Sequential selection of observations in randomly generated experiments

Rok, strany: 1999, 167 - 175

We consider a situation where experimental conditions are observed sequentially but are uncontrolled. The length of the observed sequence is $N$, and only $np$ have to be estimated in a regression model with observations $y_k=η(oldsymbol{θ},ξ_k)+ε_k$, with ${ε_k}$ a sequence of i.i.d. errors. The experimental conditions $ξ_k$ are i.i.d. random variables, independent of ${ε_k}$ and are observed sequentially. The decision of observing $y_k$ or not must be made on-line. The criterion to be maximized is $det[oldsymbolΩ^-1+Mat M]$, with $Mat M$ the Fischer information matrix, calculated at a prior nominal value $hat{oldsymbolθ}⁰$ of the model parameters, and $oldsymbolΩ$ a given positive-definite matrix. When $θ$ is a scalar parameter, the problem corresponds to the maximization of the sum of $n$ i.i.d. random variables $x_k=[z(hat{θ}⁰, ξ_k)]²$, with $z(θ, ξ)=∂η(θ,ξ)/∂θ$, which are observed sequentially in a sequence of length $N$. The optimal decision rule, which is solution of a stochastic dynamic-programming problem, is compared to the simpler (but suboptimal) open-loop feedback-optimal rule. The situation is far more complex when $oldsymbolθ$ is $p$-dimensional, because the information is not additive. However, the computation of the expected value of the determinant of the information matrix allows to construct the open-loop feedback-optimal rule. Forced certainty-equivalence is proposed for the case where the distribution of the $ξ_k's$ is unknown and estimated on-line. Certainty equivalence is also forced in the case where $θ$ can be estimated on-line: at the step $k$, the prior nominal value $hat{θ}⁰$ is replaced by an estimate $hat{oldsymbolθ}^k-1$. Illustrative examples are presented.

ISO 690:

Pronzato, L. 1999. Sequential selection of observations in randomly generated experiments. In Tatra Mountains Mathematical Publications, vol. 17, no.3, pp. 167-175. 1210-3195.

APA:

Pronzato, L. (1999). Sequential selection of observations in randomly generated experiments. Tatra Mountains Mathematical Publications, 17(3), 167-175. 1210-3195.