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

optimal eigen expansions and uniform bounds let x k k in mathbb z in mathbb l 2 mathcal t be a stationary process with associated lag operators varvec mathcal c h uniform asymptotic expansions of the corresponding empirical eigenvalues and eigenfunctions are established under almost optimal conditions on the lag operators in terms of the eigenvalues spectral gap in addition the underlying dependence assumptions are optimal in a certain sense including both short and long memory processes this allows us to study the relative maximum deviation of the empirical eigenvalues under very general conditions among other things convergence to an extreme value distribution is shown we also discuss how the asymptotic expansions transfer to the long run covariance operator varvec mathcal g in a general framework

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

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

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

limit laws for the maximum and minimum of stationary sequences the class of non degenerate joint limiting distributions for the maximum and minimum of stationary mixing sequences is determined these limit distributions are of the form h x h x y where h x y is a bivariate extreme value distribution unlike the classical result for i i d sequences the maximum and minimum of stationary mixing sequences may be asymptotically dependent we also give a sufficient condition for the asymptotic independence of the maximum and minimum finally an example of a stationary sequence satisfying the mixing condition d in leadbetter but which is not weakly mixing is constructed

monopoles on asymptotically euclidean 3 manifolds on considere une generalisation de la theorie de yang mills higgs sur r 3 euclidien a des sommes convexes de r 3 avec des 3 varietes closes compactes

hyperbolic calorons monopoles and instantons we construct families of so 3 symmetric charge 1 instantons and calorons on the space mathbb h 3 times mathbb r we show how the calorons include instantons and hyperbolic monopoles as limiting cases we show how euclidean calorons are the flat space limit of this family

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

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

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

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

geometry of linear pairs for self dual gauge fields a linear pair for self dual gauge fields is constructed for the metric ds2 gzzdzdz gyydyd it is shown that for consistency gzz and gyy apart from a possible overall conformal factor are given in terms of two liouville fields of equal and opposite curvatures the null surface corresponding to the pair and the homogeneous solutions playing a fundamental role are constructed explicitly the five dimensional space of y y z z and the spectral parameter is studied the proper transformation of corresponding to holomorphic ones of y and z is found known monopole instanton and quasi periodic solutions are all shown to emerge systematically as particular cases of our formalism as examples of new possibilities the case of accelerated observers and that of cosmic string backgrounds are presented

markov chain models for threshold exceedances in recent research on extreme value statistics there has been an extensive development of threshold methods first in the univariate case and subsequently in the multivariate case as well in this paper an alternative methodology for extreme values of univariate time series is developed by assuming that the time series is markovian and using bivariate extreme value theory to suggest appropriate models for the transition distributions a new likelihood representation for threshold methods is presented which we apply to a markovian time series an important motivation for developing this kind of theory is the possibility of calculating probability distributions for functionals of extreme events we address this issue by showing how a theory of compound poisson limits for additive functionals can be combined with simulation to obtain numerical solutions for problems of practical interest the methods are illustrated by application to temperature data

the hypervolume under the roc hypersurface of near guessing and near perfect observers in n class classification tasks we express the performance of the n class guessing observer in terms of the n sup 2 n conditional probabilities which make up an n class receiver operating characteristic roc space in a formulation in which sensitivities are eliminated in constructing the roc space equivalent to using false negative fraction and false positive fraction in a two class task we then show that the guessing observer s performance in terms of these conditional probabilities is completely described by a degenerate hypersurface with only n 1 degrees of freedom as opposed to the n sup 2 n 1 required in general to achieve a true hypersurface in such a roc space it readily follows that the hypervolume under such a degenerate hypersurface must be zero when n 2 we then consider a near guessing task that is a task in which the n underlying data probability density functions pdfs are nearly identical controlled by n 1 parameters which may vary continuously to zero at which point the pdfs become identical with this approach we show that the hypervolume under the roc hypersurface of an observer in an n class classification task tends continuously to zero as the underlying data pdfs converge continuously to identity a guessing task the hypervolume under the roc hypersurface of a perfect ideal observer in a task in which the n data pdfs never overlap is also found to be zero in the roc space formulation under consideration this suggests that hypervolume may not be a useful performance metric in n class classification tasks for n 2 despite the utility of the area under the roc curve for two class tasks

ideal observers and optimal roc hypersurfaces in n class classification the likelihood ratio or ideal observer decision rule is known to be optimal for two class classification tasks in the sense that it maximizes expected utility or equivalently minimizes the bayes risk furthermore using this decision rule yields a receiver operating characteristic roc curve which is never above the roc curve produced using any other decision rule provided the observer s misclassification rate with respect to one of the two classes is chosen as the dependent variable for the curve i e an inversion of the more common formulation in which the observer s true positive fraction is plotted against its false positive fraction it is also known that for a decision task requiring classification of observations into n classes optimal performance in the expected utility sense is obtained using a set of n 1 likelihood ratios as decision variables in the n class extension of roc analysis the ideal observer performance is describable in terms of an n sup 2 n 1 parameter hypersurface in an n sup 2 n dimensional probability space we show that the result for two classes holds in this case as well namely that the roc hypersurface obtained using the ideal observer decision rule is never above the roc hypersurface obtained using any other decision rule where in our formulation performance is given exclusively with respect to between class error rates rather than within class sensitivities

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

limit theorems for empirical processes of cluster functionals let x n i 1 i n m is an element of n be a triangular array of row wise stationary r d valued random variables we use a blocks method to define clusters of extreme values the rows of x n i are divided into m n blocks y n j and if a block contains at least one extreme value the block is considered to contain a cluster the cluster starts at the first extreme value in the block and ends at the last one the main results are uniform central limit theorems for empirical processes z n f 1 root nv n sigma mn j 1 f y n y j ef y n y j for v n p x n x i not equal 0 and f belonging to classes of cluster functionals that is functions of the blocks y n y j which only depend on the cluster values and which are equal to 0 if y n y j does not contain a cluster conditions for finite dimensional convergence include beta mixing suitable lindeberg conditions and convergence of covariances to obtain full uniform convergence we use either bracketing entropy or bounds on covering numbers with respect to a random semi metric the latter makes it possible to bring the powerful vapnik cervonenkis theory to bear applications include multivariate tail empirical processes and empirical processes of cluster values and of order statistics in clusters although our main field of applications is the analysis of extreme values the theory can be applied more generally to rare events occurring for example in nonparametric curve estimation

inference for clusters of extreme values inference for clusters of extreme values of a time series typically requires the identification of independent clusters of exceedances over a high threshold the choice of declustering scheme often has a significant effect on estimates of cluster characteristics we propose an automatic declustering scheme that is justified by an asymptotic result for the times between threshold exceedances the scheme relies on the extremal index which we show may be estimated before declustering and supports a bootstrap procedure for assessing the variability of estimates

summarising censored survival data using the mean residual life function the mean residual life function provides a clear and simple summary of the effect of a treatment or a risk factor in units of time avoiding hazard ratios or probability scales which require careful interpretation estimation of the mean residual life is complicated by the upper tail of the survival distribution not being observed as for example patients may still be alive at the end of the follow up period various approaches have been developed to estimate the mean residual life in the presence of such right censoring in this work a novel semi parametric method that combines existing non parametric methods and an extreme value tail model is presented where the limited sample information in the tail prior to study termination is used to estimate the upper tail behaviour this approach will be demonstrated with simulated and real life examples copyright 2015 john wiley sons ltd

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

confidence intervals around bayes cost in multi state diagnostic settings to estimate optimal performance a critical feature of diagnostic testing is correctly classifying subjects based upon specified thresholds of some measure the commonly employed youden index determines a test s optimal thresholds by maximizing the correct classification rates for a diagnostic scenario an alternative to the youden index is the cost function bayes cost bc bc determines a test s optimal setting by minimizing the sum of all misclassification rates from the test unlike the youden index bc can consider a priori costs of all the diagnostic outcomes including class specific misclassifications regardless of the number of classes delta method approximate confidence intervals around bc are derived under the assumption of normally distributed classes as a means for quantifying a test s performance and comparing classifiers at their optimal settings in a multi state diagnostic framework a simulation study is conducted to demonstrate the performance of the derived confidence intervals that are found to perform well especially for sample sizes of 50 or larger in each diagnostic class finally the proposed methods are applied to a four class breast tissue classification problem where four possible discriminatory features are compared under varying decision cost structures using the confidence intervals around bc the best feature for classification is selected and the optimal thresholds and their 95 confidence intervals are determined

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

roc graphs for assessing the ability of a diagnostic marker to detect three disease classes with an umbrella ordering receiver operating characteristic roc curves and the area under these curves are commonly used to assess the ability of a continuous diagnostic marker e g dna methylation markers to correctly classify subjects as having a particular disease or not e g cancer these approaches however are not applicable to settings where the gold standard yields more than two disease states or classes roc surfaces and the volume under the surfaces have been proposed for settings with more than two disease classes these approaches however do not allow one to assess the ability of a marker to differentiate two disease classes from a third disease class without requiring a monotone order for the three disease classes under study that is existing approaches do not accommodate an umbrella ordering of disease classes this article proposes the construction of an roc graph that is applicable for an umbrella ordering furthermore this article proposes that a summary measure for this umbrella roc graph can be used to summarize the classification accuracy and corresponding variance estimates can be obtained using u statistics theory or bootstrap methods the proposed methods are illustrated using data from a study assessing the ability of a dna methylation marker to correctly classify lung specimens into three histologic classes squamous cell carcinoma large cell carcinoma and nontumor lung

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

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

limiting entry and return times distribution for arbitrary null sets we describe an approach that allows us to deduce the limiting return times distribution for arbitrary sets to be compound poisson distributed we establish a relation between the limiting return times distribution and the probability of the cluster sizes where clusters consist of the portion of points that have finite return times in the limit where random return times go to infinity in the special case of periodic points we recover the known polya aeppli distribution which is associated with geometrically distributed cluster sizes we apply this method to several examples the most important of which is synchronisation of coupled map lattices for the invariant absolutely continuous measure we establish that the returns to the diagonal is compound poisson distributed where the coefficients are given by certain integrals along the diagonal

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

a limit theorem for the maximum of autoregressive processes with uniform marginal distributions a class of first order autoregressive processes is given for which the extreme value limit theorems of loynes and leadbetter do not apply a limit theorem is derived for these processes that depends on the parameter r an integer greater than or equal to 2

construction of hyperbolic monopoles methods are presented for explicit construction of hyperbolic monopoles as defined by atiyah m f atiyah commun math phys 93 437 1984 and magnetic monopoles in hyperbolic space unpublished lecture notes 1984 the techniques permit the extraction of flat space self dual monopoles trivially through a simple rescaling relationships to previous works of the present author are indicated

towards estimating extremal serial dependence via the bootstrapped extremogram abstract davis and mikosch 2009a introduced the extremogram as a flexible quantitative tool for measuring various types of extremal dependence in a stationary time series there we showed some standard statistical properties of the sample extremogram a major difficulty was the construction of credible confidence bands for the extremogram in this paper we employ the stationary bootstrap to overcome this problem the use of the stationary bootstrap for the extremogram and the resulting interpretations are illustrated with several financial time series

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

extreme values for stationary and markov sequences soit x n n 1 2 une suite strictement stationnaire de variables aleatoires a valeurs reelles soit m i j max x i 1 x j et m n m 0 n soit c n une suite de nombres reels on montre sous des circonstances generales que p m n c n p x 1 c n np mu 1 u u p u n cu n xu 1 cu n o pour toute suite pn satisfaisant certaines conditions de taux de croissance

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

singularly perturbed chern simons theory the chern simons theory of an su 2 gauge theory in three dimensions is looked at from a perturbative point of view the pure chern simons action is generalized by adding a conventional yang mills action term this acts as a singular perturbation the resulting theory has a moduli space containing that of the pure chern simons version for certain discrete values of the perturbation parameter lying in the spectrum of an appropriate elliptic operator the enlargement of the moduli space can be made explicit the extrema can be classified by a hessian with a finite index and nullity without recourse to spectral flow corrections to the resultant quantum theory are also calculated also the quantum theory of the present model should be better behaved than in the unperturbed case

local gaussian correlation a new measure of dependence abstract it is a common view among finance analysts and econometricians that the correlation between financial objects becomes stronger as the market is going down and that it approaches one when the market crashes having the effect of destroying the benefit of diversification the purpose of this paper is to introduce a local dependence measure that gives a precise mathematical description and interpretation of such phenomena we propose a new local dependence measure a local correlation function based on approximating a bivariate density locally by a family of bivariate gaussian densities using local likelihood at each point the correlation coefficient of the approximating gaussian distribution is taken as the local correlation existence uniqueness and limit results are established a number of properties of the local gaussian correlation and its estimate are given along with examples from both simulated and real data this new method of modelling carries with it the prospect of being able to do locally for a general density what can be done globally for the gaussian density in a sense it extends gaussian analysis from a linear to a non linear environment

instantons in two and four dimensions it is shown that yang mills instantons in four dimensions can naturally be identified with the instantons of a two dimensional theory with values in the loop group

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

three class roc analysis a decision theoretic approach under the ideal observer framework receiver operating characteristic roc analysis is well established in the evaluation of systems involving binary classification tasks however medical tests often require distinguishing among more than two diagnostic alternatives the goal of this work was to develop an roc analysis method for three class classification tasks based on decision theory we developed a method for three class roc analysis in this method the objects were classified by making the decision that provided the maximal utility relative to the other two by making assumptions about the magnitudes of the relative utilities of incorrect decisions we found a decision model that maximized the expected utility of the decisions when using log likelihood ratios as decision variables this decision model consists of a two dimensional decision plane with log likelihood ratios as the axes and a decision structure that separates the plane into three regions moving the decision structure over the decision plane which corresponds to moving the decision threshold in two class roc analysis and computing the true class 1 2 and 3 fractions defined a three class roc surface we have shown that the resulting three class roc surface shares many features with the two class roc curve i e using the log likelihood ratios as the decision variables results in maximal expected utility of the decisions and the optimal operating point for a given diagnostic setting set of relative utilities and disease prevalences lies on the surface the volume under the three class surface vus serves as a figure of merit to evaluate different data acquisition systems or image processing and reconstruction methods when the assumed utility constraints are relevant

nonparametric sequential change point detection by a vertically trimmed box method this paper examines a new method for sequential detection of a sudden and unobservable change in a sequence of independent observations with completely unspecified distribution functions a nonparametric detection rule is proposed which relies on the concept of a moving vertically trimmed box as such it will be coined as the vertical box control chart v box chart its implementation requires merely to count the number of data points which fall into the box attached to the last available observation no a priori knowledge of data distributions is required and proper tuning of the box size provides a quick detection technique this is supported by establishing statistical properties of the method which explain the role of the tuning parameters used in the v box chart these theoretical results are verified by simulation studies which indicate that the v box chart may provide quick detection with zero delay for jumps of moderate sizes its averaged run length to detection is more favorable than the one for the classical ewma method by comparison with the classical shewhart chart which was optimized for normal errors our method provides comparable or better performance

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

ordered multiple class roc analysis with continuous measurements receiver operating characteristic roc curves have been useful in two group classification problems in three and multiple class diagnostic problems an roc surface or hyper surface can be constructed the volume under these surfaces can be used for inference using bootstrap techniques or u statistics theory in this article roc surfaces and hyper surfaces are defined and their behaviour and utility in multi group classification problems is investigated the formulation of the problem is equivalent to what has previously been proposed in the general multi category classification problem but the definition of roc surfaces here is less complex and addresses directly the narrower problem of ordered categories in the three class and by extension the multi class problem applied to continuous and ordinal data non parametric manipulation of both continuous and discrete test data and comparison between two diagnostic tests applied to the same subjects are considered a three group classification example in the context of hiv neurological disease is presented and the results are discussed

extremal analysis of processes sampled at different frequencies the observed extremes of a discrete time process depend on the process itself and the sampling frequency we develop theoretical results which show how to account for the effect of sampling frequency on extreme values thus enabling us to analyse systematically extremal data from series with different sampling rates we present statistical methodology based on these results which we illustrate though simulations and by applications to sea waves and rainfall data

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 graphical analysis of cost sensitive regression problems several efforts have been done to bring roc analysis beyond binary classification especially in regression however the mapping and possibilities of these proposals do not correspond to what we expect from the analysis of operating conditions dominance hybrid methods etc in this paper we present a new representation of regression models in the so called regression roc rroc space the basic idea is to represent over estimation on the x axis and under estimation on the y axis the curves are just drawn by adjusting a shift a constant that is added or subtracted to the predictions and plays a similar role as a threshold in classification from here we develop the notions of optimal operating condition convexity dominance and explore several evaluation metrics that can be shown graphically such as the area over the rroc curve aoc in particular we show a novel and significant result the aoc is equal to the error variance multiplied by a factor which does not depend on the model the derivation of rroc curves with non constant shifts and soft regression models and the relation with cost plots is also discussed

roc curves for regression receiver operating characteristic roc analysis is one of the most popular tools for the visual assessment and understanding of classifier performance in this paper we present a new representation of regression models in the so called regression roc rroc space the basic idea is to represent over estimation against under estimation the curves are just drawn by adjusting a shift a constant that is added or subtracted to the predictions and plays a similar role as a threshold in classification from here we develop the notions of optimal operating condition convexity dominance and explore several evaluation metrics that can be shown graphically such as the area over the rroc curve aoc in particular we show a novel and significant result the aoc is equivalent to the error variance we illustrate the application of rroc curves to resource estimation namely the estimation of software project effort

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 roc manifold for classification systems we define the roc manifold and cc manifold as duals in a given sense their analysis is required to describe the classification system we propose a mathematical definition based on vector space methods to describe both the roc manifolds for n class classification systems fully describe each system in terms of its misclassifications and by conjunction its correct classifications optimal points which minimize misclassifications can be identified even when costs and prior probabilities differ these manifolds can be used to determine the usefulness of a classification system based on a given performance criterion many performance functionals such as summary statistics preferred for cc manifolds can also be evaluated using the roc manifold under certain constraints examples using the roc manifold and performance functionals to compete classification systems are demonstrated with simulated and applied disease detection data

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

roc representation for the discriminability of multi classification markers in this paper the receiver operating characteristic roc representation and its accuracy measures are well defined and meaningful assessments for the discriminability of multi classification markers are shown given a set of classifiers c a parameterized system can be used to characterize the corresponding optimal roc manifold a connection with the decision set further leads to a better understanding of some geometric features of optimal roc manifolds and preserves the simplicity in computing the hypervolume under the roc manifold hum in addition it motivates us to address the necessary and sufficient conditions for the existence of the hum to sum up this work provides working scientists with an extension of the two class roc analysis to the multi classification roc analysis in a theoretically sound manner highlightsa proper assessment of performance probabilities is shown to be the optimal roc manifold the necessary and sufficient conditions are addressed for the existence of the hum the scientific groundwork is extended to a more general multi class roc analysis

platonic hyperbolic monopoles we construct a number of explicit examples of hyperbolic monopoles with various charges and often with some platonic symmetry the fields are obtained from instanton data in mathbb r 4 that are invariant under a circle action and in most cases the monopole charge is equal to the instanton charge a key ingredient is the identification of a new set of constraints on adhm instanton data that are sufficient to ensure the circle invariance unlike for euclidean monopoles the formulae for the squared higgs field magnitude in the examples we construct are rational functions of the coordinates using these formulae we compute and illustrate the energy density of the monopoles we also prove for particular monopoles that the number of zeros of the higgs field is greater than the monopole charge confirming numerical results established earlier for euclidean monopoles we also present some one parameter families of monopoles analogous to known scattering events for euclidean monopoles within the geodesic approximation

a large deviations approach to limit theory for heavy tailed time series in this paper we propagate a large deviations approach for proving limit theory for generally multivariate time series with heavy tails we make this notion precise by introducing regularly varying time series we provide general large deviation results for functionals acting on a sample path and vanishing in some neighborhood of the origin we study a variety of such functionals including large deviations of random walks their suprema the ruin functional and further derive weak limit theory for maxima point processes cluster functionals and the tail empirical process one of the main results of this paper concerns bounds for the ruin probability in various heavy tailed models including garch stochastic volatility models and solutions to stochastic recurrence equations

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

measuring and estimating diagnostic accuracy when there are three ordinal diagnostic groups this article studies the problem of measuring and estimating the diagnostic accuracy when there are three ordinal diagnostic groups we use a receiver operating characteristic roc surface to describe the probabilities of correct classifications into three diagnostic groups based on various sets of diagnostic thresholds of a test and propose to use the entire and the partial volume under the surface to measure the diagnostic accuracy mathematical properties and probabilistic interpretations of the proposed measure of diagnostic accuracy are discussed under the assumption of normal distributions of the diagnostic test from three diagnostic groups we present the maximum likelihood estimate to the volume under the roc surface and give the asymptotic variance to the estimate we further propose several asymptotic confidence interval estimates to the volume under the roc surface the performance of these confidence interval estimates is evaluated in terms of attaining the nominal coverage probability based on a simulation study in addition we develop a method of sample size determination to achieve an adequate accuracy of the confidence interval estimate finally we demonstrate the proposed methodology by applying it to the clinical diagnosis of early stage alzheimer s disease based on the neuropsychological database of the washington university alzheimer s disease research center

monopoles and harmonic maps recently jarvis has proved a correspondence between su n monopoles and rational maps of the riemann sphere into flag manifolds furthermore he has outlined a construction to obtain the monopole fields from the rational map in this paper we examine this construction in some detail and provide explicit examples for spherically symmetric su n monopoles with various symmetry breakings in particular we show how to obtain these monopoles from harmonic maps into complex projective spaces the approach extends in a natural way to monopoles in hyperbolic space and we use it to construct new spherically symmetric su n hyperbolic monopoles

extremes and local dependence in stationary sequences extensions of classical extreme value theory to apply to stationary sequences generally make use of two types of dependence restriction a a weak mixing condition restricting long range dependence b a local condition restricting the clustering of high level exceedances

diagnostics for dependence within time series extremes summary the analysis of extreme values within a stationary time series entails various assumptions concerning its long and short range dependence we present a range of new diagnostic tools for assessing whether these assumptions are appropriate and for identifying structure within extreme events these tools are based on tail characteristics of joint survivor functions but can be implemented by using existing estimation methods for extremes of univariate independent and identically distributed variables our diagnostic aids are illustrated through theoretical examples simulation studies and by application to rainfall and exchange rate data on the basis of these diagnostics we can explain characteristics that are found in the observed extreme events of these series and also gain insight into the properties of events that are more extreme than those observed

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

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

implementation of an iterative map in the construction of quasi periodic instantons chaotic aspects and discontinuous rotation numbers an iterative map of the unit disk in the complex plane is used to explore certain aspects of self dual four dimensional gauge fields quasi periodic in the euclidean time these fields are characterized by two topological numbers and contain standard instantons and monopoles as different limits the iterations do not correspond directly to a discretized time evolution of the gauge fields they are implemented in an indirect fashion first t r being the standard coordinates the r t half plane is mapped on the unit disk in an appropriate way this provides an r t parametrization of z0 the starting point of the iterations and makes the iterates increasingly complex functions of r and t these are then incorporated as building blocks in the generating function of the fields we explain in what sense and to what extent some remarkable features of our map indicated in the title are thus carried over into the continuous time development of the fields special features for quasiperiodicity are stu

the stationary bootstrap abstract this article introduces a resampling procedure called the stationary bootstrap as a means of calculating standard errors of estimators and constructing confidence regions for parameters based on weakly dependent stationary observations previously a technique based on resampling blocks of consecutive observations was introduced to construct confidence intervals for a parameter of the m dimensional joint distribution of m consecutive observations where m is fixed this procedure has been generalized by constructing a blocks of blocks resampling scheme that yields asymptotically valid procedures even for a multivariate parameter of the whole i e infinite dimensional joint distribution of the stationary sequence of observations these methods share the construction of resampling blocks of observations to form a pseudo time series so that the statistic of interest may be recalculated based on the resampled data set but in the context of applying this method to stationary data it is natural

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

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

models for the extremes of markov chains the modelling of extremes of a time series has progressed from the assumption of independent observations to more realistic forms of temporal dependence in this paper we focus on markov chains deriving a class of models for their joint tail which allows the degree of clustering of extremes to decrease at high levels overcoming a key limitation in current methodologies theoretical aspects of the model are examined and a simulation algorithm is developed through which the stochastic properties of summaries of the extremal txhaviour of the chain are evaluated the approach is illustrated through a simulation study of extremal events of gaussian autoregressive processes and an application to temperature data