# Compounding a class of Rayleigh distributions : an objective Bayesian approach

Van Rooyen, Renier (2015-12)

Thesis (MSc)--Stellenbosch University, 2015.

Thesis

ENGLISH ABSTRACT: In this work, Bayesian estimation in the context of parametric survival analysis is con- sidered. A class of models derived by compounding and generalising the Rayleigh dis- tribution is regarded. These models are well suited to survival analysis settings where the hazard rate is characterised by a sharp increase over time. An objective Bayesian approach is followed, whereby non-informative prior distribution selection leads to the use of the Je reys, the reference and the probability matching priors. Bayesian point estimators are derived using two symmetric loss functions, namely absolute error and squared error, as well as two asymmetric loss functions, namely linear exponential and general entropy. The resulting models and estimators are showcased in a simulation study by generating right censored lifetime data from the various compound models and utilising the Metropolis-Hastings algorithm to draw realisations from the corresponding posterior distributions, since closed-form expressions for these cannot be found. Obtain- ing the Fisher information plays a crucial part in deriving the non-informative priors. In cases where it cannot be analytically evaluated, an adaptive quadrature routine is used for the numerical approximation of some of the elements in the Fisher information. An application to data sets from practice concludes the exposition of the compound Rayleigh models of interest.

AFRIKAANSE OPSOMMING: In hierdie tesis, word Bayes-beraming beskou in die konteks van parametriese oorle- wingsanalise. 'n Klas modelle wat afgelei is deur samestelling en veralgemening van die Rayleigh-verdeling, word beskou. Hierdie modelle is toepaslik in oorlewingsanalise- scenarios waar die gevaarfunksie beskryf word deur 'n skerp toename oor tyd. 'n Ob- jektiewe Bayes-benadering word gevolg en die toepaslike keuse van nie-inligtinggewende prior-verdelings lei na die gebruik van die Je reys-, die verwysings- en die waarskyn- likheidspassende priors. Bayes puntberamers word afgelei met inagneming van twee simmetriese verliesfunksies, naamlik absolute fout en kwadratiese fout, sowel as twee asimmetriese verliesfunksies, naamlik line^er eksponensieel en algemene entropie. Die gevolglike modelle en beramers word ten toon gestel in 'n simulasiestudie deur regs- gesensoreerde leeftyd-data te genereer vanuit die verskeie saamgestelde modelle en dan die Metropolis-Hastings algoritme te gebruik om realiserings vanuit die ooreenstem- mende posterior-verdelings te verkry, aangesien oplossings vir hierdie funksies nie in geslote vorm gevind kan word nie. Die bepaling van die Fisher-inligting speel `n kar- dinale rol in die a eiding van die nie-inligtinggewende priors. In gevalle waar dit nie analities evalueer kan word nie, word 'n aanpassende kwadratuurroetine gebruik vir die numeriese benaderings van sommige elemente in die Fisher-inligting. Laastens word die uiteensetting van die saamgestelde Rayleigh modelle afgesluit deur die toepassing op twee datastelle uit die praktyk.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/97767

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