Гущин Александр Александрович

Asymptotic inference for a linear stochastic differential equation with time delay

1999 · ARTICLE · en

For the stochastic differential equation dX(t)=a X ( t ) + b X ( t - 1 )dt+dW(t),t≥0, the local asymptotic properties of the likelihood function are studied. They depend strongly on the true value of the parameter ϑ=(a,b) * . Eleven different cases are possible if ϑ runs through ℝ 2 . Let ϑ ^ T be the maximum likelihood estimator of ϑ based on (X(t), t≥T). Applications to the asymptotic behaviour of ϑ ^ T as T→∞ are given.

On convergence to exponential-type statistical models

1998 · CHAPTER · en

О существовании стационарного решения стохастического дифференциального уравнения с запаздыванием, управляемого процессом Леви

1998 · ARTICLE · ru

On taking limits under the compensator sign

1996 · CHAPTER · en

On an information-type inequality for the Hellinger process

1996 · CHAPTER · en

Let X be a semimartingale which is locally square integrable and admitting the canonical decompositions X=M+A and X=M ' +A ' with respect to measures P and P ' . Let γ be the density of A-A ' with respect to C=(〈M〉+〈M ' 〉) in the Lebesgue decomposition. Then there is a version h of the Hellinger process h(1/2;P,P ' ) such that (1-Δh) -2 ·h⪰(1/8)γ 2 ·C P- and P ' -a.s. This inequality is related with a generalization of the Cramér-Rao inequality to the case of filtered space. The author gives some applications to a continuous-time linear regression model as well as to a discrete-time autoregression model with martingale errors.

On some inequalities of Cramér–Rao type

1996 · CHAPTER · en

Let ℰ=(Ω,ℱ,𝐏 ϑ , ϑ∈Θ⊆ℝ 1 ) be a statistical experiment and let (X ϑ , ϑ∈Θ) be a family of random variables on (Ω,ℱ) such that 𝐄 ϑ X ϑ =0 for all ϑ. Formal differentiation under the integral sign leads to the relation 𝐄 ϑ (∂X ϑ /∂ϑ)=-𝐄 ϑ (X ϑ V ϑ ), where V ϑ is the score function, and thus to generalizations of the Cramér-Rao inequality. Our main goal is to give rigorous proofs for these relations under rather minimal regularity conditions and some additional assumptions on the structure of (X ϑ ). We also prove an asymptotic information-type inequality which can be considered as an asymptotic version of the above mentioned results.

Об асимптотической оптимальности оценок параметров при условии LAQ

1995 · ARTICLE · ru

В работе рассматривается вопрос об асимптотической оптимальности оценок параметров при условии локальной асимптотической квадратичности. Показано, что, если нормировать величину отклонения оценок от истинного значения специально выбранным случайным множителем, то так называемые асимптотически центрированные оценки асимптотически допустимы относительно квадратической функции потерь и имеют наименьшую дисперсию среди оценок с асимптотически постоянным сносом.

On efficiency bounds for estimating the offspring mean in a branching process

1995 · PREPRINT · en

On semiparametric estimation for functionals of stochastic processes

1995 · CHAPTER · en

A lower bound for the variation distance between distributions of stochastic processes

1995 · CHAPTER · en