Purpose -The purpose of this paper is to analyze mathematical aspects of the q-Weibull model and explore the influence of the parameter q. Design/methodology/approach -The paper uses analytical developments with graph illustrations and an application to a practical example. Findings -The q-Weibull distribution function is able to reproduce the bathtub shape curve for the failure rate function with a single set of parameters. Moments of the distribution are also presented. Practical implications -The generalized q-Weibull distribution unifies various possible descriptions for the failure rate function: monotonically decreasing, monotonically increasing, unimodal and U-shaped (bathtub) curves. It recovers the usual Weibull distribution as a particular case. It represents a unification of models usually found in reliability analysis. Q-Weibull model has its inspiration in nonextensive statistics, used to describe complex systems with long-range interactions and/or long-term memory. This theoretical background may help the understanding of the underlying mechanisms for failure events in engineering problems. Originality/value -Q-Weibull model has already been introduced in the literature, but it was not realized that it is able to reproduce a bathtub curve using a unique set of parameters. The paper brings a mapping of the parameters, showing the range of the parameters that should be used for each type of curve.

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This paper studies time durations between extreme returns with the aim of testing whether they follow power-law behaviour. Using the Hill estimator to identify extreme returns and estimate time durations, empirical evidence for intraday returns for the S&P 500, DAX and IBEX-35 stock market indexes indicates that the time durations between extreme events are well characterized by a qWeibull density with power-law behaviour tails. We also characterize the conditional time duration for an autoregressive conditional duration model with a q-Weibull distribution.