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8/13/2019 Elsevier [Reliability Engineering and System Safety] Weibull and Inverse Weibull Mixture Models Allowing Negative

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Weibull and inverse Weibull mixture models allowing negative weights

R. Jianga, M.J. Zuob,*, H.-X. Lic

aDepartment of Mechanical Engineering, University of Saskatchewan, 57 Campus Drive, Saskatoon, SK, Canada S7N 5A9bDepartment of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G8

cDepartment of Manufacturing Engineering and Engineering Management, City University of Hong Kong, 83 Tat Chee Avenue, Hong Kong

Received 15 October 1998; accepted 30 April 1999

Abstract

This article presents the finding that when components of a system follow the Weibull or inverse Weibull distribution with a common

shape parameter, then the system can be represented by a Weibull or inverse Weibull mixture model allowing negative weights. We also usean example to illustrate that the proposed mixture model can be used to approximate the reliability behaviours of the consecutive-k-out-of-n

systems. The example also shows data analysis procedures when the parameters of the component life distributions are either known or

unknown. 1999 Elsevier Science Ltd. All rights reserved.

Keywords: Mixture model; Weibull distribution; Inverse Weibull distribution; k-out-of-nsystem; Consecutive-k-out-of-nsystem

1. Introduction

Suppose that a random variable, X, takes values in a

sample space, S, and that its distribution can be represented

by a probability density function (pdf) of the form

fx 1f1x 2f2x kfkx x S

1

where

j 0 j 1 2 k

1 2 k1

fj 0

Sfjxdx 1 j 1 2 k

In such a case,Xis said to have a finite mixture distribution

and that f() defined in Eq. (1) is a finite mixture pdf. The

parameters12 kare called the mixing weights and

f1f2 fkthe component pdf of the mixture, see [1].

It can be easily verified that the pdfs in Eq. (1) can be

replaced by the corresponding cumulative distribution func-

tions (cdf) and the corresponding reliability functions.

Titterington et al. [1] point out that mixture distributions

have been used as models throughout the history of modern

statistics and give a detailed list of references on applica-

tions of mixture models. Jiang [2] gives a detailed literature

review on Weibull mixture models. Jiang and Murthy [34]

characterise the 2-fold Weibull mixture model in terms of

the Weibull probability plotting and failure rate of the

model.

A basic feature of the finite mixture models showing

above is that the mixing weights are positive. However,Ref. [1] mention that this is not necessary in principle.

Thus, there exist mixtures allowing negative weights as

long asfxremains to be a valid pdf.

Mixture, series systems, parallel systems, k-out-of-n

systems, and consecutive-k-out-of-n systems models are

different reliability models. In this paper, we will show

that they are related to each other under certain conditions.

The paper is organised as follows. Firstly, we derive

Weibull and inverse Weibull mixture models with the

same shape parameter in Section 2. We extend these two

models to more general forms in Section 3. Section 4 gives a

numerical example. Finally a brief summary is given inSection 5.

2. Weibull and inverse Weibull mixture models with a

common shape parameter

The reliability function of a 2-parameter Weibull distri-

bution is given by

Rt expt 2

while the unreliability function of a 2-parameter inverse

Reliability Engineering and System Safety 66 (1999) 227234

0951-8320/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved.

PII: S0951-8320(99) 00037-X

www.elsevier.com/locate/ress

* Corresponding author. Tel.: 780-492-4466; fax: 780-492-2200.

E-mail address:[email protected] (M.J. Zuo)


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Weibull distribution is given by

Ft expt 3

The parameters and are called shape and scale para-

meter, respectively. For such distributions we have the

following lemmas.

Lemma 1. The product of n Weibull reliability functions

with the same shape parameter is still a Weibull reliability

function with the same shape parameter.

Proof. Assume

Rit expti

i 1 2 n

Then, we have

ni1

Rit expt0

where

0 ni1

i

1

It is obvious thatn

i1Ritis a Weibull reliability function

with shape parameter and scale parameter0. (QED)

If we assume thatm is a positive real number (not neces-

sarily an integer) and Rt is a Weibull reliability function

given by Eq. (2), then we have

Rmt expmt exptm

where

m m1

In other words, the positive power of a Weibull reliability

function is still a Weibull reliability function with the same

shape parameter.

Lemma 2. The product of n inverse Weibull unreliability

functions with the same shape parameter is still an inverseWeibull unreliability function with the same shape para-

meter.

Proof. Assume

Fit expit

i 1 2 n

Then, we have

ni1

Fit exp0t

where

0 ni1

i

1

It is obvious thatn

i1Fit is an inverse Weibull unrelia-

bility function with shape parameter and scale parameter

0. (QED)If we assume thatm is a positive real number (not neces-

sarily an integer) andFtis an inverse Weibull unreliability

function given by Eq. (3), then we have

Fmt expmt expmt

where

m m1

In other words, the positive power of an inverse Weibull

unreliability function is still an inverse Weibull unreliability

function with the same shape parameter.

Based on the lemmas, we have the following theorem.

Theorem. If

1. the reliability (or unreliability) function of a system can

be expressed as a weighted sum of the products of the

reliabilities (or unreliabilities) of its components, i.e.,

Rt

i

pi

j

Rjt 4

or

Ft i qi j Fjt 52. all components

(a) follow Weibull (or inverse Weibull) distributions;

and

(b) have the same shape parameter;

then the reliability (or unreliability) of the system can

be expressed as a Weibull (or inverse Weibull) mixture

with some weights being negative.

Proof. From Lemma 1 and condition 2, Eq. (4) can be

written as

Rt

i

piRit 6

withRit

jRjtbeing a Weibull distribution. From Eq.

(6) letting t 0 yields

ni1

pi 1

R. Jiang et al. / Reliability Engineering and System Safety 66 ( 1999) 227 234228


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Sections 2.12.3 will show thatpis are not all positive for

some structures. Hence Eq. (6) is a Weibull mixture model

allowing negative weights with a common shape parameter.

Similarly, From Lemma 2 and condition 2, Eq. (5) can be

written as

Ft i qiFit 7withFit

jFjtbeing an inverse Weibull distribution.

From Eq. (7) letting t yields

ni1

qi 1

Sections 2.12.3 will show that qis are not all positive for

some structures. Hence Eq. (7) is an inverse Weibull

mixture model allowing negative weights with a common

shape parameter. (QED)

Note that condition 1 in Theorem holds for many system

structures, while condition 2 is more restrictive as the data

will determine if the Weibull or inverse Weibull distribution

can be used (condition 2(a)) and if the shape parameters are

the same when condition 2(a) holds. The models to be intro-

duced in Section 3 can be used in those situations where the

common shape parameter assumption does not hold. We

consider several well-known structures to show that: (1)

condition 1 holds; and (2) some weights are negative.

2.1. Series system

In some context, the series model is called the competing

risk model. The reliability and unreliability of the system

are, respectively,

Rt ni1

Rit

and

Ft nk1

1k1 n k1

i1ijln

FiFjFlk

whereFi Fitand the subscriptkon the RHS of the Ft

expression implies the k-fold product. These imply that a

series system satisfies condition 1 in the Theorem.

If all components follow the Weibull distribution withthe same shape parameter (in this case, both condition

2(a) and condition 2(b) are satisfied), thenRtis a single

Weibull distribution as an extreme case of the mixture

model.

If all components follow the inverse Weibull distribution

with the same shape parameter (in this case, both condi-

tion 2(a) and condition 2(b) are satisfied), then Ftis an

inverse Weibull mixture model allowing negative

weights with a common shape parameter. For example,

whenn 2 the cdf is given by

Ft F1t F2t F3t

with F3t F1tF2t being an inverse Weibull distri-

bution andq1 q2 1 q3 1.

2.2. Parallel system

In some context, the parallel model is called the comple-

mentary competing risk model. The unreliability and relia-

bility of the system are, respectively,

Ft ni1

Fit

and

Rt nk1

1k1 nk1

i1ijln

RiRjRlk

These imply that a parallel system satisfies condition 1 in

the Theorem.

If all components follow the inverse Weibull distribution

with the same shape parameter (in this case, both condi-

tion 2(a) and condition 2(b) are satisfied), then Ft is a

single inverse Weibull distribution as an extreme case of

the mixture.

If all components follow the Weibull distribution with

the same parameter (in this case, both condition 2(a) and

condition 2(b) are satisfied), then Rt is a Weibull

mixture model allowing negative weights with a

common shape parameter. For example, when n 2

the reliability function is given by

Rt R1t

R2t

R3twithR3t R1tR2tbeing a Weibull distribution and

p1 p2 1p3 1.

2.3. k-out-of-n:G and consecutive k-out-of-n:G systems

In ak-out-of-n:G system, the system works if and only if

at least kcomponents are working. If all n components are

identical, the reliability and unreliability of the system are,

respectively,

Rt nik

CinRi01R0n

i nik

CinRi0 n ij0

1jCjniRj0

and

Ft 1 nik

Cin1 F0iFni0

1 nik

CinFni0

ij0

Cji 1

jFj0

If the components are non i.i.d.,Ft orRtcan be written

as the sum of the products of reliability or unreliability of

R. Jiang et al. / Reliability Engineering and System Safety 66 ( 1999) 227 234 229


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components with all possible combinations. It is obvious

that the k-out-of-nsystem satisfies condition 1 of the Theo-

rem too.

A consecutive-k-out-of-n:G system consists of an ordered

sequence ofncomponents such that the system works if and

only if at least kconsecutive components in the system are

good [5]. Since the failure combinations of a consecutive

k-out-of-n:G system are a sub-set of the failure combina-

tions of a k-out-of-n:G system, the reliability and unrelia-

bility of a consecutive k-out-of-n:G system also satisfy

condition 1 of the Theorem. For example, the reliability of

a consecutive-2-out-of-5:G system with identical compo-

nent is given by

RSt R20t 3R

30t 4R

40tR

50t 8

whereR0tis the reliability of the iid components. IfR0t

follows Weibull distribution, then Eq. (8) is a mixture model

allowing negative weights.

3. (Inverse) Weibull mixture model allowing negative

weights with different shape parameters

Since the common shape parameter assumption is very

restrictive, the models given by Eqs. (6) and (7) are limited

in application. Therefore, we propose the following two

models to approximate the reliability or unreliability of a

system if the common shape parameter assumption for

(inverse) Weibull components in Theorem is not satisfied.

As will be seen, the models given by Eqs. (6) and (7) are

special cases of the following models.

(a) The Weibull mixture model allowing negativeweights with different shape parameters

Rt ni1

piexptii

ni1

pi 1 pi

9

(b) The inverse Weibull mixture model allowing negative

weights with different shape parameters:

Ft ni1

piexpiti

ni1

pi 1 pi

10

The only requirement is thatRtin Eq. (9) andFtin Eq.

(10) must remain to be a valid reliability function and

unreliability function, respectively. One of the methods

to examine this is to see if the Weibull Probability Plots

(WPP) of the models are increasing curves. We require

that, for i j in each model, the following relations

should not be simultaneously satisfied:

i j i j

Otherwise, the two terms i and j can be merged into a

single term. Similarly, we limit pi 0.Without loss of

generality, we can assume

1 2 n

ifi jfor i j theni j

These mixture models (given by Eqs. (6), (7), (9) and

(10)) can be applied in the following situations:

The structure of a system is unknown. In this case, we

cannot use component life distributions to derive the

systems life distribution. The models given in this

section have the flexibility in fitting the life distribu-

tion of the system from a given data set on the

systems failure behaviour because the mixingweights are allowed to be negative. There is no need

to know the life distributions of the components.

The structure of a system is partially known. In many

k-out-of-n or consecutive-k-out-of-n systems, the

components very often perform the same function.

For example, a power generating unit in a power

plant may be considered a component in a k-out-of-

nsystem. A microwave relay station may be consid-

ered a component in a consecutive-k-out-of-nsystem

[6]. The proposed models will also be useful when one

does not know the exact structure function of a

system. For example, do we have a 2-out-of-5 system

or a 2.5-out-of -5 system? This question may come up

when the system structure of a brand new system is

designed to be a 2-out-of-5 configuration. After the

system is in operation for a while, it may still be a k-

out-of-n structure, but the k value may be different

because of degradation in the system and the compo-

nents. In this case, the partial information on the

system can help the engineer to determine the specific

form of the model, for example, the number of foldsn

and so on.

In the following section, we give a numerical example to

illustrate applications of these models.

4. A numerical example

Assume that a consecutive-2-out-of-5:G system has iid

components. The failure of each component follows

Weibull distribution with the parameters 0 25

0 10. Then, the reliability function of the system is

given by Eq. (8).

A simulated lifetime data set with 50 data points is gener-

ated based on the above parameters and Eq. (8) and is given

in Table 1. Suppose that we view the data in Table 1 as field



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data. Now we need to model the system failure by using the

data and the proposed Weibull mixture model allowing

negative weights. Section 4.1 will deal with the case

where a prior distribution ofR0t is known. We deal with

the case where R0t is unknown in Section 4.2.

4.1. The system structure and prior distribution of R0tare

known

In this case, we assume the posterior distribution of the

system is no longer Eq. (8) due to degradation and/or prior

distribution differing from practical situation, and can beexpressed as

Rt 5i2

piRi0

5i2

pi 1 11

The objective of modelling is to determine values of

p2p3p4. Once p2p3p4 are determined, then we have

p5 1 p2 p3 p4. Let

y RtR50 xi R

i0 R

50 i 2 3 4

Then,

y 4i2

pixi

Using the data given in Table 1 and the multivariate

regression method yields

p2 029066p3 4429647p4 4397831

p5 0677524

Note that the weights of the corresponding theoretical model

as shown in Eq. (8) are

p2 1p3 3p4 4p5 1

Now we check if the model obtained from the linear regres-

sion technique is reasonable. The empirical, theoretical, and

the fitted relations ofR(t) tare shown in Fig. 1. As can be

seen, the fitted curve is as close to the empirical curve as the

theoretical curve. In fact, we haveRempirical Rtheoretical

200637

Rempirical Rfitted

200356

The square sum of errors between the fitted and the empiri-cal is actually smaller than that between the theoretical and

the empirical. Table 2 compares the fitted model with the

theoretical model from the viewpoint of observed and

expected frequencies. From Table 2 we havenobserved ntheoretic

23164

nobserved nfitted2

2801

This implies that the fitted model agrees with the observed

data a little bit better than the theoretical model. We can

conclude that the fitted model based on the Weibull mixture

model allowing negative weights provides a good fit to thesystems failure data.

We can check results from the viewpoint of the WPP

plots which can be obtained by transforming data pair


Table 1

Simulated lifetime data (n 50)

3.7 5.1 6.4 7 7.4 7.7 8.4 9.2 10.2 10.9

4 5.6 6.7 7 7.4 7.8 8.7 9.4 10.2 11.1

4.2 5.8 6.7 7 7.5 7.8 9 9.4 10.2 12.9

4.7 6.1 6.8 7.1 7.5 8.1 9.1 9.6 10.4 13.4

4.8 6.3 6.9 7.2 7.7 8.4 9.1 9.9 10.5 14.2

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20

t

R(t)

theoritic

fitting

Fig. 1. The fitted, the theoretical, and the observedR(t) tcurves.


6/8

(tiRti) into (xiyi) by the Weibull transformation:

x lnt y ln{ lnRt}

Fig. 2 shows the WPP plots of observed data and the theo-

retical and the fitted models. The agreement between themshows that the fitted model is reasonable.

4.2. The system structure is known and R 0t is unknown

In this case we need to determine values of

00p2p3p4 in Eq. (11). We will firstly estimate

00. Once they are determined, the same procedure as

shown in Section 4.1 can be used to determine the weight

parameters.

We use the WPP plot to estimate 00. The Taylor

expansion ofRk0t can be expressed as:

Rk0t exp k

00

t0

1 k

00

t0 1

2k2

200

t20

fork 1 2 12

For small t(i.e. t 0), we take the first three terms of the

Taylor expansion in Eq. (12) to approximate Rk0t:

Rk0t 1k

00

t0

1

2

k2

200

t20 fork 1 2

13

Substituting the approximations in Eq. (13) into Eq. (8), we

get

RSt 14

200

t20 14

Noting ln1x xfor small x, we have

lnRS 4

200

t20 15

lnlnRS ln4 20lnt ln0 16

This gives left asymptote of the WPP plot of Eq. (8). It is

easily seen thatRStis over-estimated by Eq. (14) for small

t. This implies that the asymptote locates below the WPP

plot at the left end. We can fit a straight line to the left

several points of the WPP plot of the data, and denote the

Weibull parameters corresponding to the line as 0L 0L.

Geometrically, we have

0L 0 0L 0 17

On the other hand, when tis large, we have

R20tq R

30tq R

40tq R

50t

From Eq. (8), we have

RSt R20t 18

and in this way RStis under-estimated. From Eq. (18) we

get the right asymptote of the WPP plot:

y ln{ lnRSt} ln2 0lnt ln0 19

Because Eq. (18) under-estimates RSt, the asymptote in

Eq. (19) is above the real WPP curve at the right end (i.e.

ast ). We can use the few data points at the right end to

fit a straight line, and denote the Weibull parameters corre-

sponding to the line as 0Rand 0R. Then geometrically we


Table 2

Frequency distribution for the simulated data

Times to

failure

Observed

frequency

Expected frequency

(theoretical)

Expected frequency

(fitted, Section 4.1)

Expected frequency

(fitted, Section 4.2)

04 2 1.7 0.9 0.57

45 3 2.9 3 2.31

56 3 5 5.5 5.7867 10 6.9 7.7 8.79

78 10 8.1 8.8 8.68

89 5 7.9 8.3 7.57

910 7 6.6 6.6 7.50

1011 6 4.8 4.4 5.71

11 4 6.1 4.8 3.09

-6

-5

-4

-3

-2

-1

0

1

2

3

0 0.5 1 1.5 2 2.5 3

x

y

fitting

theoretic

Fig. 2. The WPP plot for the example system.


7/8

have

0R 0 0R 0 20

From Eqs. (17) and (20), we have the following:

0 min0L 0R

0 0L 0R

Thus, we can round down min0L 0Rand take that as an

estimate of0. The estimated 0can be taken equal to the

average value of estimates 0Land 0R.For the data given in Table 1, we used the left three points

and the right three points to fit the said straight lines and

obtained (refer to Fig. 2):

0L 5438 0L 6216

0R 3955 0R 11720

We can take 0 39 and 0 6216 117202 90.

Using these estimated parameters of the component life

distribution and the multivariate linear regression technique

gives:

p2 6748273p3 1872376p4 2224918

p5 9273685

Robserved Rfitted

20005855

nobserved nfitted2

2124

Please refer to Table 2 for a comparison of the observed and

fitted frequencies. The square sums of errors here are much

less than those obtained in Section 4.1. This illustrates the

appropriateness of the approach.

4.3. Discussions

The inverse Weibull model is a life distribution model

which is different from the Weibull model. It has found

many practical applications, for example, see Refs. [7]

and [8]. It has some similar characteristics to the Weibull

model. For example, both are flexible and convenient for

mathematical treatment. One can use the graphical method

or the linear regression method to estimate the parameters of

the inverse Weibull model in a similar way as used for the

Weibull model. This is done by the following transforma-

tions:

x lnt y ln{ lnFt} 21

Under the transformation given in Eq. (21), Eq. (3) becomes

y x ln

Fig. 3 shows the plot of the data listed in Table 1 under the

transformation given in Eq. (21).

If the components follow the Weibull distribution, the

system would follow the proposed Weibull mixture model

allowing negative weights. If the components follow the

inverse Weibull distribution, the system would follow the

proposed inverse Weibull mixture model allowing negativeweights. When component life distributions are unknown

and we only have the system failure data, we can try both

the Weibull mixture model and the inverse Weibull mixture

model to see which one fits the data better.

5. Summary

In this paper, we have shown that the negative weight

Weibull or inverse Weibull mixture model can be derived

to represent the output of a system under certain situations.

We have alsoproposed two generalised models as alternatives


-2

-1

0

1

2

3

4

5

0 0.5 1 1.5 2 2.5 3

x

y

Fig. 3. The inverse Weibull plot for the example system.


8/8

to approximate the output of a system. Two topics for

further study are:

to develop a methodology to determine specific forms

of the model for various application situations;

to develop efficient parameter estimation method for

the proposed models.

Acknowledgements

This research was partially supported by an Earmarked

Research Grant from the UGC of Hong Kong Government.

The constructive comments from the referees and editors are

very much appreciated.

References

[1] Titterington DM, Smith AFM, Makov UE. Statistical analysis of finite

mixture distribution, New York: Wiley, 1985.

[2] Jiang R, Failure models involving two Weibull distributions, PhD

Thesis, The University of Queensland, Australia, 1996.

[3] Jiang R, Murthy DNP. Modelling failure data by mixture of two

Weibull distributions: a graphical approach. IEEE Transactions on

Reliability 1995;44(3):477488.

[4] Jiang R, Murthy DNP. Mixture of Weibull distributions-parametric

characterisation of failure rate function. Applied Stochastic Models

and Data Analysis 1998;14:4765.

[5] Kuo W, Zhang W, Zuo M. A consecutive-k-out-of-n:G system: the

mirror image of a consecutive-k-out-of-n:F system. IEEE Transactions

on Reliability 1990;39(2):244253.

[6] Chiang DT, Niu SC. Reliability of consecutive-k-out-of-n: F system.

IEEE Transactions on Reliability 1981;R30:8789.

[7] Keller AZ, Giblin MT. Reliability analysis of commercial vehicle

engines. Reliability Engineering 1985;10:1525.

[8] Calabria R, Pulcini G. Confidence limits for reliability and tolerance

limits in the inverse Weibull distribution. Reliability Engineering and

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