nonlinear three point third order boundary value problems abstract in this paper we consider three point boundary value problems for third order differential equations we provide simple sufficient conditions in order to obtain a priori bounds on solutions we then rely on the topological transversality theorem of granas or on schaefer s theorem to prove existence of solutions

third order differential equations with integral boundary conditions abstract in this paper we study nonlinear third order differential equations with integral boundary conditions we provide sufficient conditions on the nonlinearity and the functions appearing in the boundary conditions that guarantee the existence of at least one solution to our problem our technique is based on a priori bounds and fixed point theorems

forecasting using principal components from a large number of predictors this article considers forecasting a single time series when there are many predictors n and time series observations t when the data follow an approximate factor model the predictors can be summarized by a small number of indexes which we estimate using principal components feasible forecasts are shown to be asymptotically efficient in the sense that the difference between the feasible forecasts and the infeasible forecasts constructed using the actual values of the factors converges in probability to 0 as both n and t grow large the estimated factors are shown to be consistent even in the presence of time variation in the factor model

nonparametric estimate of spectral density functions of sample covariance matrices a first step the density function of the limiting spectral distribution of general sample covariance matrices is usually unknown we propose to use kernel estimators which are proved to be consistent a simulation study is also conducted to show the performance of the estimators

existence uniqueness and iterative methods for third order boundary value problems abstract in this paper we shall provide necessary and sufficient conditions for the existence and uniqueness of solutions of third order nonlinear differential equations satisfying three point boundary conditions for the linear case we propose a constructive method which is a variation of the method of chasing for the nonlinear problems sufficient conditions are provided to ensure the convergence of a general class of iterative methods several examples are also included

high dimensional covariance matrix estimation using a factor model abstract high dimensionality comparable to sample size is common in many statistical problems we examine covariance matrix estimation in the asymptotic framework that the dimensionality p tends to as the sample size n increases motivated by the arbitrage pricing theory in finance a multi factor model is employed to reduce dimensionality and to estimate the covariance matrix the factors are observable and the number of factors k is allowed to grow with p we investigate the impact of p and k on the performance of the model based covariance matrix estimator under mild assumptions we have established convergence rates and asymptotic normality of the model based estimator its performance is compared with that of the sample covariance matrix we identify situations under which the factor approach increases performance substantially or marginally the impacts of covariance matrix estimation on optimal portfolio allocation and portfolio risk assessment are studied the asymptotic results are supported by a thorough simulation study

broken solutions of homogeneous variational problems a real valued function l on the tangent bundle of rn gives rise to variational problems as follows for two points x0 x1 in rn and a time interval 0 t to determine a curve o t rn connecting x0 with x1 which minimizes 0tl t gg t dt we consider the associated hamiltonian vectorfield on the cotangent bundle if l is not convex on each fibre then the corresponding hamiltonian vectorfield is not continuous for homogeneous l and n 2 restriction to an energy level gives an essentially three dimensional vectorfield in this case we list the possible discontinuities for generic l then we observe that there exits an open class of such variational problems which admit no minimizing solution

existence result for impulsive third order periodic boundary value problems this paper is devoted to the study of periodic boundary value problems for nonlinear third order differential equations subjected to impulsive effects we provide sufficient conditions on the nonlinearity and the impulse functions that guarantee the existence of at least one periodic solution our approach is based on a priori estimates the method of upper and lower solutions combined with an iterative technique which need not be monotone

on three point boundary value problems associated with third order differential equations introduction in this paper we shall be concerned with the existence and uniqueness of solutions to three point boundary value problems associated with the differential equation y f x y y s y 1 1 where f x y z w is assumed to be continuous on a subset of r4 initial value problems associated with 1 1 exist and are unique in the interval x1 x a monotonicity restriction on f x y z w ensures that the following boundary value problems y j x y y y y xj y y xj y y x m i 1 2 12i y j x y y y y x y y x m2 2 2 9 y x y i 421 le3i

theoretical and empirical estimates of mean variance portfolio sensitivity abstract this paper studies properties of an estimator of mean variance portfolio weights in a market model with multiple risky assets and a riskless asset theoretical formulas for the mean square error are derived in the case when asset excess returns are multivariate normally distributed and serially independent the sensitivity of the portfolio estimator to errors arising from the estimation of the covariance matrix and the mean vector is quantified it turns out that the relative contribution of the covariance matrix error depends mainly on the sharpe ratio of the market portfolio and the sampling frequency of historical data theoretical studies are complemented by an investigation of the distribution of portfolio estimator for empirical datasets an appropriately crafted bootstrapping method is employed to compute the empirical mean square error empirical and theoretical estimates are in good agreement with the empirical values being in general higher

determining the number of factors in approximate factor models in this paper we develop some econometric theory for factor models of large dimensions the focus is the determination of the number of factors r which is an unresolved issue in the rapidly growing literature on multifactor models we first establish the convergence rate for the factor estimates that will allow for consistent estimation of r we then propose some panel criteria and show that the number of factors can be consistently estimated using the criteria the theory is developed under the framework of large cross sections n and large time dimensions t no restriction is imposed on the relation between n and t simulations show that the proposed criteria have good finite sample properties in many configurations of the panel data encountered in practice

eigenvectors of some large sample covariance matrix ensembles we consider sample covariance matrices s n frac 1 p sigma n 1 2 x nx n sigma n 1 2 where xn is a n p real or complex matrix with i i d entries with finite 12th moment and n is a n n positive definite matrix in addition we assume that the spectral measure of n almost surely converges to some limiting probability distribution as n and p n 0 we quantify the relationship between sample and population eigenvectors by studying the asymptotics of functionals of the type frac 1 n text tr g sigma n s n zi 1 where i is the identity matrix g is a bounded function and z is a complex number this is then used to compute the asymptotically optimal bias correction for sample eigenvalues paving the way for a new generation of improved estimators of the covariance matrix and its inverse

boundaries of the risk aversion coefficient should we invest in the global minimum variance portfolio due to estimation risk the portfolios on the efficient frontier can be statistically indistinguishable from the global minimum variance portfolio we provide a methodology for determining a bound on the risk aversion coefficient which separates portfolios that are equivalent or significantly different from the global minimum variance gmv portfolio we conclude that investing in the gmv portfolio is statistically justified for investors with a very wide range of the risk aversion coefficients

existence of triple positive solutions for a third order three point boundary value problem in this paper we investigate the existence of triple positive solutions for the nonlinear third order three point boundary value problem u t a t f t u t u t u t 0 0 q 0 1 0 are continuous first green s function for the associated linear boundary value problem is constructed and then by using a fixed point theorem due to avery and peterson we establish results on the existence of triple positive solutions to the boundary value problem

the distribution of the sample minimum variance frontier in this paper we present a finite sample analysis of the sample minimum variance frontier under the assumption that the returns are independent and multivariate normally distributed we show that the sample minimum variance frontier is a highly biased estimator of the population frontier and we propose an improved estimator of the population frontier in addition we provide the exact distribution of the out of sample mean and variance of sample minimum variance portfolios this allows us to understand the impact of estimation error on the performance of in sample optimal portfolios

semiparametric dynamic portfolio choice with multiple conditioning variables abstract dynamic portfolio choice has been a central and essential objective for investors in active asset management in this paper we study the dynamic portfolio choice with multiple conditioning variables where the dimension of the conditioning variables can be either fixed or diverging to infinity at certain polynomial rate of the sample size we propose a novel data driven method to estimate the optimal portfolio choice motivated by the model averaging marginal regression approach suggested by li et al 2015 more specifically in order to avoid the curse of dimensionality associated with the multivariate nonparametric regression problem and to make it practically implementable we first estimate the marginal optimal portfolio choice by maximizing the conditional utility function for each univariate conditioning variable and then construct the joint dynamic optimal portfolio through the weighted average of the marginal optimal portfolio across all the conditioning variables under some regularity conditions we establish the large sample properties for the developed portfolio choice procedure both the simulation study and empirical application well demonstrate the finite sample performance of the proposed methodology

existence of solutions for a class of third order nonlinear boundary value problems abstract in this paper we are concerned with the following third order ordinary differential equation x t f t x t x t x t 0 0 with the nonlinear boundary conditions x 0 0 g x 0 x 0 a h x 1 x 1 b where a b r f 0 1 r 3 r is continuous g h r 2 r are continuous the existence result is given by using a priori estimate nagumo condition upper and lower solutions and leray schauder degree and we give an example to demonstrate our result

nonlinear shrinkage estimation of large dimensional covariance matrices many statistical applications require an estimate of a covariance matrix and or its inverse when the matrix dimension is large compared to the sample size which happens frequently the sample covariance matrix is known to perform poorly and may suffer from ill conditioning there already exists an extensive literature concerning improved estimators in such situations in the absence of further knowledge about the structure of the true covariance matrix the most successful approach so far arguably has been shrinkage estimation shrinking the sample covariance matrix to a multiple of the identity by taking a weighted average of the two turns out to be equivalent to linearly shrinking the sample eigenvalues to their grand mean while retaining the sample eigenvectors our paper extends this approach by considering nonlinear transformations of the sample eigenvalues we show how to construct an estimator that is asymptotically equivalent to an oracle estimator suggested in previous work as demonstrated in extensive monte carlo simulations the resulting bona fide estimator can result in sizeable improvements over the sample covariance matrix and also over linear shrinkage

adaptive thresholding for sparse covariance matrix estimation in this article we consider estimation of sparse covariance matrices and propose a thresholding procedure that is adaptive to the variability of individual entries the estimators are fully data driven and demonstrate excellent performance both theoretically and numerically it is shown that the estimators adaptively achieve the optimal rate of convergence over a large class of sparse covariance matrices under the spectral norm in contrast the commonly used universal thresholding estimators are shown to be suboptimal over the same parameter spaces support recovery is discussed as well the adaptive thresholding estimators are easy to implement the numerical performance of the estimators is studied using both simulated and real data simulation results demonstrate that the adaptive thresholding estimators uniformly outperform the universal thresholding estimators the method is also illustrated in an analysis on a dataset from a small round blue cell tumor microarray experiment a supplement to this ar

singularities of the hamiltonian vectorfield in nonautonomous variational problems abstract variational problems with n degrees of freedom give rise by pontriaguine maximum principle to a hamiltonian vectorfield in t r n that presents singularities nonsmoothness points when the lagrangian is not convex for one degree of freedom nonautonomous problems of the calculus of variations where the hamiltonian vectorfield in t r depends explicitly on the time we consider the associated autonomous vectorfield in t r r and classify its singularities up to an equivalence that takes into account the special role played by the time coordinate i e that respects the foliation of t r r into planes of constant time

mean variance approximations to expected utility abstract it is often asserted that the application of mean variance analysis assumes normal gaussian return distributions or quadratic utility functions this common mistake confuses sufficient versus necessary conditions for the applicability of modern portfolio theory if one believes as does the author that choice should be guided by the expected utility maxim then the necessary and sufficient condition for the practical use of mean variance analysis is that a careful choice from a mean variance efficient frontier will approximately maximize expected utility for a wide variety of concave risk averse utility functions this paper reviews a half century of research on mean variance approximations to expected utility the many studies in this field have been generally supportive of mean variance analysis subject to certain initially unanticipated caveats

dominating estimators for minimum variance portfolios abstract in this paper we derive two shrinkage estimators for minimum variance portfolios that dominate the traditional estimator with respect to the out of sample variance of the portfolio return the presented results hold for any number of assets d 4 and number of observations n d 2 the small sample properties of the shrinkage estimators as well as their large sample properties for fixed d but n and n d but n d q are investigated furthermore we present a small sample test for the question of whether it is better to completely ignore time series information in favor of naive diversification

nonlinear boundary value problems in hilbert spaces abstract in this paper we examine nonlinear two point boundary value problems with sturm liouville type boundary conditions in a hilbert space our technique involves using the topological transversality theorem of a granas which relies on the notions of an essential map and a priori bounds on solutions we obtain existence results for a wide class of problems with our nonlinear term satisfying a bernstein nagumo growth condition

a dynamic structure for high dimensional covariance matrices and its application in portfolio allocation abstractestimation of high dimensional covariance matrices is an interesting and important research topic in this article we propose a dynamic structure and develop an estimation procedure for high dimensional covariance matrices asymptotic properties are derived to justify the estimation procedure and simulation studies are conducted to demonstrate its performance when the sample size is finite by exploring a financial application an empirical study shows that portfolio allocation based on dynamic high dimensional covariance matrices can significantly outperform the market from 1995 to 2014 our proposed method also outperforms portfolio allocation based on the sample covariance matrix the covariance matrix based on factor models and the shrinkage estimator of covariance matrix supplementary materials for this article are available online

a flexible semiparametric forecasting model for time series in this paper we propose a semiparametric procedure called the model averaging marginal regression mamar that is flexible for forecasting of time series this procedure considers approximating a multivariate regression function by an affine combination of one dimensional marginal regression functions the weight parameters involved in the approximation are estimated by least squares on the basis of the first stage nonparametric kernel estimates of the marginal regressions under some mild conditions we have established asymptotic normality for the estimated weights and the regression function in two cases case i considers that the number of the covariates is fixed while case ii allows the number of the covariates depending on the sample size and diverging as the observations are assumed to be stationary and near epoch dependent the approach developed is applicable to both the estimation and forecasting issues in time series analysis furthermore the method and result are augmented by a simulation study and illustrated by an application in forecasting the high frequency volatility of the ftse100 index

existence of solution for discontinuous third order boundary value problems abstract in this paper we obtain existence results for the problem u q u f t u with boundary conditions u a a u b b u a c and u a u b u a u b u a c we assume f a caratheodory function q l r 0 such that 1 q l loc r 0 and suppose the existence of lower and upper solutions the existence of solution for the first considered conditions is obtained as a consequence of the fixed points theorems we obtain the solution of the second problem as a limit of solutions of the first case for the first problem the monotone method is developed

sparsistency and rates of convergence in large covariance matrix estimation this paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions here sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probability tending to one depending on the case of applications sparsity priori may occur on the covariance matrix its inverse or its cholesky decomposition we study these three sparsity exploration problems under a unified framework with a general penalty function we show that the rates of convergence for these problems under the frobenius norm are of order s n log p n n 1 2 where s n is the number of nonzero elements p n is the size of the covariance matrix and n is the sample size this explicitly spells out the contribution of high dimensionality is merely of a logarithmic factor the conditions on the rate with which the tuning parameter n goes to 0 have been made explicit and compared under different penalties as a result for the l 1 penalty to guarantee the sparsistency and optimal rate of convergence the number of nonzero elements should be small sn o pn at most among o pn2 parameters for estimating sparse covariance or correlation matrix sparse precision or inverse correlation matrix or sparse cholesky factor where sn is the number of the nonzero elements on the off diagonal entries on the other hand using the scad or hard thresholding penalty functions there is no such a restriction

estimation of the global minimum variance portfolio in high dimensions abstract we estimate the global minimum variance gmv portfolio in the high dimensional case using results from random matrix theory this approach leads to a shrinkage type estimator which is distribution free and optimal in the sense of minimizing the out of sample variance its asymptotic properties are investigated assuming that the number of assets p depends on the sample size n such that p n c 0 as n tends to infinity the results are obtained under weak assumptions imposed on the distribution of the asset returns only the existence of the fourth moments is required furthermore we make no assumption on the upper bound of the spectrum of the covariance matrix as a result the theoretical findings are also valid if the dependencies between the asset returns are described by a factor model which appears to be very popular in the financial literature nowadays this is also documented in a numerical study where the small and large sample behavior of the derived estimator is compared with existing estimators of the gmv portfolio the resulting estimator shows significant improvements and it turns out to be robust if the assumption of normality is violated

singularities of the hamiltonian vectorfield in optimal control problems abstract variational problems with n degrees of freedom give rise by the pontriaguine maximum principle to a hamiltonian vectorfield in t r n that presents singularities non smoothness points when the lagrangean is not convex for the problems of the calculus of variations the singularities that occur are points where the hamiltonian vectorfield is not c0 for optimal control problems we show that besides these singularities there appear other ones points where the hamiltonian vectorfield is c0 but not c1 and we classify them

inferential theory for factor models of large dimensions this paper develops an inferential theory for factor models of large dimensions the principal components estimator is considered because it is easy to compute and is asymptotically equivalent to the maximum likelihood estimator if normality is assumed we derive the rate of convergence and the limiting distributions of the estimated factors factor loadings and common components the theory is developed within the framework of large cross sections n and a large time dimension t to which classical factor analysis does not apply we show that the estimated common components are asymptotically normal with a convergence rate equal to the minimum of the square roots of n and t the estimated factors and their loadings are generally normal although not always so the convergence rate of the estimated factors and factor loadings can be faster than that of the estimated common components these results are obtained under general conditions that allow for correlations and heteroskedasticities in both dimensions stronger results are obtained when the idiosyncratic errors are serially uncorrelated and homoskedastic a necessary and sufficient condition for consistency is derived for large n but fixed t

on asymptotics of eigenvectors of large sample covariance matrix let x ij i j be a double array of i i d complex random variables with ex 11 0 e x 11 2 1 and e x 11 4 and let an 1 n t 1 2 n x n x n t 1 2 n where t 1 2 n is the square root of a nonnegative definite matrix t n and x n is the n x n matrix of the upper left comer of the double array the matrix an can be considered as a sample covariance matrix of an i i d sample from a population with mean zero and covariance matrix t n or as a multivariate f matrix if t n is the inverse of another sample covariance matrix to investigate the limiting behavior of the eigenvectors of an a new form of empirical spectral distribution is defined with weights defined by eigenvectors and it is then shown that this has the same limiting spectral distribution as the empirical spectral distribution defined by equal weights moreover if x ij and t n are either real or complex and some additional moment assumptions are made then linear spectral statistics defined by the eigenvectors of an are proved to have gaussian limits which suggests that the eigenvector matrix of an is nearly haar distributed when t n is a multiple of the identity matrix an easy consequence for a wishart matrix

a new modification of the adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations abstract in this paper we propose a new modified recursion scheme for the resolution of multi order and multi point boundary value problems for nonlinear ordinary and partial differential equations by the adomian decomposition method adm our new approach including duan s convergence parameter provides a significant computational advantage by allowing for the acceleration of convergence and expansion of the interval of convergence during calculations of the solution components for nonlinear boundary value problems in particular for such cases when one of the boundary points lies outside the interval of convergence of the usual decomposition series we utilize the boundary conditions to derive an integral equation before establishing the recursion scheme for the solution components thus we can derive a modified recursion scheme without any undetermined coefficients when computing successive solution components whereas several prior recursion schemes have done so this modification also avoids solving a sequence of nonlinear algebraic equations for the undetermined coefficients fraught with multiple roots which is required to complete calculation of the solution by several prior modified recursion schemes using the adm

distributional properties of portfolio weights in this paper we prove several distributional properties for optimal portfolio weights the weights are estimated by replacing the parameters with the sample counterparts all results for finite samples are made assuming normally distributed returns we calculate the exact covariances for the weights obtained by the expected quadratic utility additionally we derive the multivariate density function of the global minimum variance portfolio and the univariate density of the tangency portfolio we obtain the conditional density for the sharpe ratio optimal weights and show that the expectations of the sharpe ratio optimal weights do not exist moreover we determine the asymptotic distributions of the estimated weights assuming that the returns follow a multivariate stationary gaussian process

statistical data analysis and inference prologue c radhakrishna rao s contributions to statistics y dodge nonlinear estimation in linear models f hampel on the maximum number of factors in two construction methods for orthogonal arrays a s hedayat j stufken some geometrical aspects of data analysis and statistics j m oller inferences and test of hypotheses on bayes and empirical bayes two stage allocation procedures for selection problems s s gupta t c liang multi sample functional statistical data analysis e parzen discussing truth or falsity by computing a q value w schaafsma j tolboom b van der meulen linear pivotals and the bayes non bayes compromise g a barnard bayes modified minimax experimental design r v canfield a simple class of tests locally better than the score test t k chandra comparison of tests in the presence of a nuisance parameter r mukerjee tests for redundancy of some variables in multivariate analysis y fujikoshi model choice in the context of simultaneous inference t havranek o soudsky first zeros of empirical characteristic functions and extreme values of gaussian processes j husler parameter and variance estimation comparison of experiments with weighted distributions m j bayarri m h de groot optimal estimation for weighted distributions semi parametric models v p godambe m b rajarshi on efficiency for quasi likelihood and composite quasi likelihood methods c c heyde improved estimators of dispersion of an inverse gaussian distribution n pal b k sinha applications of edgeworth expansions to bootstrap a review g j babu dual poincare type inequalities via the cramer rao and the cauchy schwarz inequalities and related characterizations th cacoullos some optimality results on stein s two stage sampling j k ghosh r mukerjee the cramer frechet rao inequality for sequential estimation in non regular case z govindarajulu i vincze whither delete k jackknifing for smooth statistical functionals p k sen distributions weighted distributions and characterization three useful expressions for expectations involving a wishart matrix and its inverse g p h styan characterizations of distributions via moments of order statistics a survey and comparison of methods g d lin constancy of regression of a polynomial of sample average of positive random variables on their ratios characterizes gamma distribution a kagan probing encountered data meta analysis and weighted distribution methods g p patil c taillie further results on identification when the parameters are partially unknown s el khattabi f streit linear models and matrices weighted least squares estimation in the general gauss markov model j k baksalary s puntanen evaluating pre test predictors of success in linear regression models a cohen h b sackrowitz optimum invariant tests in mixed linear models with two variance components th mathew

generalized thresholding of large covariance matrices we propose a new class of generalized thresholding operators that combine thresholding with shrinkage and study generalized thresholding of the sample covariance matrix in high dimensions generalized thresholding of the covariance matrix has good theoretical properties and carries almost no computational burden we obtain an explicit convergence rate in the operator norm that shows the tradeoff between the sparsity of the true model dimension and the sample size and shows that generalized thresholding is consistent over a large class of models as long as the dimension p and the sample size n satisfy log p n 0 in addition we show that generalized thresholding has the sparsistency property meaning it estimates true zeros as zeros with probability tending to 1 and under an additional mild condition is sign consistent for nonzero elements we show that generalized thresholding covers as special cases hard and soft thresholding smoothly clipped absolute deviation and adaptive lasso and comp

the dual variational principle and equilibria for a beam resting on a discontinuous nonlinear elastic foundation the dual variational principle and equilibria for a beam resting on a discontinuous nonlinear elastic foundation m r grossinho a b 1 st a tersian c 2 a departamento de matem atica iseg universidade t ecnica de lisboa rua do quelhas 6 1200 lisboa portugal b cmaf universidade de lisboa av prof gama pinto 2 1699 lisboa codex portugal c center of applied mathematics and informatics university of rousse 8 studentska str 7017 rousse bulgaria

on piecewise smooth vector fields tangent to nested tori abstract in this paper we present a number of results involving 3d nonsmooth dynamical systems tangent to a foliation we study one parameter families of systems z e passing through a specific model z 0 whose phase portrait is foliated by invariant nested tori for each positive integer k we explicitly construct a family z e k possessing exactly k nested tori

nonparametric estimation of large covariance matrices of longitudinal data summary estimation of an unstructured covariance matrix is difficult because of its positivedefiniteness constraint this obstacle is removed by regressing each variable on its predecessors so that estimation of a covariance matrix is shown to be equivalent to that of estimating a sequence of varying coefficient and varying order regression models our framework is similar to the use of increasing order autoregressive models in approximating the covariance matrix or the spectrum of a stationary time series as an illustration we adopt fan zhang s 2000 two step estimation of functional linear models and propose nonparametric estimators of covariance matrices which are guaranteed to be positive definite for parsimony a suitable order for the sequence of auto regression models is found using penalised likelihood criteria like aic and bic some asymptotic results for the local polynomial estimators of components of a covariance matrix are established two longitudinal datasets are analysed to illustrate the methodology a simulation study reveals the advantage of the nonparametric covariance estimator over the sample covariance matrix for large covariance matrices

the picard boundary value problem for nonlinear second order vector differential equations the existence of solutions is studied for the picard boundary value problem b x b f b t b x b x 0 b x b a b x b b 0 where b b b a and b f b a b b b r sup n b r sup n rarr b r sup n is a continuous function

high dimensionality effects in the markowitz problem and other quadratic programs with linear constraints risk underestimation we study the properties of solutions of quadratic programs with linear equality constraints whose parameters are estimated from data in the high dimensional setting where p the number of variables in the problem is of the same order of magnitude as n the number of observations used to estimate the parameters the markowitz problem in finance is a subcase of our study assuming normality and independence of the observations we relate the ecient frontier computed empirically to the true ecient frontier our computations show that there is a separation of the errors induced by estimating the mean of the observations and estimating the covariance matrix in particular the price paid for estimating the covariance matrix is an underestimation of the variance by a factor roughly equal to 1 p n therefore the risk of the optimal population solution is underestimated when we estimate it by solving a similar quadratic program with estimated parameters we also characterize the statistical behavior of linear functionals of the empirical optimal vector and show that they are biased estimators of the corresponding population quantities we investigate the robustness of our gaussian results by extending the study to certain elliptical models and models where our n observations are correlated in time we show a lack of robustness of the gaussian results but are still able to get results concerning rst order properties of the quantities of interest even in the case of relatively heavy tailed data we require two moments risk underestimation is still present in the elliptical case and more pronounced that in the gaussian case we discuss properties of the non parametric and parametric bootstrap in this context we show several results including the interesting fact that standard applications of the bootstrap generally yields inconsistent estimates of bias finally we propose some strategies to correct these problems and practically validate them in some simulations in all the paper we will assume that p n and n p tend to innity and p n

second and higher order systems of boundary value problems abstract existence results for systems of kth order boundary value problems are presented our analysis is based on a nonlinear alternative of a granas and use of a priori bounds

a new semiparametric estimation approach for large dynamic covariance matrices with multiple conditioning variables this paper studies the estimation of large dynamic covariance matrices with multiple conditioning variables we introduce an easy to implement semiparametric method to estimate each entry of the covariance matrix via model averaging marginal regression and then apply a shrinkage technique to obtain the dynamic covariance matrix estimation under some regularity conditions we derive the asymptotic properties for the proposed estimators including the uniform consistency with general convergence rates we further consider extending our methodology to deal with the scenarios i the number of conditioning variables is divergent as the sample size increases and ii the large covariance matrix is conditionally sparse relative to contemporaneous market factors we provide a simulation study that illustrates the finite sample performance of the developed methodology we also provide an application to financial portfolio choice from daily stock returns