Nonparametric estimation of the antimode and the minimum of a density function
Abstract
The study of the estimation of the antimode and the minimum of a density function
has been neglected in the li terature, in spite of their useful applications. The main
objective of this thesis is to propose and study nonparametric estimators for these parameters.
Strong consistency and limiting distributions are derived. The estimators depend
on unknown smoothing parameters. Data-based choices of these smoothing parameters
are proposed, using the bootstrap and kernel density estimation techniques. A critical
review of data-driven bandwidth selection procedures for kernel density estimation is presented.
An extensive Monte Carlo study shows that the small sample behaviour of the
newly proposed estimators is very satisfactory. Finally, some applications to real data
are discussed. Die bestudering van die antimodus en die minimum van 'n digtheidsfunksie het nog
weinig aandag in die literatuur geniet. Die hoof doe! van hierdie proefskrif is die voorstel en
bestudering van nie-parametriese beramers vir hierdie parameters. Sterk konsekwentheid
word aangetoon en limietverdelings word afgelei. Die beramers is afhanklik van onbekende
gladstrykingsparameters. Keuses van hierdie gladstrykingsparameters, gegrond op die
data, word voorgestel, deur van skoenlus- en kerndigtheidsberamingstegnieke gebruik te
maak. 'n Kritiese oorsig van data-gebaseerde gladstrykingsprosedures vir kernberaming
word gegee. 'n Uitgebreide Monte Carlo-studie toon <lat die kleinsteekproefgedrag van die
nuwe voorgestelde beramers baie bevredigend is. Laastens word toepassings op werklike
data bespreek