Inference for the Mode of the
Log-Logistic Hazard Function


Emílio Augusto COELHO-BARROS[2]

Jorge Alberto ACHCAR[3]

§    ABSTRACT: In many applications of lifetime data analysis, it is important to perform inferences about the mode of the hazard function in situations of lifetime data modeling with unimodal hazard functions. For lifetime distributions where the mode of the hazard function can be analytically calculated, its maximum likelihood estimator is easily obtained from the invariance properties of the maximum likelihood estimators. From the asymptotical normality of the maximum likelihood estimators, confidence intervals can be obtained. However, these results might not be very accurate for small sample sizes and/or large proportion of censored observations. Considering the log-logistic distribution for the lifetime data with shape parameter b>1, we present and compare the accuracy of asymptotical confidence intervals with two confidence intervals based on bootstrap simulation. The alternative methodology of confidence intervals for the mode of the log-logistic hazard function is illustrated in three numerical examples.

§    KEYWORDS: Survival analysis; log-logistic distribution; hazard function; Bootstrap confidence intervals.

[1] Departamento de Estatística, Universidade Estadual de Maringá -- UEM, CEP: 87020-900, Maringá, PR, Brasil, E-mail:

[2] Departamento de Medicina Social, Faculdade de Medicina de Ribeirão Preto, Universidade de São Paulo -- USP, CEP: 14049-900, Ribeirão Preto, SP, Brasil, E-mail:

[3] Departamento de Estatística, Universidade Federal de São Carlos – UFSCar, CEP: 13565-905, São Carlos, SP, Brasil, E-mail: