On the parameter estimation problem of the q-Exponential distribution for reliability applications
This work involved the q-Exponential distribution, which can be used to model each of the three phases of the bathtub curve and is an alternative to the Weibull distribution. The q-Exponential has two parameters (𝑞 – shape; 𝜂 – scale) and it stems from the Tsallis’ non-extensive entropy. This model...
| Main Author: | NEGREIROS, Ana Claúdia Souza Vidal de |
|---|---|
| Other Authors: | LINS, Isis Didier |
| Format: | masterThesis |
| Language: | eng |
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Universidade Federal de Pernambuco
2019
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https://repositorio.ufpe.br/handle/123456789/30391 |
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ir-123456789-303912019-10-26T05:10:53Z On the parameter estimation problem of the q-Exponential distribution for reliability applications NEGREIROS, Ana Claúdia Souza Vidal de LINS, Isis Didier http://lattes.cnpq.br/3480755550348791 http://lattes.cnpq.br/5632602851077460 Engenharia de Produção Distribuição q-Exponencial Confiabilidade Verossimilhança monótona Método de Firth This work involved the q-Exponential distribution, which can be used to model each of the three phases of the bathtub curve and is an alternative to the Weibull distribution. The q-Exponential has two parameters (𝑞 – shape; 𝜂 – scale) and it stems from the Tsallis’ non-extensive entropy. This model does not have the limitation of a constant hazard rate like the Exponential one, thus allowing the modeling of either system improvement (1<𝑞<2) or degradation (𝑞<1). Besides, it has more flexibility regarding the decay of the Probability Density Function (PDF) curve and it can model very well data sets with extreme values (power law characteristic). This feature is interesting in the reliability context because many equipment can work for long time until the first failure. However, when data sets are related to the degradation phase of systems, the application of the q-Exponential distribution becomes difficult due to convergence problems in the estimation process via the maximum likelihood (ML) method. This difficulty is due to the monotone behavior of the q-Exponential log-likelihood function when 𝑞<1, which is generally known as “monotone likelihood problem”. Because of that, it is almost impossible to obtain good estimates for the parameters considering the original log-likelihood function. In this sense, this research applied the Firth’s penalization method to solve this problem. We also verified that one of the regularity conditions imposed by the ML method is not satisfied by the q-Exponential distribution. Then, with the objective of satisfying this condition, it was also proposed a variable change, which partially solved just the problems of this distribution. Nevertheless, the Firth’s method yielded satisfactory results even for small samples. Comparisons of the results were performed via Monte Carlo simulations for the original and penalized q-Exponential distribution. Additionally, bootstrap confidence intervals were constructed for the parameters and comparisons were made between the fit provided by the q-Exponential and Weibull distributions. Application examples involving failure data of complex equipment using the Firth’s penalization method are presented and discussed. The obtained results indicate that the penalized log-likelihood enables the use of the q-Exponential distribution in the modeling of data sets related to degrading systems. CNPq Este trabalho envolveu a distribuição q-Exponencial, a qual pode ser usada para modelar as três fases da curva da banheira e é uma alternativa para a distribuição Weibull. A distribuição q-Exponencial tem dois parâmetros (𝑞 – forma; 𝜂 – escala) e é oriunda da entropia não-extensiva de Tsallis. Este modelo não tem a limitação de uma taxa de falhas constante como a distribuição Exponencial, assim permite modelar tanto a fase de melhoramento (1<𝑞<2) quanto a de degradação (𝑞<1). Além disso, tem mais flexibilidade quanto ao decaimento da curva da Função Densidade de Probabilidade (FDP) e consegue modelar muito bem conjuntos de dados com grandes valores (característica de power law). Esta característica é interessante no contexto de confiabilidade porque muitos equipamentos podem trabalhar por muito tempo até que ocorra a primeira falha. No entanto, quando os conjuntos de dados estão relacionados à fase de degradação dos sistemas, a aplicação da distribuição q-Exponencial se torna difícil devido a problemas de convergência no processo de estimação pelo método da máxima verossimilhança (MV). Este problema acontece por causa de uma condição chamada de “verossimilhança monótona”. Por causa disso, é praticamente impossível obter estimativas plausíveis para os parâmetros através da função de verossimilhança original. Neste sentido, esta pesquisa aplicou o método de penalização de Firth para corrigir este problema. Também foi verificado que uma condição de regularidade imposta pelo método de MV não é satisfeita pela distribuição q-Exponencial. Então, com o objetivo de satisfazer também esta condição, uma mudança de variável foi proposta, a qual solucionou apenas parcialmente os problemas desta distribuição. Todavia, o método de Firth produziu resultados satisfatórios mesmo para amostras pequenas. Comparações dos resultados foram realizadas através de simulações Monte Carlo para as distribuições q-Exponencial original e penalizada. Além disso, intervalos de confiança bootstrap foram construídos para os parâmetros e comparações foram feitas entre o ajuste alcançado pelas distribuições q-Exponencial e Weibull. Aplicações envolvendo dados de falhas de equipamentos complexos usando o método de penalização de Firth são apresentadas e discutidas. Os resultados obtidos indicam que a log-verossimilhança penalizada permite o uso da distribuição q-Exponencial na modelagem de dados de falhas na fase de degradação dos sistemas. 2019-04-29T20:26:42Z 2019-04-29T20:26:42Z 2018-02-23 masterThesis https://repositorio.ufpe.br/handle/123456789/30391 eng openAccess Attribution-NonCommercial-NoDerivs 3.0 Brazil http://creativecommons.org/licenses/by-nc-nd/3.0/br/ application/pdf Universidade Federal de Pernambuco UFPE Brasil Programa de Pos Graduacao em Engenharia de Producao |
| institution |
REPOSITORIO UFPE |
| collection |
REPOSITORIO UFPE |
| language |
eng |
| topic |
Engenharia de Produção Distribuição q-Exponencial Confiabilidade Verossimilhança monótona Método de Firth |
| spellingShingle |
Engenharia de Produção Distribuição q-Exponencial Confiabilidade Verossimilhança monótona Método de Firth NEGREIROS, Ana Claúdia Souza Vidal de On the parameter estimation problem of the q-Exponential distribution for reliability applications |
| description |
This work involved the q-Exponential distribution, which can be used to model each of the three phases of the bathtub curve and is an alternative to the Weibull distribution. The q-Exponential has two parameters (𝑞 – shape; 𝜂 – scale) and it stems from the Tsallis’ non-extensive entropy. This model does not have the limitation of a constant hazard rate like the Exponential one, thus allowing the modeling of either system improvement (1<𝑞<2) or degradation (𝑞<1). Besides, it has more flexibility regarding the decay of the Probability Density Function (PDF) curve and it can model very well data sets with extreme values (power law characteristic). This feature is interesting in the reliability context because many equipment can work for long time until the first failure. However, when data sets are related to the degradation phase of systems, the application of the q-Exponential distribution becomes difficult due to convergence problems in the estimation process via the maximum likelihood (ML) method. This difficulty is due to the monotone behavior of the q-Exponential log-likelihood function when 𝑞<1, which is generally known as “monotone likelihood problem”. Because of that, it is almost impossible to obtain good estimates for the parameters considering the original log-likelihood function. In this sense, this research applied the Firth’s penalization method to solve this problem. We also verified that one of the regularity conditions imposed by the ML method is not satisfied by the q-Exponential distribution. Then, with the objective of satisfying this condition, it was also proposed a variable change, which partially solved just the problems of this distribution. Nevertheless, the Firth’s method yielded satisfactory results even for small samples. Comparisons of the results were performed via Monte Carlo simulations for the original and penalized q-Exponential distribution. Additionally, bootstrap confidence intervals were constructed for the parameters and comparisons were made between the fit provided by the q-Exponential and Weibull distributions. Application examples involving failure data of complex equipment using the Firth’s penalization method are presented and discussed. The obtained results indicate that the penalized log-likelihood enables the use of the q-Exponential distribution in the modeling of data sets related to degrading systems. |
| author2 |
LINS, Isis Didier |
| format |
masterThesis |
| author |
NEGREIROS, Ana Claúdia Souza Vidal de |
| author_sort |
NEGREIROS, Ana Claúdia Souza Vidal de |
| title |
On the parameter estimation problem of the q-Exponential distribution for reliability applications |
| title_short |
On the parameter estimation problem of the q-Exponential distribution for reliability applications |
| title_full |
On the parameter estimation problem of the q-Exponential distribution for reliability applications |
| title_fullStr |
On the parameter estimation problem of the q-Exponential distribution for reliability applications |
| title_full_unstemmed |
On the parameter estimation problem of the q-Exponential distribution for reliability applications |
| title_sort |
on the parameter estimation problem of the q-exponential distribution for reliability applications |
| publisher |
Universidade Federal de Pernambuco |
| publishDate |
2019 |
| url |
https://repositorio.ufpe.br/handle/123456789/30391 |
| _version_ |
1648654622719475712 |
| score |
13.657419 |