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Last edited on
Oct 18, 2021 by JJ
.
Thesis:
Bachelor in Mathematics

Author:
Maximilian Siebel

Title:
A Bernstein-von Mises Theorem for smooth functionals in semi-parametric models

Supervisor:
Jan JOHANNES

Abstract:
In mathematical statistics, Bernstein-von Mises Theorems are regarded as the link between frequentistic statistics and the Bayesian approach, since under special conditions asymptotically equivalent results can be obtained. This thesis first introduces the different approaches of frequentistic statistics and of Bayesian statistics and motivates the consideration of Bernstein-von Mises Theorems in semi-parametric models. A generalised Bernstein-von-Mises Theorem is proved under the condition that the considered functional of interest satisfies certain regularity conditions. After specific adaptions the generalised Theorem is applied to the White Noise Model and a corresponding Theorem is proved. Based on the theoretical foundations, the occurring effects of the Bernstein-von Mises Theorem are visualised in a slightly modified White Noise Model using a Monte-Carlo-Simulation and considering a linear functional of interest. The applicability of the Bernstein-von Mises Theorem is finally motivated in other statistical models, in particular in the Nonlinear Autoregressive Model and in the Density Model, respectively. Different mathematical principles used in this thesis are explained in detail at the beginning.
References:
Ismaël Castillo and Judith Rousseau. A Bernstein–von Mises theorem for smooth functionals in semiparametric models. The Annals of Statistics, 43(6):2353–2383, 2015.