time domain procedures for testing that a stationary time series is gaussian a class of time domain procedures for testing that a stationary time series is gaussian is presented and analyzed these tests are based on the deviations of the sample value of finite memory nonlinear transformations of the process from their ensemble averaged counterparts asymptotic distributions of these tests are derived under the null hypothesis of gaussianity and under a class of local and fixed alternatives specific tests are then developed based respectively on higher order moments and on the characteristic functions practical construction of the test statistics is discussed with a special emphasis on the estimation of the covariance of the sample statistics which appears to play a key role in the performance of the tests when dealing with small samples

on the asymptotic expansion of the empirical process of long memory moving averages let x n i 1 a i e n i where the e i are iid with mean 0 and finite fourth moment and the a i are regularly varying with index where 1 2 1 so that x n has long range dependence this covers an important class of the fractional arima process for r 0 let y n r n n 1 1 j1 jr r s 1 a js e n js y n 0 n 2 n r var y n r and f r the rth derivative of the distribution function of x n the y n r are uncorrelated and are stochastically decreasing in r for any positive integer p 2 1 1 it is shown under mild regularity conditions that with probability 1 n n 1 i x n x p r 0 1 r f r x y n r o n n p uniformly for all x r o 1 2 1 2 p 1 2 this generalizes a host of existing results and provides the vehicle for a number of statistical applications

construction algorithms for polynomial lattice rules for multivariate integration we introduce a new construction algorithm for digital nets for integration in certain weighted tensor product hilbert spaces the first weighted hilbert space we consider is based on walsh functions dick and pillichshammer calculated the worst case error for integration using digital nets for this space here we extend this result to a special construction method for digital nets based on polynomials over finite fields this result allows us to find polynomials which yield a small worst case error by computer search we prove an upper bound on the worst case error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights we extend the results for the weighted hilbert space based on walsh functions to weighted sobolev spaces in this case we use randomly digitally shifted digital nets the construction principle is the same as before only the worst case error is slightly different again digital nets obtained from our search algorithm yield a worst case error achieving the optimal rate of convergence and as before strong tractability holds under some condition on the weights these results show that such a construction of digital nets yields the until now best known results of this kind and that our construction methods are comparable to the construction methods known for lattice rules we conclude the article with numerical results comparing the expected worst case error for randomly digitally shifted digital nets with those for randomly shifted lattice rules

scrambled net variance for integrals of smooth functions hybrids of equidistribution and monte carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter one version randomized t m s nets has the property that the integral estimates are unbiased and that the variance is o 1 n for any square integrable integrand stronger assumptions on the integrand allow one to find rates of convergence this paper shows that for smooth integrands over s dimensions the variance is of order n 3 log n 3 1 compared to n 1 for ordinary monte carlo thus the integration errors are of order n 3 2 log n s 1 2 in probability this compares favorably with the rate n 1 log n s 1 for unrandomized t m s nets

on korobov lattice rules in weighted spaces this paper studies the error bounds of multivariate integration in weighted function spaces using lattice rules of the korobov form in which the generating vector for an n point rule with n prime has the form 1 a ad 1 mod n with the parameter a chosen optimally we establish new error bounds for korobov lattice rules in weighted korobov spaces in particular we prove that if the weights decay sufficiently fast the optimal korobov lattice rule has an error bound of order o n alpha 2 delta for arbitrary delta 0 with the implied constant depending at worst polynomially on the dimension here alpha 1 is the smoothness parameter of the weighted korobov spaces we generalize the construction to the case where n is a product of arbitrary distinct prime numbers with the purpose of reducing the construction cost without sacrificing much of the quality of the lattice rules a corresponding result is deduced for weighted sobolev spaces of nonperiodic functions using randomly shifted optimal korobov lattice rules a comparison of the worst case errors for korobov lattice rules and the recent component by component constructions is presented the investigation establishes the usefulness of shifted optimal korobov lattice rules for integration even in high dimensions if the weights which characterize the weighted spaces are suitably chosen

wkbj type approximation for finite moments perturbations of the differential equation y 0 and the analogous difference equation abstract wkbj or liouville green type approximation theorems are proved for differential equations like y g x y 0 x 1 and their discrete counterparts 2 y n g n y n 0 n v the coefficients g x g n having a finite second moment precise error bounds are obtained as in the already existing theory due to f w j olver for differential equations in other classes and in a previous work by the authors concerning certain difference equations when only the first moment is finite the same asymptotic approximation with bound still holds for recessive solutions an effort has been made to present both aspects the continuous and the discrete in a unified way basing the error analysis on a volterra stieltjes integral equation examples are given to illustrate the theory

the tent transformation can improve the convergence rate of quasi monte carlo algorithms using digital nets in this paper we investigate multivariate integration in reproducing kernel sobolev spaces for which the second partial derivatives are square integrable as quadrature points for our quasi monte carlo algorithm we use digital t m s nets over mathbb z 2 which are randomly digitally shifted and then folded using the tent transformation for this qmc algorithm we show that the root mean square worst case error converges with order 2 m 2 varepsilon for any 0 where 2m is the number of points a similar result for lattice rules has previously been shown by hickernell

testing for symmetry summary we introduce the characteristic symmetry function based on the characteristic function of the underlying distribution whose behaviour is indicative of symmetry or its absence a statistic is proposed for testing symmetry about an unspecified centre derived from the empirical characteristic symmetry function the statistic is readily computible it utilizes information in the empirical characteristic function over an interval and does not require the estimation of the centre of symmetry under general symmetry the asymptotic null distribution of the statistic is folded normal the empirical power for selected alternatives is studied by a small scale simulation and a numerical illustration is given

liouville green approximations for a class of linear oscillatory difference equations of the second order abstract an asymptotic approximation theorem is proved for the solutions of linear oscillatory three term recurrence equations in a certain class this result represents a discrete analogue of the well known liouville green or wkbj theorem rigorously proved by olver for second order linear differential equations as in olver s theorem precise estimates are obtained for the error terms and the double asymptotic nature of the approximation is made clear applications to certain orthogonal polynomials are also given

hypothesis testing in time series via the empirical characteristic function a generalized spectral density approach abstract the standardized spectral density completely describes serial dependence of a gaussian process for non gaussian processes however it may become an inappropriate analytic tool because it misses the nonlinear processes with zero autocorrelation by generalizing the concept of the standardized spectral density i propose a new spectral tool suitable for both linear and nonlinear time series analysis the generalized spectral density is indexed by frequency and a pair of auxiliary parameters it is well defined for both continuous and discrete random variables and requires no moment condition introduction of the auxiliary parameters renders the spectrum able to capture all pairwise dependencies including those with zero autocorrelation the standardized spectral density can be derived by properly differentiating the generalized spectral density with respect to the auxiliary parameters at the origin the consistency of a class of parzen s kernel type estimators for the generalized spectral dens

sublinear perturbations of the differential equation y n 0 and of the analogous difference equation abstract according to a result of a ghizzetti for any solution y t of the differential equation where y n t i 0 n 1 g i t y i t 0 t 1 1 g i x x n i 1 dx 0 i n 1 either y t 0 for t 1 or there is an integer r with 0 r n 1 such that lim t y t t r exists and 0 related results are obtained for difference and differential inequalities a special case of the former has interesting applications in the study of orthogonal polynomials

donsker type theorems for nonparametric maximum likelihood estimators let mathcal p be a nonparametric probability model consisting of smooth probability densities and let hat p n be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law mathbb p with hat mathbb p n denoting the measure induced by the density hat p n define the stochastic process hat nu n f longmapsto sqrt n int fd hat mathbb p n mathbb p where f ranges over some function class mathcal f we give a general condition for donsker classes mathcal f implying that the stochastic process hat nu n is asymptotically equivalent to the empirical process in the space ell infty mathcal f of bounded functions on mathcal f this implies in particular that hat nu n converges in law in ell infty mathcal f to a mean zero gaussian process we verify the general condition for a large family of donsker classes mathcal f we give a number of applications convergence of the probability measure hat mathbb p n to mathbb p at rate sqrt n in certain metrics metrizing the topology of weak star convergence a unified treatment of convergence rates of the mle in a continuous scale of sobolev norms sqrt n efficient estimation of nonlinear functionals defined on mathcal p limit theorems at rate sqrt n for the maximum likelihood estimator of the convolution product mathbb p ast p

a new approach to function based hypothesis testing in location scale families motivated by two applications in the mining industry we introduce a new approach to testing the hypothesis that two sampled distributions are simply location and scale changes of one another the test applicable to both paired data and two sample data is based on the empirical characteristic function more conventional techniques founded on the empirical distribution function suffer from serious drawbacks when used to test for location scale families in the motivating applications knowing that the distributions differ only in location and scale has significant operational and economic advantages enabling protocols for one type of data to be applied directly to another supplementary material in the form of matlab code is available online

examination of percentile residual life we initiate a nonparametric large sample estimation theory for the percentile residual lifetime we introduce the empirical percentile residual life process and determine its asymptotic distribution resulting in asymptotically distribution free confidence intervals and confidence bands for the percentile residual lifetime we illustrate the methods using strike duration data from the work of d c schmittlein and d g morrison

a test for normality based on the empirical characteristic function our aim in the present paper is to suggest a simple test of the composite hypothesis of normality against the alternative that the underlying distribution is long tailed the test is based on the behaviour of the empirical characteristic function in the neighbourhood of the origin and is very competitive with several well known tests for normality only a table of critical points is required for its implementation we begin by describing the philosophy behind the test let q be the characteristic function of the sampling distribution and set

strong approximations of some biometric estimates under random censorship the empirical distribution function has been widely used as an estimator for the distribution function of the elements of a random sample it is not however appropriate when the observations are incomplete this is the case for example when the observations have been randomly censored on the right in the latter case the so called product limit estimate of kaplan and meier 16 has been generally accepted as a substitute for the empirical distribution function and it possesses many of the same properties cf meier 19 developing the corresponding statement of efron 8 weak convergence over a finite interval of the kaplan meier product limit estimate to a gaussian process when there is censorship on the right has been obtained by breslow and crowley 3 cf also meier 19 breslow and crowley obtained their result via the limiting distribution of the closely related cumulative hazard process under random censorship a generalisation of the efron breslow crowley theorem was formulated by yang 24 for the case when censorship on the right is performed by more competing risks these results have numerous statistical applications in areas such as medical follow up studies life testing actuarial sciences and demography koziol and green 18 have computed tables for the limiting distribution of the cram6r von mises statistic based on the kaplan meier product limit estimate under a specific random censorship model a point will be stressed in this paper however that by a certain transformation which seems to have escaped many researchers attention and is due to efron 8 it is always possible to obtain the standard wiener process in the limit the distributions of many functionals of which are well known this transformation was only recently used in a stimulating note by gillespie and fisher 14 to construct asymptotic confidence bands for the unknown distribution function in the kaplan meier model a general scheme of random censorship can be defined in the following way let x be a real random variable with distribution function f t

estimating coefficient distributions in random coefficient regressions random coefficient regression models are important in representing linear models with heteroscedastic errors and in unifying the study of classical fixed effects and random effects linear models for prediction intervals and for bootstrapping in random coefficient regressions it is necessary to estimate the distributions of the random coefficients consistently we show that this is often possible and provide practical representative estimators of these distributions

a problem of hartman and wintner approximation for discrete perturbations abstract consider the second order self adjoint difference equation cn xn an fn xn 1 0 as a perturbation of the eventually disconjugate difference equation cn zn an zn 1 0 where cn 0 asymptotic approximation for the fundamental system of solutions of the perturbed equation are expressed explicitly in terms of the coefficients and the principal or recessive solution of the unperturbed equation in particular the coefficient cn is allowed to be oscillatory and we do not assume absolute summability conditions on fn results obtained are applied to a second order poincare difference equation whose unperturbed equation has a double characteristic root nonlinear and nonhomogeneous perturbations are also considered an example is given for illustration

a test of location for data with slowly decaying serial correlations summary slowly decaying serial correlations have if ignored disastrous effects on classical tests and confidence intervals we propose a modified t test for the location parameter which takes into account such correlations an approximate distribution function of the test statistic is given under the assumption of fractional gaussian noise the results are generalized to prediction intervals for a future average an example is given to illustrate the methods

the effect of dependence on chi squared tests of fit tests of fit that assume i i d observations may be affected by dependence among the observations this effect is studied for the pearson chi squared test when the data form a stationary stochastic process the general form of asymptotic distribution theory under the null hypothesis is outlined and examples are given for testing fit to a specified normal law it is shown that when observations come from a quite general class of gaussian stationary processes positive correlation among the observations is confounded with lack of normality

on some fourier methods for inference abstract common statistical procedures such as maximum likelihood and m estimation admit generalized representations in the fourier domain the fourier domain provides fertile ground for approaching a number of difficult problems in inference in particular the empirical characteristic function and its extension for stationary time series are shown to be fundamental tools which support numerically simple inference procedures having arbitrarily high asymptotic efficiency and certain robustness features as well a numerical illustration involving the symmetric stable laws is given

approximation of quasi monte carlo worst case error in weighted spaces of infinitely times smooth functions abstract in this paper we consider quasi monte carlo qmc worst case error of weighted smooth function classes in c 0 1 s by a digital net over f 2 we show that the ratio of the worst case error to the qmc integration error of an exponential function is bounded above and below by constants this result provides us with a simple interpretation that a digital net with small qmc integration error for an exponential function also gives the small integration error for any function in this function space

construction of interlaced polynomial lattice rules for infinitely differentiable functions we study multivariate integration over the s dimensional unit cube in a weighted space of infinitely differentiable functions it is known from a recent result by suzuki that there exists a good quasi monte carlo qmc rule which achieves a super polynomial convergence of the worst case error in this function space and moreover that this convergence behavior is independent of the dimension under a certain condition on the weights in this paper we provide a constructive approach to finding a good qmc rule achieving such a dimension independent super polynomial convergence of the worst case error specifically we prove that interlaced polynomial lattice rules with an interlacing factor chosen properly depending on the number of points n and the weights can be constructed using a fast component by component algorithm in at most o sn log n 2 arithmetic operations to achieve a dimension independent super polynomial convergence the key idea for the proof of the worst case error bound is to use a variant of jensen s inequality with a purposely designed concave function

a numerical method for evaluating zeros of solutions of second order linear differential equations a numerical algorithm for computing real zeros of solutions of 2ndorder linear differential equations y q x y 0 in the oscillatory case on a half line is studied the method applies to the class q x a b x o x p with a o ber p 1 this procedure is based on a certain nonlinear 3rd order equation kummer s equation which plays a role in the theory of transformations of 2nd order differential equations into each other and was earlier introduced by f w j olver in 1950 to compute zeros of cylinder functions a rigorous asymptotic and numerical analysis is developed by combining boruvka s approach to the study of kummer s equation and olver s original idea numerical examples are presented

quasi monte carlo point sets with small t values and wafom the t value of a t m s net is an important criterion of point sets for quasi monte carlo integration and many point sets are constructed in terms of the t values as this leads to small integration error bounds recently matsumoto saito and matoba proposed the walsh figure of merit wafom as a quickly computable criterion of point sets that ensures higher order convergence for function classes of very high smoothness in this paper we consider a search algorithm for point sets whose t value and wafom are both small so as to be effective for a wider range of function classes for this we fix digital t m s nets with small t values e g sobol or niederreiter xing nets in advance apply random linear scrambling and select scrambled digital t m s nets in terms of wafom experiments show that the resulting point sets improve the rates of convergence for smooth functions and are robust for non smooth functions

a test for normality based on the empirical characteristic function summary an omnibus test of normality is proposed which has high power against many alternative hypotheses the test uses a weighted integral of the squared modulus of the difference between the characteristic functions of the sample and of the normal distribution together with the one to one correspondence between t and f x suggest utilizing kn t in statistical inference here we use n t to test the composite hypothesis that f x is normal let 00 t exp it1 1t2 a2 be the characteristic function under the null hypothesis with 1u and u2 the unspecified mean and variance the test introduced here is based on a weighted integral over t of the squared modulus of kn t o t where 0 t depends on sample estimates of ju and a2 we compare the power of the test with that of prominent tests based on order statistics and on sample moments heathcote 1972 and feigin heathcote 1977 have previously considered using either the real or imaginary part alone of 4n t in tests of simple hypotheses their tests relied on the fact that for given t either component of 4n t is asymptotically normal with mean given by its population counterpart if the alternative hypothesis is also simple it may be possible to find a value of t that maximizes the power of the test in large samples epps singleton pulley 1982 showed how the sample moment generating function may be employed to test composite hypotheses that the data come from one or the other of two separate families of distributions murota takeuchi 1981 have recently proposed a location and scale invariant test based on the statistic an t i n t s 12 where s is the sample standard deviation applied to the problem of testing normality an t was found to have high power in the vicinity of t 10 against members of six families of symmetric alternatives but its

the stability test for symmetric alpha stable distributions symmetric alpha stable distributions are a popular statistical model for heavy tailed phenomena encountered in communications radar biomedicine and econometrics the use of the symmetric alpha stable model is often supported by empirical evidence where qualitative criteria are used to judge the fit leading to subjective decisions objective decisions can only be made through quantitative statistical tests here a goodness of fit hypothesis test for symmetric alpha stable distributions is developed based on their unique stability property critical values for the test are found using both asymptotic theory and from bootstrap estimates experiments show that the stability test using bootstrap estimates of the critical values is better able to discriminate between symmetric alpha stable distributions and other heavy tailed distributions than classical tests such as the kolmogorov smirnov test

asymptotic approximations for second order linear difference equations in banach algebras ii abstract a liouville green wkb asymptotic approximation theory is developed for some classes of linear second order difference equations in banach algebras the special case of linear matrix difference equations or equivalently of second order systems is emphasized rigorous and explicitly computable bounds for the error terms are obtained and this when both the sequence index and some parameter that may enter the coefficients go to infinity a simple application is made to orthogonal matrix polynomials in the nevai class

testing for normality in arbitrary dimension on etend le theoreme de convergence faible univariable de murota et takeuchi 1981 pour la transformee de mahalanobis de la fonction caracteristique empirique a d variables d 1

graphical exploratory data analysis 1 the role of graphics in data exploration 1 introduction 2 historical background 3 content of the book 4 central data sets 5 different types of data 6 computer programs 2 graphics for univariate and bivariate data 1 introduction 2 graphics for univariate data 3 stem and leaf plots 4 graphics for bivariate data 5 graphical perception 3 graphics for selecting a probability model 1 introduction 2 discrete models 3 continuous models 4 general 4 visual representation of multivariate data 1 introduction 2 scatterplots in more than two dimensions 3 profiles 4 star representations 5 glyphs 6 boxes 7 andrews curves 8 chernoff faces 9 general 5 cluster analysis 1 introduction 2 the probability approach 3 measures of distance and similarity 4 hierarchical cluster analysis 5 computer programs for hierarchical cluster analysis 6 digraphs 7 spanning trees 8 cluster analysis of variables 9 application of cluster analysis to fitness cholesterol data 10 other graphical techniques of cluster analysis 11 general 6 multidimensional scaling 1 introduction 2 the biplot 3 principal component analysis 4 correspondence analysis 5 classical metric scaling 6 non metric scaling 7 three way multidimensional scaling indscal 8 guttman s techniques 9 facet theory 10 partial order scalogram analysis 11 general 7 graphical representations in regression analysis 1 introduction 2 the scatterplot 3 residual plots 4 mallows q statistic 5 confidence and forecast bands 6 the ridge trace 7 general 8 chaid and xaid exploratory techniques for analyzing extensive data sets 1 introduction 2 chaid an exploratory technique for analyzing categorical data 3 applying a chaid analysis 4 xaid an exploratory technique for analyzing a quantitative dependent variable with categorical predictors 5 application of xaid analysis 6 general 9 control charts 1 introduction 2 process capability 3 control charts for items with quantitative characteristics 4 control charts for dichotomous measurements p chart 5 cumulative sum charts 6 cumulative sine charts 7 general 10 time series representations 1 representations in the time domain 2 representations in the frequency domain 11 further useful graphics 1 graphics for the two sample problem 2 graphical techniques in analysis of variance 3 four fold circular display of 2 x 2 contingency tables references inde

optimal quasi monte carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions we investigate quasi monte carlo rules for the numerical integration of multivariate periodic functions from besov spaces s r p q b mathbb t d sp qrb td with dominating mixed smoothness 1 p

a high performance computing and sensitivity analysis algorithm for stochastic many particle wave scattering we present a high performance computing framework for quantifying uncertainties in the propagation of acoustic waves through a stochastic media comprising a large number of three dimensional particles we subsequently describe an efficient postprocessing approach using our framework to statistically quantify the sensitivity of the uncertainties with respect to input parameters that govern the stochasticity in the model the stochasticity arises through the random positions and orientations of the component particles in the media simulation even for a single deterministic three dimensional configuration is inherently difficult because of the large number of particles the stochasticity leads to a larger dimensional model involving three spatial variables and additional stochastic variables and accounting for uncertainty in key parameters of the input probability distributions leads to prohibitive computational complexity in the first part of our paper we describe a high performance computing framework f

time domain tests for gaussianity and time reversibility statistical signal processing algorithms often rely upon gaussianity and time reversibility two important notions related to the probability structure of stationary random signals and their symmetry parametric models obtained via second order statistics sos are appropriate when the available data is gaussian and time reversible on the other hand evidence of nonlinearity non gaussianity or time irreversibility favors the use of higher order statistics hos in order to validate gaussianity and time reversibility and quantify the tradeoffs between sos and hos consistent time domain chi squared statistical tests are developed exact asymptotic distributions are derived to estimate the power of the tests including a covariance expression for fourth order sample cumulants a modification of existing linearity tests in the presence of additive gaussian noise is discussed briefly the novel gaussianity statistic is computationally attractive leads to a constant false alarm rate test and is well suited for parametric modeling because it employs the minimal hos lags which uniquely characterize arma processes simulations include comparisons with an existing frequency domain approach and an application to real seismic data time reversibility tests are also derived and their performance is analyzed both theoretically and experimentally

time varying periodicity in intraday volatility abstractwe develop a nonparametric test for whether return volatility exhibits time varying intraday periodicity using a long time series of high frequency data our null hypothesis commonly adopt

an explicit construction of optimal order quasi monte carlo rules for smooth integrands in a recent paper by the authors it is shown that there exists a quasi monte carlo qmc rule which achieves the best possible rate of convergence for numerical integration in a reproducing kernel hilbert space consisting of smooth functions in this paper we provide an explicit construction of such an optimal order qmc rule our approach is to exploit both the decay and the sparsity of the walsh coefficients of the reproducing kernel simultaneously this can be done by applying digit interlacing composition due to dick to digital nets with large minimum hamming and niederreiter rosenbloom tsfasman metrics due to chen and skriganov to our best knowledge our construction gives the first qmc rule which achieves the best possible convergence in this function space

duality theory and propagation rules for generalized digital nets digital nets are used in quasi monte carlo algorithms for approximating high dimensional integrals over the unit cube hence one wants to have explicit constructions of digital nets of high quality in this paper we consider the so called propagation rules for digital nets which state how one can obtain a new digital net of different size from existing digital nets this way one often can generate digital nets of higher quality than were previously known here we generalize existing propagation rules for classical digital nets to generalized digital nets as introduced by dick

slowly decaying correlations testing normality nuisance parameters abstract slowly decaying serial correlations can cause goodness of fit tests for a distribution to reject the null hypothesis with probability tending to one with increasing sample size when the null distribution is completely specified simple null hypothesis the problem is actually worse than for the case where there are nuisance parameters to be estimated composite null hypothesis in particular we consider here testing normality we discuss limit theorems and propose correction terms to be incorporated in certain goodness of fit statistics to improve the rates of convergence for the simple hypothesis case we show how the problem is solved automatically for the composite null hypothesis case when the nuisance parameters are estimated from the data simulations illustrate the results

the studentized empirical characteristic function and its application to test for the shape of distribution the empirical characteristic function is found to be effectively applied to test for the shape of distribution the squared modulus of the studentized empirical characteristic function is suggested for testing the composite hypothesis that mu sigma x is subject to a known distribution for unknown constants mu and sigma it is shown that the studentized empirical characteristic function if properly normalized converges weakly to a complex gaussian process asymptotic considerations as well as computer simulation reveal that the proposed statistic when applied to test normality is more efficient than or as efficient as the test by the sample kurtosis for certain types of alternatives

a comparison of optimized higher order spectral detection techniques for non gaussian signals using the gaussian noise rejection property of higher order spectra hos hos based detectors have been proposed that outperform conventional second order techniques in certain scenarios based on statistical tests proposed by suhha rao and gabr 1980 as well as hinich 1982 kletter and messer 1990 and hinich and wilson 1990 have developed similar bifrequency domain detectors that are dependent on bispectral estimates of the observation process formalizing the estimate consistency requirements and the asymptotics for these detectors we derive a new f test statistic we consider the detrimental effects of using spectral estimates in the denominator of hinich s test we determine refined conditional distributions for third and fourth order versions of his detector we also modify his test for colored scenarios extending the bispectral detectors to their kth order counterparts we calculate the optimal smoothing bandwidth to use in constructing the hos estimates producing the best detection performances for both our f test and hinich s test with our refined distributions these new bandwidths yield significant improvements in detector performance over previous results for the finite sample case our calculations characterize the tradeoff between the two detectors and demonstrate that a larger smoothing bandwidth than the one suggested by previous researchers should be used our calculations are verified using simulations for both white and colored cases

good interlaced polynomial lattice rules for numerical integration in weighted walsh spaces quadrature rules using higher order digital nets and sequences are known to exploit the smoothness of a function for numerical integration and to achieve an improved rate of convergence as compared to classical digital nets and sequences for smooth functions a construction principle of higher order digital nets and sequences based on a digit interlacing function was introduced in dick 2007 which interlaces classical digital nets or sequences whose number of components is a multiple of the dimension in this paper we study the use of polynomial lattice point sets for interlaced components we call quadrature rules using such point sets interlaced polynomial lattice rules we consider weighted walsh spaces containing smooth functions and derive two upper bounds on the worst case error for interlaced polynomial lattice rules both of which can be employed as a quality criterion for the construction of interlaced polynomial lattice rules we investigate the component by component construction and the korobov construction as a means of explicit constructions of good interlaced polynomial lattice rules that achieve the optimal rate of the worst case error through this approach we are able to obtain a good dependence of the worst case error bounds on the dimension under certain conditions on the weights while significantly reducing the construction cost as compared to higher order polynomial lattice rules

testing that a stationary time series is gaussian on propose une classe de procedures pour tester les hypotheses composites selon lesquelles un processus stochastique stationnaire est gaussien

a goodness of fit test based on the empirical characteristic function when parameters must be estimated summary the goodness of fit test using the empirical characteristic function proposed by koutrouvelis 1980 is extended to the case where parameters must be estimated the proposed test statistic is shown to have an asymptotic chi squared distribution with degrees of freedom reduced by the number of parameters estimated

higher order qmc petrov galerkin discretization for affine parametric operator equations with random field inputs we construct quasi monte carlo methods to approximate the expected values of linear functionals of petrov galerkin discretizations of parametric operator equations which depend on a possibly infinite sequence of parameters such problems arise in the numerical solution of differential and integral equations with random field inputs we analyze the regularity of the solutions with respect to the parameters in terms of the rate of decay of the fluctuations of the input field if p in 0 1 denotes the summability exponent corresponding to the fluctuations in affine parametric families of operators then we prove that deterministic interlaced polynomial lattice rules of order alpha lfloor 1 p rfloor 1 in s dimensions with n points can be constructed using a fast component by component algorithm in mathcal o alpha s n log n alpha 2 s 2 n operations to achieve a convergence rate of mathcal o n 1 p with the implied constant independent of s this dimension independen

testing normality a gmm approach abstract in this paper we consider testing marginal normal distributional assumptions more precisely we propose tests based on moment conditions implied by normality these moment conditions are known as the stein proceedings of the sixth berkeley symposium on mathematics statistics and probability vol 2 pp 583 602 equations they coincide with the first class of moment conditions derived by hansen and scheinkman econometrica 63 1995 767 when the random variable of interest is a scalar diffusion among other examples stein equation implies that the mean of hermite polynomials is zero the gmm approach we adopt is well suited for two reasons it allows us to study in detail the parameter uncertainty problem i e when the tests depend on unknown parameters that have to be estimated in particular we characterize the moment conditions that are robust against parameter uncertainty and show that hermite polynomials are special examples this is the main contribution of the paper the second reason for using gmm is that our tests are also valid for time series in this case we adopt a heteroskedastic autocorrelation consistent approach to estimate the weighting matrix when the dependence of the data is unspecified we also make a theoretical comparison of our tests with jarque and bera econom lett 6 1980 255 and opg regression tests of davidson and mackinnon estimation and inference in econometrics oxford university press oxford finite sample properties of our tests are derived through a comprehensive monte carlo study finally two applications to garch and realized volatility models are presented

the empirical process of some long range dependent sequences with an application to u statistics let x j infty j 1 be a stationary mean zero gaussian process with covariances r k ex k 1 x 1 satisfying r 0 1 and r k k d l k where d is small and l is slowly varying at infinity consider the two parameter empirical process for g x j bigg f n x t frac 1 n sum lbrack nt rbrack j 1 lbrack 1 g x j leq x p g x 1 leq x rbrack infty x infty 0 leq t leq 1 bigg where g is any measurable function noncentral limit theorems are obtained for f n x t and they are used to derive the asymptotic behavior of some suitably normalized von mises statistics and u statistics based on the g x j s the limiting processes are structurally different from those encountered in the i i d case

some new tests for multivariate normality a family of statistics is presented that can be used for testing goodness of fit to a parametric family these statistics include mardia s measure of multivariate kurtosis and moore and stubblebine s test for multivariate normality the asymptotic distribution of the statistics is found under mild hypotheses on the parametric family and in the case of multivariate normality the distribution is shown to be independent of the true parameter a class of tests for multivariate normality is presented and the performance of two such tests in the bivariate case is found in simulations

explicit constructions of quasi monte carlo rules for the numerical integration of high dimensional periodic functions in this paper we give explicit constructions of point sets in the s dimensional unit cube yielding quasi monte carlo algorithms which achieve the optimal rate of convergence of the worst case error for numerically integrating high dimensional periodic functions in the classical measure p alpha of the worst case error introduced by korobov the convergence for every even integer alpha ge 1 is of cal o n min alpha d log n s alpha 2 where d is a parameter of the construction which can be chosen arbitrarily large and n is the number of quadrature points this convergence rate is known to be the best possible up to some log n factors we prove the result for the deterministic and also a randomized setting the construction is based on a suitable extension of digital t m s nets over the finite field mathbb z b

applications of robust distances for regression the dd plot introduced by rousseeuw and van driessen 1999 is a plot of classical vs robust mahalanobis distances mdi vs rdi the dd plot can be used as a diagnostic for multivariate normality and elliptical symmetry and to assess the success of numerical transformations towards elliptical symmetry in the regression context many procedures can be adversely aected if strong nonlinearities are present in the predictors even if strong nonlinearities are present the robust distances can be used to help visualize important regression models such as generalized linear models

the mean square discrepancy of scrambled t s sequences the discrepancy arises in the worst case error analysis for quasi monte carlo quadrature rules low discrepancy sets yield good quadrature rules this article considers the mean square discrepancies for scrambled lam t m s nets and t s sequences in base b it is found that the mean square discrepancy for scrambled nets and sequences is never more than a constant multiple of that under simple monte carlo sampling if the reproducing kernel defining the discrepancy satisfies a lipschitz condition with respect to one of its variables separately then the asymptotic order of the root mean square discrepancy is o n 1 log n s 1 2 for scrambled nets if the reproducing kernel satisfies a lipschitz condition with respect to both of its variables then the asymptotic order of the root mean square discrepancy is o n 3 2 log n s 1 2 for scrambled nets for an arbitrary number of points taken from a t s sequence the root mean square discrepancy appears to be no better than o n 1 log n s 1 2 regardless of the smoothness of the reproducing kernel

constructing randomly shifted lattice rules in weighted sobolev spaces shifted rank 1 lattice rules a special class of quasi monte carlo methods have recently been proposed by the present authors for the integration of functions belonging to certain weighted sobolev spaces the shifts in these rules were generated in a deterministic manner in contrast in this paper we generate these shifts randomly this allows probabilistic estimates for the error in a given integral it also reduces the number of operations required to find the generating vectors for the underlying lattice rules component by component the rules thus constructed achieve a worst case strong tractability error bound in an average or probabilistic sense

a new graphical tool to detect non normality a new graphical tool for checking normality of independently and identically distributed observations based on the third derivative t 3 of the logarithm of the empirical moment generating function is proposed a significant departure of this function from the horizontal zero line is indicative of non normality behaviour in the neighbourhood of 0 indicates the type of departure from the normal distribution examples show that the t 3 method when used in conjunction with the classical probability plot can provide valuable insight into the data and be a powerful graphical tool

on bootstrap confidence intervals in nonparametric regression priate in this paper we point out some of the theoretical advantages of pivoting they include a reduction in the error of the bootstrap distribution function estimate from n 1 2 to n lh 1 2 where h denotes bandwidth and a reduction in coverage error of confidence intervals from either n l 1 2h 1 2 or n 1 2h1 2 depending on which nonpivotal method is used to n 1 several comparisons are drawn with the case of nonparametric density estimation where a pivotal approach also reduces errors associated with confidence intervals but where the orders of magnitude of the respective errors are quite different from their counterparts for nonparametric regression

normality based validation for crisp clustering we introduce a new validity index for crisp clustering that is based on the average normality of the clusters unlike methods based on inter cluster and intra cluster distances this index emphasizes the cluster shape by using a high order characterization of its probability distribution the normality of a cluster is characterized by its negentropy a standard measure of the distance to normality which evaluates the difference between the cluster s entropy and the entropy of a normal distribution with the same covariance matrix the definition of the negentropy involves the distribution s differential entropy however we show that it is possible to avoid its explicit computation by considering only negentropy increments with respect to the initial data distribution where all the points are assumed to belong to the same cluster the resulting negentropy increment validity index only requires the computation of covariance matrices we have applied the new index to an extensive set of artificial and real problems where it provides in general better results than other indices both with respect to the prediction of the correct number of clusters and to the similarity among the real clusters and those inferred

on the efficiency of empirical characteristic function procedures summary the asymptotic normality and arbitrarily high efficiency of some new statistical procedures based on the empirical characteristic function is established under general conditions

on residual empirical processes of stochastic regression models with applications to time series motivated by gaussian tests for a time series we are led to investigate the asymptotic behavior of the residual empirical processes of stochastic regression models these models cover the fixed design regression models as well as general ar q models since the number of the regression coefficients is allowed to grow as the sample size increases the obtained results are also applicable to nonlinear regression and stationary ar models in this paper we first derive an oscillation like result for the residual empirical process then we apply this result to autoregressive time series in particular for a stationary ar process we are able to determine the order of the number of coefficients of a fitted ar q n model and obtain the limiting gaussian processes for an unstable ar q process we show that if the characteristic polynomial has a unit root 1 then the limiting process is no longer gaussian for the explosive case one of our side results also provides a short proof for the brownian bridge results given by koul and levental

lattice rules for nonperiodic smooth integrands the aim of this paper is to show that one can achieve convergence rates of n alpha delta n for alpha 1 2 1 2 and for delta 0 0 arbitrarily small for nonperiodic alpha smooth cosine series using lattice rules without random shifting the smoothness of the functions can be measured by the decay rate of the cosine coefficients for a specific choice of the parameters the cosine series space coincides with the unanchored sobolev space of smoothness 1 we study the embeddings of various reproducing kernel hilbert spaces and numerical integration in the cosine series function space and show that by applying the so called tent transformation to a lattice rule one can achieve the almost optimal rate of convergence of the integration error the same holds true for symmetrized lattice rules for the tensor product of the direct sum of the korobov space and cosine series space but with a stronger dependence on the dimension in this case

riccati techniques and approximation for a second order poincare difference equation approximation results are obtained for second order self adjoint difference equations and applied to a second order poincare difference equation whose unperturbed equation has a double characteristic root

on the step by step construction of quasi monte carlo integration rules that achieve strong tractability error bounds in weighted sobolev spaces we develop and justify an algorithm for the construction of quasi monte carlo qmc rules for integration in weighted sobolev spaces the rules so constructed are shifted rank 1 lattice rules the parameters characterising the shifted lattice rule are found component by component the d 1 th component of the generator vector and the shift are obtained by successive 1 dimensional searches with the previous d components kept unchanged the rules constructed in this way are shown to achieve a strong tractability error bound in weighted sobolev spaces a search for n point rules with n prime and all dimensions 1 to d requires a total cost of o n3d2 operations this may be reduced to o n3d operations at the expense of o n2 storage numerical values of parameters and worst ease errors are given for dimensions up to 40 and n up to a few thousand the worst case errors for these rules are found to be much smaller than the theoretical bounds

bootstrapping bispectra an application to testing for departure from gaussianity of stationary signals we propose a bootstrap version of a bispectrum based test for departure from gaussianity that achieves high power while maintaining the level of significance even for small sample sizes the proposed procedure can be also used to set confidence bands for a measure of the bicoherence of stationary random signals

estimation of regression coefficients with the help of characteristic functions abstract in this paper some easily applicable estimators of the regression coefficient in a simple regression model with errors in variables are derived usually one uses sample moments to estimate the regression coefficient however we use empirical characteristic functions in several points besides theoretical derivations a small study with generated data is discussed this study compares our method with estimating methods which use higher order moments

on the koziol green model for random censorship summary in the koziol green 1976 model of random censorship the survival distribution of the censoring variables is some power the censoring parameter of the survival distribution of the lifetimes using a strong approximation result of burke csorgo and horvath we construct empirical exact confidence bands for the life distribution of the censored variable and a randomly efron transformed variant of the usual product limit process is shown to converge weakly to the brownian motion process on any interval 0 t when the censoring parameter is also estimated from the sample the technique automatically gives rates of convergence for a wide class of functionals including the cramer von mises functional whose limit theory is worked out in detail the latter goodness of fit statistic is applied to reexamine the hospital data of koziol green 1976 to test whether oestrogen treatment for prostatic cancer was effective or not for the cancer itself while it caused censoring deaths of cardiovascular diseases