Maximum Likelihood Estimates of a Class of One‐Dimensional Stochastic Differential Equation Models From Discrete Data

Cleur, Eugene M.

doi:10.1111/1467-9892.00238

Cited by 6 publications

(5 citation statements)

References 10 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%

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%

Statistical aspects of the fractional stochastic calculus

Tudor¹,

Viens²

2007

Ann. Statist.

140

113

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“…SDEs are more realistic than traditional approaches in longitudinal data analysis and other applications, but wider acceptance has been hampered by mathematical and computational complexity. Linear SDEs allow explicit solutions, and are suitable for compartmental and other models (Cleur 2000, Kristensen et al 2005, Klim et al 2009, Favetto and Samson 2010, Cuenod et al 2011, Driver et al 2017, Seber and Wild 2003). However, they may not be appropriate in more general situations.…”

Section: Furnivalmentioning

confidence: 99%

“…7.3) and pharmacokinetics (Kristensen et al 2005, Klim et al 2009, Donnet and Samson 2013, and are increasingly used in fields like econometrics (Paige and Allen 2010, Bu et al 2010, 2016, animal growth (Lv and Pitchford 2007, Strathe et al 2009, Filipe et al 2010, oncology (Sen 1989, Favetto and, and forestry (García 1983, Broad and Lynch 2006, Batho and García 2014. Other applications include Artzrouni and Reneke (1990), Cleur (2000), Driver et al (2017). In particular, deterministic growth curves are commonly fitted to repeated measurements assuming that observation error is the only source of randomness (Davidian and Giltinan 2003), but an SDE growth model can be more realistic (Hotelling 1927, Donnet et al 2010.…”

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confidence: 99%

Estimating reducible stochastic differential equations by conversion to a least-squares problem

Garćıa¹

2018

Comput Stat

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Stochastic differential equations (SDEs) are increasingly used in longitudinal data analysis, compartmental models, growth modelling, and other applications in a number of disciplines. Parameter estimation, however, currently requires specialized software packages that can be difficult to use and understand. This work develops and demonstrates an approach for estimating reducible SDEs using standard nonlinear least squares or mixed-effects software. Reducible SDEs are obtained through a change of variables in linear SDEs, and are sufficiently flexible for modelling many situations. The approach is based on extending a known technique that converts maximum likelihood estimation for a Gaussian model with a nonlinear transformation of the dependent variable into an equivalent least-squares problem. A similar idea can be used for Bayesian maximum a posteriori estimation. It is shown how to obtain parameter estimates for reducible SDEs containing both process and observation noise, including hierarchical models with either fixed or random group parameters. Code and examples in R are given. Univariate SDEs are discussed in detail, with extensions to the multivariate case outlined more briefly. The use of well tested and familiar standard software should make SDE modelling more transparent and accessible.

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“…Examples and comparisons of these techniques can be found in [7,8]. Another way of constructing the likelihood for discrete observations utilizes the Markov property of solutions to (1):…”

Section: Likelihood Estimationmentioning

confidence: 99%

Interaction between stock indices via changepoint analysis

Lenardon

Amirdjanova

2006

Appl Stoch Models Bus & Ind

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Stock market indices from several countries are modelled as discretely sampled diffusions whose parameters change at certain times. To estimate these times of parameter changes we employ both a sequential likelihood-ratio test and a non-parametric, spectral algorithm designed specifically for time series with multiple changepoints. Finally, we use point-process techniques to model relationships between changepoints of different financial time series.Suppose for simplicity we have observations of X t at times t ¼ 1; . . . ; n: Our goal will be to detect times at which the parameters y 1 and y 2 change. Intuitively, to decide if there is a change at time k we can maximize the log likelihood on t ¼ 1; . . . ; k À 1 and on t ¼ k; . . . ; n and M. J. LENARDON AND A. AMIRDJANOVA 576 Both the likelihood-based and the spectral-based sliding window methods of changepoint detection provide an entire time series of test statistics. To decide to what extent changes in one STOCK INDICES 577

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Maximum Likelihood Estimates of a Class of One‐Dimensional Stochastic Differential Equation Models From Discrete Data

Cited by 6 publications

References 10 publications

Statistical aspects of the fractional stochastic calculus

Statistical aspects of the fractional stochastic calculus

Estimating reducible stochastic differential equations by conversion to a least-squares problem

Interaction between stock indices via changepoint analysis

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