A Reliability Distribution With Increasing, Decreasing, Constant and Bathtub-Shaped Failure Rates

Abstract

A distribution with one scale and two shape parameters is studied. The distribution can describe increasing (I), decreasing (D), constant and bathtub-shaped (B) failure rates. This motivates the working name, IDB distribution. The IDB distribution is motivated by mixtures of a set of IFR distributions but can also be given a competing risk interpretation. For mixed distributions a more general result on the initial slope of the failure rate is given. Asymptotic results for the ML estimation of survival probabilities are given, and when possible compared with ML estimation based on the Weibull, Rayleigh and exponential distributions. Also, the asymptotic gain from classifying failures into two categories is illustrated. One application to real data is given.

KEY WORDS:

• Reliability distribution
• Failure rate
• Bathtub curve
• Mixed distribution
• Mixtures
• Competing risks
• Fisher's information matrix
• Maximum likelihood estimation
• Classified failures