Markovian representation of a bilinear time series model and maximum likelihood estimation of the parameters

A bilinear time series (BLTS) model is expressed in the form of Akaike's Markovian representation in order to use the Kalman recursive estimation approach. It is shown that Akaike's Markovian representation of autoregressive moving average models of orderp and q (ARMA(p,q)) and that of the bilinear model are equivalent. This equivalence facilitates the maximum likelihood estimation of the parameters involved in the bilinear model, which otherwise is an unwieldy problem. The present approach can easily be extended to take into account missing observations     

Empirical likelihood for mixed-effects error-in-variables model

This paper mainly introduces the method of empirical likelihood and its applications on two different models. We discuss the empirical likelihood inference on fixed-effect parameter in mixed-effects model with error-in-variables. We first consider a linear mixed-effects model with measurement errors in both fixed and random effects. We construct the empirical likelihood confidence regions for the fixed-effects parameters and the mean parameters of random-effects. The limiting distribution of the empirical log likelihood ratio at the true parameter is X2p＋q, where p, q are dimension of fixed and random effects respectively. Then we discuss empirical likelihood inference in a semi-linear error-in-variable mixed-effects model. Under certain conditions, it is shown that the empirical log likelihood ratio at the true parameter also converges to X2p＋q. Simulations illustrate that the proposed confidence region has a coverage probability more closer to the nominal level than normal approximation based confidence region.     

Identification of Refined ARMA Echelon Form Models for Multivariate Time Series

In the present article, we are interested in the identification of canonical ARMA echelon form models represented in a “refined” form. An identification procedure for such models is given by Tsay (J. Time Ser. Anal.10(1989), 357-372). This procedure is based on the theory of canonical analysis. We propose an alternative procedure which does not rely on this theory. We show initially that an examination of the linear dependency structure of the rows of the Hankel matrix of correlations, with originkin (i.e., with correlation at lagkin position (1, 1)), allows us not only to identify the Kronecker indicesn₁, …, n_d, whenk=1, but also to determine the autoregressive ordersp₁, …, p_d, as well as the moving average ordersq₁, …, q_dof the ARMA echelon form model by settingk>1 andk<1, respectively. Successive test procedures for the identification of the structural parametersn_i,p_i, andq_iare then presented. We show, under the corresponding null hypotheses, that the test statistics employed asymptotically follow chi-square distributions. Furthermore, under the alternative hypothesis, these statistics are unbounded in probability and are of the formNδ{1+o_p(1)}, whereδis a positive constant andNdenotes the number of observations. Finally, the behaviour of the proposed identification procedure is illustrated with a simulated series from a given ARMA model.     

Efficient and accurate spectral method for the time-fractional dual-phase-lag heat transfer model and its parameter estimation

In the current paper, a heat transfer model is suggested based on a time-fractional dual-phase-lag (DPL) model. We discuss the model in two parts, the direct problem and the inverse problem. Firstly, for solving it, the finite difference/Legendre spectral method is constructed. In the temporal direction, we employ the weighted and shifted Grünwald approximation, which can achieve second order convergence. In the spatial direction, the Legendre spectral method is used, it can obtain spectral accuracy. The stability and convergence are theoretically analyzed. For the inverse problem, the Bayesian method is used to construct an algorithm to estimate the four parameters for the model, namely, the time-fractional order α, the time-fractional order β, the delay time τ_T, and the relaxation time τ_q. Next, numerical experiments are provided to demonstrate the effectiveness of our scheme, with the values of τ_q and τ_T for processed meat employed. We also make a comparison with another method. The data obtained for the direct problem are used in the parameter estimation. The paper provides an accurate and efficient numerical method for the time-fractional DPL model.     

Returns of Eastern European financial markets:α -stable distributions,measures of risk

The daily returns of financial market indices of nineteen Eastern European countries are modelled, linear trend or ARMA(p, q) for the levels and GARCH(p ′, q ′) for the residuals. For the non Gaussian residuals -stable distributions are proposed. Then, risk measures, like the STARR and the R-ratio are used to analyse the risk in the cases of Gaussian and -stable distribution models of the residuals. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Guaranteed consistency of surface intersections and trimmed surfaces using a coupled topology resolution and domain decomposition scheme

We describe a method that serves to simultaneously determine the topological configuration of the intersection curve of two parametric surfaces and generate compatible decompositions of their parameter domains, that are amenable to the application of existing perturbation schemes ensuring exact topological consistency of the trimmed surface representations. To illustrate this method, we begin with the simpler problem of topology resolution for a planar algebraic curve F(x,y)=0 in a given domain, and then extend concepts developed in this context to address the intersection of two tensor-product parametric surfaces p(s,t) and q(u,v) defined on (s,t)∈[0,1]² and (u,v)∈[0,1]². The algorithms assume the ability to compute, to any specified precision, the real solutions of systems of polynomial equations in at most four variables within rectangular domains, and proofs for the correctness of the algorithms under this assumption are given. Mathematics subject classification (2000)  65D17     

Multivariate CARMA processes

A multivariate Lévy-driven continuous time autoregressive moving average (CARMA) model of order (p,q$p,q$), q<p$q<p$, is introduced. It extends the well-known univariate CARMA and multivariate discrete time ARMA models. We give an explicit construction using a state space representation and a spectral representation of the driving Lévy process. Furthermore, various probabilistic properties of the state space model and the multivariate CARMA process itself are discussed in detail.     

Central Units in ℤCp,q

Let C_{p, q} be the semi-direct product of a cyclic group of order q by a cyclic group of order p, and ?C_{p, q} the integral group ring of C_{p, q}. In this article, firstly, we describe the group of normalized central units of ?C_{p, q} as a direct product of two subgroups that we call units of first kind and of second kind. For a class of prime numbers that we call good primes, we construct a multiplicatively independent set which generates the group of units of first kind. Finally, we construct a set of multiplicatively independent units which generates the units of second kind for a larger class of primes.     

The Maximum Lq-Likelihood Method: An Application to Extreme Quantile Estimation in Finance

Estimating financial risk is a critical issue for banks and insurance companies. Recently, quantile estimation based on extreme value theory (EVT) has found a successful domain of application in such a context, outperforming other methods. Given a parametric model provided by EVT, a natural approach is maximum likelihood estimation. Although the resulting estimator is asymptotically efficient, often the number of observations available to estimate the parameters of the EVT models is too small to make the large sample property trustworthy. In this paper, we study a new estimator of the parameters, the maximum Lq-likelihood estimator (MLqE), introduced by Ferrari and Yang (Estimation of tail probability via the maximum Lq-likelihood method, Technical Report 659, School of Statistics, University of Minnesota, 2007 ). We show that the MLqE outperforms the standard MLE, when estimating tail probabilities and quantiles of the generalized extreme value (GEV) and the generalized Pareto (GP) distributions. First, we assess the relative efficiency between the MLqE and the MLE for various sample sizes, using Monte Carlo simulations. Second, we analyze the performance of the MLqE for extreme quantile estimation using real-world financial data. The MLqE is characterized by a distortion parameter q and extends the traditional log-likelihood maximization procedure. When q→1, the new estimator approaches the traditional maximum likelihood estimator (MLE), recovering its desirable asymptotic properties; when q ≠ 1 and the sample size is moderate or small, the MLqE successfully trades bias for variance, resulting in an overall gain in terms of accuracy (mean squared error).      

The Graded Algebra Generated by Two Eulerian Derivatives

We study the algebra R _p,q generated by the Eulerian derivatives for two parameters p and q. Subject to certain conditions on the parameters, we show that R _p,q is a finitely presented N-graded algebra of Gelfand–Kirillov dimension 3. We establish a criterion for the cyclic module R _p,q /R _p,q f to be Noetherian, where f is homogeneous of degree 1. For some choices of the parameters, this criterion always holds and we know of no situation where it fails. It is not known whether R _p,q is Noetherian. We classify the point modules for R _p,q and determine the normal elements and graded automorphisms for R _p,q .     

Influence asymptotique de la correction par la moyenne sur l'estimation d'un modèle AR(1) périodique

This Note studies asymptotic influence of mean-correction on the parameter least squares estimation for a periodic AR(1) model. Unlike the stationary ARMA case, we show that fitting a periodic ARMA model with intercepts to the observed series can provide substantial gains in terms of asymptotic accuracy for the parameter least squares estimators compared with fitting a periodic ARMA model without intercepts to the mean-corrected series. To cite this article: A. Gautier, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Linear system approximation via covariance equivalent realizations

A truncation technique for model reduction to simultaneously match a specified number of output covariance derivatives is described. These COVariance Equivalent Realizations which match q + 1 covariance derivatives are called “q-COVERs.” In general q-COVERs are not unique. The additional freedom is used herein so that the q-COVER obtained also matches q Markov parameters. The truncation technique uses a form of the observability matrices of the full-order system to determine a priori the order required of the reduced order model to match a specified number of output covariance derivatives and Markov parameters. The resulting realization is shown to be independent of the basis of the complete model to within a unitary transformation. Stability conditions for the reduced order model are also described, and the relationships are established between stochastically equivalent realizations and the q-COVERs.     

A Class of Imprimitive Association Schemes

From Burnside's p^αq^β-Theorem, it follows that any nonabelian group of order p^αq^β, where p and q are primes, cannot be simple. As a main result of this article, we state and prove an analog of the mentioned theorem for commutative association schemes.     

On the recursive fitting of subset autoregressive-moving average process

For fitting data with an ARMA model, the major topic we come cross is how to determine the order (p, q). More over we may wish to search for a subset ARMA model. This paper offers a recursive procedure for this purpose, it is computationally very cheap. If the data are generated truly by a subset ARMA model, then the procedure can be proved to give a consistent identification.Institute of Applied Mathematics, Academia Sinica     

Infinite families of 2- and 3-designs with parameters v = p + 1, k = (p − 1)/2i + 1, where p odd prime, 2e T (p − 1), e ≥ 2, 1 ≤ i ≤ e

Let p be an odd prime number such that p − 1 = 2^em for some odd m and e ≥ 2. In this article, by using the special linear fractional group PSL(2, p), for each i, 1 ≤ i ≤ e, except particular cases, we construct a 2-design with parameters v = p + 1, k = (p − 1)/2ⁱ + 1 and λ = ((p − 1)/2ⁱ+1)(p − 1)/2 = k(p − 1)/2, and in the case i = e we show that some of these 2-designs are 3-designs. Likewise, by using the linear fractional group PGL(2,p) we construct an infinite family of 3-designs with the same v k and λ = k(k − 2). These supplement a part of [4], in which we gave an infinite family of 3-designs with parameters v = q + 1, k = (q + 1)/2 = (q − 1)/2 + 1 and λ = (q + 1)(q − 3)/8 = k(k − 2)/2, where q is a prime power such that q − 1 = 2m for some odd m and q > 7. Some of the designs given in this article and in [4] fill in a few blanks in the table of Chee, Colbourn, and Kreher [2]. © 1997 John Wiley & Sons, Inc.     

A Class of Multiderivative Hybrid One-step Methods

This paper presents a class of hybrid one-step methods that are obtained by using Cramer's rule and rational approximations to function exp(q). The algorithms fall into the catalogue of implicit formula, which involves sth order derivative and s 1 free parameters. The order of the algorithms satisfies s 1≤p≤2s 2. The stability of the methods is also studied, necessary and sufficient conditions for A-stability and L-stability are given. In addition, some examples are also given to demonstrate the method presented.     

Complexity of Weighted Approximation over

We study approximation of univariate functions defined over the reals. We assume that the rth derivative of a function is bounded in a weighted L_p norm with a weight ψ. Approximation algorithms use the values of a function and its derivatives up to order r−1. The worst case error of an algorithm is defined in a weighted L_q norm with a weight ρ. We study the worst case (information) complexity of the weighted approximation problem, which is equal to the minimal number of function and derivative evaluations needed to obtain error . We provide necessary and sufficient conditions in terms of the weights ψ and ρ, as well as the parameters r, p, and q for the weighted approximation problem to have finite complexity. We also provide conditions which guarantee that the complexity of weighted approximation is of the same order as the complexity of the classical approximation problem over a finite interval. Such necessary and sufficient conditions are also provided for a weighted integration problem since its complexity is equivalent to the complexity of the weighted approximation problem for q=1.     

The autotopism group of ap-primitive semifield plane of orderq 4

Let κ be a semifield plane of order q⁴ with kernel K≅GF(q²), where q=p^r, p is prime. Previously, Johnson determined the form of p-primitive semifield planes of order q⁴, q=p^r, and Cordero specified the form of autotopisms and proved the solvability of an autotopism group for the particular case q=p. The goal of the present article is to give an explication of the form of autotopisms and prove the solvability of an autotopism group in the general case. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 334–344, May–June, 1996.     

Trigonometric and orthoprojection widths of classes of periodic functions of many variables

We obtain exact order estimates for trigonometric and orthoprojection widths of the Besov classes B ^r _p,θ and Nikol’skii classes Hr p of periodic functions of many variables in the space L _q for certain relations between the parameters p and q.