Abstract
Estimation for discrete time observations of an affine stochastic delay differential equation is considered. The delay measure is assumed to be concentrated on a finite set. A simple estimator is obtained by discretization of the continuous-time likelihood function, and its asymptotic properties are investigated. The estimator is very easy to calculate and works well at high sampling frequencies, but it is shown to have a significant bias when the sampling frequency is low.
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References
Barndorff-Nielsen OE, Sørensen M (1994) A review of some aspects of asymptotic likelihood theory for stochastic processes. Int Stat Rev 62: 133–165
Bradley RC (2007) Introduction to strong mixing conditions. Kendrick Press, Heber City, UT
Buckwar E (2000) Introduction to the numerical analysis of stochastic delay differential equations. J Comput Appl Math 125: 297–307
Doukhan P (1994) Mixing, properties and examples. Springer, New York. Lecture Notes in Statistics 85
Gushchin AA, Küchler U (1999) Asymptotic properties of maximum-likelihood-estimators for a class of linear stochastic differential equations with time delay. Bernoulli 5: 1059–1098
Gushchin AA, Küchler U (2000) On stationary solutions of delay differential equations driven by a lévy process. Stoch Proc Appl 88: 195–211
Gushchin AA, Küchler U (2003) On parametric statistical models for stationary solutions of affine stochastic delay differential equations. Math Methods Stat 12: 31–61
Küchler U, Kutoyants Y (2000) Delay estimation for some stationary diffusion-type processes. Scand J Stat 27: 405–414
Küchler U, Mensch B (1992) Langevin’s stochastic differential equation extended by a time-delayed term. Stoch Stoch Rep 40: 23–42
Küchler U, Platen E (2000) Strong discrete time approximation of stochastic differential equations with time delay. Math Comput Simul 54: 189–205
Küchler U, Sørensen M (1997) Exponential families of stochastic processes. Springer, New York
Küchler U, Sørensen M (2009) Statistical inference for discrete-time samples from affine stochastic delay differential equations. Preprint, Department of Mathematical Sciences, University of Copenhagen.
Küchler U, Vasil’jev VA (2005) Sequential identification of linear dynamic systems with memory. Stat Inference Stoch Process 8: 1–24
Reiß M (2002a) Minimax rates for nonparametric drift estimation in affine stochastic delay differential equations. Stat Inference Stoch Process 5: 131–152
Reiß M (2002b) Nonparametric estimation for stochastic delay differential equations. PhD thesis, Institut für Mathematik, Humboldt-Universität zu Berlin
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Küchler, U., Sørensen, M. A simple estimator for discrete-time samples from affine stochastic delay differential equations. Stat Inference Stoch Process 13, 125–132 (2010). https://doi.org/10.1007/s11203-010-9042-y
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DOI: https://doi.org/10.1007/s11203-010-9042-y
Keywords
- Asymptotic normality
- Discrete time observation of continuous time models
- Stochastic delay differential equation