subharmonic solutions in singular systems abstract we consider the problem of bifurcation from homoclinic towards periodic orbits for a periodic singular system of differential equations we use a functional analytic approach and the lyapunov schmidt method to obtain a bifurcation function for this problem which tends as e 0 to the well known melnikov function for the bifurcation towards transverse homoclinic orbits we also study the behavior of the bifurcation functions as the period of the periodic orbit tends to infinity

bayesian model monitoring on presente une approche bayesienne basee sur des comparaisons des predictions a partir du modele standard avec celles d un modele alternatif simple

rank based tests for randomness against first order serial dependence abstract optimal rank based procedures were derived in hallin ingenbleek and puri 1985 1987 and hallin and puri 1988 for some fundamental testing problems arising in time series analysis the optimality properties of these procedures are of an asymptotic nature however whereas much of the attractiveness of rank based methods lies in their small sample applicability and robustness features accordingly the objective of this article is twofold a a study of the finite sample behavior of the asymptotically optimal tests for randomness against first order autoregressive moving average dependence proposed in hallin et al 1985 both under the null hypothesis tables of critical values and under alternatives of serial dependence evaluation of the power function and b an heuristic investigation of the robustness properties of the proposed procedures with emphasis on the identification problem in the presence of outliers we begin sec 2 with a brief description of the rank based measu

robust parametric approach for impulse response estimation a parametric technique for estimating the impulse response of a linear system using input output observations in an outlier and distributionally uncertain environment is presented the use of various cost functions for fitting the chosen output error model are discussed by simulation it is shown that the parametric approach based on the use of huber s function as a criterion for fitting the model is robust it is also shown that even though the parametric model for the impulse response is only an approximation to the true impulse response the estimates from this approach still outperform the nonparametric approach in the presence of contaminated noise and low snr

on a mixture autoregressive conditional heteroscedastic model we propose a mixture autoregressive conditional heteroscedastic mar arch model for modeling nonlinear time series the models consist of a mixture of k autoregressive components with autoregressive conditional heteroscedasticity that is the conditional mean of the process variable follows a mixture ar mar process whereas the conditional variance of the process variable follows a mixture arch process in addition to the advantage of better description of the conditional distributions from the mar model the mararch model allows a more flexible squared autocorrelation structure the stationarity conditions autocorrelation function and squared autocorrelation function are derived construction of multiple step predictive distributions is discussed the estimation can be easily done through a simple em algorithm and the model selection problem is addressed the shape changing feature of the conditional distributions makes these models capable of modeling time series with multimodal conditional distr

on a mixture autoregressive model we generalize the gaussian mixture transition distribution gmtd model introduced by le and co workers to the mixture autoregressive mar model for the modelling of non linear time series the models consist of a mixture of k stationary or non stationary ar components the advantages of the mar model over the gmtd model include a more full range of shape changing predictive distributions and the ability to handle cycles and conditional heteroscedasticity in the time series the stationarity conditions and autocorrelation function are derived the estimation is easily done via a simple em algorithm and the model selection problem is addressed the shape changing feature of the conditional distributions makes these models capable of modelling time series with multimodal conditional distributions and with heteroscedasticity the models are applied to two real data sets and compared with other competing models the mar models appear to capture features of the data better than other competing models do

recursive estimation of autoregressions this paper is concerned with the recursive estimation of autoregressive models in particular the realtime determination of the order of such a specification a model selection criterion based on the minimum description length principle due to rissanen is discussed and strong consistency in the stationary situation is shown alternative criteria are also considered and modifications required to introduce forgetting when the procedures are implemented in the non stationary case are presented some simulation evidence on the performance of the criteria when applied to stationary and non stationary processes is given

the jackknife and the bootstrap for general stationary observations we extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence we do not attempt a reduction to i i d values the jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of l consecutive data once in the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations at zero under appropriate conditions consistency is obtained if l l n rightarrow infty and l n n rightarrow 0 general statistics are approximated by an arithmetic mean in regular cases this approximation determines the asymptotic behavior bootstrap replicates are constructed by selecting blocks of length l randomly with replacement among the blocks of observations the procedures are illustrated by using the sunspot numbers and some simulated data

outlier detection and time series modeling some statistics used in regression analysis are considered for detection of outliers in time series approximations and asymptotic distributions of these statistics are considered a method is proposed for distinguishing an observational outlier from an innovational one a four step procedure for modeling time series in the presence of outliers is also proposed and an example is presented to illustrate the methodology

further analysis of the remedian algorithm we present an analysis of the remedian an efficient known algorithm for the approximate median selection problem that is easy to implement the algorithm can be used for data in an array as well as for streaming data in an array it performs in place recursively dividing the candidate values into sets of size b from which exact medians are selected for the next phase on streaming data it performs a filter operation requiring by the time n items are processed the storage of log bn candidate entries the contribution of the article is a precise characterization combinatorial and asymptotic of the accuracy of the algorithm showing explicitly the role of the critical design parameter b in addition we compute the time and space costs of the algorithm and present experimental illustrations of its accuracy

comparison theory in lorentzian and riemannian geometry abstract a survey is given of selected aspects of comparison theory for lorentzian and riemannian manifolds in which both jacobi equation and riccati equation techniques have been employed specifically the existence of conjugate points on a complete geodesic in the presence of positive ricci curvature and the topic of volume comparison are treated

the existence of transverse homoclinic solutions for higher order equations abstract differential equations are considered which contain a small parameter when the parameter is zero the equation is autonomous with a hyperbolic equilibrium and a homoclinic solution no restriction is placed on the dimension of the phase space or the dimension of intersection of the stable and unstable manifolds sufficient conditions on the perturbation terms for producing a transverse homoclinic solution and hence chaos are obtained the technique follows the work of palmer using exponential dichotomies and the method of lyapunov schmidt

on surfaces with no conjugate points a complete riemannian manifold m has no conjugate points if any two points in its universal cover are joined by a unique geodesic if the sectional curvature of m is nonpositive then m has no conjugate points the converse is not true even for compact surfaces a natural question is to what extent properties of manifolds of nonpositive sectional curvature are valid for manifolds with no conjugate points for example by the gauss bonnet theorem any metric of nonpositive curvature on the torus t is flat in 1943 e hopf 4 proved theorem any metric ont with no conjugate points is flat the best way to explain the purpose of our paper and to introduce the necessary notations is to give an outline of hopf s argument he considers the riccati equation

perturbation of homoclinics and subharmonics in duffing s equation abstract for lambda mu and g f small the equation x x 2x cubed g x lambda x mu cos t f t is considered near the separatrix s of the unperturbed equation lambda mu 0 0 for g f in a neighborhood of the zero functions a complete description is given of the bifurcation curves in lambda mu space to homoclinic points the perturbation of subharmonics outside s is also considered in a symmetric and nonsymmetric case

robust methods for recursive autoregressive moving average estimation in this paper we consider the problem of fitting recursively in time an autoregressive moving average arma model to time series data where outliers may be present although many recursive estimation procedures are available for fitting arma models they are based on the recursive least squares algorithm which is known to be badly affected by additive outliers to minimize the impact of these sort of outliers we investigate some robustified versions of the recursive maximum likelihood procedure the problem of incorporating robust scale recursion is discussed and asymptotic properties of the procedures are examined

robust estimation for arma models this paper introduces a new class of robust estimates for arma models they are m estimates but the residuals are computed so the effect of one outlier is limited to the period where it occurs these estimates are closely related to those based on a robust filter but they have two important advantages they are consistent and the asymptotic theory is tractable we perform a monte carlo where we show that these estimates compare favorably with respect to standard m estimates and to estimates based on a diagnostic procedure

robust multiple time series modelling summary a robust estimation procedure for multiple time series is proposed based on robustifying the residual autocovariances in the estimating equation the asymptotic distribution of these estimators is derived a robustified multivariate portmanteau statistic is also obtained which can be useful in model diagnostic checking an illustrative example based on the mink muskrat data is presented

bifurcation from a heteroclinic solution in differential delay equations on etudie une classe d equations differentielles fonctionnelles x t af x t 1 a non linearite periodique f r r o f dans ao et f o dans o b f a f o f b o on demontre que pour une certaine valeur critique a a o il existe une solution heterocline allant de la solution d equilibre t a a l equilibre t b

the limiting behavior of lms a realization oriented analysis is given of the gradient noise misadjustment and lag misadjustment performance of the lms least mean square algorithm new formulas are given for both of these components of excess mean square error it is shown that the traditional formula for lag misadjustment needs to be modified by adding further terms due to gradient noise and noise variance to perform the analysis it is necessary to study the convergence with probability one of the noise free fixed parameter lms algorithm convergence is found under simple conditions that improve on those previously obtained

subharmonic solutions of systems of difference equations with periodic perturbations part i existence we consider the existence of subharmonic solutions of systems of difference equations with periodic perturbations the theory of coincidence degree coupled with some detailed a priori estimates is applied to show that the perturbed system admits subharmonic solutions near a hyperbolic periodic orbit of the unperturbed system if the perturbation is sufficiently small

exponential dichotomies and homoclinic orbits in functional differential equations abstract suppose an autonomous functional differential equation has an orbit which is homoclinic to a hyperbolic equilibrium point the purpose of this paper is to give a procedure for determining the behavior of the solutions near of a functional differential equation which is a nonautonomous periodic perturbation of the original one the procedure uses exponential dichotomies and the fredholm alternative it is also shown that any smooth function p t defined on the reals which approaches zero monotonically as t is the solution of a scalar functional differential equation and generates an orbit homoclinic to zero examples illustrating the results are also given

infinitesimal robustness for diffusions we develop infinitesimally robust statistical procedures for the general diffusion processes we first prove the existence and uniqueness of the times series influence function of conditionally unbiased m estimators for ergodic and stationary diffusions under weak conditions on the martingale estimating function used we then characterize the robustness of m estimators for diffusions and derive a class of conditionally unbiased optimal robust estimators to compute these estimators we propose a general algorithm which exploits approximation methods for diffusions in the computation of the robust estimating function monte carlo simulation shows a good performance of our robust estimators and an application to the robust estimation of the exchange rate dynamics within a target zone illustrates the methodology in a real data application

subharmonic solutions bifurcated from homoclinic orbits for weakly coupled singular systems abstract the problem of bifurcation from homoclinic solution towards periodic solution was considered for weekly coupled singular systems by using functional analytic approach based on the lyapunov schmidt reduction we obtained some functions h r d 1 r r d the simple roots of the equations h 0 correspond to the existence of subharmonic solutions and if the vector field is 2 period then for any integer m the weakly coupled singular system has 2 m period solution

robust estimation without positive real condition the strictly positive real spr condition on the noise model is necessary for a discrete time linear stochastic control system with unmodeled dynamics even so for a time invariant armax system in the past robust analysis of parameter estimation however this condition is hardly satisfied for a high order and or multidimensional system with correlated noise the main work in this paper is to show that for robust parameter estimation and adaptive tracking as well as closed loop system stabilization the spr condition is replaced by a stable matrix polynomial the main method is to design a two step recursive least squares algorithm with or without a weighted factor and with a fixed lag regressive vector and to define an adaptive control with bounded external excitation and with randomly varying truncation

robust inference with gmm estimators the local robustness properties of generalized method of moments gmm estimators and of a broad class of gmm based tests are investigated in a unified framework gmm statistics are shown to have bounded influence if and only if the function defining the orthogonality restrictions imposed on the underlying model is bounded since in many applications this function is unbounded it is useful to have procedures that modify the starting orthogonality conditions in order to obtain a robust version of a gmm estimator or test we show how this can be obtained when a reference model for the data distribution can be assumed we develop a flexible algorithm for constructing a robust gmm rgmm estimator leading to stable gmm test statistics the amount of robustness can be controlled by an appropriate tuning constant we relate by an explicit formula the choice of this constant to the maximal admissible bias on the level or and the power of a gmm test and the amount of contamination that one can reasonably assume given some information on the data finally we illustrate the rgmm methodology with some simulations of an application to rgmm testing for conditional heteroscedasticity in a simple linear autoregressive model in this example we find a significant instability of the size and the power of a classical gmm testing procedure under a non normal conditional error distribution on the other side the rgmm testing procedures can control the size and the power of the test under non standard conditions while maintaining a satisfactory power under an approximatively normal conditional error distribution

does median filtering truly preserve edges better than linear filtering image processing researchers commonly assert that median filtering is better than linear filtering for removing noise in the presence of edges using a straightforward large n decision theory framework this folk theorem is seen to be false in general we show that median filtering and linear filtering have similar asymptotic worst case mean squared error mse when the signal to noise ratio snr is of order 1 which corresponds to the case of constant per pixel noise level in a digital signal to see dramatic benefits of median smoothing in an asymptotic setting the per pixel noise level should tend to zero i e snr should grow very large we show that a two stage median filtering using two very different window widths can dramatically outperform traditional linear and median filtering in settings where the underlying object has edges in this two stage procedure the first pass at a fine scale aims at increasing the snr the second pass at a coarser scale correctly exploits the nonlinearity of the median image processing methods based on nonlinear partial differential equations pdes are often said to improve on linear filtering in the presence of edges such methods seem difficult to analyze rigorously in a decision theoretic framework a popular example is mean curvature motion mcm which is formally a kind of iterated median filtering our results on iterated median filtering suggest that some pde based methods are candidates to rigorously outperform linear filtering in an asymptotic framework

bounded influence propagation tau estimation a new robust method for arma model estimation a new robust and statistically efficient estimator for arma models called the bounded influence propagation estimator is proposed the estimator incorporates an auxiliary model which prevents the propagation of outliers strong consistency and asymptotic normality of the estimator for arma models that are driven by independently and identically distributed iid innovations with symmetric distributions are established to analyze the infinitesimal effect of outliers on the estimator the influence function is derived and computed explicitly for an ar 1 model with additive outliers to obtain estimates for the ar p model a robust durbin levinson type and a forward backward algorithm are proposed an iterative algorithm to robustly obtain arma p q parameter estimates is also presented the problem of finding a robust initialization is addressed which for orders p q 2 is a nontrivial matter numerical experiments are conducted to compare the finite sample performance of the proposed estimator to existing robust methodologies for different types of outliers both in terms of average and of worst case performance as measured by the maximum bias curve to illustrate the practical applicability of the proposed estimator a real data example of outlier cleaning for r r interval plots derived from electrocardiographic data is considered the proposed estimator is not limited to biomedical applications but is also useful in any real world problem whose observations can be modeled as an arma process disturbed by outliers or impulsive noise

alternatives to the median absolute deviation abstract in robust estimation one frequently needs an initial or auxiliary estimate of scale for this one usually takes the median absolute deviation mad n 1 4826 med xi med j x j because it has a simple explicit formula needs little computation time and is very robust as witnessed by its bounded influence function and its 50 breakdown point but there is still room for improvement in two areas the fact that mad n is aimed at symmetric distributions and its low 37 gaussian efficiency in this article we set out to construct explicit and 50 breakdown scale estimators that are more efficient we consider the estimator sn 1 1926 med med j xi xj and the estimator qn given by the 25 quantile of the distances xi x j i j note that sn and qn do not need any location estimate both sn and qn can be computed using o n log n time and o n storage the gaussian efficiency of sn is 58 whereas qn attains 82 we study sn and qn by means of their influence functions their b

rank based autoregressive order identification abstract optimal rank based procedures have been derived for testing arbitrary linear restrictions on the parameters of autoregressive moving average arma models with unspecified innovation densities the finite sample performances of these procedures are investigated here in the context of ar order identification and compared to those of classical partial correlograms and lagrange multipliers methods the results achieved by rank based methods are quite comparable in the gaussian case to those achieved by the traditional ones which under gaussian assumptions are asymptotically optimal however under non gaussian innovation densities especially heavy tailed or nonsymmetric or when outliers are present the percentages of correct order selection based on rank methods are strikingly better than those resulting from traditional approaches even in the case of very short n 25 series these empirical findings confirm the often ignored theoretical fact that the gaussian case in the arma context

a new robust estimation method for arma models the autoregressive moving average arma modeling of time series is popular and used in many applications in this paper we introduce a new robust method to estimate the parameters of a gaussian arma model contaminated by outliers this method makes use of the median and is termed ratio of medians estimator rme the ratios of medians are used to estimate robustly the autocorrelation function and thus estimate the parameters its theoretical robustness is analyzed by the computation of robust measures such as maximum bias breakdown point and influence function the rme estimator is shown to be asymptotically normal and its asymptotic variance is computed under gaussian autoregressive models of order p p 1 the new method is evaluated and compared with other robust methods via simulations its effectiveness in terms of parameter estimation and forecasting is demonstrated on an example of the french daily electricity consumptions the new approach improves the load forecasting quality for normal days and presents several interesting properties such as good robustness fast execution simplicity and easy online implementation

a robust approach for high resolution frequency estimation a robust estimation method for frequencies of received signals is considered the influence function of a robust estimate is derived for the sinusoidal signals the variance of the robust estimate is also derived using the influence function it is found that the robust estimate attains the cramer rao lower bound crlb for the contaminated gaussian distribution it is of order o n sup 3 and is close to the crlb for the perfect gaussian distribution the authors introduce some basic definitions for the high resolution frequency estimation method and prove that the robust estimate has the high resolution property this property is also confirmed by numerical simulations

higher order infinitesimal robustness using the von mises expansion we study the higher order infinitesimal robustness of a general m functional and characterize its second order properties we show that second order robustness is equivalent to the boundedness of both the estimator s estimating function and its derivative with respect to the parameter it implies at the same time i variance robustness and ii robustness of higher order saddlepoint approximations to the estimator s finite sample density the proposed construction of second order robust m estimators is fairly general and potentially useful in a variety of relevant settings besides the theoretical contributions we discuss the main computational issues and provide an algorithm for the implementation of second order robust m estimators finally we illustrate our theory by monte carlo simulation and in a real data estimation of the maximal losses of nikkei 225 index returns our findings indicate that second order robust estimators can improve on other widely applied robus

adaptive spectral factorization an on line spectral factorization algorithm is used to devise a globally convergent self tuning identifier that does not suffer from restrictions that amount to knowledge of the true system e g the positive real condition the method developed uses two ideas one idea an old one which might be called the method of split recursions is used to estimate the parameters in blocks thus one block might get the transfer function parameters while the other gets the noise parameters the other idea is to use spectral factorization to estimate moving average parameters the algorithm does have its own weaknesses e g transient behavior may not be good and it relies on a condition that is only generically true but it does not need a positive real condition to be satisfied for global convergence

two dimensional robust spectrum estimation robust estimation is studied of two dimensional 2 d power spectra of signals which are adequately represented by gaussian random field models but for which there are imperfect observations two situations of particular interest occur when the contaminating noise is additive and when the contaminating noise appears in the innovations in these cases the observed data are not gaussian and conventional procedures are no longer efficient to estimate the parameters of the signal model from the contaminated data two procedures are described which were originally proposed for estimation of scale and location from independent data and adapted to one dimensional autoregression parameter estimation by previous researchers the first algorithm is a robustification of least squares and equivalent to an iterated weighted least squares problem where the weights are data dependent the second algorithm is an iterative procedure known as a filter cleaner experiments using the robust procedures with synthetic data are reported and the results compared to a conventional method of model based spectrum estimation

robust real time identification of linear systems with correlated noise the problem of recursive robust identification of linear discrete time single input single output dynamic systems with correlated disturbances is considered problems related to the construction of optimal robust stochastic approximation algorithms in the min max sense are demonstrated since the optimal solution cannot be achieved in practice several robustified stochastic approximation algorithms are derived on the basis of a suitable non linear transformation of normalized residuals as well as step by step optimization with respect to the weighting matrix of the algorithm the convergence of the developed algorithms is established theoretically using the ordinary differential equation approach monte carlo simulation results are presented for the quantitative performance evaluation of the proposed algorithms the results indicate the most suitable algorithms for applications in engineering practice

bifurcation from homoclinic to periodic solutions in singular ordinary differential equations abstract problems of bifurcations from homoclinic to periodic orbits are considered for periodic singular differential equations a functional analytic approach together with the method of lyapunov schmidt is used to obtain bifurcation functions for these problems which yield periodic solutions of all integer multiples of the period of the vector field

fuzzy order statistics and their application to fuzzy clustering the median and the median absolute deviation mad are robust statistics based on order statistics order statistics are extended to fuzzy sets to define a fuzzy median and a fuzzy mad the fuzzy c means fcm clustering algorithm is defined for any p norm pfcm including the l sub 1 norm 1fcm the 1fcm clustering algorithm is implemented via the alternating optimization ao method and the clustering centers are shown to be the fuzzy median the resulting ao 1fcm clustering algorithm is called the fuzzy c medians fcmed clustering algorithm an example illustrates the robustness of the fcmed

the change of variance function for dependent data the infinitesimal stability of the asymptotic variance is considered form estimators of a location parameter when the nominal sample with i i d data is contaminated by a possibly dependent process it is shown that the resulting change of variance function can be expressed as a sum of two terms one corresponding contamination of the univariate distribution and one to contamination of the bivariate distributions a change of variance sensitivity is introduced the form of which is closely related to the average patch length of the outliers finally optimalv robust and mostv robust score functions are derived the resulting family of estimators is the same as for independent data in the general case but the truncation point approaches zero when dependency is accounted for for redescending score functions the family of estimators is changed

dynamic generalized linear models and bayesian forecasting abstract dynamic bayesian models are developed for application in nonlinear non normal time series and regression problems providing dynamic extensions of standard generalized linear models a key feature of the analysis is the use of conjugate prior and posterior distributions for the exponential family parameters this leads to the calculation of closed standard form predictive distributions for forecasting and model criticism the structure of the models depends on the time evolution of underlying state variables and the feedback of observational information to these variables is achieved using linear bayesian prediction methods data analytic aspects of the models concerning scale parameters and outliers are discussed and some applications are provided dynamic bayesian models are developed for application in nonlinear non normal time series and regression problems providing dynamic extensions of standard generalized linear models a key feature of the analysis is the use of conjugate prior and

boundary and lens rigidity of lorentzian surfaces let g be a lorentzian metric on the plane 2 that agrees with the standard metric g0 dx2 dy2 outside a compact set and so that there are no conjugate points along any time like geodesic of 2 g then 2 g and 2 g0 are isometric further if m g and m p are two dimensional compact time oriented lorentzian manifolds with space like boundaries and so that all time like geodesies of m g maximize the distances between their points and m g and m g are boundary isometric then there is a conformal diffeomorphism between m g and m g and they have the same areas similar results hold in higher dimensions under an extra assumption on the volumes of the manifolds

codimension 3 bifurcations of a class of homoclinic loop with saddle point abstract in this work bifurcations in the class that the homoclinic orbit connects the strong stable and strong unstable manifolds of a saddle are investigated for four dimensional vector fields the existence numbers coexistence and incoexistence of 1 homoclinic orbit 1 periodic orbit 2 n homoclinic orbit and 2 n periodic orbit are obtained the approximate expressions of the corresponding bifurcation surfaces and the bifurcation diagrams are also presented

recursive estimation of linear systems on compare par simulation deux algorithmes bien connus avec une methode liee a l algorithme de hannan et rissanen 1982

strong approximation of the recursive prediction error estimator of the parameters of an arma process abstract let n denote the recursive prediction error estimator of the parameters of an arma process and let n denote the off line prediction error estimator we show under reasonable assumptions that for any q 1 n n lq o m log n n this theorem enables us to show that most of the properties known for the off line estimator n are inherited by the on line estimator n

homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations nonautomonous ordinary differential equations depending on two parameters 1 and 2 are considered in rn it is assumed that when both parameters are zero the differential equation is autonomous with a hyperbolic equilibrium and a homoclinic solution no restriction is placed on the dimension of the phase space rn or on the dimension of intersection of the stable and unstable manifolds by means of the method of lyapunov schmidt a bifurcation function h is constructed between two finite dimensional spaces where the zeros of h correspond to homoclinic solutions at nonzero parameter values the independent variables of h consist of scalars 1 2 and a vector where is a phase angle and corresponds to directions other than along the original homoclinic solution tangent to both the stable and unstable manifolds when is fixed the equation h 0 yields in general several bifurcation curves through the origin in the 1 2 plane along which there exists a homoclinic solution when is varied these become a number of wedge shaped regions the theory is applied to two examples one in r6 where the invariant manifolds meet in dimension three and a second in r4 where these manifolds agree

the remedian a robust averaging method for large data sets abstract it is often assumed that to compute a robust estimator on n data values one needs at least n storage elements contrary to the sample average that may be calculated with an updating mechanism this is one of the main reasons why robust estimators are seldom used for large data sets and why they are not included in most statistical packages we introduce a new estimator that takes up little storage space investigate its statistical properties and provide an example on real time curve averaging in a medical context the remedian with base b proceeds by computing medians of groups of b observations and then medians of these medians until only a single estimate remains this method merely needs k arrays of size b where n bk so the total storage is o log n for fixed b or alternatively o nl k for fixed k its storage economy makes it useful for robust estimation in large data bases for real time engineering applications in which the data themselves are not stored and for resistant

modeling flat stretches bursts and outliers in time series using mixture transition distribution models abstract the class of mixture transition distribution mtd time series models is extended to general non gaussian time series in these models the conditional distribution of the current observation given the past is a mixture of conditional distributions given each one of the last p observations they can capture non gaussian and nonlinear features such as flat stretches bursts of activity outliers changepoints in a single unified model class they can also represent time series defined on arbitrary state spaces univariate or multivariate continuous discrete or mixed which need not even be euclidean they perform well in the usual case of gaussian time series without obvious nonstandard behaviors the models are simple analytically tractable easy to simulate readily estimated the stationarity and autocorrelation properties of the models are derived a simple em algorithm is given and shown to work well for estimation the models are applied to several real and simulated datasets with satisfacto

least squares identification for armax models without the positive real condition recursive identification problems of linear stochastic feedback control systems described by armax models are studied without imposing the strictly positive real condition on the noise model an increasing lag least squares is used in the first step to estimate the noise process while the parameter estimate is formed in the second step by an extended least squares in the algorithm there is an increase in computational cost in comparison to traditional algorithms however the results of this note do not need any a priori information or conditions on the noise model except that of stability and they are applicable to the identification of general feedback control systems

robust recursive estimation in the presence of heavy tailed observation noise a first order approximation is derived for the conditional prior distribution of the state of a discrete time stochastic linear dynamic system in the presence of contaminated normal observation noise this distribution is then used to derive a first order approximation of the conditional mean minimum variance estimator if the observation noise distribution can be represented as a member of the e contaminated normal neighborhood then the conditional prior is also to first order an analogous perturbation from a normal distribution whose first two moments are given by the kalman filter moreover the perturbation is itself of a special form combining distributions whose parameters are given by banks of parallel kalman filters and optimal smoothers

leave k out diagnostics for time series abstract the authors propose diagnostics for arima model fitting for time series formed by deleting observations from the data and measuring the change in the estimates of the parameters the use of leave one out diagnostics is a well established tool in regression analysis this document demonstrates the efficacy of observation deletion based diagnostics for arima models addressing issues special to the time diagnostics based on the innovations variance it is shown that the dependency aspect of time series data gives rise to a smearing effect which confounds the diagnostics for the coefficients it is also shown that the diagnostics based on the innovations variance are much clearer and more sensitive than those for the coefficients a leave k out diagnostics approach is proposed to deal with patches of outliers and problems caused by masking are handled by use of iterative deletion an overall strategy for arima model fitting is given and applied to two data sets

robust indirect inference in this article we develop robust indirect inference for a variety of models in a unified framework we investigate the local robustness properties of indirect inference and derive the influence function of the indirect estimator as well as the level and power influence functions of indirect tests these tools are then used to design indirect inference procedures that are stable in the presence of small deviations from the assumed model although indirect inference was originally proposed for statistical models whose likelihood is difficult or even impossible to compute and or to maximize we use it here as a device to robustify the estimators and tests for models where this is not possible or is difficult with classical techniques such as m estimators examples from financial applications time series and spatial statistics are used for illustration

detecting over influential observations in time series summary the purpose of this paper is to develop a tool for identifying over influential observations in time series when they are viewed in the time domain we present a method for obtaining various measures of influence for the autocorrelation function as well as thresholds for declaring an observation over influential an example of the use of these thresholds is also presented

transversal bounded solutions in systems with normal and slow variables abstract bifurcation of transversal bounded solutions is investigated for a pair of autonomous ordinary differential equations with periodic perturbations when the first unperturbed equation has a homoclinic solution and the second unperturbed equation is vanishing such systems of equations naturally arise in certain weakly coupled systems and in certain systems with slowly varying coefficients

on a gradient like integro differential equation under appropriate conditions on b and a function g with 2 k 1 simple zeros the equation has a maximal compact invariant set a b g in c 1 0 r consisting of the zeros of g and the one dimensional unstable manifolds of these zeros for k 2 it is shown that there may be a saddle connection in the flow on a b g for some g this implies that the zeros of g as elements of the flow on a b g cannot be given the natural order of the reals

bifurcation of a unique stable periodic orbit from a homoclinic orbit in infinite dimensional systems sous certaines conditions generiques on montre comment une orbite periodique stable unique peut bifurquer a partir d une orbite homocline pour des equations paraboliques semi lineaires et des equations differentielles fonctionnelles a retard

review of median stable distributions and schroder s equation abstract median stable distributions are an extension of traditional mean stable distributions the traditional definition of stability in terms of sums of iid random variables is recast as a condition on the sampling distribution of an estimator for the traditional mean stable distribution the sample mean s rescaled sampling distribution is identical to the distribution of the iid data median stable distributions are defined similarly by replacing the sample mean with the sample median since the sampling distribution of the median is a functional its stable distribution is the solution to a functional equation it turns out that this defining functional equation is an instance of a famous equation due to schroder from 1870 the fame of the equation is due to the way it incorporates iteration of functions a key feature of what many years later would become dynamic systems analysis the current paper reviews median stable distributions in light of its connection to schroder s functional equation

on tori without conjugate points in this paper we consider riemannian metrics without conjugate points on an n torus recent work of j heber established that the gradient vector fields of busemann functions on the universal cover of such a manifold induce a natural foliation akin to the weak stable foliation for a riemannian manifold with negative sectional curvature on the unit tangent bundle the main result in the paper is that the metric is flat if this foliation is lipschitz we also prove that this foliation is lipschitz if and only if the metric has bounded asymptotes this confirms a conjecture of e hopf in this case

assessing the predictive influence of cases in a state space process an important inferential objective in state space modelling is to recover unobserved states using fixed interval smoothing thus the identification of cases which have a substantial influence on the smoothers is a relevant practical problem to facilitate this identification we propose a case deletion diagnostic which can be easily computed using the outputs of the standard filtering and smoothing algorithms our diagnostic is defined as the kullback leibler directed divergence between two versions of the conditional density which determines the smoothers one based on all the data the other based on all the data except for the case or cases in question we investigate the detection performance of the diagnostic in a practical application

a goodness of fit test in robust time series modelling summary the problem of testing the adequacy of a time series model in the presence of outliers is considered the classical portmanteau statistic is generalized to an important class of robust estimators some monte carlo results suggest that the proposed generalization possesses good robustness properties over the classical statistic a robustified version of a result of newbold 1980 is also obtained

blockwise bootstrapped empirical process for stationary sequences we apply the bootstrap for general stationary observations proposed by kunsch to the empirical process for p dimensional random vectors it is known that the empirical process in the multivariate case converges weakly to a certain gaussian process we show that the bootstrapped empirical process converges weakly to the same gaussian process almost surely assuming that the block length l for constructing bootstrap replicates satisfies l n o n 1 2 varepsilon 0 varepsilon frac 1 2 and l n rightarrow infty an example where the multivariate setup arises are the robust gm estimates in an autoregressive model we prove the asymptotic validity of the bootstrap approximation by showing that the functional associated with the gm estimates is frechet differentiable

complex immersions in kahler manifolds of positive holomorphic k ricci curvature the main purpose of this paper is to prove several connectedness theorems for complex immersions of closed manifolds in kahler manifolds with positive holomorphic k ricci curvature in particular this generalizes the classical lefschetz hyperplane section theorem for projective varieties as an immediate geometric application we prove that a complex immersion of an n dimensional closed manifold in a simply connected closed kahler m manifold m with positive holomorphic k ricci curvature is an embedding provided that 2n m k this assertion for k 1 follows from the fulton hansen theorem 1979

riemannian rigidity of the parallel postulate in total curvature without any symmetry assumptions and under natural integrability conditions on the gaussian curvature we show that the euclidean plane is unique among all the complete 2 dimensional riemannian manifolds that satisfy the euclid s fifth postulate namely we prove that the euclidean plane is the only riemannian surface free of conjugate points and admitting total curvature that satisfies playfair s version of the parallel postulate

linearly independent homoclinic bifurcations parameterized by a small function abstract in this paper we discuss a small nonautonomous perturbation of an autonomous system on r n which has a homoclinic solution regarding the small perturbation as a parameter in an appropriate space of functions we discuss various situations of co existence of homoclinic orbits those conditions of various co existence actually define bifurcation manifolds in the space of functions for linearly independent homoclinic bifurcations