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