A NOVEL METHOD TO GENERATE A FAMILY OF BATHTUB-SHAPED FAILURE RATES FROM A FAMILY OF UPSIDE DOWN BATHTUB-SHAPED FAILURE RATES AND VICE-VERSA Текст научной статьи по специальности «Математика»

A NOVEL METHOD TO GENERATE A FAMILY OF BATHTUB-SHAPED FAILURE RATES FROM A FAMILY OF UPSIDE DOWN BATHTUB-SHAPED FAILURE RATES AND VICE-VERSA

R.L. Giri1, Subarna Bhattacharjee2*Suchandan Kayal3, S. K. Misra3

1,2 Department of Mathematics, Ravenshaw University, Cuttack-753003, Odisha, India 3 Department of Mathematics, National Institute of Technology Rourkela, Odisha, India 4 Department of Mathematics, KIIT University, Bhubaneswar, Odisha, India [email protected], [email protected], [email protected],

[email protected]

Abstract

It is indeed a matter of great significance for system engineers and scientists to derive new classes of lifetime distributions for providing a better statistical model which will fit a given lifetime data set. It is known that many real time data have varied characteristics and can be modeled by distributions with bathtub and upside down bathtub failure rates viz., Weibull, Modified Weibull, Inverse Weibull. This paper proposes a method which generates a family of distributions having bathtub (BT)-shaped failure rate from a distribution having upside down bathtub (UBT)-shaped failure rate and vice-versa. The proposed method is validated with the help of a few statistical distributions. The closure properties of the proposed model under various reliability operations are studied.

Keywords: Aging phenomenon, hazard rate, bathtub-shaped failure rate, upside down bathtub-shaped failure rate.

AMS 2020 Subject Classification: Primary 60E15, Secondary 62N05, 60E05

1. Introduction

Lifetime distributions are usually categorized based on their failure pattern. Given that a device has survived till time t > 0, the hazard (failure) rate provides instantaneous failure rate in a very small (future) time interval. The shape of the hazard rate function can be strictly decreasing, strictly increasing, constant, BT and UBT. Increasing failure rate often occurs in the real life situations, where devices are more likely to fail with respect to age. Decreasing hazard rate appears when materials become harder with respect to time. The concept of bathtub (resp. upside bathtub) hazard rate distribution is discussed in the literature based on whether the corresponding hazard rate is decreasing (resp. increasing) in the region (0, T0], constant in [T0, T1], and increasing (resp. decreasing) in [T1, <xi) where T0 and T1 are non-negative real numbers. In that case,

* Corresponding author : E-mail: [email protected]

the random variable X is said to be BT (resp. UBT). Here, Tq and Ti are considered as change points of the hazard rate function. This concept holds even if Tq = Ti. BT-shaped hazard rate is a combination of three different types of shapes, which usually appears in the study of life cycle of an industrial product or in the whole life span of a biological entity. Due to design error or installation problem, there is a high chance that a device has high likelihood of failure in first few weeks of operation. After initial period, the failure rate becomes relatively low, known as normal wear period. Then, the device reaches at the end of its life and the failure probability becomes very high with respect to time due to ageing. We refer to Rajarshi and Rajarshi [6] and Lai et al.[4] for some discussions on such kind of distributions. There are some other situations, related to the study of lifetime of a patient after major surgery, where models having unimodal hazard rate or having UBT-shaped hazard rate are useful. In biological science, it is observed in the course of a disease whose mortality reaches a peak after some finite period and then declines gradually. The commonly used distributions with UBT-shaped hazard rate are inverse Gaussian distribution, log-normal distribution, etc.

There are various transformations used by researchers to convert a baseline distribution into a new statistical distribution to get better flexibility. For example, inverted family of distributions can be obtained from a baseline distribution after using an inverse transformation. It has been shown by Keller et al.[3] that for pistons, crankshaft, main bearings failure data sets, the inverse Weibull distribution provides a better fit than the exponential and Weibull distributions. Akgwl et al. [1] explored that the wind speed data can be modelled by inverse Weibull distribution, which gives a better output than Weibull distribution. This paper aims to provide a new method for the generation of a family of BT-shaped failure rates from a family of UBT-shaped failure rates and vice-versa. It is well-known that a series system formed with independent components each having BT-shaped failure rate with different change points has a BT-shaped failure rate with an arbitrary change point. In this article, we propose a new transformation so as to have a common (specific) change point of the resulting BT-shaped failure rate. Some of the resulting mathematical avenues are also explored for reverse model. Let X be a non-negative absolutely continuous random variable with probability density function (PDF) f (.) and cumulative distribution function (CDF) F(.). Then, the hazard rate of X is denoted by r(t) = f (t)/F(t), where F(t) = 1 — F(t), t > 0. Throughout the paper, we assume that the derivative exists whenever, it is implemented.

The rest of the paper is arranged as follows. Section 2 provides a transformation/method for the generation of the BT-shaped failure rate distribution from UBT-shaped failure rate distribution. In addition, various properties of the resulting BT-shaped failure rate distribution are explored. Section 3 discusses a method of generating UBT-shaped failure rates from BT-shaped failure rate with some notable consequences. Finally, Section 4 concludes the paper.

2. A method to generate a BT-shaped failure rate using UBT-shaped

failure rate

Let U be a non-negative absolutely continuous lifetime random variable with CDF Fu(■) having UBT-shaped failure rate rU(t), for t € (lU, uU), where lU and uU denote respectively the lower and the upper bounds of the support of the random variable U. In this section, we introduce an interesting method to generate a distribution with corresponding lifetime random variable B with a BT-shaped failure rate rB (t), t € (lB, uB), where lB and uB denote respectively the lower and the upper bounds of the support of B using the distribution with UBT-shaped failure rate rU(t). Throughout the paper, we assume lU = 0 and uU =

Theorem 1. Denote by rB (t) and rU(t) the BT-shaped failure rate and UBT-shaped failure rate, respectively. Then, the UBT-shaped failure rate can be obtained using an equation given by

rB (t) = kM - rU(t), for t > 0, (2.1)

where k > 1 is a real number and M = max ru (t).

t>0

Proof. The proof is clear from the following discussion. Note that the graphs of rB (t) and rU(t) are geometrically equivalent because one is obtained from other by reflection about horizontal axis and then by vertical translation of kM units. For a given UBT-shaped failure rate function rU(t), -rU(t) represents its vertical reflected image (or reflection about t-axis) lying in fourth quadrant, which is eventually a BT-shaped failure rate function. To shift up and to drag -rU(t), for all t > 0, back to first quadrant, we give a positive (up) shift by kM units, k being greater than or equal to one, the minimum required factor being M, where M = ma0< ru(t). This completes the proof of the result. ■

Remark 1. Clearly, {rU(■),k} completely describes the aforementioned model which satisfies the hypothesis of Theorem 1. This notation will be used throughout the article wherever required. The parameter M is derivable from {rU(■),k}.

The next theorem provides the survival function FB(■) and the density function fB(■) corresponding to the newly generated distribution with BT-shaped failure rate, which is obtained from the UBT-shaped failure rate model by the method discussed in Theorem 1. The proof is omitted since it easily follows from Theorem 1 and the well-known relationship

- i rB(u)du

Fb (t) = e J0 . (2.2)

Theorem 2. The survival and density functions of the random variable B are respectively given

by FB(,) = ^SFnM1, > > 0 (2-3)

FU (t)

and

1

fB(t) = TY^(kM - ru(t)) exp(-kMt), t > 0. (2.4)

FU (t)

The method, discussed in Theorem 1 can be implemented to generate a family of BT-shaped failure rate models using a single UBT-shaped failure rate model as stated (without proof) in the next theorem.

Theorem 3. For i = 1,..., n, the random variables B/s have BT-shaped failure rates as given by

rBi (t) = kiM — ru(t), for t > 0, (2.5)

where ru(t) is the UBT-shaped failure rate, ki's are real constants satisfying ki > 1, and M =

max ru(t).

t>0

Proof. The proof follows using similar arguments as in the proof of Theorem 1, and thus it is omitted. ■

Next, we consider an example to illustrate the result in Theorem 3.

Example 2.1. Let a random variable U follow inverse Weibull distribution (see Jiang et al. [2]) with survival function Fu (t) = 1 — exp ( — K), t > 0, a > 0, $ > 0. This distribution has UBT-shaped failure rate for a < 1. Taking a = 0.5, $ = 2, the corresponding failure rate ru (t) = —x(t/t')-- can

t{ exp(t/t)a—1)

be shown to be uBT. Further, M = maxru(t) = 0.35536. Now, the corresponding rB. (t)'s are plotted in Figure 1(a), where rBi(t) = kiM — ru(t), with ki = i, for i = 1,... ,5.

Below, we compare two random variables Bi and Bj having UBT-shaped hazard rates in the sense of the hazard rate order. Let X and Y be two non-negative random variables with hazard rate functions rX(.) and rY(.), respectively. Then, X is said to be smaller than Y in the sense of the hazard rate order, denoted by X <hr Y, if rX(x) > rY(x), for all x > 0. For various other stochastic orders, we refer to Shaked and Shanthikumar [7]. From Theorem 3, we can write rBn (t) = knM — ru(t), for n = i, j.

Corollary 1. Let Bi and Bj be two random variables with rBn (t) = knM — ru(t), for n = i, j and t > 0. Then, Bi <hr Bj, if and only if ki > kj.

Proof. The proof is straightforward, and thus it is omitted. ■

2.1. Properties of the resulting BT-shaped failure models

In this subsection, we establish an interesting property of the resulting BT-shaped failure models. The following theorem shows that a series system formed by n number of independent components each having BT-shaped failure rate obtained from a common UBT-shaped failure rate model possesses BT-shaped failure rate model. In other words, a series system is closed under the specified BT transformation as given by (2.1) and (2.5).

Theorem 4. Consider a series system formed by n components with independent lifetimes denoted by Bi, i = 1,..., n. Further, let Bi have BT-shaped failure rate, say rBi (t) generated from a single component with UBT-shaped failure rate ru(t) satisfying rBi (t) = kiM — ru(t), i =

1,..., n, for all t > 0, ki > 1, and M = max ru(t). Then, the system has BT-shaped failure rate,

te(0,™)

denoted by rBs (t).

Proof. Note that

n n / \—-n k \

rBs (t) = E rBi (t) = L kiM — nru(t) = nMl ^nHl — nru(t).

i=1 i=1 v n '

(2.6)

yn k-

Further, max nru(t) = nM and yi=f ' > 1. Thus, it follows from (2.1) and (2.6) that rBs (t) rep-

te(0,w)

resents a BT-shaped failure rate, implies the system has BT-shaped failure rate. This completes the proof. ■

The next remark gives an interesting fact about Theorem 4, and may be noted for independent interest.

Remark 2. For i = 1,...,n, if rB.(t) = kiM - ru(t), then rB.(t) and ru(t) have the same change points as given by the roots of ddt(rBi(t)) = -dtt{rU(t)) = 0, for all t E [0, This leads to the fact that all of rBi(t), for i = 1,...,n have the same change point as a result of which rBs (t) = y i<i<nrBi (t) has a change point equal to that of rB. (t) (or rU (t)) since from (2.6) we find

dft^Bs (t» = -n^u (t» = (t)) =

and hence BS has a BT-shaped failure rate.

1.75 1.5 1.25 1

0.75 0.5 0.25

(a)

(b)

Figure 1: (a) Plots of ru (t) (the black curve) and rBi (t), for i = 1,..., 5, respectively from bottom to top as in Example 2.1. (b) Plot ofrx (t) versus t as in Counterexample 2.1

We now state a lemma with an outline of its proof, which will be used in proving upcoming theorem.

1

2

3

4

5

Lemma 1. Let S be a non-empty set and let gi(t) be defined on S such that maxtesgi(t) exists for each i = 1,..., n. Then, we have

n n

max Eg^(t) < E maxgi(t). (2-7)

teS i=1 i=1 teS Proof. We know that gi(t) < maxgi(t), for i = 1,..., n and t > 0. Thus,

nn

Egi(t) < E maxgi(t).

i=i i=i teS

nn

Further, since E maxgi(t) is an upper bound of E gi(t), it follows that

i=i teS i=i

nn

max Egi(t) < Emaxgi(t).

tes i=i i=ites

Thus, the proof is completed. ■

Now, we present an example in the light of the above lemma, where strict inequality holds. It is quite easy to construct examples where equality holds.

Example 2.2. Let g1, g2 : R ^ R be given by

{0,

. for t e Q , N 0, for t e Q ,

gi(t) = < ~ f , ™ and g2(t) = \ ' f , ™ (2.8)

for t e Qc 1, for t e Qc

Clearly, max gi (t) = 1, fori = 1,2 so that

te[0,+™)

2

E max ,gi(t) =2.

Furthermore, since (g1 + g2)(t) = 1, for all t e [0, we have

2

max , E gi(t) = 1

Thus,

22 max, E gi(t) < E max ,gi(t)

is established.

In the upcoming theorem, we will observe that even though rBi (t) possesses different change points yet

2

rBst (t) = E rBi (t)

i=1

is BT-shaped. We pause for a while and read the next remark before going to Theorem 5.

Remark 3. For i = 1,..., n, if rB. (t) = kiMi — rU. (t), then rB. (t) possesses different change points given by the roots of ddi(rBi (t)) = — ddt(rUi(t)) = 0, for all t e [0, provided that each rUi(t) is differentiable. This leads to the fact that all of rBi(t), for i = 1,...,n have different change points.

Theorem 5. If B$* is a random variable denoting the lifetime of a series system formed by n independent components with lifetimes Bi, for i = 1,..., n having BT-shaped failure rates rBi (t) generated from independent components with UBT-shaped failure rates rUi (t) satisfying Tbi(t) = kjMi - rUi(t), for all t > 0, kj > 1, and Mj = maxte(0,+m) rUi(t), then tbs* (t) yields a distribution having BT-shaped failure rate.

Proof. Note that

n n n n k-M-) n

TBS. (t) = E TB{(t) = E (kiMi) - E TUi(t) = M E ('-M) - E TUi(t), (2.9)

i=1 i=1 i=1 i=1 i=1

where M = maxfe(0,M) E"=1 rUi (t). Further, since each of ru. (t), for i e {1,..., n} is concave, En=i rUi (t) is also concave. Moreover, the local maximizer of a concave function defined over a convex set (here R) is the global maximizer. Thus, En=1 rUi (t) possesses a UBT-shaped failure rate with unique maximizer. So, it suffices to show that En=i kMr > 1 to establish our claim that rB (t) represents a BT failure rate as discussed in Theorem 1. Clearly,

nn

E (kiMi) > mm(ki) E Mi (2.10)

i=1 <"<"

Again, using Lemma 1, one can show that

i=1 1<i<n i=1

(E Mi = ) Et max , TU (t) > no ax E U (t). (2.11)

i=1 i=1 te(0,+~) te(0,+~) i=1

Thus, from (2.10) and (2.11), we conclude that

n n n

E (kiMi) > Imin (ki) E max Tu (t) > max E Tu{ (t) = (M)

i=1 1<i<n i=1 te(0,«>) te(0,<x>) i=1

as ki > 1, for all i = 1,..., n, that is En=i ~Mr > 1. This completes the proof. ■

Since this special type of construction allows the BT-shaped failure rate system to be closed under the formation of series system, a natural question that arises is whether this result can be generalized to the formation of k-out-of-n system. We recall that k-out-of-n system works if atleast k components of n number of components work. In the following counterexample, we notice that the answer of this question in negative. It shows that the BT-shaped failure rate system is not closed under the formation of parallel system.

Counterexample 2.1. Consider a parallel system with lifetime X comprised of two components having failure rates, rB. (t) = kiM - ru(t), t > 0 with ki = i + 1, for i = 1,2, and ru(t) = —P(x/t)—),

' f( exp(a/t)P-1j

a = 0.5, $ = 2, M = maxt>0 ru (t) = 0.35536. By Theorem 2, it follows that FB. (t) = exp-)Mt), for i = 1,2 so that FX(t) = 1 - (1 - FB1 (t))(1 - FB2(t)), for all t > 0. The plot of rX(t) for t > 0 given in Figure 1(b) shows that it is roller coaster.

3. A method to generate a UBT-shaped failure rate using BT-shaped

failure rate

Let B* be a continuous non-negative random variable with CDF Fb* (■) having BT-shaped failure rate rB* (t) for t e [0, On a similar line as discussed in the earlier section, we generate a distribution with corresponding random variable U* having UBT-shaped failure rate as given in the next theorem. The proof is omitted for the sake of conciseness.

Theorem 6. A distribution with UBT-shaped failure rate denoted by rU* (t) obtained from a distribution having BT-shaped failure rate rB* (t) is generated by the following equation

0 for 0 < t < ti

ru* (t)={ km — rB* (t) for ti < t < t2 (3.12)

0 for t > t2,

where t1 and t2 are the positive roots of km — rB* (t) = 0 with t1 < t2 and m = min rB* (t) and

fe[0,+~)

k is a real number satisfying k > 2.

The next corollary is useful to obtain the survival function Fu* (•), and the density function fu* (•) of the newly generated UBT-shaped failure rate model obtained from BT-shaped failure rate model by the approach as discussed in Theorem 6.

Corollary 2. With reference to the hypothesis as in Theorem 6, it is easy to note that

(i) the survival function of the random variable U* is

{1 for 0 < t < t1

exp(—km(t — t1 )) jBjjy for t1 < t < t2

exp(—km(t2 — t1 )) exp(— fh rB* (u)du) j^f) for t > tr,

(ii) the density function of the random variable U* can be obtained by simply differentiating —FU* (t) with respect to t.

The following proposition, which is useful to generate a family of UBT-shaped failure rate models using a single BT-shaped model, can be easily established from Theorem 3. The proof is omitted for the sake of brevity.

Proposition 3.1. A family of random variables Ui, for i = 1,..., n each with UBT-shaped failure rate, given by

{0 for 0 < t < t^

km — rB* (t) for 4° < t < t(i (3.13)

0 for t > t^),

(i) (i)

is generated from a random variable B* with BT-shaped failure rate rB* (t), where ty and t2' are the

positive roots of km — rB* (t) = 0, with t1!) < t^, m = min rB* (t) and ki is a real number satisfying

1 fe[0,+~)

ki > 2.

The following corollary presents condition, under which the hazard rate order between U* and U* exists. We omit the proof since it is a consequence of Proposition 3.1.

Corollary 3. We have U** >hr U*, if and only if ki < kj.

Let us use the notation Si = {t e R | rUf (t) > 0}. Clearly, it follows from (3.13) that Si = (tv t{2). Next, we state a strong result in the form of a lemma, which will be used later.

Lemma 2. If ki < kj, then Si C Sj, for any i, j e {1,..., n}.

rB versus t

0.04944

versus t

25

20

15

10

5

(a)

0.4

12 10 8 6 4 2

0.00717 0.1

(b)

0.00386 0.1

0.29105

versus t

35

30

25

20

15

10

5

0.0 D257

0.21376

0.1

0.34557

(c)

(d)

Figure 2: Plots of (a) tb* (t), (b) ru* (t), (c) ru* (t), and (d) ru* (t) versus t as in Example 3.1.

rD1 versus t

U

U

Proof. Since ki < kj, thus we have ru* (t) < ru* (t), for t > 0. We claim that S' C Sj for any i, j e {1,..., n}. If x e S' but x e Sj, then ru* (x) > 0 and ru* (x) = 0, i.e., ru* (x) > ru* (x), a contradiction. Hence the result follows. ■

Example 3.1. Let B* have failure rate, given by rB* (t) = a(At + b)eAttb-1, t > 0. Taking a = 2, b = 0.2, A = 5, it can be seen that B* has bathtub-shaped failure rate (as 0 < b < 1, (cf. Pham and Lai [5]). Here

m = min rB* (t) = rB* (tm) = 12.6948,

te(0,+

where tm = 0.0494427. We construct ru* (t) = kim - rB* (t), for t e [t(1), and = 0 otherwise, given that ki e {2,3,4}, for i = 1,2 and 3, respectively. Here, [t(1), t(^] = [0.00717441,0.21376], [t21), t(,2)] = [0.00386556,0.291059], and t32)] = [0.00257858,0.345577]. This example has been implemented and plotted in Figure 2 and Figure 3(a).

3.1. Properties of the new UBT-shaped failure models

In this subsection, we show that the nature of the failure rate of a series system constituted by n independent components each having UBT-shaped failure rate, obtained from a single bathtub-shaped failure rate distribution can be derived using the concept in Proposition 3.1. If US* is a random variable denoting the lifetime of a series system formed by n independent components with lifetimes U* for i = 1,..., n with corresponding UBT-shaped failure rate functions ru* (t), all generated from a component with BT-shaped failure rate rB* (t), then tUs* (t) can be derived as given in the following.

Let ru* (t) = kim - rB* (t), for t e Si and ru* (t) = 0 otherwise, where Si i e A = {1,...,n}. Let B = (ki,...,kn}. Let us define k*, for all i e A as

{t | ru (t) = 0},

ki = min ki, k* = min

1 ki eB ' kieB-(ki,kt2,...,k'j_1}

k, j e A -{1}.

Further, consider the roots 11p) and t^ of ru* (t), for p e A with t< t2p), where

r'u* (t) = kpm - rB* (t), pe A, t e Sp,

with Sp = (t I ru* (t) = 0}. Now, one can prove that r*u* (t) < r*u* (t) < ... < r*u* (t), for all t e R as k* < k*+1, for all j e A. Here, the two finite sets {ru*,..., ru*} and (ru*,..., ru *} are equal,

i.e., its elements are rearrangement of each other. Each r^* (t) is of UBT-shaped, for t e (t^, t^j)).

From lemma 2, we know that (tj,t^) Q (ti+1,4j+1)), for all j e A, as shown in Figure 3(b) giving

0

ru

for t e [0, t[n)]

,Wn(t) for t e [t1n), tt1)]

u(t)+ ru* 1 (t) for t e [t^, tt2)]

n n 1

ruS (t) =

TLn-jrh* (t) for t e [ttj), ttj-1)]

En=1 ru* (t) for t e [t11), t21)]

Tn=2 ru* (t)

TLn-jru** (t)

so that

ruS (t)

u (t) 0

mk n - rB (t)

for t e [t21), t22)]

for t e [t2n-j-1), tfj

for t e [t2n-1), t{n)] for t e [t2n), +~),

for t e [0, t{n)]

m(kn + k*n_ 1) - 2rB* (t)

for t e lt(n, t^] for t e [t^n-1), t(n-2)]

m Yin=n-j k* - (j + 1)rB* (t) for t e [t^, t(1t-j-1)]

m En=1 k* - nrB* (t) for t e [t^, t21)]

m En=2 k* - (n - 1)rB* (t) for t e [t21), t2]

m En=n-i k* - (j + 1)rB* (t) for t e [t2n-j-1), t^-j)]

mk*n - rB* (t)

for t e [t2n-1), t^] for t e [t(n,

0

Clearly, (j + 1)rB* (t) represents failure rate of a bathtub distribution with

( minn . 1) (j + 1)rB*(t) = (j + l)m

te[t[n-j) /1n-j-1)]

where m = minfe(0,+M) rB* (t). One can note that rUgt (t) represents a UBT-shaped failure model as in rus (t) = m ^n-jM ~ (j + 1)rB* (t), for t e [t1n-j), tin-j-1)] and ruSt (t) = m En=n-jki -(j + 1)rB* (t), for t e [ttj-1), t(2-j)]- We find that m En=n-j k* > 2(j + 1), ki being a real number satisfying ki > 2 for all i.

(a)

(b)

Figure 3: (a) Plot ofrB* (t) (blue color curve) and ru* (t) versus t, for i = 1,2,3 (bottom to top) as in Example 3.1. (b) Plot ofru* (t) for i = 1,..., n versus t.

4. Concluding remarks

In this paper, we have proposed a novel method which yields a family of distributions with BT shaped failure rate model from a distribution having UBT-shaped failure rate and vice-versa. Few examples have been presented for the validation of the newly proposed method. In addition, the closure properties of the proposed model have been studied under various reliability operations.

Conflict of interest statement On behalf of all authors, the corresponding author states that there is no conflict of interest.

Acknowledgements

The corresponding author would like to thank Odisha State Higher Education Council for providing support to carry out the research project under OURIIP, Odisha, India (Grant No. 73/Seed Fund/2022/Mathematics).

References

[1] AkgUl, F. G., Senoglu, B. and Arslan, T. (2016). An alternative distribution to Weibull for modeling the wind speed data: Inverse Weibull distribution. Energy Conversion and Management, 114: 234—240.

[2] Jiang, R., Murthy, D. and Ji, P. (2001). Models involving two inverse Weibull distributions. Reliability Engineering and System Safety, 73(1): 73—81.

[3] Keller, A., Goblin, M. and Farnworth, N. (1985). Reliability analysis of commercial vehicle engines. Reliability Engineering, 10(1): 15-25.

[4] Lai, C., Xie, M. and Murthy, D. (2001). Ch. 3 Bathtub-shaped failure rate life distributions. Handbook of statistics, 20: 69-104.

[5] Pham, H. and Lai, C.-D. (2007). On recent generalizations of the Weibull distribution. IEEE transactions on reliability, 56(3): 454-458.

[6] Rajarshi, S. and Rajarshi, M. (1988). Bathtub distributions: A review. Communications in Statistics-Theory and Methods, 17(8): 2597-2621.

[7] Shaked, M. and Shanthikumar, J. G. Stochastic Orders, Springer, New York, 2007.