C H A P T E R 10

Type III(c) Weibull Models

10.1 INTRODUCTION

A Type III(c) model involves n distributions and is derived as follows. Let Ti denote

an independent random variable with a distribution function FiðtÞ, 1 � i � n and let

Z ¼ maxfT1; T2; . . . ; Tng. Then the distribution function for Z is given by

GðtÞ ¼Yn

i¼1

FiðtÞ ð10:1Þ

This model is commonly referred to as the multiplicative model for obvious

reasons. In contrast to the competing risk model, this model has received very little

attention. The model has received some attention in the reliability literature where it

arises in the context of modeling a functionally parallel system with independent

components [see, for example, Shooman (1968, p. 206) and Elandt-Johnson and

Johnson (1980)]. Basu and Klein (1982) call this the complementary risk model.

The multiplicative risk model given by (10.1) is characterized by n (the number

of subpopulations) and the form of the distribution function for each of the subpo-

pulations. Some special cases of the model are as follows:

Model III(c)-1Here the n subpopulations are either the two-parameter Weibull distributions given

by (8.2) or the three-parameter Weibull distributions given by (8.3).

Model III(c)-2Here all the subpopulations are inverse Weibull distributions given by (8.4).

The outline of the chapter is as follows. Section 10.2 deals with Model III(c)-1.

The results for the general case are limited. In contrast, the special case n ¼ 2 has

Weibull Models, by D.N.P. Murthy, Min Xie, and Renyan Jiang.

ISBN 0-471-36092-9 # 2004 John Wiley & Sons, Inc.

197

been studied thoroughly. Section 10.3 deals with Model III(b)-2 and examines the

special case n ¼ 2.

10.2 MODEL III(c)-1: MULTIPLICATIVE WEIBULL MODEL

10.2.1 Model Structure

For the general n-fold Weibull multiplicative model, when all the subpopulations

are two-parameter Weibull distributions, we can assume, without loss of generality,

that b1 � b2 � � � � � bn, and ai � aj for i < j if bi ¼ bj. Finally, when all the

FiðtÞ’s are identical, the multiplicative model gets reduced to an exponentiated

Weibull model given by (7.24) and discussed in Section 7.5 with n ¼ n.

In this section we consider the general n with all the subpopulations being two-

parameter Weibull distributions. The results presented are from Jiang et al. (2001b).

Later, we look at the special case n ¼ 2.

10.2.2 Model Analysis

Distribution FunctionFor small t, GðtÞ can be approximated by

GðtÞ F0ðtÞ ð10:2Þ

where F0ðtÞ is a two-parameter Weibull distribution with parameters b0 and a0

given by

b0 ¼Xn

i¼1

bi a0 ¼Yn

i¼1

abi=b0

i ð10:3Þ

For large t, GðtÞ can be approximated by

GðtÞ ð1 kÞ þ kF1ðtÞ ð10:4Þ

where k is the number of subpopulations with distribution identical to F1ðtÞ.These results are used in establishing the asymptotic properties of the density

and hazard functions and the WPP plot.

Density FunctionThe density function is given by

gðtÞ ¼ GðtÞXn

i¼1

fiðtÞFiðtÞ

ð10:5Þ

The density function can be one of the following (n þ 1) shapes:

� Type 1: Decreasing

� Type (2k): k Modal (1 � k � n)

198 TYPE III(c) WEIBULL MODELS

Special Case ðn ¼ 2ÞJiang and Murthy (1997d) carry out a detailed parametric study in the three-

dimensional parameter space.

Hazard FunctionThe hazard function is given by

hðtÞ ¼ GðtÞ1 GðtÞ

Xn

i¼1

fiðtÞFiðtÞ

ð10:6Þ

The asymptotes of the hazard function are given by

hðtÞ h0ðtÞ ¼ b0

a0

ta0

� �b01

as t ! 0

h1ðtÞ ¼ b1

a1

ta1

� �b11

as t ! 1

8><>: ð10:7Þ

where h0ðtÞ and h1ðtÞ are the hazard functions associated with F0ðtÞ and F1ðtÞ,respectively.

The different possible shapes for the hazard function can be grouped into four

groups:

� Type 1: Monotonically decreasing

� Type 3: Monotonically increasing

� Type (4k þ 2): k modal followed by increasing (1 � k � n 1Þ� Type (4k þ 3): Decreasing followed by k modal (1 � k � n 1)

It is worth noting that failure rate can never be decreasing for small t if it is

increasing for large t, hence ruling out the possibility for the failure rate to have

a bathtub shape.

Special Case ðn ¼ 2ÞJiang and Murthy (1997d) carry out a detailed parametric study in the three-

dimensional parameter space. The boundaries separating the regions with different

shapes are complex.

WPP PlotUnder the Weibull transformation given by (1.7) and FiðtÞ given by (8.2), (10.1)

gets transformed into

y ¼ lnf ln½1 GðexÞ�g ð10:8Þ

The WPP plot is analytically intractable. Jiang et al. (2001b) propose an approxi-

mation (see Section 10.2.4 for more details), which makes the WPP analytically

MODEL III(c)-1: MULTIPLICATIVE WEIBULL MODEL 199

tractable. Based on this approximation, they show that the WPP plot is concave and

verify this through extensive numerical studies.

Based on this approximation, we have the following asymptotic results for the

WPP plot. As x ! 1 (or t ! 0), the left asymptote is a straight-line yL given by

yL ¼ b0ðx ln ða0ÞÞ ð10:9Þ

where b0 and a0 given by (10.3). As x ! 1 (or t ! 1), the right asymptote is

given by another straight line L1, which is the WPP plot for subpopulation 1, and

is given by

y ¼ b1ðx ln ða1ÞÞ ð10:10Þ

Special Case ðn ¼ 2ÞThe WPP plot is discussed in detail in Jiang and Murthy (1995b). In this case

b0 ¼ b1 þ b2 and a0 ¼ ða1Þb1=b0ða2Þb2=b0 ð10:11Þ

As a result, the left asymptote has a slope ðb1 þ b2Þ and the right asymptote has a

slope b1. Figure 10.1 shows a typical WPP plot along with the two asymptotes.

10.2.3 Parameter Estimation

Graphical MethodIf n is specified (see Section 10.2.4 regarding how to estimate n based on the data),

then Jiang et al. (2001b) propose a method for estimating the model parameters

based on the WPP plot. It involves estimating the parameters of one subpopulation

based on the fit to the right (or left) asymptote. The data set is modified to remove

0.5 1 1.5 2 2.5 3

−6

−4

−2

0

2

4 WPP Plot:Right Asymptote:Left Asymptote:

Figure 10.1 WPP plot for the Weibull multiplicative model ða1 ¼ 3:0; b1 ¼ 1:5; a2 ¼ 5:0; b2 ¼ 2:5Þ.

200 TYPE III(c) WEIBULL MODELS

the data from this subpopulation following a procedure similar to that used in Sec-

tion 9.2.3. The WPP plot of the modified data is carried out and the process is

repeated. The parameters of the last subpopulation are estimated by fitting a straight

line to the WPP plot of the last modified data.

Note that this procedure only yields crude estimates. These are useful as they can

be used as initial values for parameter estimation using more refined methods.

Special Case ðn ¼ 2ÞJiang and Murthy (1995b) discuss the estimation for this case. It involves the

following steps:

Steps 1–5: As in Section 4.5.1 to obtain the WPP plot of the data.

Step 6: Fit the left asymptote to the WPP plot. The slope and intercept of this

line yields estimates of b0 and a0.

Step 7: Fit the right asymptote to the WPP plot. The slope and intercept of this

line yields the estimates of b1 and a1.

Step 8: Obtain estimates of b2 and a2 from (10.11).

Optimization MethodLing and Pan (1998) consider a twofold multiplicative model involving three-

parameter Weibull distributions. The model parameters are obtained by minimizing

the maximum absolute difference between the observed probability of failure and

the expected probability of failure. The Minimax algorithm in the Matlab Toolbox

Optimization is used to obtain the estimates.

10.2.4 Modeling Data Set

Given a data set, one plots the data on the WPP plot. The plotting procedure

depends on the type of data (complete, censored, grouped, etc.) as discussed in

Chapter 5. If the WPP plot of the data is roughly concave, then an n-fold Weibull

multiplicative model can be considered a potential model for the data set. Jiang et al.

(2001b) suggest a two-part method, based on the WPP plot, as indicated below.

Part 1: Estimation of nThe basis for this is that for small t (and applicable for intermediate values) the

Weibull multplicative model can be approximated by an exponentiated Weibull

model with parameters a, b, and n ¼ n. The WPP plot for the exponentiated

Weibull model has the left asymptote given by

yE ¼ nbðx ln ðaÞÞ ð10:12Þ

The WPP plot of the multplicative Weibull model has the left asymptote given by

(10.9). If the selected value of n is appropriate, then the two straight lines should be

close to each other.

MODEL III(c)-1: MULTIPLICATIVE WEIBULL MODEL 201

Note that a0 and b0 in (10.9) can be estimated from the slope and intercept of the

fit to the left asymptote of the WPP plot. For a given n, b and a in (10.12) can be

obtained from the following linear regression equation using the data set:

lnð lnf1 ½FðtÞ�1=ngÞ ¼ bðx ln ðaÞÞ ð10:13Þ

The following quantity measures the closeness between the lines yL and yE:

DðnÞ ¼ðx2

x1

½yLðxÞ yEðxÞ�2 dx ð10:14Þ

The term D(n) is evaluated for small values of x1 and x2. Jiang et al. (2001b) suggest

x1 ¼ lnðtð1ÞÞ 0:5 and x2 ¼ lnðtð1ÞÞ þ 0:5, where t(1) is the smallest value in the

order data set. Using (10.9) and (10.12) in (10.14) yields

DðnÞ ¼ A2=12 þ ðA ln ðtð1ÞÞ þ BÞ2 ð10:15Þ

where

A ¼ b0 nb B ¼ lnðanb=ab0

0 Þ ð10:16Þ

The term DðnÞ is small if the selected value of n is close to the true value of n. As

such, we can use the following procedure to determine n:

Step 1: Fit a tangent line to the left end of the WPP plot of the data and obtain

estimates of a0 and b0 using (10.9).

Step 2: For a specified value of n (¼ 2, 3, 4, . . .) and using data points (ti, F(ti))

such that F(ti) � 0.8 obtain the estimates (aa, bb), which are functions of n,

from the following regression:

yðiÞ ¼ bðxðiÞ ln ðaÞÞ ð10:17Þ

where xðiÞ ¼ ln ðtiÞ; yðiÞ ¼ lnð lnf1 ½FFðtiÞ�1=ngÞ.Step 3: Compute the DðnÞ from (10.16) for n ¼ 2, 3, 4, . . . , using the estimates

from steps 1 and 2. The value of n that yields the minimum for D(n) is the

estimate of n.

Part 2: Estimation of Subpopulation ParametersThis is done using the approach discussed in Section 10.2.3.

Ling and Pan (1998) propose another approach for model selection. They tenta-

tively assume several models as candidate models and estimate the model para-

meters. Then they use a goodness-of-fit test to choose the most appropriate

model from the set of models.

202 TYPE III(c) WEIBULL MODELS

10.2.5 Applications

Jiang and Murthy (1995b) use a twofold Weibull multiplicative model to model

several different data sets. These include failure times for transistors, shear strength

of brass rivets, and pull strength of welds.

10.3 MODEL III(c)-2: INVERSE WEIBULL MULTIPLICATIVE MODEL

10.3.1 Model Structure

For the general n-fold inverse Weibull multiplicative model, if bi ¼ bj ¼ b; i 6¼ j,

then the two subpopulations can be merged into one subpopulation with the com-

mon shape parameter b and with a new scale parameter ðabi þ abj Þ1=b

. Hence, we

can assume, without loss of generality, that bi < bj for i < j. Note that this differs

from the n-fold Weibull multiplicative model where the parameters are not so

constrained.

The general case is yet to be studied. The special case n ¼ 2 has been studied

by Jiang et al. (2001a), and we discuss this model in this section. The model is

given by

FðtÞ ¼ F1ðtÞF2ðtÞ t � 0 ð10:18Þ

with the two subpopulations are the two-parameter inverse Weibull distribution

given by (8.4).

10.3.2 Model Analysis

Distribution FunctionFor small t we have

FðtÞ F2ðtÞ ð10:19Þ

and for large t we have

FðtÞ F1ðtÞ ð10:20Þ

Density FunctionThe density function is given by

f ðtÞ ¼ f1ðtÞF2ðtÞ þ f2ðtÞF1ðtÞ ð10:21Þ

Since the density functions for the two subpopulations are unimodal, one should

expect the possible shapes of the density function to be (i) unimodal and (ii) bimo-

dal. However, computer plotting of the density function for a range of parameter

values show that the shape is always unimodal.

MODEL III(c)-2: INVERSE WEIBULL MULTIPLICATIVE MODEL 203

An explanation for this is as follows. For the density function to be bimodal, the

two subpopulations must be well separated. Without loss of generality, assume that

tm1 � tm2, where tm1 and tm2 are the modes for the two subpopulations. Over the

interval t � tm1 and in the region close to t ¼ tm1, F2ðtÞ 0 and f2ðtÞ 0, and as

a result f ðtÞ 0. Thus, the density function cannot have a peak in a region close to

t ¼ tm1. This implies that f ðtÞ is unimodal.

The above results are in contrast to the twofold Weibull multiplicative model,

which exhibits bimodal shape for the density function. This is because the Weibull

model does not have a light left tail so that the condition F2ðtÞ 0 and f2ðtÞ 0 for

t � tm1 do hold.

Hazard FunctionThe failure rate function is given by

rðtÞ ¼ r1ðtÞpðtÞ þ r2ðtÞqðtÞ ð10:22Þ

where

pðtÞ ¼ F2ðtÞ½1 F1ðtÞ�1 F1ðtÞF2ðtÞ

qðtÞ ¼ F1ðtÞ½1 F2ðtÞ�1 F1ðtÞF2ðtÞ

ð10:23Þ

Since the failure rates of two subpopulations are unimodal, the possible shapes of

the hazard functions are expected to be unimodal and bimodal. However, computer

analysis indicate that the hazard function shape is always unimodal (type 5).

IWPP PlotUnder the inverse Weibull transform given by (6.57), the IWPP plot is given by

y ¼ ln½z1ðxÞ þ z2ðxÞ� ð10:24Þ

where

ziðxÞ ¼ ðt=aiÞbi ¼ ðex=aiÞbi ¼ exp½biðx lnðaiÞÞ� ð10:25Þ

It is easily shown that

y0ðxÞ ¼ b1z1 þ b2z2

z1 þ z2

y00 ¼ ðb1 b2Þ2z1z2

ðz1 þ z2Þ2< 0 ð10:26Þ

so that the IWPP plot is concave.

204 TYPE III(c) WEIBULL MODELS

The two asymptotes for the IWPP plot are given by L2 as x ! 1 (or t ! 0)

and by L1 as x ! 1 (or t ! 1) where L2 and L1 are the IWPP plots for the two

subpopulations. Let (xI , yI) denote the coordinates of the intersection of L1 and L2.

Then we have

yðxIÞ ¼ yI ln ð2Þ y0ðxIÞ ¼b1 þ b2

2ð10:27Þ

Figure 10.2 shows a typical IWPP plot along with the two asymptotes.

An important observation is that the IWPP plot for the inverse Weibull multipli-

cative model is similar to the WPP plot for the twofold Weibull multiplicative

model. Also, the left (right) asymptote for IWPP plot for the inverse Weibull

multiplicative model is the same as the right (left) asymptote for the WPP plot

for the Weibull competing risk model.

10.3.3 Parameter Estimation

Graphical ApproachThe estimation based on the IWPP plot involves the following steps:

Steps 1–5: Use data to plot the IWPP plot.

Step 6: Fit the left asymptote to the IWPP plot. The slope and intercept of this

line yields estimates of b2 and a2.

Step 7: Fit the right asymptote to the IWPP plot. The slope and intercept of this

line yields the estimates of b1 and a1.

Note: One should ensure that the two asymptotes are selected so as to satisfy

(10.27).

1.5 2 2.5 3 3.5 4−1

0

1

2

3

4

WPP Plot:Left Asymptote:Right Asymptote:

Figure 10.2 IWPP plot inverse Weibull for multiplicative model ða1 ¼ 3:0;b1 ¼ 1:5;a2 ¼ 5:0;

b2 ¼ 3:5Þ.

MODEL III(c)-2: INVERSE WEIBULL MULTIPLICATIVE MODEL 205

10.3.4 Modeling Data Set

If the IWPP plot of a given data set has a shape similar to that shown in Figure 10.2,

then one can model the data by a twofold inverse Weibull multiplicative model. The

parameters can be estimated following the procedure outlined in Section 10.3.3.

EXERCISES

Data Set 10.1 Complete Data: All 50 Items Put into Use at t ¼ 0 and Failure Times Arein Weeks

1.578 1.582 1.858 2.595 2.710 2.899 2.940 3.087 3.669 3.848

4.740 4.848 5.170 5.783 5.866 5.872 6.152 6.406 6.839 7.042

7.187 7.262 7.466 7.479 7.481 8.292 8.443 8.475 8.587 9.053

9.172 9.229 9.352 10.046 11.182 11.270 11.490 11.623 11.848 12.695

14.369 14.812 15.662 16.296 16.410 17.181 17.675 19.742 29.022 29.047

Data Set 10.2 Failure Times of 50 Itemsa

0.061 0.073 0.075 0.084 0.086 0.087 0.088 0.089 0.089 0.089

0.099 0.102 0.117 0.118 0.119 0.120 0.123 0.135 0.143 0.168

0.183 0.185 0.191 0.192 0.199 0.203 0.213 0.215 0.257 0.258

0.275 0.297 0.297 0.298 0.299 0.308 0.314 0.315 0.330 0.374

0.388 0.403 0.497 0.714 0.790 0.815 0.817 0.859 0.909 1.286

a Unit: 1000 h.

Data Set 10.3 The data is the censored data from Data Set 9.2 with the data

collection stopped after 15 weeks.

Data Set 10.4 The data is the censored data from Data Set 9.2 with the data

collection stopped after 300 h.

10.1. Carry out WPP and IWPP plots of Data Set 10.1. Can the data be modeled

by one of Type III(a), Type III(b), or Type III(c) models involving either two

Weibull or two inverse Weibull distributions?

10.2. Suppose that Data Set 10.1 can be modeled by a twofold Weibull multi-

plicative model. Estimate the model parameters based on (i) the WPP plot,

(ii) the method of moments, and (iii) the method of maximum likelihood.

Compare the estimates.

10.3. Suppose that the data in Data Set 10.1 can be adequately modeled by the

Weibull multiplicative model with the following parameter values:

206 TYPE III(c) WEIBULL MODELS

a1 ¼ 2:0, b1 ¼ 0:6, a2 ¼ 8:0, and b2 ¼ 1:6. Plot the P–P and Q–Q plots and

discuss whether the hypothesis should be accepted or not?

10.4. How would you test the hypothesis of Exercise 10.3 based on A–D and K–S

tests for goodness of fit?

10.5. Repeat Exercise 10.1 with Data Set 10.2.

10.6. Suppose that Data Set 10.2 can be modeled by an inverse Weibull multi-

plicative model. Estimate the model parameters based on (i) the IWPP plot,

(ii) the method of moments, and (iii) the method of maximum likelihood.

Compare the estimates.

10.7. Repeat Exercises 10.1 to 10.4 with Data Set 10.3.

10.8. Repeat Exercises 10.5 and 10.6 with Data Set 10.4.

10.9. How would you generate a set of simulated data from a twofold Weibull

multiplicative model? Can the method be used for a general n-fold Weibull

multiplicative model?

10.10. Consider a twofold multiplicative model involving a two-parameter Weibull

distribution and an inverse Weibull distribution. Study the WPP and IWPP

plots for the model.*

10.11. For the model in Exercise 10.10 derive the maximum-likelihood estimator

for (i) complete data and (ii) censored data.

10.12. Consider a twofold multiplicative model involving a two-parameter Weibull

distribution and an exponentiated Weibull distribution. Study the WPP plots

for the model.*

* Research problem.

EXERCISES 207