Reliability properties of generalized mixtures of Weibull distributions with a common shape parameter

Journal of Statistical Planning and Inference 141 (2011) 2600–2613

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Reliability properties of generalized mixtures of Weibull distributions with a common shape parameter M. Franco a,, J.M. Vivo b, N. Balakrishnan c a b c

Department of Statistics and Operations Research, University of Murcia, 30100 Murcia, Spain Department of Quantitative Methods for Economy, University of Murcia, 30100 Murcia, Spain Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1

a r t i c l e in f o

abstract

Article history: Received 12 November 2010 Received in revised form 2 February 2011 Accepted 8 February 2011 Available online 16 February 2011

Weibull mixtures have been considered in many applied problems, and they have also been generalized by allowing negative mixing weights. In this paper, we study the classification of the aging properties of generalized mixtures of two or three Weibull distributions in terms of the mixing weights, scale parameters and a common shape parameter, which extends the cases of exponential or Rayleigh distributions. We apply these general results to classify the aging properties of the minimum and maximum lifetimes of two-component systems whose component lifetimes follow one of the known bivariate Weibull distributions. & 2011 Elsevier B.V. All rights reserved.

Keywords: Generalized mixtures Bivariate Weibull distributions Failure rate Log-concavity Series systems Parallel systems Aging properties

1. Introduction The Weibull distribution plays a crucial role in reliability theory and life-testing experiments. Weibull mixtures are widely used to model lifetime and failure time data, since they exhibit a wide range of shapes for the failure rate function. For example, see Sinha (1987), Patra and Dey (1999), Kundu and Basu (2000), Tsionas (2002), Carta and Ramı´rez (2007), Mosler and Scheicher (2008), Farcomeni and Nardi (2010), Mosler and Haferkamp (2010) and Ebden et al. (2010), and some of them assume Weibull components with a common shape parameter as it is natural to assume the components in a system to have the same distribution type but with possibly different scale parameters. These mixture forms have been further generalized by allowing negative mixing weights, which arise under the formation of some structures of systems such as the series and parallel systems formed by Weibull components with the same shape parameter; see, for example, Jiang et al. (1999). In the literature, there are different bivariate extensions of the Weibull distribution. Most of them have been constructed as extensions of bivariate exponential models; see Kotz et al. (2000), Murthy et al. (2004), Balakrishnan and Lai (2009) and Hanagal (2009), and the references therein. When marginals have a common shape parameter, the series and parallel systems formed with components having such bivariate Weibull distributions turn out to be predominantly generalized mixtures of two or three Weibull distributions with the same shape parameter. Analytical and graphical methods have been used to approximate Weibull mixture models. For instance, moments, maximum likelihood, least-squares and WPP plot methods, and different numerical algorithms have been used to examine  Corresponding author.

E-mail addresses: [email protected] (M. Franco), [email protected] (J.M. Vivo), [email protected] (N. Balakrishnan). 0378-3758/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2011.02.012

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the probabilistic behavior as well as to develop inferential methods based on quasi-Newton, Newton–Raphson, conjugate gradient and EM algorithms. A brief review of these estimation methods and their applications to Weibull mixtures can be found in Bucˇar et al. (2004) and Carta and Ramı´rez (2007). Some of them may also be used to fit Weibull mixture models with negative mixing weights, which yields a more flexible model; for example, negative estimated weights of generalized mixtures of Weibull components with a common shape parameter have been obtained by Jiang et al. (1999), Bucˇar et al. (2004), and Franco and Vivo (2009a). Several authors have studied generalized mixture distributions and some distributional and structural properties of these generalized mixtures. For instance, conditions in terms of the mixing weights and the parameters to guarantee that generalized mixtures of exponential components are valid models have been analyzed by Steutel (1967), Bartholomew (1969), Baggs and Nagaraja (1996), and Franco and Vivo (2006). Franco and Vivo (2007a) further studied generalized mixtures of gamma and one or two exponential distributions. Recently, Franco and Vivo (2009a) discussed constraints for generalized mixtures of Weibull distributions with a common shape parameter. It is known that log-concavity and log-convexity are useful aging properties in many contexts including some applications in economics, political and actuarial science, biology, reliability and engineering (see Bagnoli and Bergstrom, 2005; Kokonendji et al., 2008, among others), as it is often important to make explicit assumptions on the underlying distribution. Since it is reasonable to use generalized mixtures of Weibull as underlying distributions to model heterogeneous survival data and reliability of a system when its structure is unknown or partially known, it will naturally be of interest to study the aging properties of such generalized mixtures of Weibull distributions. The aging classifications of generalized mixtures of exponential distributions have been discussed earlier by Baggs and Nagaraja (1996) and Franco and Vivo (2002, 2006, 2007b), while Franco and Vivo (2007a, 2009b) have examined generalized mixtures of gamma and exponential components in a similar vein. The main aim of this paper is to classify the aging properties of generalized mixtures of two or three Weibull distributions with a common shape parameter based on the mixing weights and scale parameters. This work can also be motivated through the reliability properties of minimum and maximum lifetimes of two-component systems whose components follow one of the known bivariate Weibull distributions. The rest of this paper is organized as follows. Section 2 contains some concepts of aging and presents the generalized mixtures of Weibull distributions, which are studied in the sequel. In Section 3, we explore the aging classification of generalized mixtures of two Weibull distributions with a common shape parameter. The case of generalized mixtures of three Weibull distributions is handled in Section 4. Finally, in Section 5, we apply our results to classify the aging properties of the series and parallel systems formed by two independent or dependent components from some known bivariate Weibull distributions.

2. Preliminaries First, we describe some aging concepts and reliability properties that will be used in the rest of the paper. Let X be a non-negative random variable with survival or reliability function SðxÞ ¼ PðX 4xÞ. Then, the conditional survival function of a unit of age x is defined by SðtjxÞ ¼ Sðt þ xÞ=SðxÞ for all x,t Z 0, and it represents the survival probability over an additional period of duration t of a unit of age x. Note that the log-concavity of the survival function is determined by the monotonicity of its conditional survival function. It is said that X has a log-concave (log-convex) survival function if log S(x) is concave (convex), i.e., SðtjxÞ is decreasing (increasing) in x for all t Z0. In the absolutely continuous case, this log-concavity property can be determined by the monotonicity of its failure rate function defined by rðxÞ ¼ ðd=dxÞlogSðxÞ ¼ S0 ðxÞ=SðxÞ, which represents the probability of instantaneous failure or death at each moment. So, X having log-concave (log-convex) survival function also corresponds to increasing (decreasing) failure rate, i.e., IFR (DFR) class; see Barlow and Proschan (1981) and Shaked and Shanthikumar (2006). We will use the closure of the DFR class under mixtures given in Barlow and Proschan (1981). The following technical lemma will be also used to discuss the log-concavity of the survival function of a generalized mixture of Weibull distributions (see Franco and Vivo, 2007a). Lemma 2.1. Let X be an absolutely continuous random variable with survival function S(x). If S00 ðxÞSðxÞS0 ðxÞ2 r 0ð Z0Þ

ð2:1Þ

for all x 40, then X is IFR (DFR). We shall now state formally a generalized mixture of Weibull distributions with a common shape parameter c 4 0. Definition 2.2. Let (X1, X2,y,Xn) be a random vector with components having Weibull distributions with a common shape c parameter c 40 and survival functions Si ðxÞ ¼ ebi x for all x 4 0 and bi 4 0, i= 1,y,n. Then, X is said to have a generalized

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mixture of Weibull distributions with common shape parameter c 4 0 if its survival function can be expressed as SðxÞ ¼

n X

ai Si ðxÞ,

for x 4 0,

ð2:2Þ

i¼1

where ai 2 R, i =1,y,n, are such that

Pn

i¼1

ai ¼ 1.

Note that Franco and Vivo (2009a) discussed constraints based on the mixing weights and parameters of the components under which a generalized mixture of Weibull distributions is a valid probability model, i.e., constraints on ai’s so that 0 rSðxÞ r 1 and S(x) is decreasing. Evidently, this generalized mixture has its distribution and density functions as FðxÞ ¼ 1

n X

c

ai ebi x

and

f ðxÞ ¼ cxc1

i¼1

n X

c

ai bi ebi x ,

for x 40:

i¼1

We shall now assume that the scale parameters of the n Weibull distributions are different and are ordered, i.e., 0 ob1 o b2 o    obn . The following two simple examples will illustrate the problems that will arise with negative mixing weights. In this setup, let (X1,X2) denote a generic random vector whose components have survival functions Si(x), i= 1,2, and Sðx1 ,x2 Þ ¼ PðX1 4x1 ,X2 4x2 Þ represents its joint survival function. Let T1 =min(X1, X2) and T2 = max(X1, X2) be its extreme order statistics, i.e., minimum and maximum, whose survival functions are given by Sð1Þ ðxÞ ¼ PðX1 4 x,X2 4xÞ ¼ Sðx,xÞ

ð2:3Þ

Sð2Þ ðxÞ ¼ S1 ðxÞ þS2 ðxÞSð1Þ ðxÞ:

ð2:4Þ

and

These expressions are useful while working with the distribution functions of minimum and maximum order statistics arising from bivariate distributions; see Baggs and Nagaraja (1996) and Franco and Vivo (2002). c

Example 2.1. If (X1,X2) has independent and identically distributed Weibull components, i.e., Si ðxÞ ¼ ebx , i= 1,2, then T1 has a Weibull distribution as in (2.3), while from (2.4), we observe that T2 is a generalized mixture of two Weibull c c distributions with survival function Sð2Þ ðxÞ ¼ 2ebx e2bx , i.e., a1 = 2 and negative weight a2 = 1. Example 2.2. If (X1,X2) has independent Weibull components with the same shape parameter, and not necessarily identically distributed, then T1 follows a Weibull distribution as in (2.3), while from (2.4), we observe that T2 is a c c c generalized mixture of three Weibull distributions with survival function Sð2Þ ðxÞ ¼ eb1 x þ eb2 x eðb1 þ b2 Þx , i.e., a1 = a2 =1 and negative weight a3 =  1. 3. Classification of generalized mixtures of two Weibull distributions In this section, we study the classification in the IFR and DFR classes of the generalized mixtures of two Weibull distributions based on the mixing weights and the scale parameters. c

c

Theorem 3.1. Let X be a generalized mixture of two Weibull distributions with survival function SðxÞ ¼ a1 eb1 x þ a2 eb2 x ðx 40Þ and scale parameters b1 o b2 and common shape parameter c 4 0. (i) If a2 40, then X is DFR iff c r1. Moreover, X is IFR iff c 41 and one of the following conditions holds: (a) c Z ða21 b1 a22 b2 Þ=ða1 b1 a2 b2 Þ and a2 o a1 b1 =b2 ; (b) w Z 0 when a2 4 a1 b1 =b2 or 1 o c oða21 b1 a22 b2 Þ=ða1 b1 a2 b2 Þ. (ii) If a2 o0, then X is IFR iff c Z1. Moreover, X is DFR iff c o1 and one of the following conditions holds: (a) c r ða21 b1 a22 b2 Þ=ða1 b1 a2 b2 Þ; (b) w Z 0. Here,

with

a1 a2 ðb2 b1 Þc tlogt c1 0 1 "  2 #1=2 ca2 ðb2 b1 Þ @ c1 A 4 1: 1 þ 1 þ 4b1 b2 t¼ 2a1 b1 ðc1Þ cðb2 b1 Þ

w ¼ a22 b2 þa21 b1 t2 þ a1 a2 ðb1 þb2 Þt

ð3:5Þ

ð3:6Þ

Proof. From the survival function of a generalized mixture (X) of two Weibull distributions and Lemma 2.1, the classification of X depends on the sign of 00

c

S ðxÞSðxÞS0 ðxÞ2 ¼ cxc2 eðb1 þ b2 Þx gðxÞ,

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where c

c

gðxÞ ¼ ðc1Þa21 b1 eðb2 b1 Þx þ ðc1Þa22 b2 eðb2 b1 Þx þðc1Þa1 a2 ðb1 þ b2 Þcxc a1 a2 ðb2 b1 Þ2 which depends on the weights ai’s, the scale parameters bi’s and the common shape parameter c. So, X has IFR (DFR) survival function if and only if g(x) is non-negative (non-positive). Otherwise, S(x) changes its log-concavity when g(x) changes its sign. Likewise, taking into account the necessary conditions presented in Theorem 2 of Franco and Vivo (2009a) for the above given generalized mixture to be a valid probability model, i.e., either a2 4 0 or fa2 o 0 and a1 b1 þa2 b2 Z0g, we see readily that the log-concavity of the survival function makes sense only when a2 4 0 or 0 4a2 Za1 b1 =b2 . So, we discuss the sign of g(x) in both these regions. Furthermore, we will consider the cases 0 o c o1 and c 4 1, since when c =1, g(x) reduces to gðxÞ ¼ xa1 a2 ðb2 b1 Þ2 and consequently S00 ðxÞSðxÞS0 ðxÞ2 ¼ a1 a2 ðb2 b1 Þ2 eðb1 þ b2 Þx is positive or negative according to the sign of a2, i.e., X is DFR if a2 40 and IFR if a2 o0. Note that the signs of g(0) and limx-1 gðxÞ depend on the value of c, where gð0Þ ¼ ðc1Þða21 b1 þa22 b2 þ a1 a2 ðb1 þb2 ÞÞ ¼ ðc1Þða1 b1 þ a2 b2 Þ: To prove Part (i), all ai’s are positive, and so X is a mixture of Weibull distributions. Moreover, a Weibull distribution with c c1 survival function SW ðxÞ ¼ ebx is log-concave if c Z1 and log-convex if c r 1, since its hazard rate is given by rW ðxÞ ¼ cbx . Therefore, by the closure property of DFR under mixtures (see Barlow and Proschan, 1981), we have that if c r1 then X is DFR, but it cannot be IFR because limx-1 gðxÞ ¼ 1. Nevertheless, if c 4 1, then S(x) cannot be DFR because gð0Þ 40 and limx-1 gðxÞ ¼ 1. Now, let us see under what conditions the sign of g(x) is positive. Its first derivative is given by g 0 ðxÞ ¼ cðb2 b1 Þxc1 hðxÞ, where c

c

hðxÞ ¼ ðc1Þa21 b1 eðb2 b1 Þx ðc1Þa22 b2 eðb2 b1 Þx ca1 a2 ðb2 b1 Þ and both functions g 0 ðxÞ and h(x) have the same sign. Moreover, h(x) is increasing because c 41 and b1 o b2 , and. therefore, hðxÞ Z hð0Þ ¼ ðc1Þða21 b1 a22 b2 Þca1 a2 ðb2 b1 Þ ¼ cða1 b1 a2 b2 Þða21 b1 a22 b2 Þ; upon taking into account the location of a2 with respect to the value a1b1/ b2, we obtain that hð0Þ Z 0 if and only if cZ

a21 b1 a22 b2 a1 a2 ðb2 b1 Þ ¼ 1þ 41 a1 b1 a2 b2 a1 b1 a2 b2

when a2 o a1 b1 =b2 , but hð0Þ cannot be positive when a2 4 a1 b1 =b2 since c 41. In the first case, when a2 o a1 b1 =b2 , the function h(x) is always positive, or equivalently, g 0 ðxÞ Z 0, i.e., g(x) is increasing, and then gðxÞ Zgð0Þ 4 0, and so the survival function of X is IFR. In the second case, when 1 oc o ða21 b1 a22 b2 Þ=ða1 b1 a2 b2 Þ or a2 4 a1 b1 =b2 , the function h(x) changes its sign from being negative to positive, i.e., g(x) decreases and then increases, i.e., g(x) attains its minimum x0 at the unique solution of h(x0)= 0 for x0 40. Moreover, upon setting x0 ¼ ðð1=ðb2 b1 ÞÞlogtÞ1=c , where t 4 1 is as given in (3.6), gðx0 Þ Z 0 is equivalent to the condition w Z0, where w is as given in (3.5). Thus, if w Z0, then g(x) is positive, and so X is IFR. However, when w o 0, then g(x) changes its sign from being positive first, then negative, and ultimately positive, and so X cannot be IFR. In order to prove Part (ii), we consider a2 2 ½a1 b1 =b2 ,0Þ since a1 b1 þa2 b2 Z 0 is the necessary condition to be a valid pdf. If c Z 1, we can see that gð0Þ Z0 and limx-1 gðxÞ ¼ 1, and so it follows that X cannot be DFR. Moreover, the first derivatives of g(x) and h(x) have the same sign, and h(x) is increasing as in the above case. Also, by taking into account that a2 b2 4 0 and a2 Za1 b1 =b2 , it is readily seen that a21 b1 a22 b2 Z a21 b1 þ a1 a2 b1 ¼ a1 b1 40 and so hð0Þ Z0. Thus, hðxÞ Z 0 and so g(x) is increasing, with gð0Þ Z0, i.e., X is IFR. Finally, when 0 oc o 1, X cannot be IFR since gð0Þ r 0 and limx-1 gðxÞ ¼ 1. In this case, the sign of g 0 ðxÞ is also the same as that of h(x), but h(x) decreases and consequently, hðxÞ r hð0Þ. Now, hð0Þ r 0 if and only if cr

a21 b1 a22 b2 a1 a2 ðb2 b1 Þ ¼ 1þ o1 a1 b1 a2 b2 a1 b1 a2 b2

since a2 o0. Hence, h(x) is always negative, or equivalently, g(x) is decreasing and so gðxÞ rgð0Þ r 0. Therefore, X is DFR. In the case when hð0Þ 40, h(x) changes its sign from positive to negative, i.e., g(x) increases and then decreases, and so g(x) attains its maximum at x0 which is the unique solution of h(x0)= 0 for x0 4 0. Moreover, gðx0 Þ r 0 is equivalent to the

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condition w Z 0, where w is as given in (3.5). Thus, if w Z 0, then g(x) is negative and so X is DFR. However, if w o0, then g(x) changes its sign from positive to negative when g(0) = 0, and from being negative first, then positive, and ultimately negative when gð0Þ o 0, and so S(x) cannot be log-convex, i.e., X cannot be DFR. This completes the proof of the theorem. & Remark 3.2. When the shape parameter c =1, Theorem 3.1 presents the classification of generalized mixtures of two exponential distributions, as given in Theorem 1 of Baggs and Nagaraja (1996) and in Corollary 2.4 of Franco and Vivo (2002). Also, when c= 2, Theorem 3.1 presents the classification of generalized mixtures of two Rayleigh distributions. Remark 3.3. From Theorem 3.1, a two-component mixture Weibull distribution with a common shape parameter can have neither bathtub shaped failure rate nor upside-down bathtub shaped failure rate function, denoted in short by U-shaped and -shaped, respectively. In detail, from the proof of Part (i) of Theorem 3.1, if w o 0 then X has N-shaped failure rate function, i.e., increasing initially, then decreasing, and ultimately increasing. Furthermore, from the proof of Part (ii) of Theorem 3.1, a generalized mixture of two Weibull distributions has -shaped failure rate iff c 2 ð1=2,1Þ and a2 ¼ b1 =ðb2 b1 Þ. Likewise, if wo 0 then X has -shaped failure rate function for a2 4 b1 =ðb2 b1 Þ, i.e., decreasing first, then increasing, and decreasing at the end.

4. Classification of generalized mixtures of three Weibull distributions In this section, we discuss the classification in the IFR and DFR classes of the generalized mixtures of three Weibull distributions with a common shape parameter based on the mixing weights and the scale parameters. For this purpose, we will use the following lemma, which displays the classification of the generalized mixtures of three exponential distributions as given in Theorem 3 of Baggs and Nagaraja (1996). Lemma 4.1. Let T be a generalized mixture of three exponential distributions with survival function ST ðtÞ ¼ a1 eb1 t þ a2 eb2 t þ a3 eb3 t for t 4 0, 0 o b1 ob2 ob3 o 1, where a1, a2 ,a3 2 R are such that a1 +a2 + a3 =1 and a1 40. Let s ¼ a1 a2 ðb2 b1 Þ2 þ a1 a3 ðb3 b1 Þ2 þ a2 a3 ðb3 b2 Þ2 . (i) (ii) (iii) (iv)

If the ai’s are all positive, ST(t) is log-convex; If a2 4 0 and a3 o 0, ST(t) is log-convex iff s Z0, and it cannot be log-concave; If a2 o0 and a3 o 0, ST(t) is log-concave iff s r 0, and it cannot be log-convex; If a2 o0 and a3 4 0, ST(t) cannot be log-convex. It is log-concave iff either sr 0 and t ¼ ð1=ðb3 b1 ÞÞ loga3 ðb3 b2 Þ=a2 ðb2 b1 Þ r0, or a1 a2 ðb2 b1 Þ2 eðb1 þ b2 Þt þ a1 a3 ðb3 b1 Þ2 eðb1 þ b3 Þt þ a2 a3 ðb3 b2 Þ2 eðb2 þ b3 Þt r 0:

Theorem 4.2. Let X be a generalized mixture of three Weibull distributions with common shape parameter c 4 0, with survival function S(x) as given in (2.2) for 0 ob1 ob2 o b3 o1, a1 4 0, and a2 ,a3 2 R are such that a1 + a2 + a3 = 1. Let T be the corresponding exponential case (i.e., when c= 1) with survival function ST(t). (i) (ii) (iii) (iv)

If If If If

ST(t) ST(t) ST(t) ST(t)

is is is is

log-concave and c Z 1, then X is IFR; log-convex and c r1, then X is DFR; log-concave and c o 1, then X cannot be IFR. Moreover, X is DFR iff c rc0 o1; log-convex and c 41, then X cannot be DFR. Moreover, X is IFR iff c Z c0 4 1.

Here, c0 ¼

1þ

ða1 a2 ðb2 b1 Þ2 tb01 þ b2 þa1 a3 ðb3 b1 Þ2 tb01 þ b3 þ a2 a3 ðb3 b2 Þ2 tb02 þ b3 Þlogðt0 Þ

!1

2b2 2b3 b1 þ b2 1 2 3 a21 b1 t2b þ a1 a3 ðb1 þ b3 Þtb01 þ b3 þ a2 a3 ðb2 þb3 Þtb02 þ b3 0 þa2 b2 t0 þ a3 b3 t0 þ a1 a2 ðb1 þb2 Þt0

ð4:7Þ

and t0 2 ð0,1Þ is solution of the equation

t1 logðt0 Þt2 ¼ 0,

ð4:8Þ

where 1 þ b2 1 þ b3 2 þ b3 t1 ¼ a31 a2 b1 ðb2 b1 Þ2 t3b þ a31 a3 b1 ðb3 b1 Þ2 t3b þ a1 a32 b2 ðb2 b1 Þ2 tb01 þ 3b2 þ a32 a3 b2 ðb3 b2 Þ2 t3b 0 0 0 2 b1 þ 3b3 2 b2 þ 3b3 2 2b1 þ 2b2 3 3 2 2 2 2 1 þ 2b3 þ a1 a3 b3 ðb3 b1 Þ t0 þa2 a3 b3 ðb3 b2 Þ t0 þa1 a2 ðb1 þ b2 Þðb2 b1 Þ t0 þ a1 a3 ðb1 þ b3 Þðb3 b1 Þ2 t2b 0 2 þ 2b3 1 þ b2 þ b3 þ a22 a23 ðb2 þ b3 Þðb3 b2 Þ2 t2b þ a21 a2 a3 ðb1 ðb3 b2 Þ2 þðb1 þ b2 Þðb3 b1 Þ2 þ ðb1 þb3 Þðb2 b1 Þ2 Þt2b 0 0

þ a1 a22 a3 ðb2 ðb3 b1 Þ2 þ ðb1 þb2 Þðb3 b2 Þ2 þðb2 þ b3 Þðb2 b1 Þ2 Þtb1 þ 2b2 þ b3 þ a1 a2 a23 ðb3 ðb2 b1 Þ2 þ ðb1 þb3 Þðb3 b2 Þ2 þ ðb2 þ b3 Þðb3 b1 Þ2 Þtb01 þ b2 þ 3b3

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and 1 þ b3 2 þ b3 t2 ¼ a31 a2 ðb2 b1 Þ3 b1 tb01 þ 3b3 a31 a3 ðb3 b1 Þ3 b1 t3b a32 a3 ðb3 b2 Þ3 b2 t3b þ a1 a32 ðb2 b1 Þ3 b2 tb01 þ 3b2 0 0 b1 þ 3b3 b2 þ 3b3 3 3 2 3 3 2 2 1 þ b2 þ b3 þa1 a3 ðb3 b1 Þ b3 t0 þ a2 a3 ðb3 b2 Þ b3 t0 þ a1 a2 a3 ðb3 b2 Þ ð5b1 b2 b3 2b1 ðb2 þ b3 ÞÞt2b 0 b þ 2b þ b 2 2 2 3 þa1 a22 a3 ðb3 b1 Þ ð5b22 b1 b3 2b2 ðb1 þ b3 ÞÞt01 þ a1 a2 a23 ðb2 b1 Þ ð5b23 b1 b2 2b3 ðb1 þ b2 ÞÞtb01 þ b2 þ 2b3 :

Proof. From the relationship between the two survival functions given by c

c

c

SðxÞ ¼ ST ðxc Þ ¼ a1 eb1 x þ a2 eb2 x þ a3 eb3 x , the first two derivatives of S(x) can be expressed as d SðxÞ ¼ cxc1 S0T ðxc Þ dx

and

d2 2

dx

SðxÞ ¼ cðc1Þxc2 S0T ðxc Þ þ ðcxc1 Þ2 S00 T ðxc Þ,

and taking into account Lemma 2.1, the log-concavity of S(x) is determined by the sign of the following expression:  2  2  d d SðxÞ SðxÞ SðxÞ ¼ ½cðc1Þxc2 S0T ðxc Þ þ ðcxc1 Þ2 S00 T ðxc Þcxc1 ST ðxc ÞðS0T ðxc Þcxc1 Þ2 : ð4:9Þ 2 dx dx By rearranging the terms, (4.9) can be rewritten as cxc2 ½ðc1ÞS0T ðxc ÞST ðxc Þ þcxc ðS00 T ðxc ÞST ðxc ÞS0T ðxc Þ2 Þ which has the same sign as A(xc), upon using the transformation t =xc, where the function A(t) is defined as AðtÞ ¼ ðc1ÞS0T ðtÞST ðtÞ þctðS00 T ðtÞST ðtÞS0T ðtÞ2 Þ:

ð4:10Þ

Note that upon setting c =1, A(t) has the same sign as S00 T ðtÞST ðtÞS0T ðtÞ2 , which incidentally is determined according to different parametric restrictions of the generalized mixture of three exponential distributions with survival function ST ðtÞ ¼ a1 eb1 t þ a2 eb2 t þ a3 eb3 t (for t 40). Moreover, it is readily seen that the first two derivatives are S0T ðtÞ ¼ a1 b1 eb1 t a2 b2 eb2 t a3 b3 eb3 t and S00 T ðtÞ ¼ a1 b21 eb1 t þa2 b22 eb2 t þ a3 b23 eb3 t : Clearly, under the parametric constraints of Lemma 4.1 in which ST(t) is log-concave and c Z 1, then AðtÞ r 0, and so S(x) is log-concave, i.e., X is IFR. Analogously, under the parametric constraints of Lemma 4.1 in which ST(t) is log-convex and c r 1, then AðtÞ Z 0, and so S(x) is log-convex, i.e., X is DFR. Nevertheless, from (4.10), the log-concavity (log-convexity) of ST(t) could not be preserved when c o1 ðc 41Þ. We shall, therefore, examine both these cases, i.e., Parts (iii) and (iv) of the statement of the theorem. From (4.10), we have Að0Þ ¼ ðc1Þða1 b1 þ a2 b2 þ a3 b3 Þ, where a1 b1 þa2 b2 þ a3 b3 Z 0 is a necessary condition for X to have a valid probability model; see Franco and Vivo (2009a). So, Að0Þ Z0 when c o 1 and Að0Þ r 0 when c 4 1. Moreover, A(t) has the same sign as that of gðtÞ ¼ e2b1 t AðtÞ, where gðtÞ ¼ ctða1 a2 ðb2 b1 Þ2 e2ðb2 b1 Þt þ a1 a3 ðb3 b1 Þ2 e2ðb3 b1 Þt þ a2 a3 ðb3 b2 Þ2 e2ðb3 þ b2 2b1 Þt Þðc1Þða21 b1 þ a22 b2 e2ðb2 b1 Þt þ a33 b3 e2ðb3 b1 Þt þa1 a2 ðb1 þ b2 Þeðb2 b1 Þt þ a1 a3 ðb1 þ b3 Þeðb3 b1 Þt þ a2 a3 ðb2 þb3 Þeðb3 þ b2 2b1 Þt Þ which converges to ðc1Þa21 b1 as t-1. Therefore, limt-1 gðtÞ 4 0 ð o0Þ when c o 1 ðc 4 1Þ, and so X cannot be IFR (DFR). In order to discuss the log-convexity of S(x) in Part (iii), we observe that A(t) defined by (4.10) can be expressed as " # S00 T ðtÞST ðtÞðS0T ðtÞÞ2 0 AðtÞ ¼ ST ðtÞST ðtÞ ðc1Þ þ ct S0T ðtÞST ðtÞ and when taking into account that S0T ðtÞST ðtÞ Z 0, we have AðtÞ Z0 for all t Z0 if and only if the following inequality holds: ! S00 T ðtÞST ðtÞðS0T ðtÞÞ2 c1 for all t Z0, ð4:11Þ 1 Z 0 3 hðtÞ Z 1 þc t c S0T ðtÞST ðtÞ where hðtÞ ¼ t

S00 T ðtÞST ðtÞðS0T ðtÞÞ2 S0T ðtÞST ðtÞ

for all t Z0:

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Note that h(t) is a function that does not depend on c, h(0) =0 and h(t) converges to zero as t-1 by applying L’Hospital’s rule. Furthermore, h(t) has the same sign as the second derivative of the logarithm of ST(t), since S0T ðtÞ Z0. Thus, hðtÞ r 0 due to the log-concavity of ST(t). Likewise, (c  1)/c increases with respect to c 2 ð0,1Þ, with limc-1 ðc1Þ=c ¼ 0 and limc-0 ðc1Þ=c ¼ 1. In this setting, (4.11) will hold when c is close to 0, i.e., for c r c0 , where c0 ¼ 1=ð1hðt0 ÞÞ o1 is determined by the minimum value of h(t). But, it will not hold as c converges to 1, i.e., for c 4 c0 , there exist at least one interval in which AðtÞ o0. Therefore, X is DFR if and only if c r c0 , where c0 is as given in (4.7) and t0 ¼ logðt0 Þ with t0 2 ð0,1Þ being the solution of (4.8). Finally, we discuss the log-concavity of S(x) when ST(t) is log-convex and c 4 1. In this case, the log-convexity of ST(t) is transformed to log-concavity of S(x) when AðtÞ r 0 for all t Z 0, i.e., the reverse of the inequality in (4.11) holds. Here, hðtÞ Z 0 due to the log-convexity of ST(t). Moreover, (c  1)/c increases with respect to c 2 ð1,1Þ, with limc-1 ðc1Þ=c ¼ 0 and limc-1 ðc1Þ=c ¼ 1. Hence, the reverse of the inequality in (4.11) cannot be satisfied as c converges to 1, i.e., for c o c0 , there exist at least one interval in which AðtÞ 40. However, it will be satisfied when c tends to infinity, i.e., if c Z c0 , where c0 ¼ 1=ð1hðt0 ÞÞ4 1 is determined by the maximum value of h(t) with hðt0 Þ o1. Therefore, X will be IFR if and only if c Zc0 41, with c0 as given in (4.7) and t0 2 ð0,1Þ being the solution of (4.8). Hence, the theorem. & Remark 4.3. When c= 1, Theorem 4.2 presents the classification of generalized mixtures of three exponential distributions in Lemma 4.1, which is in fact Theorem 3 of Baggs and Nagaraja (1996). Also, when c= 2, Theorem 4.2 presents the classification of generalized mixtures of three Rayleigh distributions. Remark 4.4. Note that Theorem 4.2 classifies a generalized mixture of three Weibull distributions with a common shape parameter from the log-concavity of the exponential case. Thus, when X cannot be classified in Parts (iii) and (iv) of Theorem 4.2, it can have neither U-shaped nor -shaped failure rate function. Further, in the case c0 o c o 1 ðc0 o c o 1Þ when ST is log-concave (log-convex), the failure rate of X decreases (increases) at the beginning and at the end, and it has at least one piecewise increasing (decreasing) in the middle, i.e., X has a single or multiple -shaped (N-shaped) failure rate. It is known that the IFR class implies the increasing failure rate average (IFRA) property. X is said to have an IFRA distribution if ð1=xÞlogSðxÞ is increasing in x 4 0, which on differentiation is equivalent to 1 S0 ðxÞ  logSðxÞ r x SðxÞ

for all x 4 0;

ð4:12Þ

see Baggs and Nagaraja (1996). Thus, the next result is useful in classifying generalized mixtures of three Weibull distributions when they are not IFR. Theorem 4.5. Let X be a generalized mixture of three Weibull distributions with common shape parameter c 4 0, scale parameters 0 o b1 o b2 ob3 o 1, and mixing weights a1 Z 1, a2 Z 1 and a3 = 1 a1  a2. Then, X is IFRA iff c Z1. Proof. Evidently, using the survival function S(x) given in (2.2), (4.12) can be rewritten as cxc1 S0T ðxc Þ 1  logST ðxc Þ r  x ST ðxc Þ

3

cS0 ðxc Þ 1 logST ðxc Þ Z T c xc ST ðx Þ

for all x 4 0,

where ST(t) denotes the survival function of the corresponding exponential case T as mentioned in Theorem 4.2. Upon using the transformation t = xc, the above inequality holds iff cS0 ðtÞ 1 logST ðtÞ Z T ST ðtÞ t

3

gðtÞ rc

for all t 4 0,

where gðtÞ ¼ ðST ðtÞ=tS0T ðtÞÞlogST ðtÞ does not depend on c. Hence, we note that gðtÞ Z 0, and its limit is 1 as t tends to infinity by applying L’Hospital’s rule. Furthermore, from Theorem 4 of Baggs and Nagaraja (1996), ST(t) is IFRA when a1 Z 1 and a2 Z1, i.e., gðtÞ r1 for all t 4 0, which completes the proof of the theorem. & 5. Reliability applications The lifetime of a system is determined by its components and structure, and most common systems in practical applications are series and parallel systems. For this reason, in this section, we classify the aging properties of such systems, i.e., minimum and maximum order statistics, from some known bivariate Weibull models, by applying the results of the preceding sections, since these order statistics are usually generalized mixtures of two or three Weibull distributions. In general, lifetimes of the components are dependent since the aging of one component affects all the rest. So, we examine here the reliability properties of the minimum and maximum lifetimes of two-component systems with independent components having Weibull distributions with a common shape parameter. Finally, we will discuss the

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classification of the aging of the same systems when the components are dependent having one of the known bivariate Weibull distributions. 5.1. Bivariate Weibull models under independence Here, we consider the two examples presented earlier in Section 2, in which generalized mixtures of Weibull distributions with a common shape parameter arise from parallel systems under independence. 5.1.1. IID components Let (X1, X2) have a bivariate Weibull distribution with independent and identically distributed (IID) components. From Example 2.1, T1 represents the lifetime of the series system formed with both components following a Weibull c distribution with Sð1Þ ðxÞ ¼ e2bx (for x 4 0). Thus, T1 is DFR if c r 1 and IFR if c Z 1. However, the lifetime of the parallel system T2 is a generalized mixture of two Weibull distributions with mixing weights a1 = 2 and a2 =  1, scale parameters 0 o b1 ¼ b o2b ¼ b2 , and a common shape parameter c. Therefore, from Part (ii) of Theorem 3.1, T2 is IFR iff c Z1. Moreover, T2 is DFR iff either c r 12, or 12 oc o 1 and w1 Z0, where w1 ¼ 1 þ2t2 3t

c tlogðtÞ 1c

with t ¼ ðc þðc2 þ 8ð1cÞ2 Þ1=2 Þ=4ð1cÞ 41. 5.1.2. INID components Now, we assume (X1,X2) to have a bivariate Weibull distribution with independent and non-identically distributed (INID) components. From Example 2.2, the series system from this model has a Weibull distribution, which is then classified as in the IID case. But, in this case, T2 is a generalized mixture of three Weibull distributions with mixing weights a1 =a2 =1 and a3 =  1, scale parameters 0 ob1 o b2 ob3 ¼ b1 þb2 , and a common shape parameter c. From Part (ii) of Lemma 4.1 and Theorem 4.2, s ¼ 2b1 b2 o 0, and so T2 cannot be classified within the two classes since it can be neither IFR nor DFR for c =1. However, by applying Theorem 4.5, we do observe that the parallel system T2 is IFRA iff c Z1. 5.2. Bivariate Weibull models under dependence In this subsection, we analyze the classification of the minimum and maximum lifetimes of two-component systems when their dependence follows one of the known bivariate Weibull distributions, since distributions of these systems are usually generalized mixtures of two or three Weibull distributions with a common shape parameter. One may refer to Kotz et al. (2000) and Balakrishnan and Lai (2009) for elaborate details on different bivariate Weibull distributions. 5.2.1. Gumbel type-I bivariate Weibull model (X1,X2) is said to have a Gumbel Type-I bivariate Weibull distribution if its survival function is given by c

c

c

c

Sðx1 , x2 Þ ¼ eðx1 þ x2 Þ ð1 þ að1ex1 Þð1ex2 ÞÞ

for x1 ,x2 4 0,

where jajr 1 and c 40, which will be denoted by GBVW-I. Substituting this expression into (2.3) and (2.4), we obtain the survival functions of T1 and T2, respectively, as c

c

c

Sð1Þ ðxÞ ¼ ð1 þ aÞex 2ae3x þ ae4x

for x 40,

and c

c

c

c

Sð2Þ ðxÞ ¼ 2ex ð1þ aÞe2x þ2ae3x ae4x

for x 4 0:

Note that if a ¼ 0, then the GBVW-I distribution simply reduces to the bivariate Weibull distribution with independent and identically distributed components. Thus, the aging classification of its series and parallel systems is determined by the IID case of the preceding subsection. Next, when a ¼ 1, the series system from the GBVW-I is a generalized mixture of two Weibull distributions given by c

c

Sð1Þ ðxÞ ¼ 2e3x e4x

for x 4 0,

i.e., a1 ¼ 2 4 0, a2 ¼ 1o 0, b1 = 3 and b2 =4. So, from Part (ii) of Theorem 3.1, we have T1 to be IFR when c Z1. Moreover, T1 is DFR iff either c r3=4, or c 2 ð3=4,1Þ and w1 Z0, where w1 ¼ 2 þ6t2 7t

c tlogt, 1c

with t ¼

c þ ðc2 þ 48ð1cÞ2 Þ1=2 4 1: 12ð1cÞ

However, the parallel system T2 is a generalized mixture of three Weibull distributions given by c

c

c

Sð2Þ ðxÞ ¼ 2ex 2e3x þ e4x

for x 40,

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i.e., a1 =2, a2 =  2, a3 = 1, b1 = 1, b2 =3 and b3 = 4. So, from Part (ii) of Lemma 4.1 and Theorem 4.2, T2 is IFR for c Z 1, since s= 0 and t ¼ 2logð2Þ=3 r 0. Moreover, T2 cannot be IFR for c o1, and it is DFR iff c r c0 o 1, where !1 t20 ðt20 þ t0 8Þlogðt0 Þ c0 ¼ 1 ðt0 1Þð1 þ2t0 Þð22t20 þ t30 Þ and t0 2 ð0,1Þ is solution of the equation 16 þ 14t0 52t20 4t30 þ47t40 20t50 3t60 þ2t70 þ 2ð16 þ 5t0 6t20 þ 4t30 34t40 þ 27t50 2t60 t70 Þlogðt0 Þ ¼ 0: In all other cases, i.e., when aa0,1, the series and parallel systems are generalized mixtures of three and four Weibull distributions, respectively, and so we cannot apply the above results to classify T2. Moreover, the mixing weights of T1 are a1 ¼ 1 þ a 4 0, a2 ¼ 2a, a3 ¼ a, and the scale parameters are b1 = 1, b2 =3 and b3 = 4. Hence, if 1 o a o 0, then a2 40 and a3 o 0, and from Part (ii) of Lemma 4.1 and Theorem 4.2, we obtain s ¼ ð1aÞa o0, and so T1 cannot be classified since it can be neither IFR nor DFR for c =1. Likewise, when 0 o a r 1, a2 o 0 and a3 4 0, and from Part (iv) of Lemma 4.1 and Theorem 4.2, T1 cannot be classified when a o1, since it can be neither IFR nor DFR for c =1. Nevertheless, when a ¼ 1, we note that the series system for a ¼ 1 is identically distributed to the parallel system for a ¼ 1, and so T1 has already been classified. 5.2.2. Gumbel type-II bivariate Weibull model A random vector (X1,X2) is said to have a Gumbel Type-II bivariate Weibull distribution if its survival function is given by 2 3  c 1=d  c 1=d !d x1 x2 4 5 for x1 ,x2 4 0, Sðx1 ,x2 Þ ¼ exp  þ

y1

y2

where y1 4 0, y2 4 0, 0 o d r 1 and c 4 0, which we shall denote by GBVW-II. Now, from (2.3) and (2.4), we obtain the survival functions of T1 and T2 as c

Sð1Þ ðxÞ ¼ elx

c

Sð2Þ ðxÞ ¼ ex

and

=y1

c

þ ex

=y2

c

elx

for x 4 0,

respectively, where l ¼ ½ð1=y1 Þ1=d þð1=y2 Þ1=d d . Under this setting, the series system from the GBVW-II model has a Weibull distribution which has been classified in the IID case. Moreover, the parallel system from this model is a generalized mixture of two or three Weibull distributions. First, when y1 ¼ y2 ¼ y, T2 is a generalized mixture of two Weibull distributions with mixing weights a1 =2 and a2 ¼ 1 o0, scale parameters b1 ¼ 1=y and b2 ¼ 2d =y, and a common shape parameter c. Therefore, from Part (ii) of Theorem 3.1, T2 is IFR iff c Z 1. Moreover, T2 is DFR iff either c rð42d Þ=ð2þ 2d Þ, or c 2 ðð42d Þ=ð2 þ2d Þ,1Þ and w1 Z0, where w1 ¼ ð2d 2tÞð12tÞ þ

c ð2d 1ÞtlogðtÞ 1c

with t ¼ ðcð12d Þ þðc2 ð12d Þ2 þð1cÞ2 22 þ d Þ1=2 Þ=4ð1cÞ 41. Next, when y1 ay2 , T2 is a generalized mixture of three Weibull distributions with mixing weights a1 ¼ a2 ¼ 1 4 0 and a3 ¼ 1 o0. Therefore, from Part (ii) of Lemma 4.1 and Theorem 4.2, we obtain s ¼ ð2=y1 y2 Þðly1 1Þðly2 1Þ o0, and so T2 cannot be classified since it can be neither IFR nor DFR for c= 1. However, by applying Theorem 4.5, we observe that the parallel system T2 from the GBVW-II model is indeed IFRA iff c Z 1. 5.2.3. Marshall and Olkin bivariate Weibull model One of the prominent bivariate Weibull distributions is the Marshall and Olkin bivariate Weibull distribution, denoted by MOBVW. (X1,X2) is said to have a MOBVW distribution if its survival function is given by c

c

c

c

Sðx1 ,x2 Þ ¼ el1 x1 l2 x2 l12 maxðx1 ,x2 Þ

for x1 ,x2 4 0,

where l1 4 0, l2 4 0, l12 Z 0 and c 40. Note that if l12 ¼ 0, the MOBVW distribution simply reduces to the INID case. Using the above expression in (2.3) and (2.4), the series and parallel systems from the MOBVW distribution have the following survival functions: c

Sð1Þ ðxÞ ¼ elx

and

c

c

c

Sð2Þ ðxÞ ¼ eðl1 þ l12 Þx þ eðl2 þ l12 Þx elx

for x 4 0,

where l ¼ l1 þ l2 þ l12 . So, T1 has already been classified in the IID case since it is simply a Weibull distribution. However, the parallel system from the MOBVW is a generalized mixture of two or three Weibull distributions. In the first case when l1 ¼ l2 , T2 is a generalized mixture of two Weibull distributions with mixing weights a1 =2 and a2 ¼ 1 o0, scale parameters b1 ¼ l1 þ l12 and b2 ¼ l, and a common shape parameter c. Hence, from Part (ii) of Theorem 3.1,

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T2 is IFR iff c Z 1. Moreover, T2 is DFR iff either c rðl þ 2l12 Þ=ð2l þ l12 Þ, or 1 4c 4 ðl þ2l12 Þ=ð2l þ l12 Þ and w1 Z0, where w1 ¼ ð2t1Þð2ðl1 þ l12 ÞtlÞ2

c l1 tlogðtÞ 1c

2

with t ¼ ðcl1 þðc2 l1 þ 4ð1cÞ2 lðl1 þ l12 ÞÞ1=2 Þ=4ð1cÞðl1 þ l12 Þ 4 1. In the second case when l1 al2 , T2 is a generalized mixture of three Weibull distributions with mixing weights a1 ¼ a2 ¼ 1 40 and a3 ¼ 1o 0. So, from Part (ii) of Lemma 4.1 and Theorem 4.2, we obtain s ¼ 2l1 l2 o 0, and so T2 cannot be classified since it can be neither IFR nor DFR for c= 1. However, by applying Theorem 4.5, we observe that the parallel system T2 from the MOBVW model is IFRA iff c Z 1. 5.2.4. Friday and Patil bivariate Weibull model We shall consider here the classification of the series and parallel systems when the dependence between their components is governed by the Friday and Patil bivariate Weibull distribution, denoted by FPBVW. A random vector (X1, X2) is said to have a FPBVW distribution if its survival function is given by ( c c c f1 eðna4 Þx1 a4 x2 þ ð1f1 Þenx2 if 0 o x1 ox2 Sðx1 ,x2 Þ ¼ c c c f2 ea3 x1 ðna3 Þx2 þ ð1f2 Þenx1 if 0 o x2 ox1 , where n ¼ a1 þ a2 , f1 ¼ a0 a1 =ðna4 Þ, f2 ¼ a0 a2 =ðna3 Þ, with 0 o a0 r 1, a1 4 0, a2 4 0, a3 4 0, a4 4 0,c 40, and the constraint naa3 , a4 . Note that the survival function of the FPBVW distribution can be obtained without the constraint naa3 , a4 from the above expression by taking limit as n-a3 and n-a4 , respectively. But in this case, the parallel system is not a generalized mixture of Weibull distributions, and hence our results are not applicable here. However, the series system follows a Weibull distribution and so the classification is as in the IID case. In this setting, substituting the above expression of the survival function into (2.4), we obtain T2 to be a generalized mixture of two or three Weibull distributions when naa3 , a4 , since its survival function is given by c

c

c

Sð2Þ ðxÞ ¼ f1 ea4 x þ f2 ea3 x þð1f1 f2 Þenx

for x 40:

ð5:13Þ

Assuming a3 ¼ a4 , the parallel system from the FPBVW model reduces to a generalized mixture of two Weibull distributions with mixing weights f1 þ f2 and 1f1 f2 , where f1 þ f2 ¼ a0 n=ðna3 Þ. Hence, if n o a3 , then a1 ¼ 1f1 f2 40 and a2 ¼ f1 þ f2 o 0. Thus, T2 from the FPBVW model is IFR iff c Z 1 from Part (ii) of Theorem 3.1. Moreover, T2 is DFR iff either c r ða3 ð1a0 Þ2 nÞ=ðð1þ a0 Þa3 ð1a0 ÞnÞ, or c 2 ðða3 ð1a0 Þ2 nÞ=ðð1 þ a0 Þa3 ð1a0 ÞnÞ,1Þ and w1 Z0, where w1 ¼ a20 a3 na0 tða3 þ nÞða3 ð1a0 ÞnÞ þ t2 ða3 ð1a0 ÞnÞ2 

c a0 ða3 nÞða3 ð1a0 ÞnÞtlogðtÞ c1

and t ¼ ða0 ða3 nÞc þða20 ða3 nÞ2 c2 þ 4a20 a3 nð1cÞ2 Þ1=2 Þ=2ða3 nð1a0 ÞÞð1cÞ 4 1. In the case when a3 o n, the mixing weights of T2 are a1 ¼ f1 þ f2 4 0 and a2 ¼ ðð1a0 Þna3 Þ=ðna3 Þ. Hence, the sign of a2 is negative for a3 2 ðð1a0 Þn, nÞ and positive for a3 oð1a0 Þn. From Part (ii) of Theorem 3.1, when ð1a0 Þn o a3 o n, the parallel system of the FPBVW model is IFR iff c Z 1. Moreover, T2 is DFR iff either c rða3 ð1a0 Þ2 nÞ=ðð1 þ a0 Þa3 ð1a0 ÞnÞ, or w2 Z 0 and c 2 ðða3 ð1a0 Þ2 nÞ=ðð1 þ a0 Þa3 ð1a0 ÞnÞ,1Þ, where w2 ¼ a23 ð1a0 tÞa3 ða20 tðt1Þ þ2a0 2Þn þð1a0 Þð1 þ a0 ðt1ÞÞn2 þ

c a0 ðna3 Þða3 ð1a0 ÞnÞtlogðtÞ 1c

with t ¼ ða3 nð1a0 ÞÞððna3 Þc þ ððna3 Þ2 c2 þ4a3 nð1cÞ2 Þ1=2 Þ=2a0 a3 nð1cÞ 4 1. Likewise, when a3 o ð1a0 Þn, the parallel system T2 is DFR iff c r 1 by Part (i) of Theorem 3.1. Moreover, T2 is IFR iff either c Z ða3 ð1a0 Þ2 nÞ=ðð1þ a0 Þae ð1a0 ÞnÞ for a3 2 ðð1a0 Þ=ð1þ a0 Þn,ð1a0 ÞnÞ, or w2 Z 0 for a3 o ð1a0 Þ=ð1þ a0 Þn, where w2 is as given above. Now, in the case when a3 aa4 , without loss of generality, we may assume that a4 o a3 , since ðf1 , a4 Þ and ðf2 , a3 Þ are interchangeable in (5.13). We shall now discuss the classification of the parallel system from the FPBVW model based on its scale parameters. First, when n ¼ b3 , the mixing weights are a1 ¼ f1 40, a2 ¼ f2 4 0 and a3 ¼ 1f1 f2 , which has the same sign as A ¼ ð1a0 Þðn2 a1 a3 a2 a4 Þa2 a3 a1 a4 þ a3 a4 : Therefore, if A Z0, then all the ai’s are positive. From Part (i) of Lemma 4.1 and Theorem 4.2, we have T2 to be DFR for c r1. Moreover, T2 cannot be DFR for c 41, and it is IFR iff c Zc0 41, where !1 a0 ðna3 Þðna4 Þlogðt0 Þða2 ðna3 ÞAta03 n þ a1 ðna4 ÞAta04 n þ a0 a1 a2 ða3 a4 Þ2 ta03 þ a4 2n Þ ð5:14Þ c0 ¼ 1 þ ðAþ a0 a1 ðna3 Þta04 n þ a0 a2 ðna4 Þta03 n ÞðnAþ a0 a1 a4 ðna3 Þta04 n þ a0 a2 a3 ðna4 Þta03 n Þ with t0 2 ð0,1Þ as given in (4.8). Similarly, if A o0, then a3 o 0, and from Part (ii) of Lemma 4.1 and Theorem 4.2, T2 cannot be classified since it can be neither IFR nor DFR for c = 1 when B ¼ ð1a0 Þn2 a1 a4 a2 a3 o 0, since B and s have the same sign. However, if B Z 0, then T2 is DFR for c r1. Moreover, T2 cannot be DFR for c 4 1, and it is IFR iff c Z c0 4 1, where c0 is as given in (5.14). Next, if n ¼ b2 , then a1 ¼ f1 4 0, a3 ¼ f2 o 0 and a2 ¼ 1f1 f2 . In this case, a2 and A have different signs, and so when A o0 and B o0, T2 cannot be classified by Part (ii) of Lemma 4.1 and Theorem 4.2, since it can be neither IFR nor DFR for c=1.

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However, when Ao 0 and B Z0, the parallel system from the FPBVW model is DFR for c r1, it cannot be DFR for c 41, and is IFR iff c Z c0 4 1 with c0 as given in (5.14), since the (ai,bi)’s for i=1,2,3 are interchangeable in (4.7). Moreover, by using Part (iii) of Lemma 4.1 and Theorem 4.2, when A 4 0 and B 40, T2 cannot be classified since it can be neither IFR nor DFR for c =1. However, when A4 0 and B r0, T2 is IFR for c Z1, it cannot be IFR for c o 1, and is DFR iff c r c0 o 1 with c0 as given in (5.14). Finally, when n ¼ b1 , it is readily seen that the mixing weights are a1 4 0, a2 ¼ f1 o 0 and a3 ¼ f2 o 0. Therefore, from Part (iii) of Lemma 4.1 and Theorem 4.2, T2 cannot be classified when B 4 0 since it can be neither IFR nor DFR for c =1. However, when B r 0, the parallel system T2 is IFR for c Z1, it cannot be IFR for c o 1, and is DFR iff c rc0 o1 with c0 as given in (5.14). Furthermore, note that Theorem 4.5 is not applicable in the last case when n ¼ b1 . However, in the first two cases, we have a1 Z1 and a2 Z 1 iff either fn r a4 þ a0 a1 and n2 Z a1 a3 þ a2 a4 when n ¼ b2 g, or fn r a4 þ a0 a1 and n r a3 þ a0 a2 when n ¼ b3 g, and so from Theorem 4.5, the parallel system T2 is IFRA iff c Z 1. 5.2.5. Raftery type-I bivariate Weibull model A random vector (X1,X2) is said to have a Raftery Type-I bivariate Weibull distribution if its survival function is given by c

c

Sðx1 ,x2 Þ ¼ elmaxðx1 ,x2 Þ þ

1p ðl=ð1pÞÞðxc þ xc Þ 1p ðl=ð1pÞÞðmaxðxc ,xc Þpminðxc ,xc ÞÞ 1 2  1 2 1 2 e e 1þ p 1þp

for x1 ,x2 4 0, where l 4 0, 0 o p o1 and c 40, and is denoted by RBVW-I. Note that the range for p can be expanded to include p ¼ 0, thus subsuming the IID setting, and the scale parameters have no role when p ¼ 1. Now, upon substituting the above survival function into (2.3) and (2.4), we see that T1 and T2 from the RBVW-I model have their survival functions as Sð1Þ ðxÞ ¼

2p lxc 1p ð2l=ð1pÞÞxc e þ e 1þp 1þp

for x 40,

Sð2Þ ðxÞ ¼

2 1p ð2l=ð1pÞÞxc c elx þ e 1þp 1þp

for x 40,

and

respectively. Thus, the series and parallel systems are both generalized mixtures of two Weibull distributions. In the first case, the mixing weights of T1 are a1 ¼ 2p=ð1 þ pÞ 4 0 and a2 ¼ ð1pÞ=ð1þ pÞ 4 0, and so due to Part (i) of Theorem 3.1, the series system from the RBVW-I model is DFR iff c r1. Moreover, T1 is DFR iff c 41 and w1 Z0, where w1 ¼ 1 þ p2 tð2t1Þ þ pð3t1Þ

c pð1 þ pÞtlogðtÞ c1

with t ¼ ðcð1 þ pÞ þ ðc2 ð3pÞ2 8ð1pÞð2c1ÞÞ1=2 Þ=4ðc1Þp 4 1. In the second case, T2 has mixing weights a1 ¼ 2=ð1þ pÞ 4 0 and a2 ¼ ð1pÞ=ð1 þ pÞ o 0. From Part (ii) of Theorem 3.1, we have T2 to be IFR iff c Z 1. Moreover, the parallel system of the RBVW-I model is DFR iff either c o 12, or 12 o c o1 and w Z 0, where w1 ¼ ðt1Þð2t þ p1Þ

c ð1þ pÞtlogðtÞ c1

with t ¼ ðcð1 þ pÞ þ ðc2 ð3pÞ2 8ð1pÞð2c1ÞÞ1=2 Þ=4ðc1Þ 4 1. 5.2.6. Raftery Type-II bivariate Weibull model (X1,X2) is said to have a Raftery Type-II bivariate Weibull distribution, denoted by RBVW-II, if its survival function is given by 8 1 c c c c > lðð1=ð1pÞÞxc1 þ xc2 Þ > þ elðx1 þ ð1=ð1pÞÞx2 Þ eðl=ð1pÞÞðx1 þ x2 Þ if p r >e > 2 > > > < 1p 1p lðxc þ ð1=ð1pÞÞxc Þ ð1pÞð2pÞ l=ð1pÞðxc þ xc Þ lðð1=ð1pÞÞxc1 þ xc2 Þ 1 2  1 2 e e þ e Sðx1 ,x2 Þ ¼ pð1 þ pÞ p p > > > > 2p1 lmaxðxc ,xc Þ ð2p1Þð1pÞ ðl=ð1pÞÞðmaxðxc ,xc Þpminðxc ,xc ÞÞ 1 > > 1 2  1 2 1 2 > e if p 4 : þ p e 2 pð1 þ pÞ for x1 ,x2 40, where l 4 0, 0 o p o 1 and c 40. The series and parallel systems from this model are generalized mixtures of two or three Weibull distributions, and from (2.3) and (2.4), their survival functions are given by 8 1 c lðð2pÞ=ð1pÞÞxc > > eð2l=ð1pÞÞx if p r < 2e 2 Sð1Þ ðxÞ ¼ 2p1 lxc 1p lðð2pÞ=ð1pÞÞxc ð1pÞð2pÞ ðð2lÞ=ð1pÞÞxc 1 > > e e þ2 e  if p 4 :2 1þp 2 pð1þ pÞ p

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and

Sð2Þ ðxÞ ¼

8 c c lxc > > 2elðð2pÞ=ð1pÞÞx þ eð2l=ð1pÞÞx < 2e 2p lxc 1p lðð2pÞ=ð1pÞÞxc ð1pÞð2pÞ ð2l=ð1pÞÞxc > > e e 2 e þ :2 1þp pð1 þ pÞ p

1 2 1 if p 4 2 if p r

for x 4 0, respectively. First, we note that T1 from the RBVW-II model when p r0:5 has mixing weights a1 = 2 and a2 ¼ 1 o 0. Using Part (ii) of Theorem 3.1, we get T1 to be IFR iff c Z 1. Moreover, the series system is DFR iff either c rð32pÞ=ð3pÞ, or c 2 ðð32pÞ=ð3pÞ,1Þ and w1 Z 0, where w1 ¼ ð2t1Þðð2pÞt1Þ þ

c ptlogðtÞ 1c

with t ¼ ðcp þðc2 ð4pÞ2 þ8ð2pÞð12cÞÞ1=2 Þ=4ð1cÞð2pÞ 4 1. Next, when p 4 0:5, the mixing weights of T1 are a1 ¼ 2ð2p1Þ=ð1 þ pÞ 4 0, a2 ¼ 2ð1pÞ=p 4 0 and a3 ¼ ð1pÞ ð2pÞ=pð1 þ pÞ o 0, since the scale parameters satisfy b1 ¼ l o b2 ¼ lð2pÞ=ð1pÞ o b3 ¼ 2l=ð1pÞ. Applying Part (ii) 2 of Lemma 4.1 and Theorem 4.2, s ¼ 2l , and so T1 cannot be classified since it can be neither IFR nor DFR for c=1. Moreover, since a1 o 1, Theorem 4.5 is not applicable. On the other hand, when 0 o p r 0:5, the parallel system from the RBVW-II model has mixing weights a1 ¼ 24 0, 2 a2 ¼ 2o 0 and a3 ¼ 1 4 0, and so taking into account that s ¼ 2l ð12pÞ=ð1pÞ2 o 0 and t ¼ ð1pÞ=lð1 þ pÞlogp=2o 0, from Part (iv) of Lemma 4.1 and Theorem 4.2, T2 is IFR for c Z 1. Moreover, T2 cannot be IFR for c o1, and it is DFR iff c r c0 o 1, where !1 gp ðð1 þ pÞ2 g2gp þ 1 p2 ÞlogðgÞ c0 ¼ 1 ð12gp þ 2gp þ 1 Þð1ð2pÞgp þ ð1pÞgp þ 1 Þ l=ð1pÞ

and g ¼ t0

4 1 is solution of the equation

ð12gp þ 2gp þ 1 Þð1ð2pÞgp þ ð1pÞgp þ 1 Þðð1þ pÞ2 g2gp þ 1 p2 Þ þ ½ð1 þ pÞ3 gp3 þ ðp2 11p6Þgp þ 1 þ 2p3 ð2pÞg2p þ 2ð3p4 2p3 7p2 þ 4p þ 6Þg2p þ 1 þ 2p2 ð72p3p2 Þg2p þ 2 þ 2ðp2 1Þð1þ pÞ2 g2p þ 3 4ð2pÞg3p þ 1 þ 4ð1pÞg3p þ 3 logðgÞ ¼ 0:

Finally, when 0:5 o p o1, T2 has mixing weights a1 ¼ 2ð2pÞ=ð1 þ pÞ 4 0, a2 ¼ 2ð1pÞ=p o0 and a3 ¼ ð1pÞð2pÞ= pð1 þ pÞ 40. Thus, from Part (iv) of Lemma 4.1 and Theorem 4.2, T2 is IFR for c Z 1, since s=0 and t ¼ ð1pÞ= lð1 þ pÞlogð1pÞ=2 o 0. Moreover, the parallel system from the RBVW-II model cannot be IFR for c o 1, and it is DFR iff c r c0 o 1, where  c0 ¼ 1þ l=ð1pÞ

and g ¼ t0

pðp þ1Þgp ðpð1pÞð2 þ pp2 Þg þ 2gp þ 1 ÞlogðgÞ ð1ð þ pÞgp þ pgp þ 1 Þðð2pÞð1pÞ2ð1p2 Þgp þ 2pð2pÞgp þ 1 Þ

1

4 1 is solution of the equation

ð1ð1 þ pÞgp þ pgp þ 1 Þðð2pÞð1pÞ2ð1p2 Þgp þ2pð2pÞgp þ 1 Þðpð1pÞ2pð1pÞg þ 2gp Þ þ logðgÞ½ð2pÞð1pÞp2 þ p2 ð7p3p2 þ p3 Þgp5 g2 p2 ð2914p þ p2 Þgp þ 1 2p2 ð12p2 Þg2p 2ð213p2 6p3 þ 10p4 Þg2p þ 1 2p3 ð1411pÞg2p þ 2 þ 2p2 ð2 þ pp2 Þ2 g2p þ 3 þ 2p5 ð43pÞg2p þ 4 2p5 ð23pÞg2p þ 5 2p6 g2p þ 6 þ 4ð3þ pp2 p3 Þg3p þ 1 4p2 ð2pÞg3p þ 3 4ð3 þ 2pÞg4p þ 1 þ4ð1 þ pÞg5p þ 1  ¼ 0:

5.2.7. Raftery type-III bivariate Weibull model A random vector (X1,X2) is said to have a Raftery Type-III bivariate Weibull distribution if its survival function is given by

p1 ð1p2 Þ2 lðð1=ð1p1 ÞÞxc1 þ ð1=ð1p2 ÞÞxc2 Þ p2 p1 lðð1=ð1p1 ÞÞxc1 þ xc2 Þ e þ e p2 ð1p1 p2 Þ p2 p1 lmaxðxc1 ,xc2 Þ p1 ð1pÞ2 ðl=ð1pÞÞðmaxðxc1 ,xc2 Þpminðxc1 ,xc2 ÞÞ e þ e  for x1 ,x2 40, p pð1p1 p2 Þ

Sðx1 ,x2 Þ ¼

where l 4 0, 0 o p1 o p2 o 1, c 40 and p ¼ pi such that xi = max(x1, x2). We will denote this model by RBVW-III. Now, substituting the above survival function into (2.3) and (2.4), we obtain the survival functions for the series and parallel systems from the RBVW-III model, respectively, as Sð1Þ ðxÞ ¼

p1 ð2p1 p2 Þ lxc p2 p1 lðð2p1 Þ=ð1p1 ÞÞxc p1 ð1p2 Þ2 lðð2p1 p2 Þ=ð1p1 Þð1p2 ÞÞxc e e þ e þ 1p1 p2 p2 p2 ð1p1 p2 Þ

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and ! ð1p1 Þ2 lxc p2 p1 lðð2p1 Þ=ð1p1 ÞÞxc p1 ð1p2 Þ2 lðð2p1 p2 Þ=ð1p1 Þð1p2 ÞÞxc e e 1þ þ e  1p1 p2 p2 p2 ð1p1 p2 Þ

Sð2Þ ðxÞ ¼

for x 40. Note that both systems are generalized mixtures of three Weibull distributions. In fact, T1 has mixing weights a1 ¼ p1 ð2p1 p2 Þ=ð1p1 p2 Þ 4 0, a2 ¼ ðp2 p1 Þ=p2 40 and a3 ¼ p1 ð1p2 Þ2 =p2 ð1p1 p2 Þ 4 0, and the scale parameters satisfy b1 ¼ l o b2 ¼ lð2p1 Þ=ð1p1 Þ ob3 ¼ lð2p1 p2 Þ=ð1p1 Þð1p2 Þ. Thus, all the ai’s are positive, and from Part (i) of Lemma 4.1 and Theorem 4.2, we have T1 to be DFR for c r 1. Moreover, it cannot be DFR for c 41, and it is IFR iff c Z c0 4 1, where 1þ

c0 ¼ 

p1 p2 ð1p2 Þð1p1 p2 Þg1p2 logðgÞ p1 ð1p2 Þ þ p2 ðp2 p1 Þð1p1 p2 Þg1p2 þ p1 p2 ð2p1 p2 Þg1p1 p2 2

ð2p2 Þp2 2p1 þ p2 ðp2 ð1p1 Þp1 ð1 þ p2 Þp31 Þg1p1 p2 þ p1 ð2p2 þ p21 p2 Þgð1p1 Þp2 p1 p2 ð1p1 Þð2p1 p2 Þ þð2p1 Þðp2 p1 Þð1p1 p2 Þg1p2 þ p1 ð1p2 Þð2p1 p2 Þg1p1 p2

1

l=ð1p Þð1p Þ

1 2 with g ¼ t0 2 ð0,1Þ as given in (4.8). Next, the mixing weights of the parallel system from the RBVW-III model are a1 ¼ 1 þ ð1p1 Þ2 =ð1p1 p2 Þ 4 0, 2 a2 ¼ ðp2 p1 Þ=p2 4 0 and a3 ¼ p1 ð1p2 Þ2 =p2 ð1p1 p2 Þ o 0. Taking into account that s ¼ ð2l ðp2 ð1p1 þ p21 Þp1 ð22p1 þ p21 ÞÞÞ=p2 ð1pÞ2 , then s o0 iff p1 2 ðp , p2 Þ, where p 2 ð0,1Þ is the unique solution of the equation

p2 ð1p þ p2 Þp ð22p þ p2 Þ ¼ 0: Thus, using Part (ii) of Lemma 4.1 and Theorem 4.2, T2 cannot be classified since it can be neither IFR nor DFR for c=1. Moreover, Theorem 4.5 is not applicable since a2 o1. However, s Z0 for p1 r p , and hence we have T2 to be DFR for c r1. Moreover, it cannot be DFR for c 4 1, and it is IFR iff c Z c0 4 1, where c0 ¼

1þ



p2 ð1p2 Þð1p1 p2 Þg1p2 logðgÞ p2 ðð1p1 Þ þ1p1 p2 Þ þðp2 p1 Þð1p1 p2 Þg1p2 p1 ð1p2 Þ2 g1p1 p2 2

ðp2 p1 Þð2ð1p1 Þp1 ðp2 p1 ÞÞp1 p2 ðp2 p1 Þð1p1 Þ2 g1p1 p2 p1 ð1p1 p2 Þðð1 þ p1 Þ2 þ1p1 p2 Þgð1p1 Þp2

!1

p2 ð1p1 Þð1p1 p2 Þðð1p1 Þ2 þ 1p1 p2 Þ þ ð2p1 Þðp2 p1 Þð1p1 p2 Þg1p2 þ p1 ð1p2 Þð2p1 p2 Þg1p1 p2

l=ð1p1 Þð1p2 Þ

with g ¼ t0

2 ð0,1Þ as given in (4.8).

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