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**Statistical inference for discrete-time samples from affine stochastic delay differential equations.**
*(English)*
Zbl 1456.62035

Summary: Statistical inference for discrete time observations of an affine stochastic delay differential equation is considered. The main focus is on maximum pseudo-likelihood estimators, which are easy to calculate in practice. A more general class of prediction-based estimating functions is investigated as well. In particular, the optimal prediction-based estimating function and the asymptotic properties of the estimators are derived. The maximum pseudo-likelihood estimator is a particular case, and an expression is found for the efficiency loss when using the maximum pseudo-likelihood estimator, rather than the computationally more involved optimal prediction-based estimator. The distribution of the pseudo-likelihood estimator is investigated in a simulation study. Two examples of affine stochastic delay equation are considered in detail.

### MSC:

62F10 | Point estimation |

62F12 | Asymptotic properties of parametric estimators |

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

### Keywords:

asymptotic normality; composite likelihood; consistency; discrete time observation of continuous-time models; prediction-based estimating functions; pseudo-likelihood; stochastic delay differential equation
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\textit{U. Küchler} and \textit{M. Sørensen}, Bernoulli 19, No. 2, 409--425 (2013; Zbl 1456.62035)

### References:

[1] | Brockwell, P.J. and Davis, R.A. (1991). Time Series : Theory and Methods , 2nd ed. Springer Series in Statistics . New York: Springer. · Zbl 0709.62080 |

[2] | Buckwar, E. (2000). Introduction to the numerical analysis of stochastic delay differential equations. J. Comput. Appl. Math. 125 297-307. · Zbl 0971.65004 |

[3] | Diekmann, O., van Gils, S.A., Verduyn Lunel, S.M. and Walther, H.O. (1995). Delay Equations : Functional- , Complex- , and Nonlinear Analysis. Applied Mathematical Sciences 110 . New York: Springer. · Zbl 0826.34002 |

[4] | Ditlevsen, S. and Sørensen, M. (2004). Inference for observations of integrated diffusion processes. Scand. J. Statist. 31 417-429. · Zbl 1062.62157 |

[5] | Doukhan, P. (1994). Mixing : Properties and Examples. Lecture Notes in Statistics 85 . New York: Springer. · Zbl 0801.60027 |

[6] | Forman, J.L. and Sørensen, M. (2008). The Pearson diffusions: A class of statistically tractable diffusion processes. Scand. J. Statist. 35 438-465. · Zbl 1198.62078 |

[7] | Godambe, V.P. and Heyde, C.C. (1987). Quasi-likelihood and optimal estimation. Internat. Statist. Rev. 55 231-244. · Zbl 0671.62007 |

[8] | Gushchin, A.A. and Küchler, U. (1999). Asymptotic inference for a linear stochastic differential equation with time delay. Bernoulli 5 1059-1098. · Zbl 0983.62049 |

[9] | Gushchin, A.A. and Küchler, U. (2000). On stationary solutions of delay differential equations driven by a Lévy process. Stochastic Process. Appl. 88 195-211. · Zbl 1045.60057 |

[10] | Gushchin, A.A. and Küchler, U. (2003). On parametric statistical models for stationary solutions of affine stochastic delay differential equations. Math. Methods Statist. 12 31-61. |

[11] | Heyde, C.C. (1997). Quasi-Likelihood and Its Application : A General Approach to Optimal Parameter Estimation. Springer Series in Statistics . New York: Springer. · Zbl 0879.62076 |

[12] | Isserlis, L. (1918). On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables. Biometrika 12 134-139. |

[13] | Jacod, J. and Sørensen, M. (2012). Asymptotic statistical theory for stochastic processes: A review. Preprint, Dept. Mathematical Sciences, Univ. Copenhagen. |

[14] | Küchler, U. and Kutoyants, Y.A. (2000). Delay estimation for some stationary diffusion-type processes. Scand. J. Statist. 27 405-414. · Zbl 0976.62083 |

[15] | Küchler, U. and Mensch, B. (1992). Langevin’s stochastic differential equation extended by a time-delayed term. Stochastics Stochastics Rep. 40 23-42. · Zbl 0777.60048 |

[16] | Küchler, U. and Platen, E. (2000). Strong discrete time approximation of stochastic differential equations with time delay. Math. Comput. Simulation 54 189-205. |

[17] | Küchler, U. and Platen, E. (2007). Time delay and noise explaining cyclical fluctuations in prices of commodities. Preprint, Inst. of Mathematics, Humboldt-Univ. Berlin. |

[18] | Küchler, U. and Sørensen, M. (2007). Statistical inference for discrete-time samples from affine stochastic delay differential equations. Preprint, Dept. Mathematical Sciences, Univ. Copenhagen. |

[19] | Küchler, U. and Sørensen, M. (2010). A simple estimator for discrete-time samples from affine stochastic delay differential equations. Stat. Inference Stoch. Process. 13 125-132. · Zbl 1209.62198 |

[20] | Küchler, U. and Vasiliev, V. (2005). Sequential identification of linear dynamic systems with memory. Stat. Inference Stoch. Process. 8 1-24. · Zbl 1062.62152 |

[21] | Reiß, M. (2002). Nonparametric estimation for stochastic delay differential equations. Ph.D. thesis, Institut für Mathematik, Humboldt-Universität zu Berlin. · Zbl 1023.62043 |

[22] | Reiß, M. (2002). Minimax rates for nonparametric drift estimation in affine stochastic delay differential equations. Stat. Inference Stoch. Process. 5 131-152. · Zbl 1006.62074 |

[23] | Reiss, M. (2005). Adaptive estimation for affine stochastic delay differential equations. Bernoulli 11 67-102. · Zbl 1059.62089 |

[24] | Sørensen, H. (2003). Simulated likelihood approximations for stochastic volatility models. Scand. J. Statist. 30 257-276. · Zbl 1053.62090 |

[25] | Sørensen, M. (2000). Prediction-based estimating functions. Econom. J. 3 123-147. · Zbl 0998.62071 |

[26] | Sørensen, M. (2011). Prediction-based estimating functions: Review and new developments. Braz. J. Probab. Stat. 25 362-391. · Zbl 1230.62111 |

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