Rates of convergence of autocorrelation estimates for autoregressive Hilbertian processes

Guillas, Serge

doi:10.1016/s0167-7152(01)00151-1

Cited by 35 publications

(39 citation statements)

References 7 publications

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“…We show how to estimate the functional parameters, with the help of B-spline expansion and sieve, and provide the asymptotic consistency of the estimator. In functional domain, Bosq (2000) has proposed Functional AutoRegressive (FAR) model for univariate functional time series and developed Yule-Walker estimation (see also Besse, Cardot and Stephenson, 2000;Kim, Chaudhuri and Shin, 2015;Guillas, 2001; Antoniadis and Sapatinas, 2003;Kokoszka and Zhang, 2010). Mourid and Bensmain (2006) proposed a maximum likelihood estimation with Fourier expansions.…”

Section: Vector Functional Autoregressive Modelmentioning

confidence: 99%

Forecasting Limit Order Book Liquidity Supply-Demand Curves with Functional Autoregressive Dynamics

2016

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Limit order book contains comprehensive information of liquidity on bid and ask sides. We propose a Vector Functional AutoRegressive (VFAR) model to describe the dynamics of the limit order book and demand curves and utilize the fitted model to predict the joint evolution of the liquidity demand and supply curves. In the VFAR framework, we derive a closed-form maximum likelihood estimator under sieves and provide the asymptotic consistency of the estimator. In application to limit order book records of 12 stocks in NASDAQ * Corresponding author: Wee Song Chua Email: a0054070@u.nus.edu Phone: +65-6516 3470 Fax: +65-6872 3919 This research was supported by the FRC grant at the National University of Singapore. Support from IRTG 1792 "High Dimensional Non Stationary Time Series", Humboldt-Universität zu Berlin, is gratefully acknowledged. 1 traded from 2 Jan 2015 to 6 Mar 2015, it shows the VAR model presents a strong predictability in liquidity curves, with R 2 values as high as 98.5 percent for insample estimation and 98.2 percent in out-of-sample forecast experiments. It produces accurate 5−, 25− and 50−minute forecasts, with root mean squared error as low as 0.09 to 0.58 and mean absolute percentage error as low as 0.3 to 4.5 percent.

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Section: Vector Functional Autoregressive Modelmentioning

confidence: 99%

Forecasting Limit Order Book Liquidity Supply-Demand Curves with Functional Autoregressive Dynamics

2016

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“…where fê k j g j2N is the system of eigenvalues associated with the empirical system of eigenfunctions f/ j g j2N : Optimal values of M, according the convergence rate (consistency) of A T ; and stability of the inversion in terms of different norms are studied, for example, in Bosq (2000) and Guillas (2001).…”

Section: Initial Estimatesmentioning

confidence: 99%

“…(8) and (9) the same information is provided for the Cauchy and Gamma autocorrelation kernels. The functional parameter estimation problem associated with ARH(p) models has been extensively studied in the last decade (see, for example, Bosq and Blanke 2007;Damon and Guillas 2005;Guillas 2001;Mas 1999Mas , 2007Mourid and Bensmain 2006;among others). Moment estimators and projection methods on the spatial covariance operator eigenfunction system are usually considered in addressing this problem.…”

Section: Truncation Order Selectionmentioning

confidence: 99%

Functional maximum-likelihood estimation of ARH(p) models

Ruiz-Medina

Gómez

2009

Stoch Environ Res Risk Assess

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In this paper the problem of functional filtering of an autoregressive Hilbertian (ARH) process, affected by additive Hilbertian noise, is addressed when the functional parameters defining the ARH(p) equation are unknown. The maximum-likelihood estimation of such parameters is obtained from the implementation of an expectation-maximization algorithm. Specifically, a finite-dimensional matrix approximation of the state equation is considered from its diagonalization in terms of the spectral decomposition of the functional parameters involved (PrincipalOscillation-Pattern-based diagonalization). The Expectation step and maximization step are then computed from the forward Kalman filtering followed by a backward Kalman smoothing recursion in terms of the Fourier coefficients associated with such a decomposition.

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“…The estimation of q is performed through several steps using the covariance operator given by C(x) = E[AEX 0 , xaeX 0 ], the cross-covariance operator given by D(x) = E[AEX 0 , xaeX 1 ], and the relation: D = qC. Since C is not invertible in general, a projection on a subspace of finite dimension k n is necessary to obtain an estimate of q. k n must be selected taking into account the decreasing rate of C's eigenvalues (Bosq 2000;Guillas 2001). After withdrawing the mean from the process, steps to estimate q are as follows:…”

Section: Autoregressive Hilbertian Model: Arh(1)mentioning

confidence: 99%

“…Several assumptions can ensure this, especially by taking k n of a small size in comparisons with the sample size n. Depending on the decay rate of C's eigenvalues, the choice of k n of type log (n) yields to a.s. convergence (Bosq 2000). The type n 1=a ; a[1 yields to L 2 mode (Guillas 2001). Our number of sample points was small (equal to 6).…”

Section: Autoregressive Hilbertian Model: Arh(1)mentioning

confidence: 99%

Boosting for real and functional samples: an application to an environmental problem

Castro¹,

González–Manteiga

2007

Stoch Environ Res Risk Assess

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In this paper, boosting techniques are given in order to forecast SO 2 levels near a power plant. We use boosting with neural networks to forecast real values of SO 2 concentration. Then, the data are considered as a time series of curves. Assuming a lag one dependence, the predictions are computed using the functional kernel and the linear autoregressive Hilbertian model. Boosting techniques are developed for those functional models. We compare results of functional boosting with different starting points and iterate models. We carry out the estimation, in real and functional cases, with the information given by a historical matrix, which is a subsample that emphasizes relevant SO 2 values.

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Rates of convergence of autocorrelation estimates for autoregressive Hilbertian processes

Cited by 35 publications

References 7 publications

Forecasting Limit Order Book Liquidity Supply-Demand Curves with Functional Autoregressive Dynamics

Forecasting Limit Order Book Liquidity Supply-Demand Curves with Functional Autoregressive Dynamics

Functional maximum-likelihood estimation of ARH(p) models

Boosting for real and functional samples: an application to an environmental problem

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