Five and four-parameter lifetime distributions for bathtub-shaped failure rate using Perks mortality equation
Introduction
Many lifetime distributions have bathtub-shapes and many real-life data exhibit this property. Most of the distributions show hazard rates increase to infinity with time passing by or only has a fixed time window for lifetime study [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [37]. The assumption that hazard rates increase infinitely with time is not always true. Electric power industry experience indicates some high voltage transformers that survive past the average life seem to have extremely long lives and the failure rate is finally a constant value [24]. The accelerated death at later age is not obvious for these transformers [25]. This suggests that lifetimes of electric transformers at later age follow the same trend as human beings, who exhibit a stable death rate after certain very old ages [26], [27]. In addition, there is also evidence that some electric devices even show decreasing in hazard rates [28].
In [29], Perks proposed a number of functions in the general format of
to model the mortality rate of human being for actuary industry to calculate annuity and disability benefits. Perks pointed out that the parameter K in Eq. (1) can model the high infant mortality rate and D can model the level-off of morality rate at very old age. Perks׳ Eq. (1) is a modification of Makeham׳s Law. The nominator part of the Eq. (1) is the same as Makeham׳s Law [30]: . With A represents ׳background risk׳ and B models the initial risk at time 0. The Makeham׳s Law is a generalization of Gompertz׳s law [31], which is . The Perks mortality equation, which was published in 1932, was earlier than the publication of Weibull distribution in 1939 in [32]. These early mortality rate functions play a very important role historically. However, with the advent of computer age, actuary industry seldom uses these simplified models for practical application.
During our work for large electric transformer study, we find, by ignoring parameter D (let D=0), Perks equation can model bathtub-shaped failure rate very well with proper parameter choices. If we keep parameter D, Perks equation can also assume bathtub shapes, but the tail of failure rate is asymptotically approaching a constant value.
In this work, we will explore the application of Perks hazard rate function to the modeling of bathtub shaped failure rates. Some nice characteristics of Eq. (1) that were not exposed before are also discussed. Sections 2 and 2.3 Shapes of Perks5, 3 Characteristics of 4-parameter Perks hazard and distribution give the derivation of the formulas and shapes of the distributions. Section 4 discusses the parameter estimation of the Perks distribution. Section 5 gives three case studies and compares with other well-known bathtub distributions.
Section snippets
Characteristics of 5-parameter Perks hazard and distribution
This section introduces the statistical characteristics of the 5-parameter distribution derived from Perks hazard rate function.
Characteristics of 4-parameter Perks hazard and distribution
As a special case of 5-parameter Perks hazard and distribution with , the 4-parameter Perks allows hazard rate to increase to infinity and the formula for PDF, CDF and reliability function of Perks4 are simpler.
Parameter estimation
It would be preferable if a graphical method can be developed for estimation of the parameters. The purpose of many graphical methods is to get an initial rough estimation. More complicated analytical methods, such as maximum likelihood and moment method, will still be applied using the rough estimation as a starting point to search for the best-fit parameters precisely. Unfortunately we are unable to find a simple graphical method for the Perks5 and Perks4 distributions.
Fig. 2, Fig. 3, Fig. 4,
Case studies and comparison
Three published lifetime data sets are chosen to test the new bathtub hazard rate function. The first data set is a collection of lifetime data of 50 devices first published in [34]. It is also tested as example of bathtub lifetime data for bathtub shaped failure rate functions in [2], [3], [4], [5], [14], [16]. The second data set is a collection of lifetime data for 18 electronic devices first published as an example to test a bathtub shaped failure rate function using an additive Burr XII
Conclusion and discussion
The Perks equations can be used to model bathtub-shaped hazard rate. The equation can be derived into a close-form in terms of PDF and CDF, namely Perks4 and Perks5. Compared with other well-known bathtub-shaped hazard rate functions, both the Perks4 and Perk5 models possess a number of nice features such as scalability and truncatability.
As a special case of Perks5, the 4-parameter Perks distribution (Perks4) can model the common bathtub shapes with an accelerated death rate at later life. The
References (37)
- et al.
A new model for a lifetime distribution with bathtub shaped failure rate
Microelectron Reliab
(1992) A new model with bathtub shaped failure rate using an additive Burr XII distribution
Reliab Eng Syst Saf
(2000)- et al.
A modified Weibull extension with bathtub-shaped failure rate function
Reliab Eng Syst Saf
(2002) A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function
Stat Probab Lett
(2000)- et al.
A new modified Weibull distribution
Reliab Eng Syst Saf
(2013) - et al.
On the upper truncated Weibull distribution and its reliability implications
Reliab Eng Syst Saf
(2011) - et al.
Choosing an optimal model for failure data analysis by graphical approach
Reliab Eng Syst Saf
(2013) - et al.
A new generalized Weibull distribution generated by gamma random variables
J Egypt Math Soc
(2015) - et al.
A new class of distribution functions for lifetime data
Reliab Eng Syst Saf
(2014) - et al.
Modifications of the Weibull distribution: a review
Reliab Eng Syst Saf
(2014)
A modified Weibull distribution
IEEE Trans Reliab
Exponentiated Weibull family for analyzing bathtub failure-rate data
IEEE Trans Reliab
Statistical analysis of a Weibull extension model
Commun Stat Methods
Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function
Reliab Eng Syst Saf
Statistical inference about the shape parameter of the new two-parameter bathtub-shaped lifetime distribution
Qual Reliab Eng Int
Weibull models
A new modified Weibull distribution
Commun Am Ceram Soc
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