The foundations of finite sample estimation in stochastic processes. (English) Zbl 0584.62135

Godambe, Vidyadhar P.

Biometrika 72, 419-428 (1985).

Let \(y_ 1,...,y_ n\) be a discrete stochastic process with distribution F belonging to some class \({\mathcal F}\). The paper considers the estimation of a parameter \(\theta =\theta (F)\) by solving the equation \(g(y_ 1,...,y_ n,\theta)=0\) where \[ g=\sum^{n}_{i=1}a_{i-1}(y_ 1,...,y_{i-1},\theta)h_ i(y_ 1,...,y_ i,\theta)\quad and\quad h_ i\quad satisfies\quad \] \[ E_ F(h_ i(y_ 1,...,y_ i,\theta (F))| y_ 1,...,y_{i-1})=0\quad for\quad all\quad F\in {\mathcal F}. \] The optimal choice of the \(a_ i's\) is determined such that \(E(g^ 2)/E(\partial g/\partial \theta)^ 2\) is minimized. This result can be viewed as a generalisation of the Gauss- Markov theorem.
Applications include linear and nonlinear autoregressions and branching processes. The extension to the multiparameter case is mentioned.
The author considers only a finite sample size, but in fact he introduces a different kind of asymptotics since the criterion for optimality is the asymptotic variance when one takes infinitely many i.i.d. copies of the sample. There are close connections with robust estimation for time series, for instance conditional unbiasedness plays a key role also in the reviewer’s paper ”Infinitesimal robustness for autoregressive processes.” Ann. Stat. 12, 843-863 (1984).
The main difference is that the present paper requests unbiasedness for a very large nonparametric class \({\mathcal F}\) which determines the functions \(h_ i\) and thus leads to a small class of estimators considered.

Reviewer: H.-R.Künsch

MSC:

62M05	Markov processes: estimation; hidden Markov models
62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M09	Non-Markovian processes: estimation
62J05	Linear regression; mixed models

Keywords:

conditional least squares; estimating function; likelihood function; observed Fisher information; score function; discrete stochastic process; generalisation of the Gauss-Markov theorem; linear and nonlinear autoregressions; branching processes; finite sample size; optimality; asymptotic variance; robust estimation for time series; conditional unbiasedness