stability results for nonlinear error correction models this paper studies analogs of granger s representation theorem in the context of a general nonlinear vector autoregressive error correction model the model allows for nonlinear autoregressive conditional heteroskedasticity and the conditional distribution involved can be a mixture distribution of a rather general type mixture models of this kind can be thought of as generalizations of threshold models and they have attracted attention in the recent time series and econometrics literature the paper develops a useful transformation which shows how the nonlinear error correction model can be transformed to a nonlinear vector autoregressive model so that available results on the stationarity or nonstationarity of the latter can be used for the former the most satisfactory results are obtained in a model in which a specific structural relation between the nonlinearity and equilibrium correction prevails without this structural relation only a lower bound for the number of long run equilibrium relations can explicitly be determined because the exact number depends on properties of the first and second moments of a nonlinear stationary component of the process

least squares temporal difference methods an analysis under general conditions we consider approximate policy evaluation for finite state and action markov de cision processes mdp with the least squares temporal difference lstd algorithm lstd in an exploration enhanced learning context where policy costs are computed from observations of a markov chain different from the one corresponding to the policy under evaluation we establish for the discounted cost criterion that lstd converges almost surely under mild minimal conditions we also analyze other properties of the iterates involved in the algorithm including convergence in mean and boundedness our analysis draws on theories of both finite space markov chains and weak feller markov chains on a topological space our results can be applied to other temporal difference algorithms and mdp models as examples we give a convergence analysis of a td algorithm and extensions to mdp with compact state and action spaces as well as a convergence proof of a new lstd algorithm with state dependent parameters

a recursive generation of local higher order sine gordon equations and their backlund transformation a new hierarchy of local nonlinear evolution equations is generated by a recursion operator and its explicit inverse it is shown that this hierarchy satisfies a canonical geometrical scheme and that it contains as special cases the sine gordon and liouville equations in laboratory coordinates a generalization of the well known backlund transformation and nonlinear superposition formula for the sine gordon equation is also obtained

ergodic theorems for discrete time stochastic systems using a stochastic lyapunov function sufficient conditions are established under which the law of large numbers and related ergodic theorems hold for nonlinear stochastic systems operating under feedback it is shown that these conditions hold whenever a moment condition is satisfied this may be interpreted as a generalizaton of the martingale property if in addition a stochastic controllability condition holds then it is shown that the underlying distributions governing the system converge to an invariant probability at a geometric rate the key assumption used is that a markov chain with stationary transition probabilities exists that serves as a state process for the closed loop system

a sharper bonferroni procedure for multiple tests of significance summary a simple procedure for multiple tests of significance based on individual p values is derived this simple procedure is sharper than holm s 1979 sequentially rejective procedure both procedures contrast the ordered p values with the same set of critical values holm s procedure rejects an hypothesis only if its p value and each of the smaller p values are less than their corresponding critical values the new procedure rejects all hypotheses with smaller or equal p values to that of any one found less than its critical value

testing linearity against smooth transition autoregressive models summary we study a general univariate smooth transition autoregressive star model it contains as a special case the self exciting threshold autoregressive setar model we present three tests for testing linearity against star models and discuss their properties the power of the tests in small samples is investigated by simulation when the alternative is the logistic star model one of the tests is identical to tsay s 1986 test statistic and is recommended only in a special case of the two remaining tests with wider applicability one seems superior to the other in small samples it is also more powerful than the cusum test recently proposed for testing linearity against setar models

stability convergence and performance of an adaptive control algorithm applied to a randomly varying system the stability and performance of a stochastic adaptive control algorithm applied to a randomly varying linear system are investigated the authors demonstrate that loss functions on the input output process converge to their expectation with respect to an invariant probability at a geometric rate and hence a form of stochastic exponential asymptotic stability is established and when the parameter variation and measurement noise are small it is shown that the performance is nearly optimal and if an excitation signal is added in the control law near consistency of the parameter estimates is obtained further results include central limit theorems and the law of large numbers of the input output and parameter processes

discrete time stochastic adaptive control this paper establishes global convergence of a stochastic adaptive control algorithm for discrete time linear systems it is shown that with probability one the algorithm will ensure the system inputs and outputs are sample mean square bounded and the conditional mean square output tracking error achieves its global minimum possible value for linear feedback control thus asymptotically the adaptive control algorithm achieves the same performance as could be achieved if the system parameters were known

backlund transformations soliton solutions and wave functions of kaup newell and wadati konno ichikawa systems using the bargmann darboux method the backlund transformations n soliton solutions and corresponding wave functions of the kaup newell and wadati konno ichikawa systems are obtained these results culminate in an algebraic recursive procedure for the determination of multisoliton solutions and their wave functions of the derivative and mixed derivative nonlinear schrodinger equations iqt qxx i q 2q x q 2q 0 0 0

an exact chi squared decomposition theorem for inverse gaussian variates on montre que la deviance d echelle d un echantillon normal inverse de dimension n peut etre exprimee comme la somme de n 1 variables khi deux independantes

theoretical explanation of observed decreasing failure rate pooled data on the times of successive failures of the air conditioning system of a fleet of jet airplanes seemed to indicate that the life distribution had a decreasing failure rate more refined analysis showed that the failure distribution for each airplane separately was exponential but with a different failure rate using the theorem that a mixture of distributions each having a non increasing failure rate e g a mixture of exponential distributions itself has a non increasing failure rate the apparent decreasing failure rate of the pooled air conditioning life distribution was satisfactorily explained the present study has implications in other areas where an observed decreasing failure rate may well be the result of mixing exponential distributions having different parameters

testing for stationarity ergodicity and for comovements between nonlinear discrete time markov processes in this paper we introduce a class of nonlinear data generating processes dgps that are rst order markov and can be represented as the sum of a linear plus a bounded nonlinear component we use the concepts of geometric ergodicity and of linear stochastic comovement which correspond to the linear concepts of integratedness and cointegratedness to characterize the dgps we show that the stationarity test due to kwiatowski et al 1992 journal of econometrics 54 159 178 and the cointegration test of shin 1994 econometric theory 10 91 115 are applicable in the current context although the shin test has a di erent limiting distribution we also propose a consistent test which has a null of linear cointegration comovement and an alternative of n c22

explicit quasi periodic solutions of the kaup newell hierarchy abstract an algebraic curve of arithmetic genus n is introduced with the aid of the lax matrix from which we construct the meromorphic function on the algebraic curve and investigate its properties further we straighten out all the flows associated with the kaup newell hierarchy under the abel jacobi coordinates finally we achieve the explicit quasi periodic solutions for the whole kaup newell hierarchy including the coupled derivative nonlinear schrodinger equations

testing the adequacy of smooth transition autoregressive models abstract smooth transition autoregressive models are a flexible family of nonlinear time series models that have also been used for modelling economic data this paper contributes to the evaluation stage of a proposed specification estimation and evaluation cycle of these models by introducing a lagrange multiplier lm test for the hypothesis of no error autocorrelation and lm type tests for the hypothesis of no remaining nonlinearity and that of parameter constancy small sample properties of the f versions of these tests and some alternative test statistics are investigated by simulation the results indicate that the proposed tests can be applied in small samples already

backlund transformations for the schrodinger equation with a spectral dependence in the potential the schrodinger equation on the line with a potential depending linearly on the spectral parameter is considered it is shown that this spectral problem admits one parameter called elementary backlund transformations and two parameter called full backlund transformations the transformations of the spectral data under all these backlund transformations bts are explicitly computed different nonlinear superposition formulae for the different bts considered are given and used to obtain soliton solutions of the evolution equations associated with the spectral problem

prediction limits for the inverse gaussian distribution prediction intervals for the inverse gaussian are obtained from both a frequentist and a bayesian viewpoint the frequentist intervals are obtained by constructing pivotals that have the x 2 and f distributions the method involves inversion of probability statements which results in two sided prediction intervals a bayesian predictive density is obtained using a vague prior from which one or two sided bayesian prediction intervals can be determined an example for which bayesian prediction limits are narrower than the frequentist is given

identifying a simplifying structure in time series abstract this article studies how to identify hidden factors in multivariate time series process this problem is important because when the series are driven by a set of common factors a a large number of parameters may be needed to obtain an adequate representation of the system and b the estimated parameters will be highly correlated therefore a complex and badly defined relationship can appear when in fact a simpler and parsimonious model in terms of a few common factors can be operating this article develops a methodology to identify the number of factors and to build a simplifying transformation to represent the series it is proved that the number of factors is equal to the rank of the covariance matrices and the parameter matrices of the infinite moving average representation of the process the eigenvectors of these matrices will provide the canonical transformation the method is illustrated with one example using series of the price of wheat in five provinces of spain in the 19th ce

the trace identity a powerful tool for constructing the hamiltonian structure of integrable systems a new approach to hamiltonian structures of integrable systems is proposed by making use of a trace identity for a variety of isospectral problems that can be unified to one model x u it is shown that both the related hierarchy of evolution equations and the hamiltonian structure can be obtained from the same solution of the equation vx u v

explicit auto backlund relation through gauge transformation application of gauge transformation for generating explicit auto backlund relations is demonstrated for a number of integrable systems e g kdv sg nls dnls mixed dnls modified dnls lle etc this method turns out to be much simpler effective and straightforward using backlund relations all conserved quantities are expressed in a novel derivative free form which simplifies the calculation of important soliton characteristics

testing and modeling threshold autoregressive processes abstract the threshold autoregressive model is one of the nonlinear time series models available in the literature it was first proposed by tong 1978 and discussed in detail by tong and lim 1980 and tong 1983 the major features of this class of models are limit cycles amplitude dependent frequencies and jump phenomena much of the original motivation of the model is concerned with limit cycles of a cyclical time series and indeed the model is capable of producing asymmetric limit cycles the threshold autoregressive model however has not received much attention in application this is due to a the lack of a suitable modeling procedure and b the inability to identify the threshold variable and estimate the threshold values the primary goal of this article therefore is to suggest a simple yet widely applicable model building procedure for threshold autoregressive models based on some predictive residuals a simple statistic is proposed to test for threshold nonlinearity and specify the

weak convergence properties of constrained emphatic temporal difference learning with constant and slowly diminishing stepsize we consider the emphatic temporal difference td algorithm etd for learning the value functions of stationary policies in a discounted finite state and action markov decision process the etd algorithm was recently proposed by sutton mahmood and white 2016 to solve a long standing divergence problem of the standard td algorithm when it is applied to off policy training where data from an exploratory policy are used to evaluate other policies of interest the almost sure convergence of etd has been proved in our recent work under general off policy training conditions but for a narrow range of diminishing stepsize in this paper we present convergence results for constrained versions of etd with constant stepsize and with diminishing stepsize from a broad range our results characterize the asymptotic behavior of the trajectory of iterates produced by those algorithms and are derived by combining key properties of etd with powerful convergence theorems from the weak convergence methods in stochastic approximation theory for the case of constant stepsize in addition to analyzing the behavior of the algorithms in the limit as the stepsize parameter approaches zero we also analyze their behavior for a fixed stepsize and bound the deviations of their averaged iterates from the desired solution these results are obtained by exploiting the weak feller property of the markov chains associated with the algorithms and by using ergodic theorems for weak feller markov chains in conjunction with the convergence results we get from the weak convergence methods besides etd our analysis also applies to the off policy td algorithm when the divergence issue is avoided by setting sufficiently large it yields for that case new results on the asymptotic convergence properties of constrained off policy td with constant or slowly diminishing stepsize

introduction to m m processes abstract in this paper we introduce a new type of nonlinear model called the min max model and analyze its properties for a pair of series the stability conditions of this system are given for a nonlinearly integrated bivariate series under these stability conditions the difference between the two series exhibits threshold type nonlinearity it is possible to construct a threshold error correction model from the min max processes neglected nonlinearity tests are applied both to the univariate series and to the bivariate system in order to detect nonlinearity and it turns out that the tests using the bivariate series have better power we apply the min max model to u s treasury bills and commercial paper interest rates the spread of these interest rates shows threshold type nonlinearity and this model outperforms a linear model in terms of its predictability for out of sample data

some generalizations on the algebra of i 1 processes abstract a number of techniques have recently appeared in the literature utilizing instantaneous nonlinear transformations of stochastic processes as in economics most variables appear to be nonstationary it is interesting to investigate the temporal properties of nonlinear transformations of these processes using a unified approach based on hermite polynomials this paper provides some insight for polynomial exponential periodic and neural network transformations of first order integrated processes

reduction technique for matrix nonlinear evolution equations solvable by the spectral transform the main purpose of this paper is to describe a technique of reduction whereby from the class of evolution equations for matrices of order n solvable via the spectral transform associated to the matrix linear schrodinger eigenvalue problem one derives subclasses of nonlinear evolution equations involving less than n2 fields to illustrate the method from the equations for matrices of order 2 two subclasses of equations for 2 fields rather than 4 are obtained the first class coincides or rather includes that solvable via the spectral transform associated to the generalized zakharov shabat spectral problem further reduction to nonlinear evolution equations for a single field reproduces a number of well known equations but also yields a novel one highly nonlinear the second class also yields highly nonlinear equations some examples are given including another novel evolution equation for a single field

nonlinear stochastic trends abstract a nonlinear growth process has its change equal to a positive function of the lagged series plus an innovation whose variance depends on the lagged series if the innovations were identically zero the resulting deterministic trends take a wide variety of shapes the balance between growth in mean and variance to guarantee eventual growth of the process is determined the relationship between stochastic and deterministic trends found parametric and nonparametric estimates of the functions considered and applications analysed the natural bivariate generalization is found to produce a form of cointegration and a nonlinear error correction model

specification estimation and evaluation of smooth transition autoregressive models abstract this article considers the application of two families of nonlinear autoregressive models the logistic lstar and exponential estar autoregressive models this includes the specification of the model based on simple statistical tests linearity testing against smooth transition autoregression determining the delay parameter and choosing between lstar and estar models are discussed estimation by nonlinear least squares is considered as well as evaluating the properties of the estimated model the proposed techniques are illustrated by examples using both simulated and real time series

testing multiple equation systems for common nonlinear components this paper provides a generalized method of moments test for common nonlinear components in multiple time series the test statistic is presented in terms of the canonical correlations between the multiple series and a judicially chosen set of test regressors and its performance in small samples is evaluated using monte carlo simulations two applications highlight the usefulness of this test in multivariate modelling the first shows that the asymmetries which characterize the us and canadian business cycles can be attributed to a common nonlinear factor while the second shows that a single nonlinear factor can account for neglected nonlinearity in an empirical real business cycle model each of the implied nonlinear factor models shows potential for forecasting

bicomplexes and backlund transformations a bicomplex is a simple mathematical structure in particular associated with completely integrable models the conditions defining a bicomplex are a special form of a parameter dependent zero curvature condition we generalize the concept of a darboux matrix to bicomplexes and use it to derive backlund transformations for several models the method also works for moyal deformed equations with a corresponding deformed bicomplex

certain approximations to achieve sharp lower and upper bounds for the mills ratio of the inverse gaussian distribution abstract in this note we prove new complete monotonicity properties of some functions associated with the inverse gaussian distribution based on these properties we present some lower and upper bounds with explicit expressions for the mills ratio of the inverse gaussian distribution finally for demonstrating the efficiency of our estimates some numerical computations are provided

discrete time stochastic control systems a continuous lyapunov function implies robustness to strictly causal perturbations abstract discrete time stochastic systems employing possibly discontinuous state feedback control laws are addressed allowing discontinuous feedbacks is fundamental for stochastic systems regulated for instance by optimization based control laws we introduce generalized random solutions for discontinuous stochastic systems to guarantee the existence of solutions and to generate enough solutions to get an accurate picture of robustness with respect to strictly causal perturbations under basic regularity conditions the existence of a continuous stochastic lyapunov function is sufficient to establish that asymptotic stability in probability for the closed loop system is robust to sufficiently small state dependent strictly causal worst case perturbations robustness of a weaker stochastic stability property called recurrence is also shown in a global sense in the case of state dependent perturbations and in a semiglobal practical sense in the case of persistent perturbations an example shows that a continuous stochastic lyapunov function is not sufficient for robustness to arbitrarily small worst case disturbances that are not strictly causal our positive results are also illustrated by examples

backlund transformations connecting different isospectral deformation equations we study singular spectral problems on riemann surfaces and isospectral deformation equations connected with them we show that for a given class of reductions all spectral problems on a given riemann surface give rise to the same class of isospectral deformation equations these isospectral deformation equations can be solved using the riemann boundary value problem all possess local conservation laws and backlund transformations

backlund transformations related to the kaup newell spectral problem abstract a hierarchy of isospectral deformation equations related to the kaup newell spectral problem is obtained the generalized gauge transformations of the eigenmatrix of the spectral problem are used to generate backlund and self backlund transformations in a special case the backlund transformation of the null solution for all the equations in the hierarchy has been explicitly solved

controllability of nonlinear discrete time systems a lie algebraic approach this paper presents a geometric study of controllability for discrete time nonlinear systems various accessibility properties are characterized in terms of lie algebras of vector fields some of the results obtained are parallel to analogous ones in continuous time but in many respects the theory is substantially different and many new phenomena appear

some asymptotic results for learning in single hidden layer feedforward network models abstract we investigate the properties of a recursive estimation procedure the method of back propagation for a class of nonlinear regression models single hidden layer feedforward network models recently developed by cognitive scientists the results follow from more general results for a class of recursive m estimators obtained using theorems of ljung 1977 and walk 1977 for the method of stochastic approximation conditions are given ensuring that the back propagation estimator converges almost surely to a parameter value that locally minimizes expected squared error loss provided the estimator does not diverge and that the back propagation estimator is asymptotically normal when centered at this minimizer this estimator is shown to be statistically inefficient and a two step procedure that has efficiency equivalent to that of nonlinear least squares is proposed practical issues are illustrated by a numerical example involving approximation of the henon map