edgeworth correction by bootstrap in autoregressions we prove that the distribution of least squares estimates in autoregressions can be bootstrapped with accuracy o n 1 2 a s thereby improving the normal approximation error of o n

relation between volatility correlations in financial markets and omori processes occurring on all scales we analyze the memory in volatility by studying volatility return intervals defined as the time between two consecutive fluctuations larger than a given threshold in time periods following stock market crashes such an aftercrash period is characterized by the omori law which describes the decay in the rate of aftershocks of a given size with time t by a power law with exponent close to 1 a shock followed by such a power law decay in the rate is here called omori process we find self similar features in the volatility specifically within the aftercrash period there are smaller shocks that themselves constitute omori processes on smaller scales similar to the omori process after the large crash we call these smaller shocks subcrashes which are followed by their own aftershocks we also show that the omori law holds not only after significant market crashes as shown by lillo and mantegna phys rev e 68 016119 2003 but also after intermediate shocks by appropriate detrending we remove the influence of the crashes and subcrashes from the data and find that this procedure significantly reduces the memory in the records moreover when studying long term correlated fractional brownian motion and autoregressive fractionally integrated moving average artificial models for volatilities we find omori type behavior after high volatilities thus our results support the hypothesis that the memory in the volatility is related to the omori processes present on different time scales

effect of nonlinear correlations on the statistics of return intervals in multifractal data sets we study the statistics of return intervals between events above a certain threshold in multifractal data sets without linear correlations we find that nonlinear correlations in the record lead to a power law i decay of the autocorrelation function of the return intervals ii increase in the conditional return period and iii decay in the probability density function of the return intervals we show explicitly that all the observed quantities depend both on the threshold value and system size and hence there is no simple scaling observed we also demonstrate that this type of behavior can be observed in real economic records and can be used to improve considerably risk estimation

indication of multiscaling in the volatility return intervals of stock markets a certain trend with the mean interval we generate surrogate records using the schreiber method and find that their cumulative distributions almost collapse to a single curve and moments are almost constant for most ranges of those substantial differences suggest that nonlinear correlations in the original volatility sequence account for the deviations from a single scaling law we also find that the original and surrogate records exhibit slight tendencies for short and long due to the discreteness and finite size effects of the records respectively to avoid as possible those effects for testing the multiscaling behavior we investigate the moments in the range 10100 and find that the exponent from the power law fitting m has a narrow distribution around 0 which depends on m for the 500 stocks the distribution of for the surrogate records are very narrow and centered around 0 this suggests that the return interval distribution exhibits multiscaling behavior due to the nonlinear correlations in the original volatility

on resampling and uncertainty estimation in linear system identification linear system identification yields a nominal model parameter which minimizes a specific criterion based on the single input output data set here we investigate the utility of various methods for estimating the probability distribution of this nominal parameter using only the data from this single experiment the results are compared to the actual parameter distribution generated by many monte carlo runs of the data collection experiment the methods considered are collectively known as resampling schemes which include subsampling the jackknife and the bootstrap the broad aim is to generate an empirical parameter distribution function via the construction of a large number of new data records from the original single set of data based on an assumption that this data is representative of all possible data and then to run the parameter estimator on each of these new records to develop the distribution function the performance of these schemes is evaluated on a difficult almost unidentifiable system and compared to the standard results based on asymptotic normality in addition to the exploration of this example as a means to evaluate the strengths and weaknesses of these resampling schemes some new theoretical results are proven and demonstrated for subsampling schemes

memory in the occurrence of earthquakes we study the statistics of the recurrence times ensuremath tau between earthquakes above a certain magnitude m in six one global and five regional earthquake catalogs we find that the distribution of the recurrence times strongly depends on the previous recurrence time ensuremath tau 0 such that small and large recurrence times tend to cluster in time this dependence on the past is reflected in both the conditional mean recurrence time and the conditional mean residual time until the next earthquake which increase monotonically with ensuremath tau 0 as a consequence the risk of encountering the next event within a certain time span after the last event depends significantly on the past an effect that has to be taken into account in any effective earthquake prognosis

positive solutions of an elliptic partial differential equation on rn abstract in this paper we discuss the existence nonexistence and uniqueness of positive solutions of a one parameter family of elliptic partial differential equations on r n n 2 these equations are of interests in mathematical biology and riemannian geometry our approach are based on variational arguments and comparison principles

existence and non existence results for a logistic type equation on manifolds we study the steady state solutions of a generalized logistic type equation on a complete riemannian manifold we provide sufficient conditions for existence respectively non existence of positive solutions which depend on the relative size of the coefficients and their mutual interaction with the geometry of the manifold which is mostly taken into account by means of conditions on the volume growth of geodesic balls

positive solutions of yamabe type equations on complete manifolds and applications we study the semilinear equation u a x u b x u 1 on a complete riemannian manifold we determine conditions on the coefficients that guarantee existence and nonexistence of positive solutions a very general uniqueness result is also established our main results are valid without explicit curvature assumptions and appear to be new even in rm

a priori estimates uniqueness and existence of positive solutions of yamabe type equations on complete manifolds abstract we study the asymptotic behaviour of non negative solutions of yamabe type equations on a complete riemannian manifold then we provide a comparison result based on a form of the weak maximum principle at infinity which together with the a priori estimates previously obtained yields uniqueness under very general ricci assumptions the paper ends with an existence result and an application to the non compact yamabe problem

the distribution of low variability periods in human heartbeat dynamics abstract we study the long term dynamics of the short time variability level of human heart rate an aspect which is not addressed by the traditional methods of non linear time series analysis the length distribution of low variability periods in human heartbeat dynamics typically follows a multi scaling power law the values of the scaling exponents are personal characteristics and depend on the daily habits of the subjects though the distribution function of the low variability periods as a whole discriminates efficiently between several heart pathologies

statistics of return intervals in long term correlated records we consider long term correlated data with several distribution densities gaussian exponential power law and log normal and various correlation exponents ensuremath gamma 0l ensuremath gamma l1 and study the statistics of the return intervals r j between events above some threshold q we show that irrespective of the distribution the return intervals are long term correlated in the same way as the original record but with additional uncorrelated noise due to this noise the correlations are difficult to observe by the detrended fluctuation analysis which exhibits a crossover behavior but show up very clearly in the autocorrelation function the distribution p q r of the return intervals is characterized at large scales by a stretched exponential with exponent ensuremath gamma and at short scales by a power law with exponent ensuremath gamma ensuremath 1 we discuss in detail the occurrence of finite size effects for large threshold values for all considered distributions we show that finite size effects are most pronounced in exponentially distributed data sets where they can even mask the stretched exponential behavior in records of up to 10 6 data points finally in order to quantify the clustering of extreme events due to the long term correlations in the return intervals we study the conditional distribution function and the related moments we find that they show pronounced memory effects irrespective of the distribution of the original data

computer intensive methods in statistical regression this is a survey of modern developments in statistical regression written for the mathematically educated nonstatistician it begins with a review of the traditional theory of least squares curve fitting modern developments in regression theory have developed in response to the practical limitations of the least squares approach recent progress has been made feasible by the electronic computer which frees statisticians from the confines of mathematical tractability topics discussed include robust regression bootstrap measures of variability local smoothing and cross validation projection pursuit mallows c p criterion stein estimation generalized regression for poisson data and regression methods for censored data all of the methods are illustrated with real life examples

estimation of model quality this paper provides an introduction to recent work on the problem of quantifying errors in the estimation of models for dynamic systems this is a very large field we therefore concentrate on approaches that have been motivated by the need for reliable models for control system design this will involve a discussion of efforts that go under the titles of estimation in th worst case estimation estimation in l1 and stochastic embedding of undermodelling a central theme of this survey is to examine these new methods with reference to the classic bias variance tradeoff in model structure selection

scaling and memory of intraday volatility return intervals in stock markets we study the return interval tau between price volatilities that are above a certain threshold q for 31 intraday data sets including the standard and poor s 500 index and the 30 stocks that form the dow jones industrial index for different threshold q the probability density function pq tau scales with the mean interval tau as formula see text similar to that found in daily volatilities since the intraday records have significantly more data points compared to the daily records we could probe for much higher thresholds and still obtain good statistics we find that the scaling function f x is consistent for all 31 intraday data sets in various time resolutions and the function is well approximated by the stretched exponential f x similar to e ax gamma with gamma 0 38 0 05 and a 3 9 0 5 which indicates the existence of correlations we analyze the conditional probability distribution pq tau tau0 for tau following a certain interval tau0 and find pq tau tau0 depends on tau0 which demonstrates memory in intraday return intervals also we find that the mean conditional interval tau tau0 increases with tau0 consistent with the memory found for pq tau tau0 moreover we find that return interval records in addition to having short term correlations as demonstrated by pq tau tau0 have long term correlations with correlation exponents similar to that of volatility records

properties of low variability periods in financial time series abstract properties of low variability periods in the time series are analysed the theoretical approach is used to show the relationship between the multi scaling of low variability periods and multi affinity of the time series it is shown that this technically simple method is capable of revealing more details about time series than the traditional multi affine analysis we have applied this scaling analysis to financial time series a number of daily currency and stock index time series the results show a good scaling behaviour for different model parameters the analysis of high frequency usd eur exchange rate data confirmed the theoretical expectations

scaling analysis of multi variate intermittent time series abstract the scaling properties of the time series of asset prices and trading volumes of stock markets are analysed it is shown that similar to the asset prices the trading volume data obey multi scaling length distribution of low variability periods in the case of asset prices such scaling behaviour can be used for risk forecasts the probability of observing next day a large price movement is super universally inversely proportional to the length of the ongoing low variability period finally a method is devised for a multi factor scaling analysis we apply the simplest two factor model to equity index and trading volume time series

probability of large movements in financial markets abstract based on empirical financial time series we show that the silence breaking probability follows a super universal power law the probability of observing a large movement is inversely proportional to the length of the on going low variability period such a scaling law has been previously predicted theoretically r kitt j kalda physica a 353 2005 480 assuming that the length distribution of the low variability periods follows a multi scaling power law

positive solutions for logistic type quasilinear elliptic equations on rn in this paper we consider positive solutions of the logistic type p laplacian equation pu a x u p 2u b x u q 1u x rn n 2 we show that under rather general conditions on a x and b x for large x the behavior of the positive solutions for large x can be determined this is then used to show that there is a unique positive solution our results improve the corresponding ones in j london math soc 2 64 2001 107 124 and j anal math in press

diffusion type operators liouville theorems and gradient estimates on complete manifolds abstract we study liouville theorems and gradient estimates for solutions of eq 1 1 with the help of a diffusion operator and the related geometry

guaranteed non asymptotic confidence regions in system identification in this paper we consider the problem of constructing confidence regions for the parameters of identified models of dynamical systems taking a major departure from the previous literature on the subject we introduce a new approach called leave out sign dominant correlation regions lscr which delivers confidence regions with guaranteed probability all results hold rigorously true for any finite number of data points and no asymptotic theory is involved moreover prior knowledge on the noise affecting the data is reduced to a minimum the approach is illustrated on several simulation examples showing that it delivers practically useful confidence sets with guaranteed probabilities

memory effects in the statistics of interoccurrence times between large returns in financial records we study the statistics of the interoccurrence times between events above some threshold q in two kinds of multifractal data sets multiplicative random cascades and multifractal random walks with vanishing linear correlations we show that in both data sets the relevant quantities probability density functions and the autocorrelation function of the interoccurrence times as well as the conditional return period are governed by power laws with exponents that depend explicitly on the considered threshold by studying a large number of representative financial records market indices stock prices exchange rates and commodities we show explicitly that the interoccurrence times between large daily returns follow the same behavior in a nearly quantitative manner we conclude that this kind of behavior is a general consequence of the nonlinear memory inherent in the multifractal data sets

large sample confidence regions based on subsamples under minimal assumptions in this article the construction of confidence regions by approximating the sampling distribution of some statistic is studied the true sampling distribution is estimated by an appropriate normalization of the values of the statistic computed over subsamples of the data in the i i d context the method has been studied by wu in regular situations where the statistic is asymptotically normal the goal of the present work is to prove the method yields asymptotically valid confidence regions under minimal conditions essentially all that is required is that the statistic suitably normalized possesses a limit distribution under the true model unlike the bootstrap the convergence to the limit distribution need not be uniform in any sense the method is readily adapted to parameters of stationary time series or more generally homogeneous random fields for example an immediate application is the construction of a confidence interval for the spectral density function of a homogeneous random field