root t consistent density estimation in garch models we consider a new nonparametric estimator of the stationary density of the logarithm of the volatility of the garch 1 1 model this problem is particularly challenging since this density is still unknown even in cases where the model parameters are given although the volatility variables are only observed with multiplicative independent innovation errors with unknown density we manage to construct a nonparametric procedure which estimates the log volatility density consistently by carefully exploiting the specific garch dependence structure of the data our iterative procedure even attains the striking parametric root t convergence rate as a by product of our main results we also derive new smoothness properties of the stationary density using numerical simulations we illustrate the performance of our estimator and we provide an application to financial data

jump tails extreme dependencies and the distribution of stock returns we provide a new framework for estimating the systematic and idiosyncratic jump tail risks in financial asset prices our estimates are based on in fill asymptotics for directly identifying the jumps together with extreme value theory evt approximations and methods of moments for assessing the tail decay parameters and tail dependencies on implementing the procedures with a panel of intraday prices for a large cross section of individual stocks and the s p 500 market portfolio we find that the distributions of the systematic and idiosyncratic jumps are both generally heavy tailed and close to symmetric and show how the jump tail dependencies deduced from the high frequency data together with the day to day variation in the diffusive volatility account for the extreme joint dependencies observed at the daily level

a two channel overlapped block transform for image compression in this correspondence a new two channel overlapped block transform is introduced by prefiltering the haar transform the prefiltering employs plane rotations with one single rotation angle to achieve high computational efficiency the resulting transform is equivalent to a periodically time varying ptv filter bank the proposed transform or ptv filter bank has several advantages for image coding first symmetric extension is very easily performed at the boundaries in spite of the fact that the filters are asymmetric second arithmetic complexity can be reduced to simple shifts and adds finally the performance for image compression is shown to be high both in terms of visual quality and psnr as a result the proposed transform is a competitive alternative to the 5 3 and the 9 7 filters which are currently popular for subband wavelet image coders

stable particle like solutions to the nonlinear wave equations of three dimensional excitable media so called excitable media support nonlinear waves that propagate as pulses e g the action potential in nerve fibers and heart muscle and the oxidation pulse in certain chemical reactions in two dimensional media the right stimulus will also create rotating vortices which radiate such pulses periodically in every direction their investigation in diverse excitable media has dramatically expanded in the usa and ussr since their discovery in both countries in 1970 in three dimensional media the pivot of the rotating structure is not a point but a curved line generically a closed ring rings linked together and knotted constitute topologically distinctive organizing centers for periodic activity throughout the medium it is of interest to understand how stimuli create organizing centers and how organizing centers persist or decay the numerically discovered stable organizing centers exhibit enough symmetries to encourage mathematical attention this paper illustrates organizing centers by computer g

new perspectives and improvements on the symmetric extension filter bank for subband wavelet image compression in subband wavelet image coding size limited subband decompositions are ordinarily used to avoid increasing the number of samples that need to be coded to reduce coding distortions that can occur at the borders the symmetric extension filter bank is typically employed this paper introduces some new perspectives and improvements to that decomposition the symmetric extension filter bank is couched in the cyclic frequency domain providing a framework that accommodates fir and iir filters in a natural way all with perfect reconstruction iir filters with both rational and irrational transfer functions can be implemented and in the context of symmetric extension can accommodate iirs that effectively have perfect stopband suppression enhancements to the filter bank at a tree structured system level are also presented and include the application of spectral reversal correction and a transition band normalization approach to designing the constituent filters of the symmetric extension wavelet packet transform

lade based inferences for autoregressive models with heavy tailed g garch 1 1 noise abstract this paper explores the least absolute deviation lad estimator of the autoregressive model with heavy tailed g garch 1 1 noise when the tail index 1 2 it is shown that the lad estimator asymptotically converges to a linear function of a series of stable random vectors with a rate of convergence n 1 1 the result is significantly different from that of the corresponding least square estimator which is not consistent and partially solves the problem on the asymptoticity of the lad estimator when the tail index is less than 2 a simulation study is carried out to assess the performance of the lad estimator and a real example is given to illustrate this approach

least absolute deviation estimation for fractionally integrated autoregressive moving average time series models with conditional heteroscedasticity summary we consider a unified least absolute deviation estimator for stationary and nonstationary fractionally integrated autoregressive moving average models with conditional heteroscedasticity its asymptotic normality is established when the second moments of errors and innovations are finite several other alternative estimators are also discussed and are shown to be less efficient and less robust than the proposed approach a diagnostic tool consisting of two portmanteau tests is designed to check whether or not the estimated models are adequate the simulation experiments give further support to our model and the results for the absolute returns of the dow jones industrial average index daily closing price demonstrate their usefulness in modelling time series exhibiting the features of long memory conditional heteroscedasticity and heavy tails

quantile autocovariances a powerful tool for hard and soft partitional clustering of time series abstract a key issue in cluster analysis is determining a proper dissimilarity measure between two data objects and many pairwise dissimilarities have been proposed to deal with time series assuming that the clustering purpose is to group series according to the underlying dependence structures a detailed study of the behavior in clustering of a dissimilarity based on comparing estimated quantile autocovariance functions qaf is carried out quantile autocovariances provide information about the serial dependence structure that other conventional features are not able to capture which suggests great potential to perform clustering of series the asymptotic behavior of the sample quantile autocovariances is studied and an algorithm to determine optimal combinations of lags and pairs of quantile levels to perform clustering is introduced the proposed metric is used to perform hard and soft partitioning based clustering first a broad simulation study examines the behavior of the proposed metric in crisp clustering with the pam procedure a novel fuzzy c medoids algorithm based on the qaf dissimilarity is then proposed and compared with other fuzzy procedures in a new simulation study conducted to cluster fuzzy scenarios involving ar and garch models in all cases the qaf based procedures outperform or are highly competitive with a range of dissimilarities reported in the literature particularly exhibiting high capability to cluster conditionally heteroskedastic time series and robustness to the distributional form of the errors two specific applications involving air quality data and financial time series illustrate the usefulness of the proposed procedures

helicity and the calugareanu invariant the helicity of a localized solenoidal vector field i e the integrated scalar product of the field and its vector potential is known to be a conserved quantity under frozen field distortion of the ambient medium in this paper we present a number of results concerning the helicity of linked and knotted flux tubes particularly as regards the topological interpretation of helicity in terms of the gauss linking number and its limiting form the c lug reanu invariant the helicity of a single knotted flux tube is shown to be intimately related to the c lug reanu invariant and a new and direct derivation of this topological invariant from the invariance of helicity is given helicity is decomposed into writhe and twist contributions the writhe contribution involving the gauss integral for definition see equation 4 8 which admits interpretation in terms of the sum of signed crossings of the knot averaged over all projections part of the twist contribution is shown to be associated with the torsion of the knot and part with what may be described as intrinsic twist of the field lines in the flux tube around the knot see equations 5 13 and 5 15 the generic behaviour associated with the deformation of the knot through a configuration with points of inflexion points at which the curvature vanishes is analysed and the role of the twist parameter is discussed the derivation of the c lug reanu invariant from first principles of fluid mechanics provides a good demonstration of the relevance of fluid dynamical techniques to topological problems

asymptotics for parametric garch in mean models abstract in this paper we develop an asymptotic theory for the quasi maximum likelihood estimator qmle of the parametric garch in mean model the asymptotics is based on a study of the volatility as a process of the model parameters the proof makes use of stochastic recurrence equations for this random function and uses exponential inequalities to localize the problem our results show why the asymptotics for this specification is quite complex although it is a rather standard parametric model nevertheless our theory does not yet treat all standard specifications of the mean function

analysis synthesis techniques for subband image coding analysis synthesis systems designed for low bit rate image coding their impact on overall system quality and their computational complexity are discussed the investigation focuses on the design of analysis synthesis systems for image coding and the perceptual impact of these systems at low bit rates two objectives are emphasized in developing these systems confining the total size of the subband images to be equal to the original image size and designing the filters so that perceptual distortion is not introduced by the analysis synthesis system methods based on circular convolution and symmetric extensions are developed and discussed in detail the theory design and implementation of both recursive and nonrecursive filtering systems are discussed methods are introduced which display advantages over conventional quadrature mirror filter based approaches

on the exceedance point process for a stationary sequence it is known that the exceedance points of a high level by a stationary sequence are asymptotically poisson as the level increases under appropriate long range and local dependence conditions when the local dependence conditions are relaxed clustering of exceedances may occur based on poisson positions for the clusters in this paper a detailed analysis of the exceedance point process is given and shows that under wide conditions any limiting point process for exceedances is necessarily compound poisson more generally the possible random measure limits for normalized exceedance point processes are characterized sufficient conditions are also given for the existence of a point process limit the limiting distributions of extreme order statistics are derived as corollaries

precise large deviations for dependent regularly varying sequences we study a precise large deviation principle for a stationary regularly varying sequence of random variables this principle extends the classical results of nagaev theory probab appl 14 51 64 193 208 1969 and nagaev ann probab 7 745 789 1979 for iid regularly varying sequences the proof uses an idea of jakubowski stoch proc appl 44 291 327 1993 68 1 20 1997 in the context of central limit theorems with infinite variance stable limits we illustrate the principle for stochastic volatility models real valued functions of a markov chain satisfying a polynomial drift condition and solutions of linear and non linear stochastic recurrence equations

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

estimating the extremal index summary the extremal index is an important parameter measuring the degree of clustering of process the extremal index a parameter in the interval 0 1 is the reciprocal of the mean cluster size apart from being of interest in its own right it is a crucial parameter for determining the limiting distribution of extreme values from the process in this paper we review current work on statistical estimation of the extremal index and consider an optimality criterion based on a bias variance trade off theoretical results are developed for a simple doubly stochastic process and it is argued that the main formula obtained is valid for a much wider class of processes the practical implications are examined through simulations and a real data example

context based entropy coding of block transform coefficients for image compression it has been well established that state of the art wavelet image coders outperform block transform image coders in the rate distortion r d sense by a wide margin wavelet based jpeg2000 is emerging as the new high performance international standard for still image compression an often asked question is how much of the coding improvement is due to the transform and how much is due to the encoding strategy current block transform coders such as jpeg suffer from poor context modeling and fail to take full advantage of correlation in both space and frequency sense this paper presents a simple fast and efficient adaptive block transform image coding algorithm based on a combination of prefiltering postfiltering and high order space frequency context modeling of block transform coefficients despite the simplicity constraints coding results show that the proposed coder achieves competitive r d performance compared to the best wavelet coders in the literature

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

image coding using wavelet transform a scheme for image compression that takes into account psychovisual features both in the space and frequency domains is proposed this method involves two steps first a wavelet transform used in order to obtain a set of biorthogonal subclasses of images the original image is decomposed at different scales using a pyramidal algorithm architecture the decomposition is along the vertical and horizontal directions and maintains constant the number of pixels required to describe the image second according to shannon s rate distortion theory the wavelet coefficients are vector quantized using a multiresolution codebook to encode the wavelet coefficients a noise shaping bit allocation procedure which assumes that details at high resolution are less visible to the human eye is proposed in order to allow the receiver to recognize a picture as quickly as possible at minimum cost a progressive transmission scheme is presented it is shown that the wavelet transform is particularly well adapted to progressive transmission

a development of symmetric extension method for subband image coding describes the development of a technique for subband image coding called the symmetric extension method which utilizes the nature of a symmetrically extended image to achieve high quality coding there are however some restrictions on the usable subband coding system the development is done based on the property of a symmetrically extended signal in the analysis process to remove the restriction the behavior of the sequence after two processes filtering and decimation is examined and formulated the development extends the application area of the technique

robust score and portmanteau tests of volatility spillover abstract this paper presents a variety of tests of volatility spillover that are robust to heavy tails generated by large errors or garch type feedback the tests are couched in a general conditional heteroskedasticity framework with idiosyncratic shocks that are only required to have a finite variance if they are independent we negligibly trim test equations or components of the equations and construct heavy tail robust score and portmanteau statistics trimming is either simple based on an indicator function or smoothed in particular we develop the tail trimmed sample correlation coefficient for robust inference and prove that its gaussian limit under the null hypothesis of no spillover has the same standardization irrespective of tail thickness further if spillover occurs within a specified horizon our test statistics obtain power of one asymptotically we discuss the choice of trimming portion including a smoothed p value over a window of extreme observations a monte carlo study shows our tests provide significant improvements over extant garch based tests of spillover and we apply the tests to financial returns data finally based on ideas in patton 2011 we construct a heavy tail robust forecast improvement statistic which allows us to demonstrate that our spillover test can be used as a model specification pre test to improve volatility forecasting

the random difference equation x sb n a sb nx sb n 1 b sb n in the critical case let b n a n n 1 be a sequence of i i d random variables with values in r d x r the markov chain on r d which satisfies the random equation x n a n x n 1 b n is studied when e log a 1 0 no density assumption on the distribution of b 1 a 1 is made the main results are recurrence of the markov chain x n stability properties of the paths existence and uniqueness of a radon invariant measure and a limit theorem for the occupation times the results rely on a renewal theorem for the process x n a n a 1

the limit distribution of the maximum increment of a random walk with dependent regularly varying jump sizes we investigate the maximum increment of a random walk with heavy tailed jump size distribution here heavy tailedness is understood as regular variation of the finite dimensional distributions the jump sizes constitute a strictly stationary sequence using a continuous mapping argument acting on the point processes of the normalized jump sizes we prove that the maximum increment of the random walk converges in distribution to a frechet distributed random variable

the quantilogram with an application to evaluating directional predictability abstract we propose a new diagnostic tool for time series called the quantilogram the tool can be used formally and we provide the inference tools to do this under general conditions and it can also be used as a simple graphical device we apply our method to measure directional predictability and to test the hypothesis that a given time series has no directional predictability the test is based on comparing the correlogram of quantile hits to a pointwise confidence interval or on comparing the cumulated squared autocorrelations with the corresponding critical value we provide the distribution theory needed to conduct inference propose some model free upper bound critical values and apply our methods to s p500 stock index return data the empirical results suggest some directional predictability in returns the evidence is strongest in mid range quantiles like 5 10 and for daily data the evidence for predictability at the median is of comparable strength to the evidence around the mean and is strongest at the daily frequency

volatility regressions with fat tails abstract nowadays a common method to forecast integrated variance is to use the fitted value of a simple ols autoregression of the realized variance however non parametric estimates of the tail index of this realized variance process reveal that its second moment is possibly unbounded in this case the behavior of the ols estimators and the corresponding statistics are unclear we prove that when the second moment of the spot variance is unbounded the slope of the spot variance s autoregression converges to a random variable as the sample size diverges the same result holds when one uses the integrated or realized variance instead of the spot variance we then consider the class of diffusion variance models with an affine drift a class which includes garch and cev processes and we prove that iv estimation with adequate instruments provide consistent estimators of the drift parameters as long as the variance process has a finite first moment regardless of the existence of the second moment in particular for the garch diffusion model with fat tails an iv estimation where the instrument equals the sign of the centered lagged value of the variable of interest provides consistent estimators simulation results corroborate the theoretical findings of the paper

point process and partial sum convergence for weakly dependent random variables with infinite variance let j be a strictly stationary sequence of random variables with regularly varying tail probabilities we consider via point process methods weak convergence of the partial sums s n 1 n suitably normalized when j satisfies a mild mixing condition we first give a characterization of the limit point processes for the sequence of point processes n n with mass at the points j a n j 1 n where a n is the 1 n 1 quantile of the distribution of 1 then for 0 t n 0 p s n t n np 1 t n tends to a constant which can in general be different from 1 applications of our main results to self norming sums m dependent sequences and linear processes are also given

spectrum decomposition for image signal coding in conventional subband wavelet image coding the subband decomposition is performed on the spatial domain image here we introduce a novel decomposition where the subband decomposition is performed on the global dct spectrum of the image that is the two dimensional spectrum rather than the image is represented by a sum of basis functions each weighted by the transform coefficients the distinct features of this decomposition are analyzed from a transform perspective this spectral subband decomposition is then used as the basis for a new image coder building on the condensed wavelet packet cwp algorithm ironically this new method is shown to have lower arithmetic complexity than conventional subband wavelet coders that directly decompose a time or spatial domain signal comparisons of the new method against conventional subband wavelet coders that use the popular 9 7 dyadic decomposition condensed wavelet packets and generalized lapped orthogonal transforms show that the new method has lower complexity and higher compression performance

efficient block based frequency domain wavelet transform implementations subband decompositions for image coding have been explored extensively over the last few decades the condensed wavelet packet cwp transform is one such decomposition that was recently shown to have coding performance advantages over conventional decompositions a special feature of the cwp is that its design and implementation are performed in the cyclic frequency domain while performance gains have been reported efficient implementations of the cwp or more generally efficient implementations of cyclic filter banks have not yet been fully explored in this paper we present efficient block based implementations of cyclic filter banks along with an analysis of the arithmetic complexity block based cyclic filter bank implementations of the cwp coder are compared with conventional subband wavelet image coders whose filter banks are implemented in the time domain it is shown that block based cyclic filter bank implementations can result in cwp coding systems that outperform the popular image coding systems both in terms of arithmetic complexity and coding performance

a strong ergodic theorem for extreme and intermediate order statistics abstract we study almost sure limiting behavior of extreme and intermediate order statistics arising from strictly stationary sequences first we provide sufficient dependence conditions under which these order statistics converges almost surely to the left or right endpoint of the population support as in the classical setup of sequences of independent and identically distributed random variables next we derive a generalization of this result valid in the class of all strictly stationary sequences for this purpose we introduce notions of conditional left and right endpoints of the support of a random variable given a sigma field and present basic properties of these concepts using these new notions we prove that extreme and intermediate order statistics from any discrete time strictly stationary process converges almost surely to some random variable we describe the distribution of the limiting variate thus we establish a strong ergodic theorem for extreme and intermediate order statistics

cyclic lti systems in digital signal processing cyclic signal processing refers to situations where all the time indices are interpreted modulo some integer l in such cases the frequency domain is defined as a uniform discrete grid as in l point dft this offers more freedom in theoretical as well as design aspects while circular convolution has been the centerpiece of many algorithms in signal processing for decades such freedom especially from the viewpoint of linear system theory has not been studied in the past in this paper we introduce the fundamentals of cyclic multirate systems and filter banks presenting several important differences between the cyclic and noncyclic cases cyclic systems with allpass and paraunitary properties are studied the paraunitary interpolation problem is introduced and it is shown that the interpolation does not always succeed state space descriptions of cyclic lti systems are introduced and the notions of reachability and observability of state equations are revisited it is shown that unlike in traditional linear systems these two notions are not related to the system minimality in a simple way throughout the paper a number of open problems are pointed out from the perspective of the signal processor as well as the system theorist

a functional limit theorem for dependent sequences with infinite variance stable limits under an appropriate regular variation condition the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non gaussian stable random variable a functional version of this is known to be true as well the limit process being a stable levy process the main result in the paper is that for a stationary regularly varying sequence for which clusters of high threshold excesses can be broken down into asymptotically independent blocks the properly centered partial sum process still converges to a stable levy process due to clustering the levy triple of the limit process can be different from the one in the independent case the convergence takes place in the space of cadlag functions endowed with skorohod s m 1 topology the more usual j 1 topology being inappropriate as the partial sum processes may exhibit rapid successions of jumps within temporal clusters of large values collapsing in the limit to a single jump the result rests on a new limit theorem for point processes which is of independent interest the theory is applied to moving average processes squared garch 1 1 processes and stochastic volatility models

generalized symmetric extension for size limited multirate filter banks a multirate mr filter bank is called size limited if the total number of output samples equals the number of input samples a method called symmetric extension improved performance in subband image compression systems compared to the earlier method of circular convolution however the symmetric extension method was developed only for two band uniform filter banks and required even length linear phase analysis filters the authors generalize the symmetric extension method to the m band possibly nonuniform filter banks where m spl ges 2 the length restriction on the analysis filters is relaxed

inference on the tail process with application to financial time series modeling abstract to draw inference on serial extremal dependence within heavy tailed markov chains drees et al 2015 proposed nonparametric estimators of the spectral tail process the methodology can be extended to the more general setting of a stationary regularly varying time series the large sample distribution of the estimators is derived via empirical process theory for cluster functionals the finite sample performance of these estimators is evaluated via monte carlo simulations moreover two different bootstrap schemes are employed which yield confidence intervals for the pre asymptotic spectral tail process the stationary bootstrap and the multiplier block bootstrap the estimators are applied to stock price data to study the persistence of positive and negative shocks

two band hybrid fir iir filters for image compression two band analysis synthesis filters or wavelet filters are used pervasively for compressing natural images both fir and iir filters have been studied in this context the former being the most popular in this paper we examine the compression performance of these two band filters in a dyadic wavelet decomposition and attempt to isolate features that contribute most directly to the performance gain then employing the general exact reconstruction condition hybrid fir iir analysis synthesis filters are designed to maximize compression performance for natural images experimental results are presented that compare performance with the popular biorthogonal filters in terms of peak snr subjective quality and computational complexity

on some estimates based on sample behavior near high level excursions let j be a stationary sequence of weakly dependent random variables and letm n k be thek th largest value of j 1 j n the estimation of the parameters of the asymptotic distribution ofm n k is considered using a procedure motivated by a limit theorem pertaining to the point process sum nolimits j delta j n n bar f xi j a number of statistical issues concerning the procedure including how to select the tuning parameters are addressed the second problem that we consider is the estimation of the filter of a moving average process with heavy tails in particular the investigation covers the moving average stable process motivated by ideas in rootzen 1978 our estimator uses information contained in the sample behavior of the process near the largest excursion

extremes of markov chains with tail switching potential summary a recent advance in the utility of extreme value techniques has been the characteri zation of the extremal behaviour of markov chains this has enabled the application of extreme value models to series whose temporal dependence is markovian subject to a limitation that prevents switching between extremely high and extremely low levels for many applications this is sufficient but for others most notably in the field of finance it is common to find series in which successive values switch between high and low levels we term such series markov chains with tail switching potential and the scope of this paper is to generalize the previous theory to enable the characterization of the extremal properties of series displaying this type of behaviour in addition to theoretical developments a modelling procedure is proposed a simulation study is made to assess the utility of the model in inferring the extremal dependence structure of autoregressive conditional heteroscedastic processes which fall within the tail switching markov family and generalized autoregressive conditional heteroscedastic processes which do not being non markov in general finally the procedure is applied to model extremal aspects of a financial index extracted from the new york stock exchange compendium

limit theory for the sample autocorrelations and extremes of a garch 1 1 process the asymptotic theory for the sample autocorrelations and extremes of a garch i 1 process is provided special attention is given to the case when the sum of the arch and garch parameters is close to 1 that is when one is close to an infinite variance marginal distribution this situation has been observed for various financial log return series and led to the introduction of the igarch model in such a situation the sample autocorrelations are unreliable estimators of their deterministic counterparts for the time series and its absolute values and the sample autocorrelations of the squared time series have nondegenerate limit distributions we discuss the consequences for a foreign exchange rate series

the cross quantilogram measuring quantile dependence and testing directional predictability between time series this paper proposes the cross quantilogram to measure the quantile dependence between two time series we apply it to test the hypothesis that one time series has no directional predictability to another time series we establish the asymptotic distribution of the cross quantilogram and the corresponding test statistic the limiting distributions depend on nuisance parameters to construct consistent confidence intervals we employ the stationary bootstrap procedure we show the consistency of this bootstrap also we consider the self normalized approach which is shown to be asymptotically pivotal under the null hypothesis of no predictability we provide simulation studies and two empirical applications first we use the cross quantilogram to detect predictability from stock variance to excess stock return compared to existing tools used in the literature of stock return predictability our method provides a more complete relationship between a predictor and stock return second we investigate the systemic risk of individual financial institutions such as jp morgan chase goldman sachs and aig this article has supplementary materials online

stable limits for sums of dependent infinite variance random variables the aim of this paper is to provide conditions which ensure that the affinely transformed partial sums of a strictly stationary process converge in distribution to an infinite variance stable distribution conditions for this convergence to hold are known in the literature however most of these results are qualitative in the sense that the parameters of the limit distribution are expressed in terms of some limiting point process in this paper we will be able to determine the parameters of the limiting stable distribution in terms of some tail characteristics of the underlying stationary sequence we will apply our results to some standard time series models including the garch 1 1 process and its squares the stochastic volatility models and solutions to stochastic recurrence equations

inference in arch and garch models with heavy tailed errors arch and garch models directly address the dependency of conditional second moments and have proved particularly valuable in modelling processes where a relatively large degree of fluctuation is present these include financial time series which can be particularly heavy tailed however little is known about properties of arch or garch models in the heavy tailed setting and no methods are available for approximating the distributions of parameter estimators there in this paper we show that for heavy tailed errors the asymptotic distributions of quasi maximum likelihood parameter estimators in arch and garch models are nonnormal and are particularly difficult to estimate directly using standard parametric methods standard bootstrap methods also fail to produce consistent estimators to overcome these problems we develop percentile t subsample bootstrap approximations to estimator distributions studentizing is employed to approximate scale and the subsample bootstrap is used to estimate shape the good performance of this approach is demonstrated both theoretically and numerically

perfect reconstruction iir digital filter banks supporting nonexpansive linear signal extensions in this paper perfect reconstruction polyphase infinite impulse response iir filter banks involving causal and anticausal inverses are examined for finite length signals first a novel and efficient nonexpansive perfect reconstruction algorithm based on the state space implementation is presented then the proposed method is extended to support linear signal extensions at the boundaries in a nonexpansive manner the powerfulness of the proposed algorithm is demonstrated with the image compression results

generation of ribbons helicoids and complex scherk surfaces in laser matter interactions motivated by a diversity of the shape instability phenomena in condensed matter physics we study the formation of elastic ribbon structures and transformation into helicoidal structures using the multipulse laser matter interaction with the co coated surface a one dimensional high density vortex filament array has been created increasing the number of pulses the oscillatory strain field causes the cascade of the shape transformations into structures of increasing topological complexity vortex filaments into ribbons into ribbon helicoids and tubular ribbon helicoids and then into short ribbon structures with the complex scherk surface being identified the cascade of transformations follows the scenario in which the topological complexity of the new structure increases with the number of pulses thus realizing configurations with more efficient energy relaxation above a critical number of pulses the system is catastrophically disintegrated into small scale wrinkled and crumpled surfaces we show that the critical parameter for the ribbon transformations is the number of laser pulses which is equivalent to some critical oscillatory frequency

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

a progressive transmission image coder using linear phase uniform filterbanks as block transforms this paper presents a novel image coding scheme using m channel linear phase perfect reconstruction filterbanks lpprfbs in the embedded zerotree wavelet ezw framework introduced by shapiro 1993 the innovation here is to replace the ezws dyadic wavelet transform by m channel uniform band maximally decimated lpprfbs which offer finer frequency spectrum partitioning and higher energy compaction the transform stage can now be implemented as a block transform which supports parallel processing and facilitates region of interest coding decoding for hardware implementation the transform boasts efficient lattice structures which employ a minimal number of delay elements and are robust under the quantization of lattice coefficients the resulting compression algorithm also retains all the attractive properties of the ezw coder and its variations such as progressive image transmission embedded quantization exact bit rate control and idempotency despite its simplicity our new coder outperforms some of the best image coders published previously in the literature for almost all test images especially natural hard to code ones at almost all bit rates

hybrid quantile regression estimation for time series models with conditional heteroscedasticity estimating conditional quantiles of financial time series is essential for risk management and many other financial applications for time series models with conditional heteroscedasticity although it is the generalized auto regressive conditional heteroscedastic garch model that has the greatest popularity quantile regression for this model usually gives rise to non smooth non convex optimization which may hinder its practical feasibility the paper proposes an easy to implement hybrid quantile regression estimation procedure for the garch model where we overcome the intractability due to the square root form of the conditional quantile function by a simple transformation the method takes advantage of the efficiency of the garch model in modelling the volatility globally as well as the flexibility of quantile regression in fitting quantiles at a specific level the asymptotic distribution of the estimator is derived and is approximated by a novel mixed bootstrapping procedure a portmanteau test is further constructed to check the adequacy of fitted conditional quantiles the finite sample performance of the method is examined by simulation studies and its advantages over existing methods are illustrated by an empirical application to value at risk forecasting

the energy and helicity of knotted magnetic flux tubes magnetic relaxation of a magnetic field embedded in a perfectly conducting incompressible fluid to minimum energy magnetostatic equilibrium states is considered it is supposed that the magnetic field is confined to a single flux tube which may be knotted a local non orthogonal coordinate system zero framed with respect to the knot is introduced and the field is decomposed into toroidal and poloidal ingredients with respect to this system the helicity of the field is then determined this vanishes for a field that is either purely toroidal or purely poloidal the magnetic energy functional is calculated under the simplifying assumptions that the tube is axially uniform and of circular cross section the case of a tube with helical axis is first considered and new results concerning kink mode instability and associated bifurcations are obtained the case of flux tubes in the form of torus knots is then considered and the ground state energy function m h where h is an internal twist parameter is obtained as expected m h which is a topological invariant of the knot increases with increasing knot complexity the function m h provides an upper bound on the corresponding function m h that applies when the above constraints on tube structure are removed the technique is applicable to any knot admitting a parametric representation on condition that points of vanishing curvature are excluded

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

moment condition tests for heavy tailed time series abstract we develop an asymptotically chi squared statistic for testing moment conditions e m t 0 0 where m t 0 may be weakly dependent scalar components of m t 0 may have an infinite variance and e m t need not exist for any under the alternative score tests are a natural application and in general a variety of tests can be heavy tail robustified by our method including white noise garch affects omitted variables distribution functional form causation volatility spillover and over identification the test statistic is derived from a tail trimmed sample version of the moments evaluated at a consistent plug in t for 0 depending on the test in question and heaviness of tails t may be any consistent estimator including sub t 1 2 convergent and or asymptotically non gaussian ones since t can be assured not to affect the test statistic asymptotically we adapt bootstrap p value occupation time and covariance determinant methods for selecting the trimming fractile in any sample and apply our statistic to tests of white noise omitted variables and volatility spillover we find it obtains sharp empirical size and strong power while conventional tests exhibit size distortions

fast multiplierless approximations of the dct with the lifting scheme we present the design implementation and application of several families of fast multiplierless approximations of the discrete cosine transform dct with the lifting scheme called the bindct these bindct families are derived from chen s 1977 and loeffler s 1989 plane rotation based factorizations of the dct matrix respectively and the design approach can also be applied to a dct of arbitrary size two design approaches are presented in the first method an optimization program is defined and the multiplierless transform is obtained by approximating its solution with dyadic values in the second method a general lifting based scaled dct structure is obtained and the analytical values of all lifting parameters are derived enabling dyadic approximations with different accuracies therefore the bindct can be tuned to cover the gap between the walsh hadamard transform and the dct the corresponding two dimensional 2 d bindct allows a 16 bit implementation enables lossless compression and maintains satisfactory compatibility with the floating point dct the performance of the bindct in jpeg h 263 and lossless compression is also demonstrated

a robust goodness of fit test for generalized autoregressive conditional heteroscedastic models summary the estimation of time series models with heavy tailed innovations has been widely discussed but corresponding goodness of fit tests have attracted less attention primarily because the autocorrelation function commonly used in constructing goodness of fit tests necessarily imposes certain moment conditions on the innovations as a bounded random variable has finite moments of all orders we address the problem by first transforming the residuals with a bounded function more specifically we consider the sample autocorrelation function of the transformed absolute residuals of a fitted generalized autoregressive conditional heteroscedastic model with the corresponding residual empirical distribution function naturally employed as the transformation a robust goodness of fit test is then constructed the asymptotic distributions of the test statistic under the null hypothesis and local alternatives are derived and monte carlo experiments are conducted to examine finite sample properties the proposed test is shown to be more powerful than existing tests when the innovations are heavy tailed

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

discrete wavelet transform based on cyclic convolutions a filterbank in which cyclic convolutions are used in place of linear convolutions will be referred to as a cyclic convolution filterbank ccfb a dyadic tree structured ccfb can be used to perform a discrete wavelet transform suitable for coding based on symmetric extension methods this paper derives two types of efficient implementation techniques for the tree structured ccfb one using complex arithmetic and one using only real arithmetic in addition the present paper analyzes in detail the perfect reconstruction pr condition for the two channel ccfb and shows that this condition is much less restrictive than that of usual two channel filterbanks when each of the two channel ccfbs constituting a tree structured ccfb is designed to be pr the whole system is pr a quadrature mirror filter qmf ccfb having the pr property is demonstrated to be easily designed using a standard filter design subroutine in contrast designing a linear phase pr fir qmf bank that has good frequency response is not possible when filters are realized by linear convolutions

the cluster index of regularly varying sequences with applications to limit theory for functions of multivariate markov chains we introduce the cluster index of a multivariate stationary sequence and characterize the index in terms of the spectral tail process this index plays a major role in limit theory for partial sums of sequences we illustrate the use of the cluster index by characterizing infinite variance stable limit distributions and precise large deviation results for sums of multivariate functions acting on a stationary markov chain under a drift condition

subband coder design incorporating recursive quadrature filters and optimum adpcm coders this paper presents the results of an analytical and experimental study of the use of iir quadrature mirror filters and forward adapting adpcm with optimum quantizers for octave band subband coders for speech it is shown that the underlying analysis synthesis systems can be designed such that there is no interband aliasing distortion in the coded speech and such that the analysis synthesis transfer function has no frequency distortion or no phase distortion but not both the experimental study showed that subband coders based on a mixture of infinite impulse response iir and finite duration impulse response fir filters resulted in higher quality coding systems using fewer multiplies than for systems based on fir filters alone it was also found that a signal to noise ratio snr gain of about 4 db could be obtained using optimum quantizers for forward adapting adpcm coders over linear quantizers in backward adapting apcm codes but no perceptual quality gain was observed

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

frequency domain design of overcomplete rational dilation wavelet transforms the dyadic wavelet transform is an effective tool for processing piecewise smooth signals however its poor frequency resolution its low q factor limits its effectiveness for processing oscillatory signals like speech eeg and vibration measurements etc this paper develops a more flexible family of wavelet transforms for which the frequency resolution can be varied the new wavelet transform can attain higher q factors desirable for processing oscillatory signals or the same low q factor of the dyadic wavelet transform the new wavelet transform is modestly overcomplete and based on rational dilations like the dyadic wavelet transform it is an easily invertible constant q discrete transform implemented using iterated filter banks and can likewise be associated with a wavelet frame for l2 r the wavelet can be made to resemble a gabor function and can hence have good concentration in the time frequency plane the construction of the new wavelet transform depends on the judicious use of both the transform s redundancy and the flexibility allowed by frequency domain filter design

lapped transform via time domain pre and post filtering this paper presents a general framework of constructing a large family of lapped transforms with symmetric basis functions by adding simple time domain pre and post processing modules onto existing block discrete cosine transform dct based infrastructures a subset of the resulting solutions is closed form fast computable modular near optimal in the energy compaction sense and leads to an elegant boundary handling of finite length data starting from these solutions a general framework for block based signal decomposition with a high degree of flexibility and adaptivity is developed several simplified models are also introduced to approximate the optimal solutions these models are based on cascades of plane rotation operators and lifting steps respectively despite tremendous savings in computational complexity the optimized results of these simplified models are virtually identical to that of the complete solution the multiplierless versions of these pre and post filters when combined with an appropriate multiplierless block transform such as the bindct which is described in an earlier paper by liang and tran see ieee trans signal processing vol 49 p 3032 44 dec 2001 generate a family of very large scale integration vlsi friendly fast lapped transforms with reversible integer to integer mapping numerous design examples with arbitrary number of channels and arbitrary number of borrowed samples are presented