IJPAM: Volume 70, No. 6 (2011)

ESTIMATION OF A PARAMETER OF A KNOWN LAW
OF A PROBABILITY BY STOCHASTIC APPROXIMATIONS

Abdelkrim Bennar$^1$, Abdelhalim Bouamaine$^2$, Youssef Benghabrit$^3$
$^{1}$Faculté des Sciences Ben M'sik
Université Hassan II
Sidi Othmane, Casablanca, MOROCCO
$^2$Ecole Nationale Supérieure d'éléctricité et de Mécanique
Laboratoire Architecture des Systèmes
Université Hassan II
Ain Chok, Casablanca, MOROCCO
$^3$Ecole Nationale Supérieure des arts et Métiers
Université Moulay Ismail
Meknès, MAROCCO


Abstract. We consider a stochastic approximation process in a non-empty closed convex set $K$ of $\mathbb{R}^{k}$: $X_{n+1}=\Pi \left(X_{n}-A_{n}(X_1,X_2,...,X_n)\Psi_{n}(Y_n;X_n)\right) $, with for each $n, \ E[\Psi_n(Y_n;\theta_n)]=0$, and $\Pi $ is the projection operator on $K$.

We denote $T_{n}$ the sub-$\sigma $-algebra generated by the events before time $n$.

We prove two theorems of almost sure convergence for the process $(X_n)$ and we give two applications for estimation of a parameter of a known law of probability.

Received: March 6, 2011

AMS Subject Classification: 62L20

Key Words and Phrases: stochastic approximation, parameter estimation, Likehood maximum

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 70
Issue: 6