Asymptotic behavior of trajectories of extreme sets of random functions and stochastic estimation. (English. Ukrainian original) Zbl 0923.60058

Anisimov, V. V.

Theory Probab. Math. Stat. 55, 1-12 (1997); translation from Teor. Jmovirn. Mat. Stat. 55, 1-12 (1996).

The following problem is considered. Let \(F_n(q,t)\), \(q \in Q\), \(t \geq 0\), be a sequence of random functions, and let \(\{ q_n(t) \} = \arg \min_{q \in Q} F_n(q,t).\) For any \(t>0\) the set \(\{ q_n(t)\}\) is a random set. There are to find conditions under which \(\{ q_n(t)\}\) \(U\)-converges to \(q_0(t)\) as \(n \to \infty.\) Note, that the sequence \(\{ q_n(t)\}\) \(U\)-converges to \(q_0(t)\) if any sequence \({\widetilde q}_n(t)\) such that \(P\{ {\widetilde q}_n(t) \in \{ q_n(t)\}\} =1\) \(U\)-converges to \(q_0(t).\) It means that the sequence of measures generated by \({\widetilde q}_n(t)\) in the Skorokhod space \(D_{[0,T]}\) converges weakly to the measure generated by \(q_0(t).\) The author finds conditions under which \(\{ q_n(t)\}\) \(U\)-converges to \(q_0(t)\), where \(F_0(q,t)\) is a continuous random function. Moreover, the author finds conditions under which \(V_n( \{ q_n(t)\} -q_0(t))\) \(U\)-converges, where \(V_n\) is a sequence such that \(V_n \to \infty\) as \(n \to \infty.\) These results are used to find the statistical estimators of parameters. Let \(p(y,\theta_0,\alpha)\) be the density of random variables \(\gamma_k(\alpha)\) and let observations \(y_{nk} = \gamma_k(S_{n,k})\) and \(S_{n,k}, {k/n} \leq T\), are known, where \(S_{n,k}\) is a trajectory of some stochastic or determined system. Denote \(\theta_n(t) =\arg \max_{\theta \in \Theta} L_n(\theta,t),\) where \[ L_n(\theta,t) ={1 \over n} \sum_{k=0}^{[nt]} \ln p(y_{n_k},\theta,S_{n,k}) \] is the function of maximum likelihood. The author finds conditions under which \(\theta_n(t)\) \(U\)-converges to \(\theta_0\) for \(t \in [t_0,T]\), \(t_0 >0\).

Reviewer: Yu.V.Kozachenko (Kyïv)

MSC:

60G70	Extreme value theory; extremal stochastic processes
62F12	Asymptotic properties of parametric estimators
60F15	Strong limit theorems
62M09	Non-Markovian processes: estimation

Keywords:

Skorokhod space; measure; statistical estimation; function of maximum likelihood; extremal set