perturbation expansions of the axial next nearest neighbour ising and asymmetric four state clock models the axial next nearest neighbour ising annni and four state asymmetric clock 4 asc models are investigated by perturbing about points at which they reduce the simple nearest neighbour ising models these limits are a for the annni model i small nearest neighbour interaction j1 in the x direction and ii small next nearest neighbour interaction j2 in the x direction and b for the 4 asc model small asymmetry delta in the limit j1 to 0 the phase boundary is located to first order in j1 and found to be ising like in the limit j2 to 0 the behaviour to second order in j2 is more complicated but is identical to that found to second order in delta for the 4 asc model these results are discussed in the light of other recent work

relations between information criteria for model structure selection part 1 the role of bayesian model order estimation model structure estimation is first accommodated in the general framework of prediction error identification the concept of the best model structure is defined a theorem is given on bayesian estimation from which the strong consistency of well known information criteria with penalty terms is inferred

nonasymptotic bounds for autoregressive time series modeling the subject of this paper is autoregressive ar modeling of a stationary gaussian discrete time process based on a finite sequence of observations the process is assumed to admit an ar representation with exponentially decaying coefficients we adopt the nonparametric minimax framework and study how well the process can be approximated by a finiteorder armodel a lower bound on the accuracy of arapproximations is derived and a nonasymptotic upper bound on the accuracy of the regularized least squares estimator is established it is shown that with a proper choice of the model order this estimator is minimax optimal in order these considerations lead also to a nonasymptotic upper bound on the mean squared error of the associated one step predictor a numerical study compares the common model selection procedures to the minimax optimal order choice 1 introduction the standard methods for estimating parameters of time series are based on the assumption that the observations come from an autoregressive ar moving average ma or mixed arma model of known orders this assumption can rarely be justified in practice and the less stringent assumption is that the time series data are observations from a linear stationary process a common approach to modeling linear stationary processes is based on an arapproximation in this framework a finite order armodel is fitted to the observations the order of the armodel should provide an optimal finite arapproximation to the process and it is usually chosen by selection procedures based on the data this nonparametric arapproach to modeling linear stationary processes has been investigated by shibata 1980 bhansali 1981 1986 an chen and hannan 1982 and hannan and kavalieris 1986 shibata 1980 considered the problem of predicting a gaussian infiniteorder arprocess by fitting a finite armodel the notion of optimality for the model selection procedure proposed by shibata 1980 is based on an asymptotic lower bound on the mean squared prediction error specifically the procedure is asymptotically efficient if it attains the lower bound asymptotically shibata 1980 also established that the final prediction error fpe akaike 1970 and the aic akaike 1974 criteria are asymptotically efficient in the above sense provided that the linear process does not degenerate to a finite order autoregression a similar result has been obtained

a method for the estimation and identification of transfer function models summary this paper proposes an approach to the simultaneous identification and estimation of transfer function models an estimation procedure is advanced consisting of a sequence of four ordinary least squares calculations and the asymptotic properties of each step are presented consistency and asymptotic efficiency being shown a strategy for the identification of the model structure based on a selection criterion that arises naturally from the last step of the estimation algorithm is also investigated consideration is given to the numerical implementation of the techniques discussed and finally a practical example is employed to illustrate the methods

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 asymmetric clock model on a cayley tree the asymmetric clock model is studied on a cayley tree and commensurate and incommensurate phases are found the transition lines are obtained from stability conditions and characteristic points in the phase diagram are analysed by numerical iteration a critical endpoint is found for the case of three states per site and a lifshitz point for the case of four states per site

a modified hannan rissanen strategy for mixed autoregressive moving average order determination summary a modification to the model selection strategy for mixed autoregressive moving average order determination suggested by hannan rissanen 1982 is proposed the modification is designed to eliminate the bias discussed by hannan kavalieris 1984 and involves the use of an alternative model selection criterion asymptotic properties of the modified procedure are established and some simulation evidence on the finite sample performance is provided

almost sure convergence analysis of autoregressive spectral estimation in additive noise the almost sure convergence properties of autoregressive spectral estimates from noisy observations are derived sharp rates of almost sure convergence are established for the estimates of the autoregressive parameters innovation variance and spectral density function of the signal process the distributions of the signal and noise processes are arbitrary

a universal prior for integers and estimation by minimum description length of the number of bits required to write down the observed data has been reformulated to extend the classical maximum likelihood principle the principle permits estimation of the number of the parameters in statistical models in addition to their values and even of the way the parameters appear in the models i e of the model structures the principle rests on a new way to interpret and construct a universal prior distribution for the integers which makes sense even when the parameter is an individual object truncated realvalued parameters are converted to integers by dividing them by their precision and their prior is determined from the universal prior for the integers by optimizing the precision 1 introduction in this paper we study estimation based upon the principle of minimizing the total number of binary digits required to rewrite the observed data when each observation is given with some precision instead of attempting at an absolutely shortest description which would be futile we look for the optimum relative to a class of parametrically given distributions this minimum description length mdl principle which we introduced in a less comprehensive form in 25 turns out to degenerate to the more familiar maximum likelihood ml principle in case the number of parameters in the models is fixed so that the description length of the parameters themselves can be ignored in another extreme case where the parameters determine the data it similarly degenerates to jaynes s principle of maximum entropy 14 but the main power of the new criterion is that it permits estimates of the entire model its parameters their number and even the way the parameters appear in the model i e the model structure hence there will be no need to supplement the estimated parameters with a separate hypothesis test to decide whether a model is adequately parameterized or perhaps over parameterized

finite lattice extrapolations for z3 and zs models a method is presented to accelerate the convergence of finite lattice sequences to their bulk limit the calculation of highly accurate estimates of the critical parameters of the bulk system is then possible applied to the hamiltonian version of the z3 model three state potts model in 1 1 dimensions these techniques yield estimates for the exponents gamma 1 444 or 0 0001 nu 0 8333 or 0 0003 and alpha 0 33 or 0 01 for the z5 model the presence of a kosterlitz thouless transition is confirmed

the effect of order estimation on estimating the peak frequency of an autoregressive spectral density summary the effect of estimating the order of an autoregressive process on estimating the peak frequency is examined for consistent order determining methods it is shown that the asymptotic results are the same as for the case when the order is known for the order criteria which asymptotically overestimate the order the peak frequency estimator is shown to be a consistent estimator of the true peak frequency

on the multiplicity of the solution set of some nonlinear boundary value problems ii on considere le probleme non lineaire lu f u s 1 h dans u 0 sur soit h l 2 f est de classe c et que f 1 et n f n 1 n 1 alors il existe s 0 tel que pour tout s s 0 l equation a au moins 2 solutions si n est impair et 3 solutions si n est pair

bayes inference in regression models with arma p q errors we develop practical and exact methods of analyzing arma p 4 regression error models in a bayesian framework by using the gibbs sampling and metropolis hastings algorithms and we prove that the kernel of the proposed markov chain sampler converges to the true density the procedures can be applied to pure arma time series models and to determine features of the likelihood function by choosing appropriate diffuse priors our results are unconditional on the initial observations we also show how the algorithm can be further simplified for the important special cases of stationary ar p and invertible ma q models recursive transformations developed in this paper to diagonalize the covariance matrix of the errors should prove useful in frequentist estimation examples with simulated and actual economic data are presented kpj ourls gibbs sampling metropolis hastings algorithm data augmentation time series arma processes markov chain bayesian statistics

precision complexity and bayesian model determination summary in this paper a new criterion for model determination is proposed an explicit decision theoretic approach to model selection is employed to derive a bayes decision rule the operational characteristics of the criterion are discussed and consistency is shown

multiple periodic oscillations in a nonlinear suspension bridge system in this paper we study periodic oscillations in a suspension bridge system governed by the coupled nonlinear wave and beam equations describing oscillations in the supporting cable and roadbed under periodic external forces by applying a variational reduction method it is proved that the suspension bridge system has at least three periodic oscillations

testing the order of a model this paper deals with order identification for nested models in the i i d framework we study the asymptotic efficiency of two generalized likelihood ratio tests of the order they are based on two estimators which are proved to be strongly consistent a version of stein s lemma yields an optimal underestimation error exponent the lemma also implies that the overestimation error exponent is necessarily trivial our tests admit non trivial underestimation error exponents the optimal underestimation error exponent is achieved in some situations the overestimation error can decay exponentially fast with respect to a positive power of the number of observations these results are proved under mild assumptions by relating the underestimation resp overestimation error to large resp moderate deviations of the log likelihood process in particular it is not necessary that the classical cramer condition be satisfied namely the log densities are not required to admit every exponential moment three benchmark examples with specific difficulties location mixture of normal distributions abrupt changes and various regressions are detailed so as to illustrate the generality of our results

stochastic complexity and modeling on demontre un theoreme fondamental qui donne une borne inferieure pour la longueur de code et donc pour les erreurs de prediction on definit les notions d information a priori et d information utile dans les donnees

twenty one ml estimators for model selection classical approaches to determine a suitable model structure from observed input output data are based on hypothesis tests and information based criteria recently the model structure has been considered as a stochastic variable and standard estimation techniques have been proposed the resulting estimators are closely related to the aforementioned methods however it turns out that there are a number of prior choices in the problem formulation which are crucial for the estimators behavior the contribution of this paper is to clarify the role of the prior choices to examine a number of possibilities and to show which estimators are consistent this is done in a linear regression framework for autoregressive models we also investigate a novel prior assumption on stability and give the estimator for the model order and the parameters themselves

sequential estimation of hidden arma processes by particle filtering part ii this is part ii of a series of two papers where we address sequential estimation of wide sense stationary autoregressive moving average arma state processes by particle filtering in part i we considered a state space model where the state was an arma process of known order and where the parameters of the process could be known or unknown in this paper we extend our work from part i by considering the same type of models with the added complexity that the arma processes are now of unknown order instead of working on a scheme that first tracks the state by operating with different assumed models and then selects the best model by using a predefined criterion we present a method that directly estimates the state without the need of knowing the model order we derive the transition density of the state for unknown arma model order and propose a particle filter based on that density and the empirical bayesian methodology we demonstrate the performance of the proposed method with computer simulations and compare it with the methods from part i

on nonlinear oscillations in a suspension bridge system in this paper we study nonlinear oscillations in a suspension bridge system governed by two coupled nonlinear partial differential equations by applying the leray schauder degree theory it is proved that the suspension bridge system has at least two solutions one is a near equilibrium oscillation and the other is a large amplitude oscillation

estimation of the interpolation error variance and an index of linear determinism summary the autoregressive and window estimators of the inverse variance ri 0 say are adopted for estimating the interpolation error variance ri o and an index of linear determinism a of a stationary process x the index a is defined as 1 r o ri 0 where r o denotes the variance of x a measures the proportion of total variance explained by the interpolator when interpolating x say from xs s t some uses of a and ri o are discussed the asymptotic distributions of the estimators are derived the order of consistency of the autoregressive and window estimators of a is different when a 0 than when 0 a 1 the finite sample behaviour of the estimators is examined by means of a simulation study

arma models prewhitening and minimum cross entropy the problem of spectral estimation on the basis of observations from a finite stretch of a stationary time series is considered in connection with knowledge of a prior estimate of the spectral density a reasonable posterior spectral density estimate would be the density that is closest to the prior according to some measure of divergence while at the same time being compatible with the data the cross entropy has often been proposed to serve as such a measure of divergence a correction of the original minimum cross entropy spectral analysis mcesa method of j e shore see ieee trans acoust speech signal process vol 29 p 230 7 1981 to traditional prewhitening techniques and to autoregressive moving average arma models is pointed out and a fast approximate solution of the minimum cross entropy problem is proposed the solution is in a standard multiplicative form that is the posterior is equal to the prior multiplied by a correction factor

complexity of simple nonlogarithmic loss functions the loss complexity for nonlogarithmic loss functions is defined analogously to the stochastic complexity for logarithmic loss functions such that its mean provides an achievable lower bound for estimation the mean taken with respect to the worst case data generating distribution the loss complexity also provides a lower bound for the worst case mean prediction error for all predictors for the important spl alpha loss functions y spl circ y sup spl alpha where y y spl circ denotes the prediction or fitting error and spl alpha is in the interval an accurate asymptotic formula for the loss complexity is given

determining a portfolio of linear time series models on applique les concepts de degre d evidence suggeres par jeffreys 1961 au probleme de la determination de l ordre dans des modeles de series temporelles lineaires

star triangle equations and multicomponent chiral potts models abstract under certain conditions lipid bilayer membranes exhibit an incommensurate ripple phase pearce and scott have proposed lattice models with either next nearest neighbor interactions or with chiral nearest neighbor interactions to study this phenomenon after briefly discussing the equivalence of these two points of view we present the results of our first search for solutions of the star triangle equations for multicomponent chiral potts models generalizing some of the biomembrane models in the nonchiral limit the results apply also to the n n model of domany and riedel describing the adsorption of molecules on the surface of a crystal

on the posterior odds of time series models summary this paper is concerned with model selection in time series analysis a bayesian approach is adopted to investigate various decision criteria previously advocated the discussion is conducted in the context of a class of nonlinear time series models of sufficient generality to encompass a wide range of processes currently of interest

fully bayesian analysis of arma time series models abstract statistical analysis of autoregressive moving average arma models is an important non standard problem no classical approach is widely accepted legitimacy for most classical approaches is based solely on asymptotic grounds while small sample sizes are common the only obstacle to the bayesian approach are designing a structure through which prior information can be incorporated and designing a practical computational method the objective of this work is to overcome these two obstacles in addition to the standard results the bayesian approach gives a different method of determining the order of the arma model that is p q

new method of order estimation for arma armax processes let the observation y k be generated by the multivariate arma process a z y k c z w k with unknown coefficients theta a theta c and orders p r where w k is a sequence of independent and identically distributed i i d random vectors with zero mean and unknown covariance matrix r w 0 a new method for estimating the orders p r is introduced in contrast to most of the existing results the new method is not based on optimizing a certain criterion and the order estimates given in the paper are rather easy to update computationally in comparison with the criterion optimization based methods when new data arrive the method is then extended to determining the orders of armax processes under reasonable conditions the estimates are proved to converge to the true orders with probability one as time tends to infinity

regression procedures for arma estimation abstract regression procedures for parameter estimation in autoregression moving average arma models are discussed mainly for providing initial estimates for iterative maximization of a gaussian likelihood an iterative procedure of spliid 1983 is compared to a procedure of hannan and rissanen 1982 and a global convergence result is established for an iterative modification of spliid s procedure spliid s iteration does not always converge when it does it has the same asymptotic distribution as the second stage of the hannan rissanen procedure this second stage iteration gives the same problems as spliid s procedure so it is preferable to go immediately to the third stage which is asymptotically efficient in the gaussian case an example is provided by a first order arma model y t y t 1 e t e t 1 under the usual regularity conditions of stationarity and invertibility spliid s procedure fits an autoregression of predetermined order i e 2 to obtain estimates of the e t a

a bayes procedure for the identification of univariate time series models on presente un critere d identification qui est asymptotiquement equivalent a une regle de decision bayesienne

subspace estimation and prediction methods for hidden markov models hidden markov models hmms are probabilistic functions of finite markov chains or put in other words state space models with finite state space in this paper we examine subspace estimation methods for hmms whose output lies a finite set as well in particular we study the geometric structure arising from the nonminimality of the linear state space representation of hmms and consistency of a subspace algorithm arising from a certain factorization of the singular value decomposition of the estimated linear prediction matrix for this algorithm we show that the estimates of the transition and emission probability matrices are consistent up to a similarity transformation and that the m step linear predictor computed from the estimated system matrices is consistent i e converges to the true optimal linear m step predictor

a method for autoregressive moving average estimation summary the problem considered is that of estimating an autoregressive moving average system including estimating the degrees of the autoregressive and moving average lag operators the basic method is that introduced by hannan rissanen 1982 however that method may sometimes overestimate the degrees and modifications are here introduced to correct this the problem is itself due to the use of a long autoregression of order clog t when t is large in the first stage of the process the effect of this is investigated and in particular its effect on the speed of convergence of the estimates

consistent estimation of the order of stochastic control systems a consistent estimate of the order of feedback control systems with unknown matrix coefficients estimated by the least squares method is derived by minimizing a modified version of the bayesian information criterion

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

a bayesian approach to nonparametric bivariate regression abstract this article outlines a general bayesian approach to estimating a bivariate regression function in a nonparametric manner it models the function using a bivariate regression spline basis with many terms binary indicator variables corresponding to these terms are introduced to explicitly model the uncertainty of whether or not the terms provide a significant contribution to the regression the regression function is estimated using an estimate of its posterior mean smoothing over the distribution of these binary indicator variables to make the computations tractable all estimates are obtained using markov chain monte carlo sampling extensive simulated comparisons are provided that demonstrate the competitive performance of this approach against other data driven bivariate surface estimators prominent in the literature it is then shown how the procedure can be extended to provide a general approach to nonparametric bivariate surface estimation in two difficult regression settings the first

a mountain pass method for the numerical solution of semilinear wave equations it is well known that periodic solutions of semilinear wave equations can be obtained as critical points of related functionals in the situation that we studied there is usually an obvious solution obtained as a solution of linear problem we formulate a dual variational problem in such a way that the obvious solution is a local minimum we then find additional non obvious solutions via a numerical mountain pass algorithm based on the theorems of ambrosetti rabinowitz and ekeland numerical results are presented

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

accumulated prediction errors information criteria and optimal forecasting for autoregressive time series the predictive capability of a modification of rissanen s accumulated prediction error ape criterion ape n is investigated in infinite order autoregressive ar oo models instead of accumulating squares of sequential prediction errors from the beginning ape n is obtained by summing these squared errors from stage n n where n is the sample size and 1 n n 1 1 n may depend on n under certain regularity conditions an asymptotic expression is derived for the mean squared prediction error mspe of an ar predictor with order determined by ape n this expression shows that the prediction performance of ape n can vary dramatically depending on the choice of n another interesting finding is that when n approaches 1 at a certain rate ape n can achieve asymptotic efficiency in most practical situations an asymptotic equivalence between ape n and an information criterion with a suitable penalty term is also established from the mspe point of view this offers new perspectives for understanding the information and prediction based model selection criteria finally we provide the first asymptotic efficiency result for the case when the underlying ar oo model is allowed to degenerate to a finite autoregression

a bayesian approach to state space multivariate time series modeling abstract time series modeling is mainly a model specification exercise recognizing this fact a new bayesian procedure is developed that takes full advantage of the subjective probability foundations of the bayesian approach allowing logical consistency in the face of an unknown model the computational demands of this new procedure have been greatly reduced relative to earlier techniques through the application of linear systems theory linear systems theory substitutes a formal model approximation based on the rank of an autocovariance matrix of the time series for the more arbitrary model selection processes of some other procedures this formalization ensures that the most important elements of the series dynamics are included in the model model selection procedures and an approach to the construction of optimal composite forecasts are derived using bayesian methodologies two empirical applications demonstrate the tractability and the accuracy of the new procedure

least informative bayesian prior distributions for finite samples based on information theory a procedure is presented based on shannon information theory for producing least informative prior distributions for bayesian estimation and identification this approach relies on constructing an optimal mixture distribution and applies in small sample sizes unlike certain approaches based on asymptotic theory the procedure is illustrated in a small scale numerical study and is contrasted with an approach based on maximum entropy

a multiple regime smooth transition heterogeneous autoregressive model for long memory and asymmetries in this paper we propose a flexible model to describe nonlinearities and long range dependence in time series dynamics the new model is a multiple regime smooth transition extension of the heterogeneous autoregressive har model which is specifically designed to model the behavior of the volatility inherent in financial time series the model is able to simultaneously approximate long memory behavior as well as describe sign and size asymmetries a sequence of tests is developed to determine the number of regimes and an estimation and testing procedure is presented monte carlo simulations evaluate the finite sample properties of the proposed tests and estimation procedures we apply the model to several dow jones industrial average index stocks using transaction level data from the trades and quotes database that covers ten years of data we find strong support for long memory and both sign and size asymmetries furthermore the new model when combined with the linear har model is viable and flexible for purposes of forecasting volatility

nonparametric estimation of transfer functions rates of convergence and adaptation the paper deals with estimating the transfer functions of stable linear time invariant systems under stochastic assumptions we adopt a nonparametric minimax approach for measuring the estimation accuracy the quality of an estimator is measured by its worst case error over a family of transfer functions the families with polynomially and exponentially decaying impulse response sequences are considered we establish nonasymptotic upper bounds on the accuracy of the least squares estimator for finite impulse response approximation it is shown that the attainable estimation accuracy is determined essentially by the rate at which the true impulse response tends to zero lower bounds on the estimation accuracy are presented an adaptive estimator which does not exploit any a priori information about the true system is developed

multiple periodic solutions for suspension bridge systems abstract this paper studies the existence of multiple periodic oscillations of the suspension bridge system governed by the coupled nonlinear beam and wave equations by using the degree theory it is proved that the system has at least three periodic solutions

a note on enforcing stationarity in autoregressive moving average models summary a simple reparameterization is given that can implicitly restrict the autoregressivemoving average parameters to the stationary and invertible region

strongly consistent online forecasting of centered gaussian processes an estimator e spl circ d sub n n of the conditional expectation e x sub n 1 x sub n x n d sub n 1 in a centered stationary and ergodic gaussian process x sub i sub i with absolutely summable wold coefficients is constructed on the basis of having observed x sub 1 x sub n for a suitable choice of the length dn spl rarr spl infin n spl rarr spl infin of the past covered by the conditional expectation it is established that e spl circ d sub n n e x sub n 1 x sub n x n d sub n 1 spl rarr 0 with probability 1 in addition sufficient conditions for e x sub n 1 x sub n x sub n 1 e x sub n 1 x sub n x n d sub n 1 spl rarr 0 to hold with probability 1 are given that is conditions under which e spl circ d sub n n can be used as a strongly consistent forecaster for e x sub n 1 x sub n x sub n 1

sieve bootstrap for smoothing in nonstationary time series we propose a sieve bootstrap procedure for time series with a deterministic trend the sieve for constructing the bootstrap is based on nonparametric trend estimation and autoregressive approximation for some noise process the bootstrap scheme itself does i i d resampling of estimated innovations from fitted autoregressive models we show the validity and indicate second order correctness of such sieve bootstrap approximations for the limiting distribution of nonparametric linear smoothers the resampling can then be used to construct nonparametric confidence intervals for the underlying trend in particular we show asymptotic validity for constructing confidence bands which are simultaneous within a neighborhood of size of the order of the smoothing bandwidth our resampling procedure yields satisfactory results in a simulation study for finite sample sizes we also apply it to the longest series of total ozone measurements from arosa switzerland and find a significant decreasing trend

a method for adaptive estimation of arma processes abstract when the noise process in adaptive identification of linear stochastic systems is correlated and can be represented by a moving average model extended least squares algorithms are commonly used and converge under a strictly positive real spr condition on the noise model in this paper we present an adaptive algorithm for the estimation of autoregressive moving average arma processes and show that it is convergent without any spr condition and has a convergence rate of o loglog t t 1 2

a fast estimation method for arma processes an improvement of a batch algorithm for parameter estimation of an arma process is presented a fast and iterative algorithm is presented and provides estimates with the same asymptotic distribution as the maximum likelihood estimates this algorithm is a modified version of the method suggested by mayne and firoozan 1982 linear identification of arma processes automatica 18 461 466 and requires one less regression than the previous method if only one pass is executed

numerical solution of the mean field theory of the annni model in an external field a comparison of two methods it is shown how the solution of the mean field equations for the annni model can be reduced to the iteration of an area preserving two dimensional mapping if an external magnetic field is applied this method is analysed in detail and compared with the exact numerical solution of the mean field equations for simple periodic magnetisations

a new order estimation technique for time series modeling a new approach to estimating the order of the autoregressive moving average model is proposed which is based on the approximate stochastic realization introduced in davis and fotopoulos 1991 the present approach is attractive because overparameterization a very common problem in order determination is avoided successfully simulation results are included to illustrate the effectiveness of the proposed order estimation approach