Particle approximations of the score and observed information matrix in state space models with application to parameter estimation. (English) Zbl 1214.62093

Poyiadjis, George; Doucet, Arnaud; Singh, Sumeetpal S.

Biometrika 98, No. 1, 65-80 (2011).

Summary: Particle methods are popular computational tools for Bayesian inference in nonlinear non-Gaussian state space models. For this class of models, we present two particle algorithms to compute the score vector and observed information matrix recursively. The first algorithm is implemented with computational complexity \(\mathcal O(N)\) and the second with complexity \(\mathcal O(N^2)\), where \(N\) is the number of particles. Although cheaper, the performance of the \(\mathcal O(N)\) method degrades quickly, as it relies on the approximation of a sequence of probability distributions whose dimension increases linearly with time. In particular, even under strong mixing assumptions, the variance of the estimates computed with the \(\mathcal O(N)\) method increases at least quadratically in time. The more expensive \(\mathcal O(N^2)\) method relies on a nonstandard particle implementation and does not suffer from this rapid degradation. It is shown how both methods can be used to perform batch and recursive parameter estimation.

MSC:

62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
65C60	Computational problems in statistics (MSC2010)
62F10	Point estimation
65Y20	Complexity and performance of numerical algorithms

Keywords:

observed information matrix; particle method; sequential Monte Carlo simulation; state space model; stochastic approximation