population distribution of the covariates we argue that in some cases this may be a more interesting parameter we derive the asymptotic variance for this parameter which is generally smaller than the eicker huber white robust variance and propose a consistent estimator for this asymptotic variance supplementary materials for this article are available online