covariance regularization by thresholding this paper considers regularizing a covariance matrix of p variables estimated from n observations by hard thresholding we show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense the variables are gaussian or sub gaussian and log p n 0 and obtain explicit rates the results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity we discuss an intuitive resampling scheme for threshold selection and prove a general cross validation result that justifies this approach we also compare thresholding to other covariance estimators in simulations and on an example from climate data 1 introduction estimation of covariance matrices is important in a number of areas of statistical analysis including dimension reduction by principal component analysis pca classification by linear or quadratic discriminant analysis lda and qda establishing independence and conditional independence relations in the context of graphical models and setting confidence intervals on linear functions of the means of the components in recent years many application areas where these tools are used have been dealing with very high dimensional datasets and sample sizes can be very small relative to dimension examples include genetic data brain imaging spectroscopic imaging climate data and many others it is well known by now that the empirical covariance matrix for samples of size n from a p variate gaussian distribution np p is not a good estimator of the population covariance if p is large many results in random matrix theory illustrate this from the classical mary law 29 to the more recent work of johnstone and his students on the theory of the largest eigenvalues 12 23 30 and associated eigenvectors 24 however with the exception of a method for estimating the covariance spectrum 11 these probabilistic results do not offer alternatives to the sample covariance matrix alternative estimators for large covariance matrices have therefore attracted a lot of attention recently two broad classes of covariance estimators have emerged those that rely on a natural ordering among variables and assume that variables

all maximally entangled four qubit states we find an operational interpretation for the 4 tangle as a type of residual entanglement somewhat similar to the interpretation of the 3 tangle using this remarkable interpretation we are able to find the class of maximally entangled four qubits states which is characterized by four real parameters the states in the class are maximally entangled in the sense that their average bipartite entanglement with respect to all possible bipartite cuts is maximal we show that while all the states in the class maximize the average tangle there are only a few states in the class that maximize the average tsillas or renyi entropy of entanglement quite remarkably we find that up to local unitaries there exists two unique states one maximizing the average tsallis entropy of entanglement for all 2 while the other maximizing it for all 0 2 including the von neumann case of 1 furthermore among the maximally entangled four qubits states there are only three maximally entangled states that have the property that for two out of the three bipartite cuts consisting of two qubits verses two qubits the entanglement is 2 ebits and for the remaining bipartite cut the entanglement between the two groups of two qubits is 1 ebit the unique three maximally entangled states are the three cluster states that are related by a swap operator we also show that the cluster states are the only states up to local unitaries that maximize the average renyi entropy of entanglement for all 2

on model diagnostics using varying coefficient models summary the paper motivates the use of varying coefficient models for diagnostics in regression models with continuous and factorial covariates the varying coefficient model which is restricted only by assumptions concerning the smoothness of the effects is considered as the alternative to a parametric generalised linear model estimation is based on local likelihood smoothing and asymptotic properties of the estimators are derived for the investigation of the discrepancy between the parametric model and the smooth alternative a simple graphical procedure is proposed moreover tests are derived which allow for overall and componentwise investigation the componentwise tests may be used to examine interaction effects of continuous and factorial regressors

nonparametric detection of correlated errors summary in regression problems it is hard to detect correlated errors since the errors are not observed in this paper a nonparametric method is proposed for the detection of correlated errors when the design points are equally spaced it turns out that the first order sample autocovariance of the residuals from the kernel regression estimates provides essential information about correlated errors and its bootstrap is quite effective in implementing such information

nonparametric comparison of several regression functions exact and asymptotic theory a new test is proposed for the comparison of two regression curves f and g we prove an asymptotic normal law under fixed alternatives which can be applied for power calculations for constructing confidence regions and for testing precise hypotheses of a weighted l 2 distance between f and g in particular the problem of nonequal sample sizes is treated which is related to a peculiar formula of the area between two step functions these results are extended in various directions such as the comparison of k regression functions or the optimal allocation of the sample sizes when the total sample size is fixed the proposed pivot statistic is not based on a nonparametric estimator of the regression curves and therefore does not require the specification of any smoothing parameter

one sided cross validation abstract a new method of selecting the smoothing parameters of nonparametric regression estimators is introduced the method termed one sided cross validation oscv has the objectivity of cross validation and statistical properties comparable to those of a plug in rule the new method may be viewed as an application of the prequential model selection method of dawid as such our results identify a situation in which the prequential method is a more efficient model selector than cross validation an example simulations and theoretical results demonstrate the utility of oscv when used with local linear and kernel estimators

efficiency of a kernel density estimator under an autoregressive dependence model abstract the problem of estimating the probability density function of a strictly stationary process is considered to study the effect of a dependence structure on the efficiency of a kernel density estimator the mean integrated squared error mise of the fourier integral estimator fie is derived on the assumption that the observed data are generated by a first order autoregressive process numerical results for the normal and cauchy densities show that even moderate departures from independence can lead to a considerable loss in efficiency of the fie in addition to efficiency considerations the issue of determining an optimal smoothing parameter for the fie under the autoregressive model is addressed

a cubic spline extension of the durbin watson test summary we propose a new test for model misspecification in linear regression based on a measure of smoothness of the regression residuals this curvature statistic is related to a property of cubic splines and leads to a convenient powerful test for patterns the distribution of the curvature statistic may be derived from assumptions of normality for equally spaced data the calculations are simplified and a table of significance points of the distribution is given smoothness considerations also lead to a natural generalization of the durbin watson or mean square successive differences statistic for unequally spaced observations this generalized mean squared successive differences statistic arises when linear splines are substituted for cubic splines in the development of the curvature statistic the power of the curvature statistic is investigated by simulation compared to several other commonly used tests for patterns the curvature test is shown to be more powerful where noise levels are small and the complexity of the underlying pattern is moderate the calculation and use of this statistic is illustrated on a problem involving nonlinear regression residuals

testing lack of fit in multiple regression summary we study lack of fit tests based on orthogonal series estimators a common feature of these tests is that they are functions of score statistics that employ data driven model dimensions the criteria used to select the dimension are score based versions of aic and bic the tests can be applied in a wide variety of settings including both continuous and discrete data with two or more covariates a model sequence i e a path in the alternative models space has to be chosen critical points and p values of the lack of fit tests can be obtained via asymptotic distribution theory or by use of the bootstrap data examples and a simulation study illustrate the applicability of the tests

testing for parallelism among trends in multiple time series this paper considers the inference of trends in multiple nonstationary time series to test whether trends are parallel to each other we use a parallelism index based on the l2 distances between nonparametric trend estimators and their average a central limit theorem is obtained for the test statistic and the test s consistency is established we propose a simulation based approximation to the distribution of the test statistic which significantly improves upon the normal approximation the test is also applied to devise a clustering algorithm finally the finite sample properties of the test are assessed through simulations and the test methodology is illustrated by a cell phone download data collected in the united states

the asymptotic spectrum of locc transformations we study the exact non deterministic conversion of multipartite pure quantum states into one another via local operations and classical communication locc and asymptotic entanglement transformation under such channels in particular we consider the maximal number of copies of any given target state that can be extracted exactly from many copies of any given initial state as a function of the exponential decay in the success probability known as the converse error exponent we give a formula for the optimal rate presented as an infimum over the asymptotic spectrum of locc conversion a full understanding of exact asymptotic extraction rates between pure states in the converse regime thus depends on a full understanding of this spectrum we present a characterization of spectral points and use it to describe the spectrum in the bipartite case this leads to a full description of the spectrum and thus an explicit formula for the asymptotic extraction rate between pure bipartite states given a converse error exponent this extends the result on entanglement concentration in 1 where the target state is fixed as the bell state in the limit of vanishing converse error exponent the rate formula provides an upper bound on the exact asymptotic extraction rate between two states when the probability of success goes to 1 in the bipartite case we prove that this bound holds with equality

nonlinear system theory another look at dependence based on the nonlinear system theory we introduce previously undescribed dependence measures for stationary causal processes our physical and predictive dependence measures quantify the degree of dependence of outputs on inputs in physical systems the proposed dependence measures provide a natural framework for a limit theory for stationary processes in particular under conditions with quite simple forms we present limit theorems for partial sums empirical processes and kernel density estimates the conditions are mild and easily verifiable because they are directly related to the data generating mechanisms

testing parametric assumptions of trends of a nonstationary time series the paper considers testing whether the mean trend of a nonstationary time series is of certain parametric forms a central limit theorem for the integrated squared error is derived and a hypothesis testing procedure is proposed the method is illustrated in a simulation study and is applied to assess the mean pattern of lifetime maximum wind speeds of global tropical cyclones from 1981 to 2006 we also revisit the trend pattern in the central england temperature series copyright 2011 oxford university press

monitoring profiles based on nonparametric regression methods the use statistical process control spc in monitoring and diagnosis of process and product quality profiles remains an important problem in various manufacturing industries the spc problem with a nonlinear profile is particularly challenging this article proposes a novel scheme to monitor changes in both the regression relationship and the variation of the profile online it integrates the multivariate exponentially weighted moving average procedure with the generalized likelihood ratio test based on nonparametric regression the proposed scheme not only provides an effective spc solution to handle nonlinear profiles which are common in industrial practice but it also resolves the latent problem in popular parametric monitoring methods of being unable to detect certain types of changes due to a misspecified out of control model our simulation results demonstrate the effectiveness and efficiency of the proposed monitoring scheme in addition a systematic diagnostic approach is provided to locate t

regularized estimation of large covariance matrices this paper considers estimating a covariance matrix of p variables from n observations by either banding the sample covariance matrix or estimating a banded version of the inverse of the covariance we show that these estimates are consistent in the operator norm as long as logp 2 n 0 and obtain explicit rates the results are uniform over some fairly natural well conditioned families of covariance matrices we also introduce an analogue of the gaussian white noise model and show that if the population covariance is embeddable in that model and well conditioned then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix the results can be extended to smooth versions of banding and to non gaussian distributions with su ciently short tails a resampling approach is proposed for choosing the banding parameter in practice this approach is illustrated numerically on both simulated and real data

validation of linear regression models a new test is proposed in order to verify that a regression function say g has a prescribed linear parametric form this procedure is based on the large sample behavior of an empirical l 2 distance between g and the subspace u spanned by the regression functions to be verified the asymptotic distribution of the test statistic is shown to be normal with parameters depending only on the variance of the observations and the l 2 distance between the regression function g and the model space u based on this result a test is proposed for the hypothesis that g is not in a preassigned l 2 neighborhood of u whichallows the verification of the model u at a controlled type i error rate the suggested procedure is very easy to apply because of its asymptotic normal law and the simple form of the test statistic in particular it does not require nonparametric estimators of the regression function and hence the test does not depend on the subjective choice of smoothing parameters

non parametric analysis of covariance an analysis of covariance model where the covariate effect is assumed only to be smooth is considered the possibility of different shapes of covariate effect in different groups is also allowed and tests of equality and of parallelism across groups are constructed these are implemented using gasser maller smoothing whose properties enable problems of bias to be avoided accurate moment based approximations are available for the distribution of each test statistic some data on spanish onions are used to contrast the non parametric approach with that of a nonlinear but parametric model a simulation study is also used to explore the properties of the non parametric tests

wavelet shrinkage using cross validation summary wavelets are orthonormal basis functions with special properties that show potential in many areas of mathematics and statistics this paper concentrates on the estimation of functions and images from noisy data by using wavelet shrinkage a modified form of twofold cross validation is introduced to choose a threshold for wavelet shrinkage estimators operating on data sets of length a power of 2 the cross validation algorithm is then extended to data sets of any length and to multidimensional data sets the algorithms are compared with established threshold choosers by using simulation an application to a real data set arising from anaesthesia is presented

smooth optimum kernel estimators near endpoints summary kernel estimators for smooth curves like density spectral density or regression functions require modifications when estimating near endpoints of the support both for practical and asymptotic reasons the construction of such boundary kernels as solutions of a variational problem is addressed and representations in orthogonal polynomials are given including explicit solutions for the most important cases based on explicit formulae for certain functionals of the kernels it is shown that local bandwidth variation might be indicated near boundaries various bandwidth variation schemes are discussed and

testing the parametric null model hypothesis in semiparametric partial and generalized spline models on etend les resultats de cox et koh 1986 aux modeles generalises de wahba 1985 et a divers autres modeles splines partiels on montre que la statistique de test a une relation etroite avec le comportement de la fonction de validation croisee generalisee

fuglede kadison determinants and entropy for actions of discrete amenable groups consider a discrete group t and an element in the integral group ring zt then t acts from the left on the discrete additive group zt ztf by automorphisms of groups dualizing we obtain a left t action on the compact pontrjagin dual group xf zt ztf by continuous automorphisms of groups by definition xf is a closed subshift of s1 1 for measurable or topological actions of amenable groups a good entropy theory is available ow m thus from now on let t be a finitely generated discrete amenable group we are interested in determining the topological entropy hf of the t action on xf it agrees with the measure theoretic entropy of the t action on xf equipped with its haar probability measure in their paper fk from 1952 fuglede and kadison introduced a determinant for units in finite factor von neumann algebras more recently l ck has general ized their theory to nonunits and to group von neumann algebras which are not necessarily factors l2 he needed this to define l2 torsion for general covering spaces the von neumann algebra a t of the amenable group t is a certain completion of cr and hence we may consider the fuglede kadison l ck determinant detjvr of we refer to section 6 for the definition of a log strong f0lner sequence it is known that finitely generated virtually nilpotent groups have log strong f0lner sequences our main result is

inference of trends in time series summary we consider statistical inference of trends in mean non stationary models a test statistic is proposed for the existence of structural breaks in trends on the basis of a strong invariance principle of stationary processes we construct simultaneous confidence bands with asymptotically correct nominal coverage probabilities the results are applied to global warming temperature data and nile river flow data our confidence band of the trend of the global warming temperature series supports the claim that the trend is increasing over the last 150 years

a frequency domain selection criterion for regression with autocorrelated errors abstract we consider the regression model yt t e t t 0 1 n 1 where y t are scalar observations t is the unknown regression function and e t are unobservable errors generated by a zero mean weakly stationary process independent of t and with completely unknown autocorrelation structure we propose a data driven method for selecting a parametric or nonparametric estimator of t the method is based on cross validation in the frequency domain and requires no assumptions about the form of the estimator or the error correlations it does however require the discrete fourier transform dft of the signal t to be a smooth complex function of frequency as is the case for example with transient signals or growth and decay curves after giving some general motivations for the method we focus on the special case of linear estimators of a nonparametric regression function including both kernel and spline estimators for these estimators we develop efficient methods of evaluating t

goodness of fit statistics for age specific reference intervals the age specific reference interval is a commonly used screening tool in medicine it involves estimation of extreme quantile curves such as the 5th and 95th centiles of a reference distribution of clinically normal individuals it is crucial that models used to estimate such intervals fit the data extremely well however few procedures to assess goodness of fit have been proposed in the literature and even fewer have been evaluated systematically here we consider procedures based on the distribution of the z scores standardized residuals from a model and on pearson chi 2 statistics for observed and expected counts in groups defined by age and the estimated reference centile curves two of the procedures q and grid tests are mainly inferential whereas the third permutation bands and b tests is essentially graphical we obtain approximations to the null distributions of several relevant test statistics and examine their size and power for a range of models based on real data sets we recommend q tests in all situations where z scores are available since they are general simple to calculate and usually have the highest power among the three classes of test considered for the cases considered the grid tests are always inferior to the q and b tests

entangled pure state transformations via local operations assisted by finitely many rounds of classical communication we consider generic pure n qubit states and a general class of pure states of arbitrary dimensions and arbitrarily many subsystems we characterize those states which can be reached from some other state via local operations assisted by finitely many rounds of classical communication locc n for n qubits with n 3 we show that this set of states is of measure zero which implies that the maximally entangled set is generically of full measure if restricted to the practical scenario of locc n moreover we identify a class of states for which any locc n protocol can be realized via a concatenation of deterministic steps we show however that in general there exist state transformations which require a probabilistic step within the protocol which highlights the difference between bipartite and multipartite locc

resource theory of entanglement with a unique multipartite maximally entangled state entanglement theory is formulated as a quantum resource theory in which the free operations are local operations and classical communication locc this defines a partial order among bipartite pure states that makes it possible to identify a maximally entangled state which turns out to be the most relevant state in applications however the situation changes drastically in the multipartite regime not only do there exist inequivalent forms of entanglement forbidding the existence of a unique maximally entangled state but recent results have shown that locc induces a trivial ordering almost all pure entangled multipartite states are incomparable i e locc transformations among them are almost never possible in order to cope with this problem we consider alternative resource theories in which we relax the class of locc to operations that do not create entanglement we consider two possible theories depending on whether resources correspond to multipartite entangled or genuinely multipartite entangled gme states and we show that they are both nontrivial no inequivalent forms of entanglement exist in them and they induce a meaningful partial order i e every pure state is transformable to more weakly entangled pure states moreover we prove that the resource theory of gme that we formulate here has a unique maximally entangled state the generalized ghz state which can be transformed to any other state by the allowed free operations

smoothing spline models with correlated random errors abstract spline smoothing techniques are commonly used to estimate the mean function in a nonparametric regression model their performances depend greatly on the choice of smoothing parameters many methods of selecting smoothing parameters such as generalized maximum likelihood gml generalized cross validation gcv and unbiased risk ubr have been developed under the assumption of independent observations they tend to underestimate smoothing parameters when data are correlated in this article i assume that observations are correlated and that the correlation matrix depends on a parsimonious set of parameters i extend the gml gcv and ubr methods to estimate the smoothing parameters and the correlation parameters simultaneously i also relate a smoothing spline model to three mixed effects models these relationships show that the smoothing spline estimates evaluated at design points are best linear unbiased prediction blup estimates and that the gml estimates of the smoothing parameters an

the moduli space of three qutrit states we study the invariant theory of trilinear forms over a three dimensional complex vector space and apply it to investigate the behavior of pure entangled three partite qutrit states and their normal forms under local filtering operations slocc we describe the orbit space of the slocc group sl 3 c 3 both in its affine and projective versions in terms of a very symmetric normal form parametrized by three complex numbers the parameters of the possible normal forms of a given state are roots of an algebraic equation which is proved to be solvable by radicals the structure of the sets of equivalent normal forms is related to the geometry of certain regular complex polytopes

kernel regression when the boundary region is large with an application to testing the adequacy of polynomial models abstract it is well known that kernel regression estimators are subject to so called boundary or edge effects a phenomenon in which the bias of an estimator increases near the endpoints of the estimation interval when the regression curve is linear or nearly linear the requisite amount of smoothing is so great that the boundary region is effectively the entire estimation interval special boundary kernels are proposed here to deal with such cases it is shown that the proposed kernel estimator has a property also enjoyed by cubic smoothing splines namely as the estimator s smoothing parameter becomes large the estimator tends to a straight line the limiting straight line is essentially the least squares line when the design points are equally spaced a simple generalization of ideas in the linear case leads to kernel estimates that are polynomials of any given degree for large bandwidths such estimates are an important component of a proposed test for the adequacy of a polynomial model the test s

covariate matched one sided tests for the difference between functional means abstract when testing hypotheses about the effects of different treatments variation among covariates can become confounded with that between treatments unless the treatments are applied using paired covariates in the context of unpaired covariates we propose implicit covariate matching methods for testing the hypothesis that one treatment effect is greater than another the methods are founded on the assumption that the mean treatment effect conditional on the covariate is a smooth function of the covariate they are implemented using new interpolation techniques for nonparametric curve estimation bootstrap arguments are used to construct critical points we show that even when the covariate distributions are identical for both treatments covariate matching of the type that we propose produces tests of greater power than methods that do not attempt matching our techniques have application to two sided hypothesis testing

testing for cubic smoothing splines under dependent data in most research on smoothing splines the focus has been on estimation while inference especially hypothesis testing has received less attention by defining design matrices for fixed and random effects and the structure of the covariance matrices of random errors in an appropriate way the cubic smoothing spline admits a mixed model formulation which places this nonparametric smoother firmly in a parametric setting thus nonlinear curves can be included with random effects and random coefficients the smoothing parameter is the ratio of the random coefficient and error variances and tests for linear regression reduce to tests for zero random coefficient variances we propose an exact f test for the situation and investigate its performance in a real pine stem data set and by simulation experiments under certain conditions the suggested methods can also be applied when the data are dependent

understanding exponential smoothing via kernel regression exponential smoothing is the most common model free means of forecasting a future realization of a time series it requires the specification of a smoothing factor which is usually chosen from the data to minimize the average squared residual of previous one step ahead forecasts in this paper we show that exponential smoothing can be put into a nonparametric regression framework and gain some interesting insights into its performance through this interpretation we also use theoretical developments from the kernel regression field to derive for the first time asymptotic properties of exponential smoothing forecasters

on the use of nonparametric regression for model checking summary the use of nonparametric regression is explored to check the fit of a parametric regression model the principal aim is to check the validity of the regression curve rather than necessarily to detect outliers a pseudo likelihood ratio test is developed to provide a global assessment of fit and simulation bands are used to indicate the nature of departures from the model the types of data considered include discrete response variables where standard diagnostic techniques are often not appropriate and first order autoregressive series several numerical examples are given nonparametric regression can be used in an informal graphical way to assess the relationship between a response and an explanatory variable in this paper we aim to develop more formal methods of assessing the assumptions of a parametric model in particular when regression diagnostics of the type developed for normal linear models are not readily available the principal aim is to check the validity of the systematic part of the model by comparing a nonparametric estimate of the regression curve with a parametric one such a comparison may also identify outliers although the distinction between outliers and model inadequacy is not always easy two techniques are used to assess the fit of a parametric model in 2 confidence bands are constructed around the fitted regression curve by simulation a comparison of these with the nonparametric curve gives an indication of the nature of any departures from the model in 3 a pseudo likelihood ratio test is developed this provides a quantitative global assessment of fit in applying these ideas special emphasis is given to discrete data and notably logistic regression because of the difficulty in applying standard residual based model checking techniques to this type of response variable a poisson regression example is discussed in 4 however the underlying ideas have wider applications autoregressive time series of order 1 are discussed in 6 sections 5 and 7 discuss general issues we first discuss the context of binary regression with a single covariate and the difficulties caused by the discreteness of the response variable the observed data are assumed to be of the form xi yi ni where xi is a covariate value and yi has a binomial

the maximally entangled set of 4 qubit states entanglement is a resource to overcome the natural restriction of operations used for state manipulation to local operations assisted by classical communication locc hence a bipartite maximally entangled state is a state which can be transformed deterministically into any other state via locc in the multipartite setting no such state exists there rather a whole set the maximally entangled set of states mes which we recently introduced is required this set has on the one hand the property that any state outside of this set can be obtained via locc from one of the states within the set and on the other hand no state in the set can be obtained from any other state via locc recently we studied locc transformations among pure multipartite states and derived the mes for three and generic four qubit states here we consider the non generic four qubit states and analyze their properties regarding local transformations as already the most coarse grained classification due to stochastic locc slo

testing for no effect when estimating a smooth function by nonparametric regression a randomization approach abstract when a linear regression function is estimated by ordinary least squares the null hypothesis of no relationship between the response and the design variable can be tested by the normal theory f test this article describes a new test for use in the case in which a smooth regression function is estimated by a nonparametric procedure such as kernel estimation or local regression the test is constructed as an approximation to an exact permutation test the test statistic is the ratio of sums of squares that are defined by analogy to the analysis of variance the null permutation distribution of the test statistic is approximated by matching its exact mean and variance to the moments of a gamma distribution a simulation study shows that the approximation is excellent for several regression procedures and for normal heavy tailed and skewed error distributions simulation results are also used to investigate the power of the test as a function of the smoothing parameter the test is applied to dat

bandwidth choice for nonparametric regression this paper is concerned with the problem of choosing a bandwidth parameter for nonparametric regression we analyze a tapered fourier series estimate and discuss the relationship of this estimate to a kernel estimate we first consider a method based on an unbiased estimate of mean square error and show that the bandwidth thus chosen is asymptotically optimal other methods are examined as well and are shown to be asymptotically equivalent a small simulation shows however that for small or moderate sample size the methods perform quite differently

comparing nonparametric versus parametric regression fits in general there will be visible differences between a parametric and a nonparametric curve estimate it is therefore quite natural to compare these in order to decide whether the parametric model could be justified an asymptotic quantification is the distribution of the integrated squared difference between these curves we show that the standard way of bootstrapping this statistic fails we use and analyse a different form of bootstrapping for this task we call this method the wild bootstrap and apply it to fitting engel curves in expenditure data analysis

goodness of fit tests for parametric regression models several new tests are proposed for examining the adequacy of a family of parametric models against large nonparametric alternatives these tests formally check if the bias vector of residuals from parametric fits is negligible by using the adaptive neyman test and other methods the testing procedures formalize the traditional model diagnostic tools based on residual plots we examine the rates of contiguous alternatives that can be detected consistently by the adaptive neyman test applications of the procedures to the partially linear models are thoroughly discussed our simulation studies show that the new testing procedures are indeed powerful and omnibus the power of the proposed tests is comparable to the f test statistic even in the situations where the f test is known to be suitable and can be far more powerful than the f test statistic in other situations an application to testing linear models versus additive models is also discussed

testing the goodness of fit of a linear model via nonparametric regression techniques abstract this article investigates the use of nonparametric regression methodology to test the adequacy of a parametric linear model the large sample properties of parametric goodness of fit tests for linearity are considered the inadequacies of such tests lead to the proposal of new tests that are constructed from nonparametric regression fits to the residuals from linear regression large sample theory is derived for two variants of this type of statistic the results demonstrate that such tests are consistent against all fixed smooth alternatives to linearity but are incapable of detecting local alternatives converging to a linear model at the parametric rate n 1 2 simulation experiments involving a test based on fitting cubic smoothing splines to residuals reveals that this test has good power properties against several reasonable alternatives

bootstrap test for difference between means in nonparametric regression abstract a bootstrap test is proposed for detecting a difference between two mean functions in the setting of nonparametric regression error distributions in the regression model are permitted to be arbitrary and unequal the test enjoys power properties akin to those in a parametric setting in the sense that it can distinguish between regression functions distant only n 1 2 apart where n is the sample size it has exceptional level accuracy with level error of only n 2 and uses a very accurate estimate of the critical point of an exact test being in error by only n 3 2 under the null hypothesis the test admits several generalizations for example to the case of testing for differences between several regression means this is a nonparametric regression analog of analysis of variance a simulation study using n as small as 15 corroborates the asymptotic result on level accuracy of the bootstrap test applications are illustrated with an example involving acid rain data

transformations of w type entangled states the transformations of w type entangled states by using local operations assisted with classical communication are investigated for this purpose a parametrization of the w type states which remains invariant under local unitary transformations is proposed and a complete characterization of the local operations carried out by a single party is given these are used for deriving the necessary and sufficient conditions for deterministic transformations a convenient upper bound for the maximum probability of distillation of arbitrary target states is also found

the use of marginal likelihood for a diagnostic test for the goodness of fit of the simple linear regression model a novel test procedure is proposed to diagnose the goodness of fit of a simple regression model by applying a marginal likelihood under the assumption of smoothness of the alternative regression curve no specific form for the alternative regression function is assumed an advantage of the proposed test procedure is that it simultaneously provides an estimated regression curve spline some existing diagnostic procedures are explored in relation to the one proposed the procedure is applied to data of a spectrophotometric analysis of mixtures of two types of flour cornstarch and casein soybean

test of significance when data are curves abstract with modern technology massive data can easily be collected in a form of multiple sets of curves new statistical challenge includes testing whether there is any statistically significant difference among these sets of curves in this article we propose some new tests for comparing two groups of curves based on the adaptive neyman test and the wavelet thresholding techniques introduced earlier by fan we demonstrate that these tests inherit the properties outlined by fan and that they are simple and powerful for detecting differences between two sets of curves we then further generalize the idea to compare multiple sets of curves resulting in an adaptive high dimensional analysis of variance called hanova these newly developed techniques are illustrated by using a dataset on pizza commercials where observations are curves and an analysis of cornea topography in ophthalmology where images of individuals are observed a simulation example is also presented to illustrate the power of the adapti

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

bandwidth selection for kernel regression with long range dependent errors summary we investigate the effect of long range dependence on bandwidth selection for kernel regression with the plug in method of herrmann gasser kneip 1992 a new bandwidth estimator is proposed to allow for long range dependence properties of the proposed estimator are investigated theoretically and via simulation we find that the proposed estimator performs well in terms of integrated squared error of the estimated trend allowing us to incorporate both deterministic nonlinear features having an unknown structure and long range dependence into a single model the method is illustrated using biweekly measurements of the volume of the great salt lake

testing for trends in high dimensional time series the article considers statistical inference for trends of high dimensional time series based on a modified l2 distance between parametric and nonparametric trend estimators we propose a de diagon

generalized likelihood ratio statistics and wilks phenomenon likelihood ratio theory has had tremendous success in parametric inference due to the fundamental theory of wilks yet there is no general applicable approach for nonparametric inferences based on function estimation maximum likelihood ratio test statistics in general may not exist in nonparametric function estimation setting even if they exist they are hard to find and can not be optimal as shown in this paper we introduce the generalized likelihood statistics to overcome the drawbacks of nonparametric maximum likelihood ratio statistics a new s wilks phenomenon is unveiled we demonstrate that a class of the generalized likelihood statistics based on some appropriate nonparametric estimators are asymptotically distribution free and follow chi 2 distributions under null hypotheses for a number of useful hypotheses and a variety of useful models including gaussian white noise models nonparametric regression models varying coefficient models and generalized varying coefficient models we further demonstrate that generalized likelihood ratio statistics are asymptotically optimal in the sense that they achieve optimal rates of convergence given by ingster they can even be adaptively optimal in the sense of spokoiny by using a simple choice of adaptive smoothing parameter our work indicates that the generalized likelihood ratio statistics are indeed general and powerful for nonparametric testing problems based on function estimation

nonparametric spline regression with autoregressive moving average errors summary we estimate by spline nonparametric regression an unknown function observed with autocorrelated errors when the errors are modelled by an autoregressive moving average model unknown parameters are estimated by either maximum likelihood cross validation or generalized cross validation by expressing the problem in state space form we obtain 0 n algorithms to estimate the function and its derivatives and evaluate the marginal likelihood and cross validation functions the finite sample properties of the function estimates are evaluated by an extensive simulation study and examples are given

nonparametric trend estimation in functional time series with application to annual mortality rates this research was supported by the king abdullah university of science and technology kaust

analysis of field experiments by least squares smoothing summary assuming a smooth trend plus independent error model for the environmental effects in the yields of a field plot experiment least squares smoothing methods are developed to estimate both the treatment effects and the unknown trend treatment estimates are closely related to those resulting from a generalized least squares analysis in which the covariance structure for the environmental effects has a particular form however the main emphases are on the accuracy of treatment estimates under a fixed smooth trend plus error model and the exploratory power of the basic method to isolate trend effects of unknown form although the detailed development is for the one dimensional case generalizations of the smoothness concept and extensions to two dimensions are also discussed application of the basic method is illustrated on three data sets and the results compared with other analyses

comparison of two bandwidth selectors with dependent errors for nonparametric regression in the case of dependent observations cross validation is known to be severely affected by dependence this effect is precisely quantified through a limiting distribution for the cross validated bandwidth the performance of two methods the leave 2e 1 out version of cross validation and partitioned cross validation which adjust for the dependence effect on bandwidth selection is investigated the bandwidths produced by these two methods are analyzed by further limiting distributions which reveal significantly different characteristics simulations demonstrate that the asymptotic effects hold for reasonable sample sizes

local unitary equivalence of multipartite pure states necessary and sufficient conditions for the equivalence of arbitrary n qubit pure quantum states under local unitary lu operations are derived first an easily computable standard form for multipartite states is introduced two generic states are shown to be lu equivalent iff their standard forms coincide the lu equivalence problem for nongeneric states is solved by presenting a systematic method to determine the lu operators if they exist which interconvert the two states

score tests for the single index model the single index model is a generalization of the linear regression model with e y x g x where g is an unknown function the model provides a flexible alternative to the linear regression model while providing more structure than a fully nonparametric approach although the fitting of single index models does not require distributional assumptions on the error term the properties of the estimates depend on such assumptions as does practical application of the model in this article score tests are derived for three potential misspecifications of the single index model heteroscedasticity in the errors autocorrelation in the errors and the omission of an important variable in the linear index these tests have a similar structure to corresponding tests for nonlinear regression models monte carlo simulations demonstrate that the first two tests hold their nominal size well and have good power properties in identifying model violations often outperforming other tests testing for the need for ad

on the use of nonparametric regression for checking linear relationships summary the problem of checking the linearity of a regression relationship is addressed through the idea of smoothing of a residual plot a pseudolikelihood ratio test statistic which measures the distance between the nonparametric and the parametric models is derived as a ratio of quadratic forms the distribution of this statistic under the null hypothesis of linearity is calculated numerically by using johnson curves a power study shows the new statistic to be more sensitive to non linearity than the durbin watson statistic

testing parametric assumptions on band or time limited signals under noise this paper considers the problem of testing parametric assumptions on signals f from which only noisy observations y sub k f spl tau k spl epsi sub k are available and where the signal is assumed to be either band limited or time limited to this end the signal is reconstructed by an estimator based on the whittaker shannon ws sampling theorem with oversampling as test statistic the minimal l sub 2 distance between the estimated signal and the parametric model is used to construct appropriate tests the asymptotic distribution of the test statistic is derived both under the hypothesis of the validity of the parametric model and under fixed local alternatives as a byproduct the asymptotic distribution of the integrated square error of the estimator is computed which is of interest by itself e g for the analysis of a cross validated bandwidth selector

kernel smoothing of data with correlated errors abstract kernel smoothing is a common method of estimating the mean function in the nonparametric regression model y f x e where f x is a smooth deterministic mean function and e is an error process with mean zero in this article the mean squared error of kernel estimators is computed for processes with correlated errors and the estimators are shown to be consistent when the sequence of error processes converges to a mixing sequence the standard techniques for bandwidth selection such as cross validation and generalized cross validation are shown to perform very badly when the errors are correlated standard selection techniques are shown to favor undersmoothing when the correlations are predominantly positive and oversmoothing when negative the selection criteria can however be adjusted to correct for the effect of correlation in simulations the standard selection criteria are shown to behave as predicted the corrected criteria are shown to be very effective when the correlation functi

everything you always wanted to know about locc but were afraid to ask in this paper we study the subset of generalized quantum measurements on finite dimensional systems known as local operations and classical communication locc while locc emerges as the natural class of operations in many important quantum information tasks its mathematical structure is complex and difficult to characterize here we provide a precise description of locc and related operational classes in terms of quantum instruments our formalism captures both finite round protocols as well as those that utilize an unbounded number of communication rounds while the set of locc is not topologically closed we show that finite round locc constitutes a compact subset of quantum operations additionally we show the existence of an open ball around the completely depolarizing map that consists entirely of locc implementable maps finally we demonstrate a two qubit map whose action can be approached arbitrarily close using locc but nevertheless cannot be implemented perfectly

using difference based methods for inference in nonparametric regression with time series errors we show that difference based methods can be used to construct simple and explicit estimators of error covariance and autoregressive parameters in nonparametric regression with time series errors when the error process is gaussian our estimators are efficient but they are available well beyond the gaussian case as an illustration of their usefulness we show that difference based estimators can be used to produce a simplified version of time series cross validation this new approach produces a bandwidth selector that is equivalent to both first and second orders to that given by the full time series cross validation algorithm other applications of difference based methods are to variance estimation and construction of confidence bands in nonparametric regression

on bandwidth choice in nonparametric regression with both short and long range dependent errors we analyse methods based on the block bootstrap and leave out cross validation for choosing the bandwidth in nonparametric regression when errors have an almost arbitrarily long range of dependence a novel analytical device for modelling the dependence structure of errors is introduced this allows a concise theoretical description of the way in which the range of dependence affects optimal bandwidth choice it is shown that provided block length or leave out number respectively are chosen appropriately both techniques produce first order optimal bandwidths nevertheless the block bootstrap has far better empirical properties particularly under long range dependence

choice of bandwidth for kernel regression when residuals are correlated summary standard techniques for selecting the bandwidth of a kernel estimator from the data in a nonparametric regression model perform badly when the errors are correlated in this paper we propose a modified version of an existing plug in bandwidth selector the method is generalized to stationary error variables by estimating a functional of the residual covariance function the proposed bandwidth selector shows good properties in asymptotic theory and in simulations without assuming a parametric model for the error process

an equality test across nonparametric regressions abstract we propose a procedure for testing the equality of several nonparametric multivariate regressions we allow the regressors designs and the number of observations to differ across subsamples the division into subsamples is defined through a variable c which can be either fixed or random for a random c our procedure is a general test of significance for qualitative variables in a nonparametric regression for a fixed c our procedure provides a nonparametric analysis of covariance which is valid for cross section or panel data in both cases the test is a one sided normal test and is consistent against all alternatives

bandwidth selection in nonparametric kernel testing we propose a sound approach to bandwidth selection in nonparametric kernel testing the main idea is to find an edgeworth expansion of the asymptotic distribution of the test concerned due to the involvement of a kernel bandwidth in the leading term of the edgeworth expansion we are able to establish closed form expressions to explicitly represent the leading terms of both the size and power functions and then determine how the bandwidth should be chosen according to certain requirements for both the size and power functions for example when a significance level is given we can choose the bandwidth such that the power function is maximized while the size function is controlled by the significance level both asymptotic theory and methodology are established in addition we develop an easy implementation procedure for the practical realization of the established methodology and illustrate this on two simulated examples and a real data example

on the use of the variogram in checking for independence in spatial data the variogram is a standard tool in the analysis of spatial data and its shape provides useful information on the form of spatial correlation that may be present however it is also useful to be able to assess the evidence for the presence of any spatial correlation a method of doing this based on an assessment of whether the true function underlying the variogram is constant is proposed nonparametric smoothing of the squared differences of the observed variables on a suitably transformed scale is used to estimate variogram shape a statistic based on a ratio of quadratic forms is proposed and the test is constructed by investigating the distributional properties of this statistic under the assumption of an independent gaussian process the power of the test is investigated reference bands are proposed as a graphical follow up an example is discussed

integrated square error properties of kernel estimators of regression functions weak laws of large numbers and central limit theorems are proyed for integrated square error of kernel estimators of regression functions the regressor is assumed to take values in rp and the regressand x to be real valued it is shown that in many cases integrated square error is asymptotically normally distributed and independent of the x sample as an application a test for the regression function such as that proposed by konakov is seen to be asymptotically independent of an arbitrary test based on the xsample the proofs involve martingale methods

curve fitting by polynomial trigonometric regression summary we consider a method of estimating an unknown regression curve by regression on a combination of low order polynomial terms and trigonometric terms estimation based on trigonometric functions alone is known to suffer from bias problems at the boundaries due to the periodic nature of the fitted functions we show that these boundary problems are alleviated by adding low order polynomial terms the utility of the method is illustrated with examples and asymptotic results show that the estimators are competitive with other nonparametric procedures applications to estimation in semiparametric models are also considered with particular emphasis on the case of analysis of covariance when the relationship between the response and the covariate is difficult to model parametrically

biased and unbiased cross validation in density estimation abstract nonparametric density estimation requires the specification of smoothing parameters the demands of statistical objectivity make it highly desirable to base the choice on properties of the data set in this article we introduce some biased cross validation criteria for selection of smoothing parameters for kernel and histogram density estimators closely related to one investigated in scott and factor 1981 these criteria are obtained by estimating l 2 norms of derivatives of the unknown density and provide slightly biased estimates of the average squared l 2 error or mean integrated squared error these criteria are roughly the analog of wahba s 1981 generalized cross validation procedure for orthogonal series density estimators we present the relationship of the biased cross validation procedure to the least squares cross validation procedure which provides unbiased estimates of the mean integrated squared error both methods are shown to be based on u statistics we compare the two metho