Esquível, M. L., N. Machado, NP Krasii, and P. Mota. "
On the Information Content of Some Stochastic Algorithms."
Recent Developments in Stochastic Methods and Applications. Eds. A. N. Shiryaev, K. E. Samouylov, and D. V. Kozyrev. Cham: Springer, 2021. 57-75.
AbstractWe formulate an optimization stochastic algorithm convergence theorem, of Solis and Wets type, and we show several instances of its application to concrete algorithms. In this convergence theorem the algorithm is a sequence of random variables and, in order to describe the increasing flow of information associated to this sequence we define a filtration – or flow of σ -algebras – on the probability space, depending on the sequence of random variables and on the function being optimized. We compare the flow of information of two convergent algorithms by comparing the associated filtrations by means of the Cotter distance of σ-algebras. The main result is that two convergent optimization algorithms have the same information content if both their limit minimization functions generate the full σ-algebra of the probability space.
Mota, Pedro, and Manuel L. Esquível. "
Pseudo Maximum Likelihood and Moments Estimators for Some Ergodic Diffusions."
Contributions to Statistics. Springer International Publishing, 2018. 335-343.
AbstractWhen (Xt)t≥0 is an ergodic process, the density function of Xt converges to some invariant density as t →∞. We will compute and study some asymptotic properties of pseudo moments estimators obtained from this invariant density, for a specific class of ergodic processes. In this class of processes we can find the Cox-Ingersoll & Ross or Dixit & Pindyck processes, among others. A comparative study of the proposed estimators with the usual estimators obtained from discrete approximations of the likelihood function will be carried out.