Rates of convergence of approximate maximum likelihood estimators in the Ornstein-Uhlenbeck process

Bishwal, Jaya P. N.; Bose, Arup

doi:10.1016/s0898-1221(01)00127-4

Cited by 24 publications

(21 citation statements)

References 5 publications

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“…When the whole trajectory of the diffusion can be observed, then the parameter estimation problem is relatively simple, but of practical contemporary interest is work in which an approximate estimator, using only information gleaned from the underlying process in discrete time, is able to do as well as an estimator that uses continuously gathered information. Several methods have been employed to construct good estimators for this challenging question of discretely observed diffusions; among these methods, we refer to numerical approximation of the likelihood function (see [1,5,32]), martingale estimating functions (see [6]), indirect statistical inference (see [16]), the Bayesian approach (see [15]), some sharp probabilistic bounds on the convergence of estimators in [7], and [10,12,31] for particular situations. We mention the survey [36] for parameter estimation in discrete cases, further details in [21,25] and the book [23].…”

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confidence: 99%

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Statistical aspects of the fractional stochastic calculus

Tudor¹,

Viens²

2007

Ann. Statist.

140

113

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We apply the techniques of stochastic integration with respect to fractional Brownian motion and the theory of regularity and supremum estimation for stochastic processes to study the maximum likelihood estimator (MLE) for the drift parameter of stochastic processes satisfying stochastic equations driven by a fractional Brownian motion with any level of Hölder-regularity (any Hurst parameter). We prove existence and strong consistency of the MLE for linear and nonlinear equations. We also prove that a version of the MLE using only discrete observations is still a strongly consistent estimator.1. Introduction. Stochastic calculus with respect to fractional Brownian motion (fBm) has recently experienced intensive development, motivated by the wide array of applications of this family of stochastic processes. For example, recent work and empirical studies have shown that traffic in modern packet-based high-speed networks frequently exhibits fractal behavior over a wide range of time scales; in quantitative finance and econometrics, the fractional Black-Scholes model has recently been introduced (see, e.g., [14,17]) and this motivates the statistical study of stochastic differential equations governed by fBm.The topic of parameter estimation for stochastic differential equations driven by standard Brownian motion is of course not new. Diffusion processes are widely used for modeling continuous time phenomena; therefore, statistical inference for diffusion processes has been an active research area over the last few decades. When the whole trajectory of the diffusion can be observed, then the parameter estimation problem is relatively simple, but of practical contemporary interest is work in which an approximate estimator,

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confidence: 99%

“…There are some strong known results concerning equation (7). In [30] strong existence and uniqueness is proved assuming only the linear growth…”

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confidence: 99%

Statistical aspects of the fractional stochastic calculus

Tudor¹,

Viens²

2007

Ann. Statist.

140

113

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“…Different values of the window size land the drift parameter 1 are considered in order to compare this scheme (based on ¡t) with the detection schemes proposed in [13,14] (t), to provide an good trade-of between robustness against false alarms and sensitivity to faults. This behavior is typical in a wide range of values of 1 e [1,10]. For smaller values of 1, the scheme ¡t may trigger some false alarms (its behavior approaching that of ¡i a )\ for very large values of X, the scheme ¡t approaches the conservative behavior of fi c , but it is a bit more sensitive to faults.…”

Section: Simulation Examplementioning

confidence: 92%

“…Most of the works provide estimators for the drift parameter X (see [10,18,35]) which in our case is known by design; in [3,30,52] maximum likelihood (ML) estimators for the mean parameter ¡A are provided whereas the estimation of a is usually referred to the computation of a limit [ 6]. In general, these results do not apply to the case of time-varying vector faults.…”

Section: The Ornstein-uhlenbeck Processmentioning

confidence: 99%

A mathematical framework for new fault detection schemes in nonlinear stochastic continuous-time dynamical systems

Zufiria

2012

Applied Mathematics and Computation

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In this work, a mathematical unifying framework for designing new fault detection schemes in nonlinear stochastic continuous-time dynamical systems is developed. These schemes are based on a stochastic process, called the residual, which reflects the system behavior and whose changes are to be detected. A quickest detection scheme for the residual is proposed, which is based on the computed likelihood ratios for time-varying statistical changes in the Ornstein-Uhlenbeck process. Several expressions are provided, depending on a priori knowledge of the fault, which can be employed in a proposed CUSUM-type approximated scheme. This general setting gathers different existing fault detection schemes within a unifying framework, and allows for the definition of new ones. A comparative simulation example illustrates the behavior of the proposed schemes.

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“…Bishwal (2001) obtained weak convergence bound of the order OðT À1=2 Þ for the MLE and Bayes estimators using two different random normings which are useful for computation of a confidence interval. Bishwal and Bose (2001) obtained weak convergence bound for two approximate maximum likelihood estimators (AMLEs). Note that in the stationary case where X 0 has Nð0; À1=2yÞ distribution, the exact log-likelihood function is given by…”

Section: New Estimatorsmentioning

confidence: 99%

Rates of weak convergence of approximate minimum contrast estimators for the discretely observed Ornstein–Uhlenbeck process

Bishwal

2006

Statistics & Probability Letters

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Rates of convergence of approximate maximum likelihood estimators in the Ornstein-Uhlenbeck process

Cited by 24 publications

References 5 publications

Statistical aspects of the fractional stochastic calculus

Statistical aspects of the fractional stochastic calculus

A mathematical framework for new fault detection schemes in nonlinear stochastic continuous-time dynamical systems

Rates of weak convergence of approximate minimum contrast estimators for the discretely observed Ornstein–Uhlenbeck process

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