a two stage estimator for probit models with structural group effects this paper outlines a two stage technique for estimation and inference in probit models with structural group effects the structural group specification belongs to a broader class of random components models in particular individuals in a given group share a common component in the specification of the conditional mean of a latent variable for a number of computational reasons existing random effects models are impractical for estimation and inference in this type of problem our two stage estimator provides an easily estimable alternative to the random effects specification in addition we conduct a monte carlo simulation comparing the performance of alternative es timators and find that the two stage estimator is superior both in terms of estimation and inference to traditional estimators

on global diffeomorphisms of euclidian space as is well known the first attempt at characterizing global diffeomorphisms of ir goes back to the 1906 work of hadamard l since that time progress in the global diffeomorphism problem has been rather slow notable contributions are those of levy 2 who generalized hadamard s theorem to banach spaces and of banach and mazur 3 and caccioppoli 4 who proved the so called properness criterion saying that a proper local homeomorphism of a banach space is a global homeomorphism the standard proofs of the theorems of hadamard and levy and of banach mazur and caccioppoli rely upon ideas of what is now called covering space theory via the important homotopy lifting property of coverings see e g 5 this approach makes it clear that it is only lack of compactness of ir that allows for the existence of noninjective local homeomorphisms when n l 2 the complex exponential when n 2 being the most famous example the theorems mentioned above were found again by various authors sometimes in special cases and sometimes in a different framework but always through some variant of the same argument a synthesis was made by rheinboldt 6 in 1969 part of his results were rediscovered five years later by plastock 7 and given a somewhat simpler exposition for the special case of banach spaces other contributions were made by meisters and olech 8 global homeomorphisms of bounded domains in i meyer 9 complementing the work of levy and radulescu and radulescu lo expanding upon plastock s results many other articles have been devoted to special cases where extra hypotheses are introduced consistently with the problem at hand for instance polynomial mappings only are relevant to the jacobian conjecture of algebraic geometry and hypotheses about the spectrum of the derivative at each point come up in a natural way in global stability questions for differential equations the new results presented in this paper have been obtained through arguments from critical point theory and differential topology here the central notion is that of admissible flow given a mapping f ii ir the main property of what we call an f admissible flow q ir x ii f o i f o is that for every x e ir f o the function t jf q t x l where 1 1 is the euclidean norm is strictly increasing and tends to 0 resp 00 as t tends to co resp 00 the basic feature we prove is that a local diffeomorphism f is a global one if and only if f o zi and there is an f admissible flow this statement makes it clear that 0 plays a special role regarding f admissibility and the condition f o 0 cannot be eliminated as examples will show the introduction of f admissibility followed the author s remark that in hilbert space hadamard s theorem could be derived from the mountain pass theorem of ambrosetti and

bootstrap inference for instrumental variable models with many weak instruments this study s main contribution is to theoretically analyze the application of bootstrap methods to instrumental variable models when the available instruments may be weak and the number of instruments goes to infinity with the sample size we demonstrate that a standard residual based bootstrap procedure cannot consistently estimate the distribution of the limited information maximum likelihood estimator or fuller 1977 estimator under many many weak instrument sequence the primary reason is that the standard procedure fails to capture the instrument strength in the sample adequately in addition we consider the restricted efficient re bootstrap of davidson and mackinnon 2008 2010 2014 that generates bootstrap data under the null restricted and uses an efficient estimator of the coefficient of the reduced form equation efficient we find that the re bootstrap is also invalid however it effectively mimics more key features in the limiting distributions of interest and thus is less distorted in finite samples than the standard bootstrap procedure finally we propose modified bootstrap procedures that provide a valid distributional approximation for the two estimators with many many weak instruments a monte carlo experiment shows that hypothesis testing based on the asymptotic normal approximation can have severe size distortions in finite samples instead our modified bootstrap procedures greatly reduce these distortions

a trace minimization algorithm for the generalized eigenvalue problem an algorithm for computing a few of the smallest or largest eigenvalues and associated eigenvectors of the large sparse generalized eigenvalue problem ax lambda bx is presented the matrices a and b are assumed to be symmetric and haphazardly sparse with b being positive definite the problem is treated as one of constrained optimization and an inverse iteration is developed which requires the solution of linear algebraic systems only to the accuracy demanded by a given subspace the rate of convergence of the method is established and a technique for improving it is discussed numerical experiments and comparisons with other methods are presented

rate of convergence for the wild bootstrap in nonparametric regression this paper concerns the distributions used to construct confidence intervals for the regression function in a nonparametric setup some rates of convergence for the normal limit its plug in approach and the wild bootstrap are obtained conditionally on the explanatory variable x and also unconditionally the bound found for the wild bootstrap approximation is slightly better by a factor n 1 45 than the bounds given by the plug in approach or the clt for the conditional probability on the contrary the unconditional bounds present a different feature the rate obtained when approximating by the clt improves the one given by the plug in approach by a factor of n8 45 while this last one performs better than the wild bootstrap approximation and the corresponding ratio is n1 45 it should be mentioned that these two sequences especially the last one tend to zero at an extremely slow rate

hypothesis testing in linear regression when k n is large abstract this paper derives the asymptotic distribution of the f test for the significance of linear regression coefficients as both the number of regressors k and the number of observations n increase together so that their ratio remains positive in the limit the conventional critical values for this test statistic are too small and the standard version of the f test is invalid under this asymptotic theory this paper provides a correction to the f statistic that gives correctly sized tests both under this paper s limit theory and also under conventional asymptotic theory that keeps k finite this paper also presents simulations that indicate the new statistic can perform better in small samples than the conventional test the statistic is then used to reexamine olivei and tenreyro s results from olivei g tenreyro s 2007 the timing of monetary policy shocks the american economic review 97 636 663 and sala i martin s results from sala i martin x x 1997 i just ran two million regressions the american economic review 87 2 178 183

heteroscedastic one way anova and lack of fit tests recent articles have considered the asymptotic behavior of the one way analysis of variance anova f statistic when the number of levels or groups is large in these articles the results were obtained under the assumption of homoscedasticity and for the case when the sample or group sizes ni remain fixed as the number of groups a tends to infinity in this article we study both weighted and unweighted test statistics in the heteroscedastic case the unweighted statistic is new and can be used even with small group sizes we demonstrate that an asymptotic approximation to the distribution of the weighted statistic is possible only if the group sizes tend to infinity suitably fast in relation to a our investigation of local alternatives reveals a similarity between lack of fit tests for constant regression in the present case of replicated observations and the case of no replications which uses smoothing techniques the asymptotic theory uses a novel application of the projection principle to obtain

quadratic linear keller maps abstract it is proved that a quadratic linear keller map c n c n is linearly triangularizable if its rank is at most 2 or corank at most 1

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

on the existence of global analytic conjugations for polynomial mappings of yagzhev type abstract consider a mappingf c n c nof the form identity plus a term with polynomial components that are homogeneous of the third degree and suppose that the jacobian determinant of this mapping is constant throughout c n polynomial mapping of yagzhev type as a stronger version of the classical jacobian conjecture the question has been posed whether for some values of c 0 there exists a global change of variables conjugation on c nthrough which the mapping fbecomes its linear part at the origin van den essen has recently produced a simple yagzhev mapping for which no suchpolynomialconjugation exists we show here that van den essen s example still admitsglobal analyticconjugations the question on the existence of global conjugations for general yagzhev maps is then still open

on the adaptive elastic net with a diverging number of parameters we consider the problem of model selection and estimation in situations where the number of parameters diverges with the sample size when the dimension is high an ideal method should have the oracle property fan and li 2001 fan and peng 2004 which ensures the optimal large sample performance furthermore the high dimensionality often induces the collinearity problem which should be properly handled by the ideal method many existing variable selection methods fail to achieve both goals simultaneously in this paper we propose the adaptive elastic net that combines the strengths of the quadratic regularization and the adaptively weighted lasso shrinkage under weak regularity conditions we establish the oracle property of the adaptive elastic net we show by simulations that the adaptive elastic net deals with the collinearity problem better than the other oracle like methods thus enjoying much improved finite sample performance

a projected preconditioned conjugate gradient algorithm for computing many extreme eigenpairs of a hermitian matrix we present an iterative algorithm for computing an invariant subspace associated with the algebraically smallest eigenvalues of a large sparse or structured hermitian matrix a we are interested in the case in which the dimension of the invariant subspace is large e g over several hundreds or thousands even though it may still be small relative to the dimension of a these problems arise from for example density functional theory dft based electronic structure calculations for complex materials the key feature of our algorithm is that it performs fewer rayleigh ritz calculations compared to existing algorithms such as the locally optimal block preconditioned conjugate gradient or the davidson algorithm it is a block algorithm and hence can take advantage of efficient blas3 operations and be implemented with multiple levels of concurrency we discuss a number of practical issues that must be addressed in order to implement the algorithm efficiently on a high performance computer

correlation and high dimensional consistency in pattern recognition abstract classical discriminant analysis breaks down when the feature vectors are of extremely high dimension for example when the basic observation is a random function observed over a fine grid alternative methods have been developed assuming a simplified form for the covariance structure we analyze the high dimensional asymptotics of some of these methods emphasizing the effects of correlations such as occur when the baseline is random for instance the euclidean distance classifier which has been proposed for generic use in high dimensional classification problems is dimensionally inconsistent under a simple repeated measurement model we provide exponential bounds for the error rates of several classifiers we develop new dimensionally consistent methods to deal with the effects of correlation in high dimensional problems

the jacobian conjecture reduction of degree and formal expansion of the inverse introduction i the jacobian conjecture 1 statement of the jacobian problem first observations 2 some history of the jacobian conjecture 3 faulty proofs 4 the use of stabilization and of formal methods ii the reduction theorem 1 notation 2 statement of the reduction theorem 3 reduction to degree 3 4 proof of the reduction theorem 5 r linearization and unipotent reduction 6 nilpotent rank 1 jacobians iii the formal inverse 1 notation 2 abhyankar s inversion formula 3 the terms gj 4 the tree expansion g lt a t lfptf 5 calculations references

effects of data dimension on empirical likelihood summary we evaluate the effects of data dimension on the asymptotic normality of the empirical likelihood ratio for high dimensional data under a general multivariate model data dimension and dependence among components of the multivariate random vector affect the empirical likelihood directly through the trace and the eigenvalues of the covariance matrix the growth rates to infinity we obtain for the data dimension improve the rates of hjort et al 2008

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

goodness of fit tests for mixed model diagnostics a simple goodness of fit test is proposed for checking distributional assumptions involved in a mixed linear model an estimated critical value of the test statistic is derived and is shown to be asymptotically correct under mild conditions as a special case the test may be applied to linear regression models to formally check distribution of the errors finite sample performance of the proposed test is examined and compared with that of a previously proposed test by simulations

nonparametric specification testing via the trinity of tests tests are developed for inference on a parameter vector whose dimension grows slowly with sample size the statistics are based on the lagrange multiplier wald and pseudo likelihood ratio principles admit standard normal asymptotic distributions under the null and are straightforward to compute they are shown to be consistent and possessing non trivial power against local alternatives the settings considered include multiple linear regression panel data models with fixed effects and spatial autoregressions when a nonparametric regression function is estimated by series we use our statistics to propose specification tests and in semiparametric adaptive estimation we provide a test for correct error distribution specification these tests are nonparametric but handled in practice with parametric techniques a monte carlo study suggests that our tests perform well in finite samples two empirical examples use them to test for correct shape of an electricity distribution cost function and linearity and equality of engel curves

foliations and polynomial diffeomorphisms of mathbb r 3 let y f g h mathbb r 3 to mathbb r 3 be a c 2 map and let spec y denote the set of eigenvalues of the derivative dy p when p varies in mathbb r 3 we begin proving that if for some 0 spec y cap epsilon epsilon emptyset then the foliation mathcal f k with k in f g h made up by the level surfaces k constant consists just of planes as a consequence we prove a bijectivity result related to the three dimensional case of jelonek s jacobian conjecture for polynomial maps of mathbb r n

tests for high dimensional regression coefficients with factorial designs we propose simultaneous tests for coefficients in high dimensional linear regression models with factorial designs the proposed tests are designed for the large p small n situations where the conventional f test is no longer applicable we derive the asymptotic distribution of the proposed test statistic under the high dimensional null hypothesis and various scenarios of the alternatives which allow power evaluations we also evaluate the power of the f test for models of moderate dimension the proposed tests are employed to analyze a microarray data on yorkshire gilts to find significant gene ontology terms which are significantly associated with the thyroid hormone after accounting for the designs of the experiment

block locally optimal preconditioned eigenvalue xolvers blopex in hypre and petsc we describe our software package block locally optimal preconditioned eigenvalue xolvers blopex recently publicly released blopex is available as a stand alone serial library as an external package to petsc portable extensible toolkit for scientific computation a general purpose suite of tools developed by argonne national laboratory for the scalable solution of partial differential equations and related problems and is also built into hypre high performance preconditioners a scalable linear solvers package developed by lawrence livermore national laboratory the present blopex release includes only one solver the locally optimal block preconditioned conjugate gradient lobpcg method for symmetric eigenvalue problems hypre provides users with advanced high quality parallel multigrid preconditioners for linear systems with blopex the same preconditioners can now be efficiently used for symmetric eigenvalue problems petsc facilitates the integration of independently developed application modules with strict attention to component interoperability and makes blopex extremely easy to compile and use with preconditioners that are available via petsc we present the lobpcg algorithm in blopex for hypre and petsc we demonstrate numerically the scalability of blopex by testing it on a number of distributed and shared memory parallel systems including a beowulf system sun fire 880 an amd dual core opteron workstation and ibm bluegene l supercomputer using petsc domain decomposition and hypre multigrid preconditioning we test blopex on a model problem the standard 7 point finite difference approximation of the 3 d laplacian with the problem size in the range of 10 5 10 8

residual bootstrap tests in linear models with many regressors abstract this paper is concerned with bootstrap hypothesis testing in linear regression models with many regressors i show that bootstrap f l r and l m tests are asymptotically valid even when the numbers of estimated parameters and tested restrictions are not asymptotically negligible fractions of the sample size one of the conditions for these results is that the regressors come from an asymptotically balanced design depending on the number of restrictions tested and on the errors distribution violation of that condition might render the bootstrap tests asymptotically invalid in that case i propose bootstrapping calhoun s 2011 g statistic or modified versions of the l r and l m statistics and show that these procedures remain asymptotically valid monte carlo simulations indicate that the bootstrap tests often outperform asymptotic ones however analyzing the approximate third cumulant of the f statistic reveals that the bootstrap test does not generally provide the usual higher order asymptotic refinements nevertheless it is found that the bootstrap third cumulant partially matches the population third cumulant which might explain the bootstrap s good finite sample performances especially when the errors come from a symmetric distribution

asymptotics and the theory of inference asymptotic analysis has always been very useful for deriving distributions in statistics in cases where the exact distribution is unavailable more importantly asymptotic analysis can also provide insight into the inference process itself suggesting what information is available and how this information may be extracted the development of likelihood inference over the past twenty some years provides an illustration of the interplay between techniques of approximation and statistical theory

asymptotics for high dimensional regression m estimates fixed design results we investigate the asymptotic distributions of coordinates of regression m estimates in the moderate p n regime where the number of covariates p grows proportionally with the sample size n under appropriate regularity conditions we establish the coordinate wise asymptotic normality of regression m estimates assuming a fixed design matrix our proof is based on the second order poincare inequality and leave one out analysis some relevant examples are indicated to show that our regularity conditions are satisfied by a broad class of design matrices we also show a counterexample namely an anova type design to emphasize that the technical assumptions are not just artifacts of the proof finally numerical experiments confirm and complement our theoretical results

the linear triangularizability of some keller maps abstract in this paper we prove that keller maps are linearly triangularizable under some conditions as a consequence we show that non homogeneous power linear keller maps with jh symmetric are linearly triangularizable

the linear dependence problem for power linear maps abstract let bl l 1 k be m nl complex matrices and let x l c n l l 1 k be complex vector variables we show that the components of the map h b 1 x 1 d 1 b k x k d k are linearly dependent over c if and only if det b 1 b 1 d 1 b k b k d k 0 where means the hadamard product x and x d denote the conjugate transpose and the dth hadamard power of a matrix x respectively connections are established between the homogenous dependence problem hdp n d which arises in the study of the jacobian conjecture and the dependence problem for power linear maps pldp n d an algorithm is given to compute counterexamples to pldp n d from those to hdp n d and counterexamples to pldp n 3 are obtained for all n 67

best subset selection persistence in high dimensional statistical learning and optimization under l1 constraint let y x 1 x m be a random vector it is desired to predict y based on x 1 x m examples of prediction methods are regression classification using logistic regression or separating hyperplanes and so on we consider the problem of best subset selection and study it in the context m n a 1 where n is the number of observations we investigate procedures that are based on empirical risk minimization it is shown that in common cases we should aim to find the best subset among those of size which is of order o n log n it is also shown that in some asymptotic sense when assuming a certain sparsity condition there is no loss in letting m be much larger than n for example m n a 1 this is in comparison to starting with the best subset of size smaller than n and regardless of the value of a we then study conditions under which empirical risk minimization subject to l 1 constraint yields nearly the best subset these results extend some recent results obtained by greenshtein and ritov finally we present a high dimensional simulation study of a boosting type classification procedure

asymptotic behavior of the empiric distribution of m estimated residuals from a regression model with many parameters on etudie le comportement asymptotique des residus d un modele de regression lorsque les m estimateurs sont utilises et que le nombre de parametres peut croitre avec la taille de l echantillon

asymptotic behavior of m estimators of p regression parameters when p 2 n is large ii normal approximation in a general linear model y x beta r with y and r n dimensional x a n times p matrix and beta p dimensional let hat beta be an m estimator of beta satisfying 0 sum x i psi y i x i beta let p rightarrow infty such that p log n 3 2 n rightarrow 0 then max i x i hat beta beta rightarrow p 0 and it is possible to find a uniform normal approximation for the distribution of hat beta under which arbitrary linear combinations a n hat beta beta are asymptotically normal when appropriately normalized and hat beta beta x x hat beta beta is approximately chi 2 p

bootstrap and wild bootstrap for high dimensional linear models in this paper two bootstrap procedures are considered for the estimation of the distribution of linear contrasts and of f test statistics in high dimensional linear models an asymptotic approach will be chosen where the dimension p of the model may increase for sample size n rightarrow infty the range of validity will be compared for the normal approximation and for the bootstrap procedures furthermore it will be argued that the rates of convergence are different for the bootstrap procedures in this asymptotic framework this is in contrast to the usual asymptotic approach where p is fixed

augmented factor models with applications to validating market risk factors and forecasting bond risk premia abstract we study factor models augmented by observed covariates that have explanatory powers on the unknown factors in financial factor models the unknown factors can be reasonably well explained by a few observable proxies such as the fama french factors in diffusion index forecasts identified factors are strongly related to several directly measurable economic variables such as consumption wealth variable financial ratios and term spread with those covariates both the factors and loadings are identifiable up to a rotation matrix even only with a finite dimension to incorporate the explanatory power of these covariates we propose a smoothed or projected principal component analysis pca i regress the data onto the observed covariates and ii take the principal components of the fitted data to estimate the loadings and factors this allows us to more accurately estimate the percentage of both explained and unexplained components in factors and thus to assess the explanatory power of covariates we show that both the estimated factors and loadings can be estimated with improved rates of convergence compared to the benchmark method the degree of improvement depends on the strength of the signals representing the explanatory power of the covariates on the factors the proposed estimator is robust to possibly heavy tailed distributions we apply the model to forecast us bond risk premia and find that the observed macroeconomic characteristics contain strong explanatory powers of the factors the gain of forecast is more substantial when the characteristics are incorporated to estimate the common factors than directly used for forecasts

on the impact of predictor geometry on the performance on high dimensional ridge regularized generalized robust regression estimators we study ridge regularized generalized robust regression estimators i e begin aligned widehat beta mathrm argmin beta in mathbb r p frac 1 n sum i 1 n rho i y i x i beta frac tau 2 beta 2 quad text where quad y i epsilon i x i beta 0 end aligned in the situation where p n tends to a finite non zero limit our study here focuses on the situation where the errors epsilon i s are heavy tailed and x i s have an elliptical like distribution our assumptions are quite general and we do not require homoskedasticity of epsilon i s for instance we obtain a characterization of the limit of widehat beta beta 0 as well as several other results including central limit theorems for the entries of widehat beta

weak convergence of the empirical process of residuals in linear models with many parameters when fitting by least squares a linear model with an intercept term with p parameters to n data points the asymptotic behavior of the residual empirical process is shown to be the same as in the single sample problem provided p 3 log 2 p n 0 for any error density having finite variance and a bounded first derivative no further conditions are imposed on the sequence of design matrices the result is extended to more general estimates with the property that the average error and average squared error in the fitted values are on the same order as for least squares

some remarks on the jacobian conjecture and druzkowski mappings abstract in this paper we first show that the jacobian conjecture is true for non homogeneous power linear mappings under some conditions secondly we prove an equivalent statement about the jacobian conjecture in dimension r 1 and give some partial results for r 2 finally for a homogeneous power linear keller map f x h of degree d 2 we give the inverse polynomial map under the condition that j h 3 0 we shall show that deg f 1 d k if k 2 and j h k 1 0 but also give an example with d 2 and j h 4 0 such that deg f 1 d 3

a note on the jacobian conjecture abstract in this note we show that if the druzkowski mappings f x x ax 3 i e f x x 1 a 11 x 1 a 1 n x n 3 x n a n 1 x 1 a nn x n 3 satisfies trj ax 3 0 then rank a 1 2 n where is the number of diagonal elements of a which are equal to zero furthermore we show the jacobian conjecture is true for the druzkowski mappings in dimension 9 in the case i 1 n a ii 0

generalized jackknife estimators of weighted average derivatives with the aim of improving the quality of asymptotic distributional approximations for nonlinear functionals of nonparametric estimators this article revisits the large sample properties of an important member of that class namely a kernel based weighted average derivative estimator asymptotic linearity of the estimator is established under weak conditions indeed we show that the bandwidth conditions employed are necessary in some cases a bias corrected version of the estimator is proposed and shown to be asymptotically linear under yet weaker bandwidth conditions implementational details of the estimators are discussed including bandwidth selection procedures consistency of an analog estimator of the asymptotic variance is also established numerical results from a simulation study and an empirical illustration are reported to establish the results a novel result on uniform convergence rates for kernel estimators is obtained the online supplemental material to this article includes details on

jackknife model averaging for quantile regressions abstract in this paper we consider model averaging for quantile regressions qr when all models under investigation are potentially misspecified and the number of parameters is diverging with the sample size to allow for the dependence between the error terms and regressors in the qr models we propose a jackknife model averaging jma estimator which selects the weights by minimizing a leave one out cross validation criterion function and demonstrate its asymptotic optimality in terms of minimizing the out of sample final prediction error we conduct simulations to demonstrate the finite sample performance of our estimator and compare it with other model selection and averaging methods we apply our jma method to forecast quantiles of excess stock returns and wages

on robust regression with high dimensional predictors we study regression m estimates in the setting where p the number of covariates and n the number of observations are both large but p n we find an exact stochastic representation for the distribution of argmin p i 1 n y i x i at fixed p and n under various assumptions on the objective function and our statistical model a scalar random variable whose deterministic limit r can be studied when p n 0 plays a central role in this representation we discover a nonlinear system of two deterministic equations that characterizes r interestingly the system shows that r depends on through proximal mappings of as well as various aspects of the statistical model underlying our study several surprising results emerge in particular we show that when p n is large enough least squares becomes preferable to least absolute deviations for double exponential errors

conditional quantile estimation for generalized autoregressive conditional heteroscedasticity models conditional quantile estimation is an essential ingredient in modern risk management although generalized autoregressive conditional heteroscedasticity garch processes have proven highly successful in modeling financial data it is generally recognized that it would be useful to consider a broader class of processes capable of representing more flexibly both asymmetry and tail behavior of conditional returns distributions in this article we study estimation of conditional quantiles for garch models using quantile regression quantile regression estimation of garch models is highly nonlinear we propose a simple and effective two step approach of quantile regression estimation for linear garch time series in the first step we use a quantile autoregression sieve approximation for the garch model by combining information over different quantiles then second stage estimation for the garch model is carried out based on the first stage minimum distance estimation of the scale process of the time series

geometric representation of high dimension low sample size data summary high dimension low sample size data are emerging in various areas of science we find a common structure underlying many such data sets by using a non standard type of asymptotics the dimension tends to while the sample size is fixed our analysis shows a tendency for the data to lie deterministically at the vertices of a regular simplex essentially all the randomness in the data appears only as a random rotation of this simplex this geometric representation is used to obtain several new statistical insights

on rank estimators in increasing dimensions abstract the family of rank estimators including han s maximum rank correlation han 1987 as a notable example has been widely exploited in studying regression problems for these estimators although the linear index is introduced for alleviating the impact of dimensionality the effect of large dimension on inference is rarely studied this paper fills this gap via studying the statistical properties of a larger family of m estimators whose objective functions are formulated as u processes and may be discontinuous in increasing dimension set up where the number of parameters p n in the model is allowed to increase with the sample size n first we find that often in estimation as p n n 0 p n n 1 2 rate of convergence is obtainable second we establish bahadur type bounds and study the validity of normal approximation which we find often requires a much stronger scaling requirement than p n 2 n 0 third we state conditions under which the numerical derivative estimator of asymptotic covariance matrix is consistent and show that the step size in implementing the covariance estimator has to be adjusted with respect to p n all theoretical results are further backed up by simulation studies

a modern maximum likelihood theory for high dimensional logistic regression students in statistics or data science usually learn early on that when the sample size n is large relative to the number of variables p fitting a logistic model by the method of maximum likelihood produces estimates that are consistent and that there are well known formulas that quantify the variability of these estimates which are used for the purpose of statistical inference we are often told that these calculations are approximately valid if we have 5 to 10 observations per unknown parameter this paper shows that this is far from the case and consequently inferences produced by common software packages are often unreliable consider a logistic model with independent features in which n and p become increasingly large in a fixed ratio we prove that i the maximum likelihood estimate mle is biased ii the variability of the mle is far greater than classically estimated and iii the likelihood ratio test lrt is not distributed as a 2 the bias of the mle yields wrong predictions for the probability of a case based on observed values of the covariates we present a theory which provides explicit expressions for the asymptotic bias and variance of the mle and the asymptotic distribution of the lrt we empirically demonstrate that these results are accurate in finite samples our results depend only on a single measure of signal strength which leads to concrete proposals for obtaining accurate inference in finite samples through the estimate of this measure

bootstrap simultaneous error bars for nonparametric regression simultaneous error bars are constructed for nonparametric kernel estimates of regression functions the method is based on the bootstrap where resampling is done from a suitably estimated residual distribution the error bars are seen to give asymptotically correct coverage probabilities uniformly over any number of gridpoints applications to an economic problem are given and comparison to both pointwise and bonferroni type bars is presented through a simulation study

pseudo maximum likelihood estimation of spatial autoregressive models with increasing dimension pseudo maximum likelihood estimates are developed for higher order spatial autoregressive models with increasingly many parameters including models with spatial lags in the dependent variables both with and without a linear or nonlinear regression component and regression models with spatial autoregressive disturbances consistency and asymptotic normality of the estimates are established monte carlo experiments examine finite sample behaviour

inference on higher order spatial autoregressive models with increasingly many parameters this paper develops consistency and asymptotic normality of parameter estimates for a higher order spatial autoregressive model whose order and number of regressors are allowed to approach infinity slowly with sample size both least squares and instrumental variables estimates are examined and the permissible rate of growth of the dimension of the parameter space relative to sample size is studied besides allowing the number of parameters to increase with the data this has the advantage of accommodating some asymptotic regimes that are suggested by certain spatial settings several of which are discussed a small empirical example is also included and a monte carlo study analyses various implications of the theory in finite samples

consistent estimation with a large number of weak instruments this paper analyzes the conditions under which consistent estimation can be achieved in instrumental variables iv regression when the available instruments are weak and the number of instruments k n goes to infinity with the sample size we show that consistent estimation depends importantly on the strength of the instruments as measured by r the rate of growth of the so called concentration parameter and also on k n in particular when k n the concentration parameter can grow even if each individual instrument is only weakly correlated with the endogenous explanatory variables and consistency of certain estimators can be established under weaker conditions than have previously been assumed in the literature hence the use of many weak instruments may actually improve the performance of certain point estimators more specifically we find that the limited information maximum likelihood liml estimator and the bias corrected two stage least squares b2sls estimator are consistent when k n r n 0 while the two stage least squares 2sls estimator is consistent only if k n r n 0 as n these consistency results suggest that liml and b2sls are more robust to instrument weakness than 2sls

asymptotics when the number of parameters tends to infinity in the bradley terry model for paired comparisons we are concerned here with establishing the consistency and asymptotic normality for the maximum likelihood estimator of a merit vector u 0 u t representing the merits of t 1 teams players treatments objects under the bradley terry model as t this situation contrasts with the well known neyman scott problem under which the number of parameters grows with t the amount of sampling and for which the maximum likelihood estimator fails even to attain consistency a key feature of our proof is the use of an effective approximation to the inverse of the fisher information matrix specifically under the bradley terry model when teams i and j with respective merits u i and u j play each other the probability that team i prevails is assumed to be u i u i u j suppose each pair of teams play each other exactly n times for some fixed n the objective is to estimate the merits u i s based on the outcomes of the nt t 1 2 games clearly the model depends on the u i s only through their ratios under some condition on the growth rate of the largest ratio u i u j 0 i j t as t the maximum likelihood estimator of u 1 u 0 u t u 0 is shown to be consistent and asymptotically normal some simulation results are provided

efficiency of projected score methods in rectangular array asymptotics summary the paper considers a rectangular array asymptotic embedding for multistratum data sets in which both the number of strata and the number of within stratum replications increase and at the same rate it is shown that under this embedding the maximum likelihood estimator is consistent but not efficient owing to a non zero mean in its asymptotic normal distribution by using a projection operator on the score function an adjusted maximum likelihood estimator can be obtained that is asymptotically unbiased and has a variance that attains the cram6r rao lower bound the adjusted maximum likelihood estimator can be viewed as an approximation to the conditional maximum likelihood estimator

power linear keller maps with ditto triangularizations abstract we show that power linear keller maps f x 1 a 1 x d x 2 a 2 x d x n a n x d are linearly triangularizable if 1 rk a 2 or 2 cork a 2 and d 3 or 3 cork a 3 d 5 and the diagonal of a is nonzero furthermore we show that the triangularizations can be chosen power linear as well

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

gmm inference when the number of moment conditions is large abstract asymptotic theory typically presumes that the dimensionality of econometric models is independent of the sample size even though this presumption is often quite unrealistic in gmm estimation whenever optimal instruments are not available it can frequently be shown that adding over identifying restrictions moment conditions will increase asymptotic precision however the conventional asymptotics which underlies this view insists that the number of moment conditions remain finite even though the number of available moment conditions may grow without bound we consider the explicit dependence of the number of moment conditions or instruments q n on the sample size n and establish that under conventional regularity conditions for the estimation of a linear model with general heteroskedasticity q n 3 n 0 is a sufficient condition for the validity of conventional asymptotic inference about the gmm estimator

asymptotic properties of bridge estimators in sparse high dimensional regression models we study the asymptotic properties of bridge estimators in sparse high dimensional linear regression models when the number of covariates may increase to infinity with the sample size we are particularly interested in the use of bridge estimators to distinguish between covariates whose coefficients are zero and covariates whose coefficients are nonzero we show that under appropriate conditions bridge estimators correctly select covariates with nonzero coefficients with probability converging to one and that the estimators of nonzero coefficients have the same asymptotic distribution that they would have if the zero coefficients were known in advance thus bridge estimators have an oracle property in the sense of fan and li j amer statist assoc 96 2001 1348 1360 and fan and peng ann statist 32 2004 928 961 in general the oracle property holds only if the number of covariates is smaller than the sample size however under a partial orthogonality condition in which the covariates of the zero coefficients are uncorrelated or weakly correlated with the covariates of nonzero coefficients we show that marginal bridge estimators can correctly distinguish between covariates with nonzero and zero coefficients with probability converging to one even when the number of covariates is greater than the sample size

on the range of a hadamard power of a positive semidefinite matrix abstract for any n n complex matrix a and any integer k 1 we prove that the span of image of the map x ax k is equal to the range of aa k where x and x k denote the conjugate transpose and the k th hadamard power of a matrix x respectively this settles a conjecture due to gorni and tutaj gasinska related to the study of the jacobian conjecture

quadratic linear keller maps of nilpotency index three let f x h cn cn be a homogeneous polynomial map of degree d 2 and fa y ay d cn cn a power linear map such that f and fa are a generalized gorni zampieri pair we discuss the relation between the nilpotency indices of jh and j ay d and we show that f is linearly triangularizable if and only if fa is linearly triangularizable as a consequence we show that a quadratic linear keller map fa y ay 2 with nilpotency index three i e j ay 2 3 0 is linearly triangularizable