analytic derivatives for estimation of linear dynamic models abstract this paper develops two algorithms algorithm 1 computes the exact gaussian log likelihood function its exact gradient vector and an asymptotic approximation of its hessian matrix for discrete time linear dynamic models in state space form algorithm 2 derived from algorithm 1 computes the exact sample information matrix of this likelihood function the computed quantities are analytic not numerical approximations and should therefore be useful for reliably quickly and accurately i checking local identifiability of parameters by checking the rank of the information matrix ii using the gradient vector and hessian matrix to compute maximum likelihood estimates of parameters with newton methods and iii computing asymptotic covariances cramer rao bounds of the parameter estimates with the hessian or the information matrix the principal contribution of the paper is algorithm 2 which extends to multivariate models the univariate results of porat and friedlander 1 by relying on the kalman filter instead of the levinson durbin filter used by porat and friedlander algorithms 1 and 2 can automatically handle any pattern of missing or linearly aggregated data although algorithm 1 is well known it is treated in detail in order to make the paper self contained

estimation prediction and interpolation for nonstationary series with the kalman filter abstract we show how our definition of the likelihood of an autoregressive integrated moving average arima model with missing observations alternative to that of kohn and ansley and based on the usual assumptions made in estimation of and forecasting with arima models permits a direct and standard state space representation of the nonstationary original data so that the ordinary kalman filter and fixed point smoother can be efficiently used for estimation forecasting and interpolation in this way the problem of estimating missing values in nonstationary series is considerably simplified the results are extended to regression models with arima errors and a computer program is available from the authors

integral estimates for null lagrangians sobolev spaces of differential forms are studied lp projections onto exact forms are introduced as a tool to obtain integral estimates for null lagrangians new results on compensated compactness are given and mean coercive variational integrals are found existence of minima of certain mean coercive functionals is established

missing observations in arima models skipping approach versus additive outlier approach optimal estimation of missing values in arma models is typically performed by using the kalman filter for likelihood evaluation skipping in the computations the missing observations obtaining the maximum likelihood ml estimators of the model parameters and using some smoothing algorithm the same type of procedure has been extended to nonstationary arima models in go mez and maravall 1994 an alternative procedure suggests filling in the holes in the series with arbitrary values and then performing ml estimation of the arima model with additive outliers ao when the model parameters are not known the two methods di er since the ao likelihood is a ected by the arbitrary values we develop the proper likelihood for the ao approach in the general non stationary case and show the equivalence of this and the skipping method finally the two methods are compared through simulation and their relative advantages assessed the comparison also includes the ao method with the uncorrected likelihood 1999 elsevier science s a all rights reserved

quasiregular mappings in even dimensions o introduction 1 notation 2 some exterior algebra 3 differential forms in lnm i2 a n 4 differential systems for quasiregular mappings 5 liouville theorem in even dimensions 6 hodge theory in lp r n 7 the beltrami equation in even dimensions 8 the beurling ahlfors operator 9 regularity theorems for quasiregular mappings 10 the caccioppoli type estimate 11 removability theorems for quasiregular mappings 12 some examples appendix the 4 dimensional case

ar λ weighted integral inequalities for a harmonic tensors abstract in this paper we prove the ar weighted caccioppoli type inequality and weak reverse holder inequality for a harmonic tensors we also obtain the ar weighted hardy littlewood inequality for conjugate a harmonic tensors these inequalities can be considered as extensions of the classical results

estimation of missing values in possibly partially nonstationary vector time series summary ljung s 1989 method for estimating missing values and evaluating the corresponding likelihood function in scalar time series is extended to the vector case the series is assumed to be generated by a possibly partially nonstationary and noninvertible vector autoregressive moving average process no particular pattern of missing data is assumed future and past values are special cases of missing data that can be estimated in the same way the method does not use kalman filter iterations and hence avoids initialisation problems it does not require the series to be differenced and thus avoids complications caused by over differencing the estimators of the missing data are provided by the normal equations of an appropriate regression problem these equations are adapted to cope with temporally aggregated data the procedure parallels a matrix treatment of contour conditions in the analysis of variance autoregressive processes are considered in some detail

weighted poincare type inequalities for differential forms inls μ averaging domains abstract we first introduce a new class of weighted functions and obtain some basic properties of this class then as applications of this class we prove local weighted integral inequalities for differential forms finally we obtain the global weighted integral inequalities for differential forms inls averaging domains which can be considered as generalizations of the poincare inequality for sobolev functions

a fast algorithm for signal extraction influence and cross validation in state space models summary a fast algorithm is developed for computing the conditional mean and variance of the signal given the observations in a signal plus noise model the resulting recursions can be applied immediately to provide new and efficient formulae for smoothing part or all of the state vector the ideas of studentized residuals and leverage from regression analysis are generalized to state space models and the new algorithm is used to compute the various measures the results are also applied to obtain a new efficient algorithm for polynomial spline smoothing suppose observations are generated by a gaussian signal plus noise process with the signal described by a state space model such a model for the signal occurs often in practice for example when the signal is the output of a stochastic difference or differential equation this paper presents a new algorithm for signal extraction that is the computa tion of the conditional mean and variance of the signal given the observations the usual method for signal extraction is to use the kalman filter see for example anderson moore 1979 p 105 followed by a smoothing step using an algorithm such as the fixed interval smoothing algorithm anderson moore 1979 p 187 our approach also uses the kalman filter but for the smoothing step introduces a new set of recursions which are more efficient than those for the fixed interval smoothing algorithm as for the fixed interval smoothing algorithm recursions for estimating the signal and its variance are carried out separately so that considerable additional savings can be made if the signal estimate alone is required the development uses ideas introduced by ansley kohn 1987a a remarkable property of the new recursions is that they can be applied immediately

estimation prediction and interpolation for arima models with missing data abstract we show how to define and then compute efficiently the marginal likelihood of an arima model with missing observations the computation is carried out by using the univariate version of the modified kalman filter introduced by ansley and kohn 1985a which allows a partially diffuse initial state vector we also show how to predict and interpolate missing observations and obtain the mean squared error of the estimate

lp averaging domains in homogeneous spaces abstract we define l p averaging domains in spaces of homogeneous type endowed with an intrinsic metric we characterize these domains in terms of the quasihyperbolic metric and prove boman chain domains are l p averaging

invariance properties of lϕ μ averaging domains under some mappings abstract in this paper we first show that l averaging domains are invariant under some mappings such as k quasi isometries and quasi isometries then we prove that john domains are l averaging domains for any satisfying the conditions in the definition of l averaging domains

estimation filtering and smoothing in state space models with incompletely specified initial conditions on definit la vraisemblance pour un modele d espace d etats avec des conditions initiales incompletement specifiees par transformation des donnees pour eliminer la dependance sur des conditions non specifiees on etend cette approche pour obtenir des estimations sur les vecteurs d etat les predicteurs et les interpolateurs pour des observations manquantes

two weight imbedding inequalities for solutions to the a harmonic equation abstract we obtain the two weight imbedding inequalities for solutions to the a harmonic equation and explore their applications to the k quasiregular mappings in r n these results can be used to study the properties of the homotopy operator t l p d l w 1 p d l 1 l 1 2 n and to establish the weighted l p estimate for solutions to the a harmonic equation

bayesian estimation of an autoregressive model using markov chain monte carlo abstract we present a complete bayesian treatment of autoregressive model estimation incorporating choice of autoregressive order enforcement of stationarity treatment of outliers and allowance for missing values and multiplicative seasonality the paper makes three distinct contributions first we enforce the stationarity conditions using a very efficient metropolis within gibbs algorithm to generate the partial autocorrelations second we show how to carry out the gibbs sampler when the autoregressive order is unknown third we show how to combine the various aspects of fitting an autoregressive model giving a more comprehensive and efficient treatment than previous work we illustrate our methodology with a real example

on the bias in estimates of forecast mean squared error abstract we examine the forecasting problem for a finite series of observations from either a nonseasonal or a seasonal autoregressive moving average process four sources of bias in the usual estimator of forecast mean squared error are identified and analyzed for particular models using maximum likelihood and least squares parameter estimates the usual estimator is found to be biased downwards especially near the boundary of a stationary or invertible region and the bias is severe for least squares estimators an alternative estimator is proposed for the maximum likelihood case which is shown generally to have reduced bias

fast evaluation of the distribution of the durbin watson and other invariant test statistics in time series regression abstract a method is given for evaluating p values in o n operations for a general class of invariant test statistics that can be expressed as the ratio of quadratic forms in time series regression residuals the best known of these is the durbin watson statistic although several others have been discussed in the literature the method is numerically exact in the sense that the user specifies the error tolerance at the outset as with existing exact methods the problem is reexpressed in terms of the distribution function of a single quadratic form in independent normals which is evaluated by numerically inverting its characteristic function existing methods however calculate the characteristic function by reducing the matrix defining the quadratic form to either eigenvalue or tridiagonal form each of which requires o n 3 operations for sample size n whereas the new method uses a modification of the kalman filter to do it in o n operations moreover the new method has minimal storage requiremen

efficient estimation and prediction in time series regression models summary for a time series regression model with correlated residuals we show how to compute efficiently the likelihood predict future values and interpolate missing values as an illustration we consider a regression model with autoregressive moving average residuals

estimating missing observations in economic time series abstract two related problems are considered the first concerns the maximum likelihood estimation of the parameters in an arima model when some of the observations are missing or subject to temporal aggregation the second concerns the estimation of the missing observations both problems can be solved by setting up the model in state space form and applying the kalman filter

ar ω weighted inequalities for a harmonic tensors and related operators abstract we first prove the local a r weighted poincare inequality for a harmonic tensors on domain and the analogue for the projection operator then we prove the global a r weighted poincare inequality for a harmonic tensors finally using the global a r weighted inequality established for a harmonic tensors we obtain the global a r weighted poincare inequality for the projection operator

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

modeling of time series arrays by multistep prediction or likelihood methods abstract an estimation theory is provided for the fitting of possibly incorrect invertible short memory models to short or long memory time series or time series arrays by multistep prediction error minimization or gaussian likelihood maximization by array we mean data y t t 1 t t that depend on the number of observations t such as regression or other estimated model residuals or the outputs of time varying filters for example seasonal adjustments our theory only requires the modeled array to have basic properties for a s i p convergence of parameter estimates the array s sample lagged second moments must converge a s i p and its end values y 1 j t and y t j t must be of order less than t 1 2 or an appropriately differenced version of the observed array must have these properties in findley et al ann statist 29 2001 815 broad classes of arrays were shown to have these properties even for the special case of autoregressive moving average models fit to stationary gaussian time series data our result on the convergence of parameter estimates minimizing p step ahead prediction error sums of squares is new

computation of the separatrix of basins of attraction in a non smooth dynamical system this paper describes a new numerical method to compute the separatrix of the basins of attraction of coexisting attractors in a forced friction oscillator numerical results show that its intersection with a poincare section is a non smooth curve

measuring influence in dynamic regression models this article presents a methodology for building measures of influence in regression models with time series data we introduce statistics that measure the influence of each observation on the parameter estimates and on the forecasts these statistics take into account the autocorrelation of the sample the first statistic can be decomposed to measure the change in the univariate autoregressive integrated moving average parameters the transfer function parameters and the interaction between both for independent data they reduced to the d statistic considered by cook in the standard regression model these statistics can be easily computed using standard time series software their performance is analyzed in an example in which they are shown to be useful in identifying important events such as additive outliers and trend shifts in time series data

l φ μ averaging domains and the quasi hyperbolic metric in this paper we first introduce email protected m averaging domains which are generalizations of existing domains such as john domains and l s m averaging domains then we characterize l f m averaging domains using the quasihyperbolic metric finally we give applications to quasiconformal mappings

arλ ω weighted imbedding inequalities for a harmonic tensors abstract we prove new versions of a r weighted imbedding inequalities for a harmonic tensors locally and globally the results are used to estimate the integrals of a homotopy operator t from the banach space l s l to the sobolev space w 1 s l 1 l 1 2 n and sobolev poincare imbedding inequality with a r weight

weighted integral inequalities for conjugate a harmonic tensors abstract we first prove a local weighted integral inequality for conjugate a harmonic tensors then as an application of our local result we prove a global weighted integral inequality for conjugate a harmonic tensors in ls averaging domains which can be considered as a generalization of the classical result finally we give applications of the above results to quasiregular mappings