a causality in variance test and its application to financial market prices abstract this paper develops a test for causality in variance the test is based on the residual cross correlation function ccf and is robust to distributional assumptions asymptotic normal and asymptotic 2 statistics are derived under the null hypothesis of no causality in variance monte carlo results indicate that the proposed ccf test has good empirical size and power properties two empirical examples illustrate that the causality test yields useful information on the temporal dynamics and the interaction between two time series

a new twist two step iterative shrinkage thresholding algorithms for image restoration iterative shrinkage thresholding 1st algorithms have been recently proposed to handle a class of convex unconstrained optimization problems arising in image restoration and other linear inverse problems this class of problems results from combining a linear observation model with a nonquadratic regularizer e g total variation or wavelet based regularization it happens that the convergence rate of these 1st algorithms depends heavily on the linear observation operator becoming very slow when this operator is ill conditioned or ill posed in this paper we introduce two step 1st twist algorithms exhibiting much faster convergence rate than 1st for ill conditioned problems for a vast class of nonquadratic convex regularizers lscrp norms some besov norms and total variation we show that twist converges to a minimizer of the objective function for a given range of values of its parameters for noninvertible observation operators we introduce a monotonic version of twist mtwist although the convergence proof does not apply to this scenario we give experimental evidence that mtwist exhibits similar speed gains over ist the effectiveness of the new methods are experimentally confirmed on problems of image deconvolution and of restoration with missing samples

joint and marginal specification tests for conditional mean and variance models this article proposes a class of joint and marginal spectral diagnostic tests for parametric conditional means and variances of linear and nonlinear time series models the use of joint and marginal tests is motivated from the fact that marginal tests for the conditional variance may lead to misleading conclusions when the conditional mean is misspecified the new tests are based on a generalized spectral approach and do not need to choose a lag order depending on the sample size or to smooth the data moreover the proposed tests are robust to higher order dependence of unknown form in particular to conditional skewness and kurtosis it turns out that the asymptotic null distributions of the new tests depend on the data generating process hence we implement the tests with the assistance of a wild bootstrap procedure a simulation study compares the finite sample performance of the proposed and competing tests and shows that our tests can play a valuable role in time series modeling finally an application to the s p 500 highlights the merits of our approach

on maximum likelihood estimation of the differencing parameter of fractionally integrated noise with unknown mean abstract there are two approaches to maximum likelihood ml estimation of the parameter of fractionally integrated noise approximate frequency domain ml fox and taqqu 1986 and exact time domain ml sowell 1992b if the mean of the process is known then a clear finite sample mean squared error ranking of the estimators emerges the exact time domain estimator is superior we show in this paper however that the finite sample efficiency of approximate frequency domain ml relative to exact time domain ml rises dramatically when the mean is unknown and so must be estimated the intuition for our result is straightforward the frequency domain ml estimator is invariant to the true but unknown mean of the process while the time domain ml estimator is not feasible time domain estimation must therefore be based upon de meaned data but the long memory associated with fractional integration makes precise estimation of the mean difficult we conclude that the frequency domain estimator is an attractive and efficient alternative for situations in which large sample sizes render time domain estimation impractical

weak convergence of the sequential empirical processes of residuals in arma models this paper studies the weak convergence of the sequential empirical process hat k n of the estimated residuals in arma p q models when the errors are independent and identically distributed it is shown that under some mild conditions hat k n converges weakly to a kiefer process the weak convergence is discussed for both finite and infinite variance time series models an application to a change point problem is considered

non negativity conditions for the hyperbolic garch model abstract in this article we derive conditions which ensure the non negativity of the conditional variance in the hyperbolic garch p d q hygarch model of davidson 2004 the conditions are necessary and sufficient for p 1 and sufficient for p 2 and emerge as natural extensions of the inequality constraints derived in nelson and cao 1992 and tsai and chan 2008 for the garch model and in conrad and haag 2006 for the figarch model as a by product we obtain a representation of the arch coefficients which allows computationally efficient multi step ahead forecasting of the conditional variance of a hygarch process we also relate the necessary and sufficient parameter set of the hygarch to the necessary and sufficient parameter sets of its garch and figarch components finally we analyze the effects of erroneously fitting a figarch model to a data sample which was truly generated by a hygarch process empirical applications of the hygarch 1 d 1 model to daily nyse and dax30 data illustrate the importance of our results

testing for arch in the presence of a possibly misspecified conditional mean abstract ever since the development of the autoregressive conditional heteroskedasticity arch model engle 1982 testing for the presence of arch has become a routine diagnostic one popular method of testing for arch is t times the r2 from a regression of the squared residuals on p of its lags this test has been shown to have a lagrange multiplier interpretation and is asymptotically distributed as a 2 p random variable underlying this test is the assumption of a correctly specified conditional mean in this paper we consider the properties of the arch test when there is a possibly misspecified conditional mean examples of misspecification include omitted variables structural change and parameter instability we show that in general misspecification will lead to overrejection of the null hypothesis of conditional homoskedasticity we demonstrate the use of recursive residuals to improve the fit of a first stage conditional mean regression we illustrate these results via monte carlo simulations and consider two empirical examples

analysis of multiresolution image denoising schemes using generalized gaussian and complexity priors research on universal and minimax wavelet shrinkage and thresholding methods has demonstrated near ideal estimation performance in various asymptotic frameworks however image processing practice has shown that universal thresholding methods are outperformed by simple bayesian estimators assuming independent wavelet coefficients and heavy tailed priors such as generalized gaussian distributions ggds in this paper we investigate various connections between shrinkage methods and maximum a posteriori map estimation using such priors in particular we state a simple condition under which map estimates are sparse we also introduce a new family of complexity priors based upon rissanen s universal prior on integers one particular estimator in this class outperforms conventional estimators based on earlier applications of the minimum description length mdl principle we develop analytical expressions for the shrinkage rules implied by ggd and complexity priors this allows us to show the equivalence between universal hard thresholding map estimation using a very heavy tailed ggd and mdl estimation using one of the new complexity priors theoretical analysis supported by numerous practical experiments shows the robustness of some of these estimates against mis specifications of the prior a basic concern in image processing applications

on estimation of a regression model with long memory stationary errors we consider estimation of a regression model with long memory stationary errors first we estimate the regression parameters by the least squares estimator lse and next those describing the correlation structure of the error terms by using the residuals obtained from the lse certain regularity conditions introduced to develop the asymptotic theory no longer hold in this model in this situation we derive asymptotic properties of the preceding estimation procedure

second p descents on elliptic curves let p be a prime and let c be a genus one curve over a number field k representing an element of order dividing p in the shafarevich tate group of its jacobian we describe an algorithm which computes the set of d in the shafarevich tate group such that pd c and obtains explicit models for these d as curves in projective space this leads to a practical algorithm for performing explicit 9 descents on elliptic curves over q

mixture densities maximum likelihood and the em algorithm the problem of estimating the parameters which determine a mixture density has been the subject of a large diverse body of literature spanning nearly ninety years during the last two decades the

inconsistency of the maximum likelihood estimator of a distribution having increasing failure rate average abstract marshall and proschan showed that the mle for a life distribution with increasing failure rate is strongly consistent ann math statist 1965 36 69 77 in this note the author shows that the mle for a life distribution with increasing failure rate average is not consistent in fact maximum likelihood estimation in the ifra case yields estimators of the average failure rate and of the distribution function which in general coverage a s to values other than the true values in the decreasing failure rate average case the mle fails to exist

garch models without positivity constraints exponential or log garch this paper provides a probabilistic and statistical comparison of the log garch and egarch models which both rely on multiplicative volatility dynamics without positivity constraints we compare the main probabilistic properties strict stationarity existence of moments tails of the egarch model which are already known with those of an asymmetric version of the log garch the quasi maximum likelihood estimation of the log garch parameters is shown to be strongly consistent and asymptotically normal similar estimation results are only available for particular egarch models and under much stronger assumptions the comparison is pursued via simulation experiments and estimation on real data

r estimation in semiparametric dynamic location scale models we propose rank based estimation r estimators as an alternative to gaussian quasi likelihood and standard semiparametric estimation in time series models where conditional location and or scale depend on a euclidean parameter of interest while the unspecified innovation density is a nuisance we show how to construct r estimators achieving semiparametric efficiency at some predetermined reference density while preserving root n consistency and asymptotic normality irrespective of the actual density contrary to the standard semiparametric estimators our r estimators neither require tangent space calculations nor innovation density estimation numerical examples illustrate their good performances on simulated and real data

image denoising via sparse and redundant representations over learned dictionaries we address the image denoising problem where zero mean white and homogeneous gaussian additive noise is to be removed from a given image the approach taken is based on sparse and redundant representations over trained dictionaries using the k svd algorithm we obtain a dictionary that describes the image content effectively two training options are considered using the corrupted image itself or training on a corpus of high quality image database since the k svd is limited in handling small image patches we extend its deployment to arbitrary image sizes by defining a global image prior that forces sparsity over patches in every location in the image we show how such bayesian treatment leads to a simple and effective denoising algorithm this leads to a state of the art denoising performance equivalent and sometimes surpassing recently published leading alternative denoising methods

inference on the cointegration rank in fractionally integrated processes abstract for univariate time series we suggest a new variant of efficient score tests against fractional alternatives this test has three important merits first by means of simulations we observe that it is superior in terms of size and power in some situations of practical interest second it is easily understood and implemented as a slight modification of the dickey fuller test although our score test has a limiting normal distribution third and most important our test generalizes to multivariate cointegration tests just as the dickey fuller test does thus it allows to determine the cointegration rank of fractionally integrated time series it does so by solving a generalized eigenvalue problem of the type proposed by johansen j econ dyn control 12 1988 231 however the limiting distribution of the corresponding trace statistic is 2 where the degrees of freedom depend only on the cointegration rank under the null hypothesis the usefulness of the asymptotic theory for finite samples is established in a monte carlo experiment

testing for multivariate volatility functions using minimum volume sets and inverse regression we propose two new types of nonparametric tests for investigating multivariate regression functions the tests are based on cumulative sums coupled with either minimum volume sets or inverse regression ideas involving no multivariate nonparametric regression estimation the methods proposed facilitate the investigation for different features such as if a multivariate regression function is i constant ii of a bathtub shape and iii in a given parametric form the inference based on those tests may be further enhanced through associated diagnostic plots although the potential use of those ideas is much wider we focus on the inference for multivariate volatility functions in this paper i e we test for i heteroscedasticity ii the so called smiling effect and iii some parametric volatility models the asymptotic behavior of the proposed tests is investigated and practical feasibility is shown via simulation studies we further illustrate our methods with real financial data

long run behaviour of the autocovariance function of arch models abstract the asymptotic properties of the memory structure of arch equations are investigated this asymptotic analysis is achieved by expressing the autocovariance function of arch equations as the solution of a linear volterra summation equation and analysing the properties of an associated resolvent equation via the admissibility theory of linear volterra operators it is shown that the autocovariance function decays subexponentially or geometrically if and only if the kernel of the resolvent equation has the same decay property it is also shown that upper subexponential bounds on the autocovariance function result if and only if similar bounds apply to the kernel

quasi maximum likelihood estimation and bootstrap inference in fractional time series models with heteroskedasticity of unknown form we consider the problem of conducting estimation and inference on the parameters of univariate heteroskedastic fractionally integrated time series models we first extend existing results in the literature developed for conditional sum of squares estimators in the context of parametric fractional time series models driven by conditionally homoskedastic shocks to allow for conditional and unconditional heteroskedasticity both of a quite general and unknown form global consistency and asymptotic normality are shown to still obtain however the covariance matrix of the limiting distribution of the estimator now depends on nuisance parameters derived both from the weak dependence and heteroskedasticity present in the shocks we then investigate classical methods of inference based on the wald likelihood ratio and lagrange multiplier tests for linear hypotheses on either or both of the long and short memory parameters of the model the limiting null distributions of these test statistics are shown to be non pivotal under heteroskedasticity while that of a robust wald statistic based around a sandwich estimator of the variance is pivotal we show that wild bootstrap implementations of the tests deliver asymptotically pivotal inference under the null we demonstrate the consistency and asymptotic normality of the bootstrap estimators and further establish the global consistency of the asymptotic and bootstrap tests under fixed alternatives monte carlo simulations highlight significant improvements in finite sample behavior using the bootstrap in both heteroskedastic and homoskedastic environments our theoretical developments and monte carlo simulations include two bootstrap algorithms which are based on model estimates obtained either under the null hypothesis or unrestrictedly our simulation results suggest that the former is preferable to the latter displaying superior size control yet largely comparable power

feature selection based on the approximation of class densities by finite mixtures of special type abstract a new method of feature selection based on the approximation of class conditional densities by a mixture of parameterized densities of a special type suitable especially for multimodal data is presented no search procedure is needed when using the proposed method its performance is tested both on real simulated data

the sure let approach to image denoising we propose a new approach to image denoising based on the image domain minimization of an estimate of the mean squared error stein s unbiased risk estimate sure unlike most existing denoising algorithms using the sure makes it needless to hypothesize a statistical model for the noiseless image a key point of our approach is that although the nonlinear processing is performed in a transformed domain typically an undecimated discrete wavelet transform but we also address nonorthonormal transforms this minimization is performed in the image domain indeed we demonstrate that when the transform is a ldquotightrdquo frame an undecimated wavelet transform using orthonormal filters separate subband minimization yields substantially worse results in order for our approach to be viable we add another principle that the denoising process can be expressed as a linear combination of elementary denoising processes linear expansion of thresholds let armed with the sure and let principles we show that a denoising algorithm merely amounts to solving a linear system of equations which is obviously fast and efficient quite remarkably the very competitive results obtained by performing a simple threshold image domain sure optimized on the undecimated haar wavelet coefficients show that the sure let principle has a huge potential

the difference between the weil height and the canonical height on elliptic curves estimates for the difference of the weil height and the canonical height of points on elliptic curves are used for many purposes both theoretical and computational in this note we give an explicit estimate for this difference in terms of the j invariant and discriminant of the elliptic curve the method of proof suggested by serge lang is to use the decomposition of the canonical height into a sum of local heights we illustrate one use for our estimate by computing generators for the mordell weil group in three examples let e be an elliptic curve defined over a number field k say given by a weierstrass equation 1 y2 x ax b with a and b in the ring of integers of k the canonical height on e is a quadratic form h e k r for the definition and basic properties of h see 10 chapter viii 9 or 6 chapter vi the canonical height is determined by this property together with the fact that the difference

stationarity of garch processes and of some nonnegative time series abstract two time series models are considered garch processes and generalized multivariate autoregressive equations x n 1 a n 1 x n b n 1 with nonnegative i i d coefficients in each case a necessary and sufficient condition ensuring the existence of a strictly stationary solution is given

inconsistency of the mle and inference based on weighted ls for larch models this paper considers a class of finite order autoregressive linear arch models the model captures the leverage effect allows the volatility to be arbitrarily close to zero and to reach its minimum for non zero innovations and is appropriate for long memory modeling when infinite orders are allowed however the quasi maximum likelihood estimator is in general inconsistent a self weighted least squares estimator is proposed and is shown to be asymptotically normal a score test for conditional homoscedasticity and diagnostic portmanteau tests are developed their performance is illustrated via simulation experiments it is also investigated whether stock market returns exhibit some of the characteristic features of the linear arch model

a new hyperbolic garch model abstract there are two commonly used hyperbolic garch processes the figarch and hygarch processes in modeling the long range dependence in volatility however the figarch process always has infinite variance and the hygarch model has a more complicated form this paper builds a simple bridge between a common garch model and an integrated garch model and hence a new hyperbolic garch model along the lines of figarch models the new model remedies the drawback of figarch processes by allowing the existence of finite variance as in hygarch models while it has a form nearly as simple as the figarch model two inference tools including the gaussian qmle and a portmanteau test for the adequacy of the fitted model are derived and an easily implemented test for hyperbolic memory is also constructed their finite sample performances are evaluated by simulation experiments and an empirical example gives further support to our new model

self weighted and local quasi maximum likelihood estimators for arma garch igarch models the limit distribution of the quasi maximum likelihood estimator qmle for parameters in the arma garch model remains an open problem when the process has infinite 4th moment we propose a self weighted qmle and show that it is consistent and asymptotically normal under only a fractional moment condition based on this estimator the asymptotic normality of the local qmle is established for the arma model with garch finite variance and igarch errors using the self weighted and the local qmles we construct wald statistics for testing linear restrictions on the parameters and their limiting distributions are given in addition we show that the tail index of the igarch process is always 2 which is independently of interest

testing for pairwise serial independence via the empirical distribution function built on skaug and tjostheim s approach this paper proposes a new test for serial independence by comparing the pairwise empirical distribution functions of a time series with the products of its marginals for various lags where the number of lags increases with the sample size and different lags are assigned different weights typically the more recent information receives a larger weight the test has some appealing attributes it is consistent against all pairwise dependences and is powerful against alternatives whose dependence decays to zero as the lag increases although the test statistic is a weighted sum of degenerate cramer von mises statistics it has a null asymptotic n 0 1 distribution the test statistic and its limit distribution are invariant to any order preserving transformation the test applies to time series whose distributions can be discrete or continuous with possibly infinite moments finally the test statistic only involves ranking the observations and is computationally simple it has the advantage of avoiding smoothed nonparametric estimation a simulation experiment is conducted to study the finite sample performance of the proposed test in comparison with some related tests

the detection and estimation of long memory in stochastic volatility abstract we propose a new time series representation of persistence in conditional variance called a long memory stochastic volatility lmsv model the lmsv model is constructed by incorporating an arfima process in a standard stochastic volatility scheme strongly consistent estimators of the parameters of the model are obtained by maximizing the spectral approximation to the gaussian likelihood the finite sample properties of the spectral likelihood estimator are analyzed by means of a monte carlo study an empirical example with a long time series of stock prices demonstrates the superiority of the lmsv model over existing short memory volatility models

a test of the null of integer integration against the alternative of fractional integration abstract this paper proposes a test of the null of integer integration against the alternative of fractional integration the null of integer integration is satisfied if the series is either i 0 or i 1 we reject the null if the kpss test rejects i 0 and a unit root test rejects i 1 we suggest a new unit root test a lower tail kpss test applied to the differenced data to use in this procedure we provide critical values under standard asymptotics and fixed b asymptotics we prove the consistency of this testing procedure against i d alternatives with 0 d 1 and simulation evidence on the size and power of the test in finite samples is provided

testing for a slowly changing level with special reference to stochastic volatility abstract a test for the presence of a stationary first order autoregressive process embedded in white noise is constructed so as to be relatively powerful when the autoregressive parameter is close to one the test statistic is shown to have a cramer von mises distribution in large samples a comparison is made with some standard tests for serial correlation in the context of a stochastic volatility model and the proposed test is shown to have relatively high power for the parameter values typically found with daily financial time series it is recommended that tests on returns be carried out using absolute values

fractionally integrated generalized autoregressive conditional heteroskedasticity abstract the new class of fractionally integrated generalized autoregressive conditionally heteroskedastic figarch processes is introduced the conditional variance of the process implies a slow hyperbolic rate of decay for the influence of lagged squared innovations unlike i d processes for the mean maximum likelihood estimates mle of the figarch parameters are argued to be t 1 2 consistent the small sample behavior of an approximate mle procedure is assessed through a simulation study which also documents how the estimation of a standard garch model tends to produce integrated or igarch like estimates an empirical example with daily deutschmark u s dollar exchange rates illustrates the practical relevance of the new figarch specification

an inconsistent maximum likelihood estimate abstract an example is given of a family of distributions on 1 1 with a continuous one dimensional parameterization that joins the triangular distribution when 0 to the uniform when 1 for which the maximum likelihood estimates exist and converge strongly to 1 as the sample size tends to infinity whatever be the true value of the parameter a modification that satisfies cramer s conditions is also given

bivariate shrinkage functions for wavelet based denoising exploiting interscale dependency most simple nonlinear thresholding rules for wavelet based denoising assume that the wavelet coefficients are independent however wavelet coefficients of natural images have significant dependencies we only consider the dependencies between the coefficients and their parents in detail for this purpose new non gaussian bivariate distributions are proposed and corresponding nonlinear threshold functions shrinkage functions are derived from the models using bayesian estimation theory the new shrinkage functions do not assume the independence of wavelet coefficients we show three image denoising examples in order to show the performance of these new bivariate shrinkage rules in the second example a simple subband dependent data driven image denoising system is described and compared with effective data driven techniques in the literature namely visushrink sureshrink bayesshrink and hidden markov models in the third example the same idea is applied to the dual tree complex wavelet coefficients

testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression abstract broad classes of diagnostics for serial correlation and or dynamic conditional heteroskedasticity of regression disturbances are considered the classes include statistics with good power against strongly dependent alternatives along with the usual ones that test against weak dependence and many others limiting null distributions are obtained under mild conditions on the dependence structure of the alternative against which the test is derived on moments of the disturbances and on the regressors the various test statistics have a similar overall structure and while tests against strongly dependent alternatives entail more computation than ones against weakly dependent alternatives the difference can be slight if the fast fourier transform is used

a semiparametric stochastic volatility model abstract in this paper the correlation structure in the classical leverage stochastic volatility sv model is generalized based on a linear spline in the new model the correlation between the return and volatility innovations is time varying and depends nonparametrically on the type of news arrived to the market theoretical properties of the proposed model are examined the model estimation and comparison are conducted by bayesian methods the performance of the estimates are examined in simulations the new model is fitted to daily and weekly us data and compared with the classical sv and garch models in terms of their in sample and out of sample performances empirical results suggest evidence in favor of the proposed model in particular the new model finds strong evidence of time varying leverage effect in individual stocks when the classical model fails to identify the leverage effect

non standard inference for augmented double autoregressive models with null volatility coefficients abstract this paper considers an augmented double autoregressive dar model which allows null volatility coefficients to circumvent the over parameterization problem in the dar model since the volatility coefficients might be on the boundary the statistical inference methods based on the gaussian quasi maximum likelihood estimation gqmle become non standard and their asymptotics require the data to have a finite sixth moment which narrows the applicable scope in studying heavy tailed data to overcome this deficiency this paper develops a systematic statistical inference procedure based on the self weighted gqmle for the augmented dar model except for the lagrange multiplier test statistic the wald quasi likelihood ratio and portmanteau test statistics are all shown to have non standard asymptotics the entire procedure is valid as long as the data are stationary and its usefulness is illustrated by simulation studies and one real example

computing rational points on rank 1 elliptic curves via l series and canonical heights let e q be an elliptic curve of rank 1 we describe an algorithm which uses the value of l e 1 and the theory of canonical heghts to efficiently search for points in e q and e z s for rank 1 elliptic curves e q of moderately large conductor say on the order of 10 7 to 10 10 and with a generator having moderately large canonical height say between 13 and 50 our algorithm is the first practical general purpose method for determining if the set e z s contains non torsion points

optimal predictions of powers of conditionally heteroscedastic processes in conditionally heteroskedastic models the optimal prediction of powers or logarithms of the absolute process has a simple expression in terms of the volatility process and an expectation involving the independent process a standard procedure for estimating this prediction is to estimate the volatility by gaussian quasi maximum likelihood qml in a first step and to use empirical means based on rescaled innovations to estimate the expectation in a second step this paper proposes an alternative one step procedure based on an appropriate non gaussian qml estimation of the model and establishes the asymptotic properties of the two approaches their performances are compared for finite order garch models and for the infinite arch for the standard garch p q and the asymmetric power garch p q it is shown that the are of the estimators only depends on the prediction problem and some moments of the independent process an application to indexes of major stock exchanges is proposed

estimating the innovation distribution in nonparametric autoregression we prove a bahadur representation for a residual based estimator of the innovation distribution function in a nonparametric autoregressive model the residuals are based on a local linear smoother for the autoregression function our result implies a functional central limit theorem for the residual based estimator

sign tests for long memory time series abstract this paper proposes sign based tests for simple and composite hypotheses on the long memory parameter of a time series process the tests allow for nonstationary hypothesis such as unit root as well as for stationary hypotheses such as weak dependence or no integration the proposed generalized lagrange multiplier sign tests for simple hypotheses on the long memory parameter are exact and locally optimal among those in their class we also propose tests for composite hypotheses on the parameters of arfima p d q processes the resulting tests statistics have a standard normal limiting distribution under the null hypothesis

nonparametric model checks for time series this paper studies a class of tests useful for testing the goodness of fit of an autoregressive model these tests are based on a class of empirical processes marked by certain residuals the paper first gives their large sample behavior under null hypotheses then a martingale transformation of the underlying process is given that makes tests based on it asymptotically distribution free consistency of these tests is also discussed briefly

statistical inference for time varying arch processes in this paper the class of arch oo models is generalized to the nonstationary class of arch oo models with time varying coefficients for fixed time points a stationary approximation is given leading to the notation locally stationary arch oo process the asymptotic properties of weighted quasi likelihood estimators of time varying arch p processes p oo are studied including asymptotic normality in particular the extra bias due to nonstationarity of the process is investigated moreover a taylor expansion of the nonstationary arch process in terms of stationary processes is given and it is proved that the time varying arch process can be written as a time varying volterra series

multiscale representations of markov random fields recently a framework for multiscale stochastic modeling was introduced based on coarse to fine scale recursive dynamics defined on trees this model class has some attractive characteristics which lead to extremely efficient statistically optimal signal and image processing algorithms the authors show that this model class is also quite rich in particular they describe how 1 d markov processes and 2 d markov random fields mrfs can be represented within this framework the recursive structure of 1 d markov processes makes them simple to analyze and generally leads to computationally efficient algorithms for statistical inference on the other hand 2 d mrfs are well known to be very difficult to analyze due to their noncausal structure and thus their use typically leads to computationally intensive algorithms for smoothing and parameter identification in contrast their multiscale representations are based on scale recursive models and thus lead naturally to scale recursive algorithms which can be substantially more efficient computationally than those associated with mrf models in 1 d the multiscale representation is a generalization of the midpoint deflection construction of brownian motion the representation of 2 d mrfs is based on a further generalization to a midline deflection construction the exact representations of 2 d mrfs are used to motivate a class of multiscale approximate mrf models based on one dimensional wavelet transforms they demonstrate the use of these latter models in the context of texture representation and in particular they show how they can be used as approximations for or alternatives to well known mrf texture models

diagnostic checking in arma models with uncorrelated errors we consider tests for lack of fit in arma models with nonindependent innovations in this framework the standard box pierce and ljung box portmanteau tests can perform poorly specifically the usual text book formulas for asymptotic distributions are based on strong assumptions and should not be applied without careful consideration in this article we derive the asymptotic covariance matrix of a vector of autocorrelations for residuals of arma models under weak assumptions on the noise the asymptotic distribution of the portmanteau statistics follows a consistent estimator of and a modification of the portmanteau tests are proposed this allows us to construct valid asymptotic significance limits for the residual autocorrelations and asymptotically valid goodness of fit tests when the underlying noise process is assumed to be noncorrelated rather than independent or a martingale difference a set of monte carlo experiments and an application to the standard poor 500 returns illustrate the p

a goodness of fit test for arch arch models a goodness of fit test in the class of conditional heteroscedastic time series models is examined due to the nonstandard limiting distribution of the test we propose to bootstrap the test showing its asymptotic validity moreover we illustrate the finite sample performance of the test by a small monte carlo study

parameter estimation and tissue segmentation from multispectral mr images a statistical method is developed to classify tissue types and to segment the corresponding tissue regions from relaxation time t sub 1 t sub 2 and proton density p sub d weighted magnetic resonance images the method assumes that the distribution of image intensities associated with each tissue type can be expressed as a multivariate likelihood function of three weighted signal intensity values t sub 1 t sub 2 p sub d at each location within that tissue regions the method further assumes that the underlying tissue regions are piecewise contiguous and can be characterized by a markov random field prior in classifying the tissue types the method models the likelihood of realizing the images as a finite multivariate mixture function the class parameters associated with the tissue types i e the weighted intensity means variances and correlation coefficients of the multivariate function as well as the number of voxels within regions of the tissue types of are estimated by maximum likelihood the estimation fits the class parameters to the image data via the expectation maximization algorithm the number of classes associated with the tissue types is determined by the information criterion of minimum description length the method segments the tissue regions given the estimated class parameters by maximum a posteriori probability the prior is constructed by the tissue region membership of the first and second order neighborhood the method is tested by a few sets of t sub 1 t sub 2 and p sub d weighted images of the brain acquired with a 1 5 tesla whole body scanner the number of classes and the associated class parameters are automatically estimated the regions of different brain tissues are satisfactorily segmented

noncentral limit theorems for quadratic forms in random variables having long range dependence we study the weak convergence in d lbrack 0 1 rbrack of the quadratic form sum lbrack nt rbrack j 1 sum lbrack nt rbrack k 1 a j k h m x j h m x k adequately normalized here a s infty s infty is a symmetric sequence satisfying sum a s infty h m is the m th hermite polynomial and x j j geq 1 is a normalized gaussian sequence with covariances r k sim k d l k as k rightarrow infty where 0 d 1 and l is slowly varying we prove that for all m geq 1 the limit is brownian motion when 1 2 d 1 and it is the non gaussian rosenblatt process when 0 d 1 2

testing for serial correlation in multivariate regression models abstract this paper considers the problem of detecting serial correlation in the disturbances of a multivariate regression model when these are known to be correlated up to a known finite lag q 0 and are possibly conditionally heteroskedastic we extend the results of cumby and huizinga 1992 to the case of a linear dynamic system of equations and derive the asymptotic distribution of a vector of sample autocovariances of the regression residuals this distribution is used to construct a test for serial correlation at lags greater than q a comparative monte carlo study of the small sample behavior of various tests in the case of purely autoregressive series reveals that the proposed test performs satisfactorily while tests that are commonly used in the literature are found to lead to serious size distortions under conditional heteroskedasticity

fitting an error distribution in some heteroscedastic time series models this paper addresses the problem of fltting a known distribution to the innovation distribution in a class of stationary and ergodic time series models the asymptotic null distribution of the usual kolmogorov smirnov test based on the residuals generally depends on the underlying model parameters and the error distribution to overcome the dependence on the underlying model parameters we propose that tests be based on a vector of certain weighted residual empirical processes under the null hypothesis and under minimal moment conditions this vector of processes is shown to converge weakly to a vector of independent copies of a gaussian process whose covariance function depends only on the fltted distribution and not on the model under certain local alternatives the proposed test is shown to have nontrivial asymptotic power the monte carlo critical values of this test are tabulated when fltting standard normal and double exponential distributions the results obtained are shown to be applicable to garch and arma garch models the often used models in econometrics and flnance a simulation study shows that the test has satisfactory size and power for flnite samples at these models the paper also contains an asymptotic uniform expansion result for a general weighted residual empirical process useful in heteroscedastic models under minimal moment conditions a result of independent interest

adaptive bayesian denoising for general gaussian distributed signals we study behavior of the bayesian estimator for noisy general gaussian distributed ggd data and show that this estimator can be well estimated with a simple shrinkage function the four parameters of this shrinkage function are functions of ggd s shape parameter and data variance the shrinkage map denoted by rigorous bayesshrink r bayesshrink models the bayesian estimator for any value of shape parameter in addition when the shape parameter is between 0 5 and 1 this shrinkage function transforms into a simple soft threshold this result places the role of soft thresholding image denoising methods such as bayesskrink in a new theoretical perspective moreover bayesshrink is shown to be a special case of r bayesshrink when the shape parameter is one laplacian distribution our simulation results confirm optimality of r bayesshrink in ggd signal denoising in the sense of peak signal to noise ratio psnr and structural similarity ssim index for a range of shape parameters

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

three points of great height on elliptic curves we give three elliptic curves whose generators have great height demonstrating along the way a moderately efficient method for finding such points

an automatic portmanteau test for serial correlation this article introduces a data driven box pierce test for serial correlation the proposed test is very attractive compared to the existing ones in particular implementation of this test is extremely simple for two reasons rst the researcher does not need to specify the order of the autocorrelation tested since the test automatically chooses this number second its asymptotic null distribution is chi square with one degree of freedom so there is no need of using a bootstrap procedure to estimate the critical values in addition the test is consistent and robust to the presence of conditional heteroskedasticity of unknown form finally the proposed test presents higher power in simulations than the existing ones for models commonly employed in empirical nance

modeling and estimation of multiresolution stochastic processes an overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing a natural framework for developing such a theory is the study of stochastic processes indexed by nodes on lattices or trees in which different depths in the tree or lattice correspond to different spatial scales in representing a signal or image in particular it is shown how the wavelet transform directly suggests such a modeling paradigm this perspective then leads directly to the investigation of several classes of dynamic models and related notions of multiscale stationarity in which scale plays the role of a time like variable the investigation of models on homogeneous trees is emphasized the framework examined here allows for consideration in a very natural way of the fusion of data from sensors with differing resolutions also thanks to the fact that wavelet transforms do an excellent job of compressing large classes of covariance kernels it is seen that these modeling paradigms appear to have promise in a far broader context than one might expect

estimating semiparametric arch models by kernel smoothing methods we investigate a class of semiparametric arch models that includes as a special case the partially nonparametric pnp model introduced by engle and ng 1993 and which allows for both flexible dynamics and flexible function form with regard to the news impact function we show that the functional part of the model satisfies a type ii linear integral equation and give simple conditions under which there is a unique solution we propose an estimation method that is based on kernel smoothing and profiled likelihood we establish the distribution theory of the parametric components and the pointwise distribution of the nonparametric component of the model we also discuss efficiency of both the parametric part and the nonparametric part we investigate the performance of our procedures on simulated data and on a sample of s p500 index returns we find evidence of asymmetric news impact functions consistent with the parametric analysis

adaptive wavelet thresholding for image denoising and compression the first part of this paper proposes an adaptive data driven threshold for image denoising via wavelet soft thresholding the threshold is derived in a bayesian framework and the prior used on the wavelet coefficients is the generalized gaussian distribution ggd widely used in image processing applications the proposed threshold is simple and closed form and it is adaptive to each subband because it depends on data driven estimates of the parameters experimental results show that the proposed method called bayesshrink is typically within 5 of the mse of the best soft thresholding benchmark with the image assumed known it also outperforms sureshrink donoho and johnstone 1994 1995 donoho 1995 most of the time the second part of the paper attempts to further validate claims that lossy compression can be used for denoising the bayesshrink threshold can aid in the parameter selection of a coder designed with the intention of denoising and thus achieving simultaneous denoising and compression specifically the zero zone in the quantization step of compression is analogous to the threshold value in the thresholding function the remaining coder design parameters are chosen based on a criterion derived from rissanen s minimum description length mdl principle experiments show that this compression method does indeed remove noise significantly especially for large noise power however it introduces quantization noise and should be used only if bitrate were an additional concern to denoising

wavelet based statistical signal processing using hidden markov models wavelet based statistical signal processing techniques such as denoising and detection typically model the wavelet coefficients as independent or jointly gaussian these models are unrealistic for many real world signals we develop a new framework for statistical signal processing based on wavelet domain hidden markov models hmms that concisely models the statistical dependencies and non gaussian statistics encountered in real world signals wavelet domain hmms are designed with the intrinsic properties of the wavelet transform in mind and provide powerful yet tractable probabilistic signal models efficient expectation maximization algorithms are developed for fitting the hmms to observational signal data the new framework is suitable for a wide range of applications including signal estimation detection classification prediction and even synthesis to demonstrate the utility of wavelet domain hmms we develop novel algorithms for signal denoising classification and detection

the memory of stochastic volatility models a valid asymptotic expansion for the covariance of functions of multivariate normal vectors is applied to approximate autocovariances of time series generated by nonlinear transformation of gaussian latent variates and nonlinear functions of these with special reference to long memory stochastic volatility models serving to identify the roles played by the underlying gaussian processes and the nonlinear transformation implications for simple stochastic volatility models are examined in detail with numerical and monte carlo calculations and applications to cyclic behaviour cross sectional and temporal aggregation and multivariate models are discussed 2001 elsevier science s a all rights reserved

on discriminating between long range dependence and changes in mean we develop a testing procedure for distinguishing between a long range dependent time series and a weakly dependent time series with change points in the mean in the simplest case under the null hypothesis the time series is weakly dependent with one change in mean at an unknown point and under the alternative it is long range dependent we compute the cusum statistic t n which allows us to construct an estimator k of a change point we then compute the statistic t n 1 based on the observations up to time k and the statistic t n 2 2 based on the observations after time k the statistic m n max t n 1 t n 2 converges to a well known distribution under the null but diverges to infinity if the observations exhibit long range dependence the theory is illustrated by examples and an application to the returns of the dow jones index

stability of random coefficient arch models and aggregation schemes in this paper we consider arch models with nonnegative random coefficients necessary and sufficient conditions for the existence of first and second moments are established the arch models with deterministic coefficients are shown to be always short memory processes the effect of small perturbations of the parameters is investigated the results obtained are applied to some aggregation schemes

out of sample forecasts of quadratic variation we compare the forecasts of quadratic variation given by the realized volatility rv and the two scales realized volatility tsrv computed from high frequency data in the presence of market microstructure noise under several different dynamics for the volatility process and assumptions on the noise we show that tsrv largely outperforms rv whether looking at bias variance rmse or out of sample forecasting ability an empirical application to all djia stocks confirms the simulation results

a goodness of fit test for arch models a goodness of fit test in the class of conditional heteroscedastic time series models is examined due to the nonstandard limiting distribution of the test we propose to bootstrap the test showing its asymptotic validity moreover we illustrate the finite sample performance of the test by a small monte carlo study

efficient inference on fractionally integrated panel data models with fixed effects a dynamic panel data model is considered that contains possibly stochastic individual com ponents and a common fractional stochastic time trend we propose four dierent ways of coping with the individual eects so as to estimate the fractional parameter like models with autoregressive dynamics ours nests a unit root but unlike the nonstandard asymptotics in the autoregressive case estimates of the fractional parameter can be asymptotically normal establishing this property is made di cult due to bias caused by the individual eects or by the consequences of eliminating them and requires the number of time series observations t to increase while the cross sectional size n can either remain xed or increase with t the biases in the central limit theorem are asymptotically negligible only under stringent conditions on the growth of n relative to t but these can be relaxed by bias correction for three of the estimates the biases depend only on the fractional parameter in hypothesis testing bias correc tion of the estimates is readily carried out we evaluate the biases numerically for a range of t and parameter values develop and justify feasible bias corrected estimates and brie y discuss simpli ed but less eective corrections a monte carlo study of nite sample performance is included

evaluating garch models in this paper a unified framework for testing the adequacy of an estimated garch model is presented parametric lagrange multiplier lm or lm type tests of no arch in standardized errors linearity and parameter constancy are proposed the asymptotic null distributions of the tests are standard which makes application easy versions of the tests that are robust against nonnormal errors are provided the finite sample properties of the test statistics are investigated by simulation the robust tests prove superior to the nonrobust ones when the errors are nonnormal they also compare favourably in terms of power with misspecification tests previously proposed in the literature

pseudo maximum likelihood estimation of arch models strong consistency and asymptotic normality of the gaussian pseudo maximum likelihood estimate of the parameters in a wide class of arch 1 processes are established we require the arch weights to decay at least hyperbolically with a faster rate needed for the central limit theorem than for the law of large numbers various rates are illustrated in examples of particular parameteriza tions in which our conditions are shown to be satis ed

on a class of elliptic curves with rank at most two in this note we consider the elliptic curves y 2 x 3 px defined over q for primes p satisfying p 1 mod 8 and review some of their properties we then compute and list in the supplement their ranks and give when the rank is positive the generators of the group of rational points and mordell weil lattice invariant for all primes p 50000 of the form m 2 64n 2

computation of the neron tate height on elliptic curves using neron s reduction theory and a method of tate we develop a procedure for calculating the local and global neron tate height on an elliptic curve over the rationals the procedure is illustrated by means of two examples of silverman and is then applied to calculate the global neron tate height of a series of rank one curves of bremner cassels and of a series of rank two curves of selmer in the latter case the regulator is also computed and a conjecture of s lang is investigated numerically in dealing with the arithmetic of elliptic curves e over a global field k the task arises of computing the ndron tate height on the group e k of rational points of e over k solving this task in an efficient manner is important for instance in view of calculations concerning the birch and swinnerton dyer conjecture see 2 or of the conjectures of serge lang 6 the purpose of this note is to suggest a procedure for performing the necessary calculations 1 multiplication formulas let the elliptic curve e over any field k be defined by a generalized weierstrass equation e y2 alxy a3y x3 a2x2 a4x a6 ai e k as usual we introduce the quantities see 10 11 b2 al 4a2 b4 ala3 2a4 b6 a3 4a6 2 2 b8 a1a6 a1a3a4 4a2a6 a2a3 a4 b2 24b4 c6 b3 36b2b4 216b6 and the discriminant a b2b8 8b2 27bw 9b2b4b6 0 as well as the absolute invariant

bootstrap score tests for fractional integration in heteroskedastic arfima models with an application to price dynamics in commodity spot and futures markets abstract empirical evidence from time series methods which assume the usual i 0 i 1 paradigm suggests that the efficient market hypothesis stating that spot and futures prices of a commodity should co integrate with a unit slope on futures prices does not hold however these statistical methods are known to be unreliable if the data are fractionally integrated moreover spot and futures price data tend to display clear patterns of time varying volatility which also has the potential to invalidate the use of these methods using new tests constructed within a more general heteroskedastic fractionally integrated model we are able to find a body of evidence in support of the efficient market hypothesis for a number of commodities our new tests are wild bootstrap implementations of score based tests for the order of integration of a fractionally integrated time series these tests are designed to be robust to both conditional and unconditional heteroskedasticity of a quite general and unknown form in the shocks we show that the asymptotic tests do not admit pivotal asymptotic null distributions in the presence of heteroskedasticity but that the corresponding tests based on the wild bootstrap principle do a monte carlo simulation study demonstrates that very significant improvements in finite sample behaviour can be obtained by the bootstrap vis a vis the corresponding asymptotic tests in both heteroskedastic and homoskedastic environments

the sample autocorrelations of heavy tailed processes with applications to arch we study the sample acvf and acf of a general stationary sequence under a weak mixing condition and in the case that the marginal distributions are regularly varying this includes linear and bilinear processes with regularly varying noise and arch processes their squares and absolute values we show that the distributional limits of the sample acf can be random provided that the variance of the marginal distribution is infinite and the process is nonlinear this is in contrast to infinite variance linear processes if the process has a finite second but infinite fourth moment then the sample acp is consistent with scaling rates that grow at a slower rate than the standard root n consequently asymptotic confidence bands are wider than those constructed in the classical theory we demonstrate the theory in full detail far an arch 1 process

asymptotic convergence rate of the em algorithm for gaussian mixtures it is well known that the convergence rate of the expectation maximization em algorithm can be faster than those of convention first order iterative algorithms when the overlap in the given mixture is small but this argument has not been mathematically proved yet this article studies this problem asymptotically in the setting of gaussian mixtures under the theoretical framework of xu and jordan 1996 it has been proved that the asymptotic convergence rate of the em algorithm for gaussian mixtures locally around the true solution is o e0 5 e where e 0 is an arbitrarily small number o x means that it is a higher order infinitesimal as x 0 and e is a measure of the average overlap of gaussians in the mixture in other words the large sample local convergence rate for the em algorithm tends to be asymptotically superlinear when e tends to zero

strict stationarity of generalized autoregressive processes in this paper we consider the multivariate equation x n 1 a n 1 x n b n 1 with i i d coefficients which have only a logarithmic moment we give a necessary and sufficient condition for existence of a strictly stationary solution independent of the future as an application we characterize the multivariate arma equations with general noise which have such a solution

wavelet thresholding via a bayesian approach we discuss a bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression a prior distribution is imposed on the wavelet coefficients of the unknown response function designed to capture the sparseness of wavelet expansion that is common to most applications for the prior specified the posterior median yields a thresholding procedure our prior model for the underlying function can be adjusted to give functions falling in any specific besov space we establish a relationship between the hyperparameters of the prior model and the parameters of those besov spaces within which realizations from the prior will fall such a relationship gives insight into the meaning of the besov space parameters moreover the relationship established makes it possible in principle to incorporate prior knowledge about the function s regularity properties into the prior model for its wavelet coefficients however prior knowledge about a function s regularity properties might be difficult to elicit with this in mind we propose a standard choice of prior hyperparameters that works well in our examples several simulated examples are used to illustrate our method and comparisons are made with other thresholding methods we also present an application to a data set that was collected in an anaesthesiological study

empirical process of the squared residuals of an arch sequence we derive the asymptotic distribution of the sequential empirical process of the squared residuals of an arch p sequence unlike the residuals of an arma process these residuals do not behave in this context like asymptotically independent random variables and the asymptotic distribution involves a term depending on the parameters of the model we show that in certain applications including the detection of changes in the distribution of the unobservable innovations our result leads to asymptotically distribution free statistics

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed