Five and four-parameter lifetime distributions for bathtub-shaped failure rate using Perks mortality equation

https://doi.org/10.1016/j.ress.2016.03.014Get rights and content

Highlights

  • Two new distributions are proposed to model bathtub shaped hazard rate.

  • Derive the close-form PDF, CDF and feature of scalability and truncatability.

  • Perks4 is verified to be good to model common bathtub shapes through comparison.

  • Perks5 has the potential to model the stabilization of hazard rate at later life.

Abstract

Two lifetime distributions derived from Perks׳ mortality rate function, one with 4 parameters and the other with 5 parameters, for the modeling of bathtub-shaped failure rates are proposed in this paper. The Perks׳ mortality/failure rate functions have historically been used for human life modeling in life insurance industry. Although this distribution is no longer used in insurance industry, considering many nice and some unique features of this function, it is necessary to revisit it and introduce it to the reliability community. The parameters of the distributions can control the scale, shape, and location of the PDF. The 4-parameter distribution can be used to model the bathtub failure rate. This model is applied to three previously published groups of lifetime data. This study shows they fit very well. The 5-parameter version can potentially model constant hazard rates of the later life of some devices in addition to the good features of 4-parameter version. Both the 4 and 5-parameter versions have closed form PDF and CDF. The truncated distributions of both versions stay within the original distribution family with simple parameter transformation. This nice feature is normally considered to be only possessed by the simple exponential distribution

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 ofh(t)=A+BctKct+1+Dct

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]: A+Bct. 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 Bct. 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 μ, 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

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