## 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)
Full Text:

### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.