Se dan estimaciones puntuales y por intervalo para la edad y el número inicial de individuos en procesos de nacimiento puro de intensidad conocida y en procesos de Galton-Watson con distribución de descencientes conocida.
Se proponen estimadores no paramétricos de la edad y de la probabilidad de extinción de un proceso de ramificación de Galton-Watson. Dichos estimadores son comparados por simulación de Monte-Carlo, con otros estimadores propuestos por Stigler (1970) y Grump and Howe (1972).
The joint estimation of both drift and diffusion coefficient parameters is treated under the situation where the data are discretely observed from an ergodic diffusion process and where the statistical model may or may not include the true diffusion process. We consider the minimum contrast estimator, which is equivalent to the maximum likelihood type estimator, obtained from the contrast function based on a locally Gaussian approximation of the transition density. The asymptotic normality of the...
The joint estimation of both drift and diffusion coefficient parameters is treated under the situation where the data are discretely observed from an ergodic diffusion process and where the statistical model may or may not include the true diffusion process. We consider the minimum contrast estimator, which is equivalent to the maximum likelihood type estimator, obtained from the contrast function based on a locally Gaussian approximation of the transition density. The asymptotic normality of...
Let {bH(t), t∈ℝ} be the fractional brownian motion with parameter 0<H<1. When 1/2<H, we consider diffusion equations of the type X(t)=c+∫0tσ(X(u)) dbH(u)+∫0tμ(X(u)) du. In different particular models where σ(x)=σ or σ(x)=σ x and μ(x)=μ or μ(x)=μ x, we propose a central limit theorem for estimators of H and of σ based on regression methods. Then we give tests of the hypothesis on σ for these models. We also consider functional estimation on σ(⋅)...
A histogram sieve estimator of the drift function in Ito processes and some semimartingales is constructed. It is proved that the estimator is pointwise and L¹ consistent and its finite-dimensional distributions are asymptotically normal. Our approach extends the results of Leśkow and Różański (1989a).
We present two data-driven procedures to estimate the transition density of an homogeneous Markov chain. The first yields a piecewise constant estimator on a suitable random partition. By using an Hellinger-type loss, we establish non-asymptotic risk bounds for our estimator when the square root of the transition density belongs to possibly inhomogeneous Besov spaces with possibly small regularity index. Some simulations are also provided. The second procedure is of theoretical interest and leads...
A seminal paper by Rissanen, published in 1983, introduced the class of Variable Length Markov Chains and the algorithm Context which estimates the probabilistic tree generating the chain. Even if the subject was recently considered in several papers, the central question of the rate of convergence of the algorithm remained open. This is the question we address here. We provide an exponential upper bound for the probability of incorrect estimation of the probabilistic tree, as a function...