Հայաստանի ատենախոսությունների բաց մատչելիության պահոց = Open Access Repository of the Armenian Electronic Theses and Dissertations (Armenian ETD-OA) = Репозиторий диссертаций Армении открытого доступа

Պատահական պրոցեսների վիճակագրության որոշ հարցեր

Գասպարյան, Սամվել Բագրատի (2015) Պատահական պրոցեսների վիճակագրության որոշ հարցեր. PhD thesis, ԵՊՀ.

[img]
Preview
PDF (Abstract)
Available under License Creative Commons Attribution.

Download (445Kb) | Preview

    Abstract

    Asymptotic efficiency in parametric models goes back to the beginning of the twentieth century and based on the works of Fisher. He proposed a program how to define asymptotic efficiency by comparing asymptotic error of an estimator in a point and proposed an estimator, which is called the maximum likelihood estimator, as an efficient estimator. Initial program of Fisher was not exactly true, since Hoges constructed an estimator which had a lower asymptotic error in a point than the asymptotic error of the maximum likelihood estimator. Such estimators were called super-efficient estimators. But, in realty these estimators were not better than the maximum likelihood estimator, since, for improving asymptotic behavior of an estimator in a point we damage the behavior of the estimator in neighbor points. To exclude such estimators Hajek proved a lower bound which compares behavior of estimators locally uniformly. Theorem was proved in the models where there was a LAM (local asymptotic normality) condition, which was introduced by Le Cam. Hence, the lower bound was called Hajek-Le Cam lower bound. For this definition the maximum likelihood estimator is asymptotically efficient for various models. Later Ibragimov and Khasminskii proved convergence of moments of the maximum likelihood estimator in the model of independent identically distributed random variables, hence they proved asymptotic efficiency of the maximum likelihood estimator for polynomial loss functions. Jeganathan generalized the notion of local asymptotic normality by defining local asymptotic mixed normality (LAMN) and for such models proved asymptotic lower bound for all possible estimators. Dohnal, for multidimensional case Gobet, proved that for a stochastic differential equation (SDE) with parameter in the diffusion coefficient, we have local asymptotic mixed normality property. Efficient estimators for this model were constructed by Genon-Catalot and Jacod. Same ideas of asymptotic efficiency transferred to the non-parametric estimation problems. First a such result was proved by Pinsker in the model of signal estimation in the presence of Gaussian white noise. In such problems the role of the inverse of the Fisher information in parametric statistics plays a constant, which is called the Pinsker constant. Later, such results where proved for other models too, particularly, for the intensity function of an inhomogeneous Poisson process the result was proved by Kutoyants. To compare asymptotic efficient estimators Golubev and Levit introduced the concept of second order efficiency for the model of independent identically distributed random variables and constructed an estimator which is asymptotically the best one among asymptotically efficient estimators, hence it is called second order asymptotically efficient estimator. Ատենախոսությունում դիտարկվում են հակադարձ ստոխաստիկ դիֆերենցիալ հավասարման լուծման մոտարկման խնդիրը, ինչպես նաև ոչ համասեռ պուասոնյան պատահական պրոցեսի միջին ֆունկցիայի համար երկրորդ կարգի էֆեկտիվության ապացուցման խնդիրը: В диссертации рассматриваются проблема аппроксимации решения обратного стохастического дифференциального уравнения и проблема оценивания второго порядка средней функции неоднородного пуассоновского процесса.

    Item Type: Thesis (PhD)
    Additional Information: Պատահական պրոցեսների վիճակագրության որոշ հարցեր:
    Uncontrolled Keywords: Գասպարյան Սամվել Բագրատի
    Subjects: Physics
    Divisions: UNSPECIFIED
    Depositing User: NLA Circ. Dpt.
    Date Deposited: 12 May 2017 15:49
    Last Modified: 18 May 2017 09:38
    URI: http://etd.asj-oa.am/id/eprint/4671

    Actions (login required)

    View Item