Parametric study of sectional models involving two Weibull distributions

Reliability Engineering and System Safety 56 (1997) 151-159

Pll:

ELSEVIER

S0951-8320(96)00114-7

© 1997 Elsevier Science Limited All rights reserved. Printed in Northern Ireland 0951-8320/97/$17.00

Parametric study of sectional models involving two Weibull distributions D. N. P. Murthy & R. Jiang* Department of Mechanical Engineering, The University of Queensland, St. Lucia, Q 4072, Australia (Received 24 April 1996; accepted 4 September 1996)

Sectional models involving two We;bull distributions are characterized by four independent parameters. The shapes of the density and failure rate functions for the models depend on the parameter values. A complete parametric characterization of these shapes is presented in this paper, t~) 1997 Elsevier Science Limited.

model. A similar study for the other three models can be found in Ref. 2. The shapes of the density and failure rate functions for such models depend on the parameter values. Parametric study of these shapes is of interest in the context of model selection when the failure data is displayed as a histogram or as an empirical failure rate plot. Jiang [3] gives a complete parametric characterization of the density function for a mixture model involving two We;bull distributions. A similar study of the failure rate function can be found in Ref. 4. A complete parametric study of the density and failure rate functions for competing risk and for multiplicative models involving two We;bull distributions can be found in Refs 5 and 6, respectively. Sectional models involving two We;bull distributions are characterized by six parameters. However, there are only four independent parameters as two are constrained to ensure that the distribution and density functions are continuous. As a result, the shapes of the density and failure rate functions depend on the two scale parameters ('q;, i = 1,2) and the two shape parameters (/3;, i = 1,2). In this paper, we carry out a complete parametric characterization of the density and failure rate functions for this model. The outline of the paper is as follows. We commence with a review of the literature on sectional models involving We;bull distributions in Section 2. Section 3 gives the parametric characterization of the density and failure rate functions for a single We;bull distribution. In Section 4 we give the mathematical details of the two sectional models examined in this paper. Sections 5 and 6 deal with the parametric characterization of the density and failure rate functions for Model 1. Similar results for Model 2 are

NOTATION

R(t) [Ri(t)]: F(t) [F/(/)]: f(t) [f(/)]: h(t) [h;(t)]: ~i, /3i:

k:

Reliability function of population [subpopulation i, i -- 1,2] Distribution function of population [sub-population i, i = 1,2] Density function of population [subpopulation i, i = 1,2] Failure rate of population [subpopulation i, i = 1,2] Scale and shape parameter for F~(t), i = 1,2 Location parameter [for Models 1 and 2] Shift parameter [for Model 1] Additional parameter [for Model 2] = 172/'!71

/3:

=/32//31

~i:

Point at which f~(t), i = 1, 2, achieves a maximum.

1 INTRODUCTION When a given data set cannot be adequately modeled by a single We;bull distribution, a natural alternative is to examine models involving two We;bull distributions. Many models involving two We;bull distributions have been developed. These include mixture, competing risk, multiplicative and sectional models. Jiang and Murthy [1] deal with the mixture model and examine the graphical plotting approach to determine if a given data set can be modeled by such a * Current address: Department of Automobile Engineering, Changsha Communications Institute, Changsha, Hunan 410076, P.R. China. 151

152

D. N. P. Murthy, R. Jiang

given in Sections 7 and 8. Finally, we conclude with some comments in Section 9.

2 REVIEW OF THE LITERATURE In the sectional model (also known as composite model, piece-wise model or step function model), the failure distribution over different time intervals is given by a different function. As a result, F(t) over n successive intervals is given by F(t) =

i - (1 - F~(t)) - k2(l - Fz(t)) -kn(1 -F,(t))

for t e (0,t~) for t ~ (t~,t2) fort e (t,_a,~)

where F,-(t) are distribution functions and ki > 0, for i = 2, 3,..-, n. This model (henceforth called a n-fold sectional model) offers greater flexibility and is suited to modelling a variety of complex failure data. If F~(t), i = 1,..., n, are Weibull distributions (with 2 or 3 parameters) then we call it a n-fold Weibull sectional model. Kao [7] uses a 2-fold 2-parameter Weibull sectional model to approximate a 2-fold Weibull mixture model. The cdf of the model is continuous but the density and failure rate functions are discontinuous. Aroian and Robison [8] deal with sequential life tests for an exponential distribution with changing parameter. In their model, the failure rate A(t) is given by A(t) = A,, R,(t) = a~exp( - Ad) fort in l i : [ t i - a , l i ) , i

= 1 .....

k + 1

with al=l, ai=exp

-

tj_ (Aj-Aj+ls)

.

The parameter ai ensures that R(t) is continuous at the partition point t = t , _ l . Obviously, f ( t ) and A(t) are not continuous. Note that this is a (k + 1)-fold sectional model. Shooman [9] discusses a piece-wise-linear model. It is based on a well known method of approximating a function by straight lines over different regions. His motivation was to develop a model with bathtub failure rate. In his model the survivor function, R(t), is given by f e x p [ - (alt - a2t2)] for 0 < t -< tl R ( t ) = ~ a 3 e x p [ - a4(t - t,)] for t, < t --
This is not a Weibull sectional model. However, the first and third expressions can be viewed as WeibuU competing risk models.

Mann et al. [10] deal with a Weibull sectional model and present results similar to that of Kao [7]. Colvert and Boardman [11] present a piece-wise constant hazard (or failure) rate model which is similar to that of Aroian and Robison [8]. They give four possible reasons for justifying the failure rate being modeled as a piece-wise constant. Firstly, it yields a bathtub shape for the failure rate function. Secondly, there are often sound physical reasons for the failure rate being discontinuous at certain time points--for example, periodic maintenance of the item involving repacking of bearings, changing of oil, or cleaning the filter and so on. Thirdly, such a model is relatively simple. Finally, the model is appropriate when the empirical failure rate, obtained from failure data sets, is roughly piece-wise constant. Elandt-Johnson and Johnson [12] discuss the problem of fitting different distribution functions over successive periods of time. They demonstrate that cancer data often indicate rather high mortality rates in the first few years after diagnosis, but the patients who survive this critical period seem to experience subsequent mortality similar to that in the general population of the same age. This implies that it is not possible to fit a single distribution function of a simple known mathematical form over the whole range of data and piece-wise function becomes a natural alternative. They define a sectional model with the cdf being continuous at all partition points. They present an example of fitting a sectional model using Weibull hazard plot. Jiang and Murthy [2] have studied the shapes of the WPP plots for the two sectional models discussed in this paper. They also examine the estimation of the model parameters based on such plots. The application of these models to model real data sets can be found in Refs 2 and 3. Finally, it is difficult to justify sectional models on any physical grounds. It is useful in reliability modelling due to its flexibility. However, this flexibility is achieved at the expense of an increased set of model parameters.

3 SINGLE WEIBULL DISTRIBUTION In this section we give a parametric characterization of the density function and failure rate for a single Weibull distribution. 3.1 Parametric characterization of density function The reliability function (rf) and the probability density function (pdf) of a 2-parameter Weibull distribution are given by R(t) = exp( - (t/rto) ~o) (1)

Parametric study o f sectional models and

f ( t ) = - dR(t)/dt = -/3oR(t)ln(R(t))/t

(2)

where /3o and 7/0 are the shape and scale parameters assuming values in the interval (0, oo). From eqn (2), we have, on differentiation f ' ( t ) =/3of(t)[ln(R)(t)) + (1 - 1//3o)]/t.

(4)

Note that the shape of the pdf is solely determined by/30 and 77o has no influence. The results for the 3-parameter Weibull distribution are similar to that for the two-parameter case except that, when/30 > 1, the peak of f(t) occurs at

tv = y + r/o(1 - 1//3o) l/&.

r'(t) = 0 and in this case fr is a constant. When 13o> 1, r'(t) is positive and the fr shape is Type B of Fig. 2. Note that again, the fr shape is solely determined by /30 and r/o has no influence. The results for the 3-parameter Weibull distribution are similar to that for the two-parameter case.

(3)

When /30 < - 1, if(t) is negative for 0<-t < ~. As a result, f(t) is decreasing and the pdf shape is Type A of Fig. 1. When/30 > 1, f ' ( t ) has one zero. In this case, f(t) is unimodal and the pdf shape is Type B of Fig. l. The peak of f ( t ) occurs at tp given by

tp = r/0(1 - 1//3o) v~o.

153

(5)

3.2 Parametric characterization of failure rate function

4 S E C T I O N A L M O D E L S I N V O L V I N G TWO W E I B U L L DISTRIBUTIONS In this section, we give the mathematical details of the two sectional models, each involving two Weibull distributions. 4.1 Model 1 In this model, over the interval [0, to), F(t) is given by a two parameter Weibull distribution F~(t) (with shape parameter /31 and scale parameter ~1) and, over the interval [to, oo), it is given by a three parameter Weibull distribution (with shape parameter /32, scale parameter T/2 and shift parameter y). Type A

Type B

Type A 1

Type B 1

Type A2

Type B2

The failure rate (fr) function is given by

r(t) = f ( t ) / R ( t ) :--/3o(t/Tqo)~O-1/rlo.

(6)

The shape of the fr can be characterized by studying its derivative. From eqn (6), we have, on differentiation,

r'(t) =/30(/30- 1)(t/~o)~-2/(TIo) 2.

(7)

/

When do < 1, r'(t) is negative for 0 <-t < ~. As a result, the fr shape is Type A of Fig. 2. When do = 1, Type A

Type B

v

t Type C

f-,

/

Type D Type C

Type D

t

Fig. 1. Plots of pdf shapes (Type A-D) for sectional models involving two Weibull distributions.

Fig. 2. Plots of fr shapes for sectional models involving two Weibull distributions.

D. N. P. Murthy, R. Jiang

154

As a result, the reliability function (R(t)), the density function (f(t)) and the failure rate function (h(t)) for the model are given by f

and

h(t) = ~ (fll/'qO(t[~l)~'-l'

O<-t<-t°

(15)

t(/321n2)(tln2) a2-t, to < t < ~.

O<---t<--to

exp[-(t/r/l)~,],

The continuity conditions at to require that exp -

,

to
(16)

k = exp[(1 -/3)(to/n2)~q.

(17)

to = -

~exp[ - (t/nl)",l,

o -< t <- tO

and

|

f(O = l(/3 2/712

)(~)'~-texp[

--

t~]

,

to
and

f (/3'/rh)(t/rh) ~'-1, h(t) = { /t - 7\ '~2-' '

O<_t<_to (lo) t0
The model is characterized by six parameters--(/3~, rh, i = 1,2), 3, and to. We assume that both R(t) and f(t) are continuous at to. This requires that the following two conditions be satisfied. t,>

Note that in this model we require fl1#/32, otherwise to = 0 or ~ and the model reduces to a single Weibull distribution. As in Model 1, there are only four independent parameters--/3~, rl,, i = 1, 2.

= [rf,l(/3irlz)m-] u(~,-ep

(11)

and

r = (1 -/3)t,,

02)

where/3 = fit~fit. As a result, the model has only four independent parameters /3~, rl~, i = 1, 2. Jiang and Murthy [2] assume fll >/32 to ensure y > 0 . This is restrictive since we only need to have to - y > 0. From eqn (12) it can be seen that this inequality is always satisfied and hence y need not be constrained to satisfy y > 0. Note that we require/3~ #/32, otherwise to = 0 or o0 and in this case the model reduces to a single Weibull distribution.

4.3 Comments

For both models, the number of independent parameters is four. The shape of density function depends only on the two shape parameters (/3t and /32) and the ratio of the scale parameters (rl2/rh). As a result, the parametric characterization will be carried out in the two dimensional plane of /31-/32 for different values of 7/. The shapes of the failure rate function is not influenced by the two scale parameters. Hence, the parametric characterization will be done in the two dimensional plane of/31 -/32. Let tm and t02 represent the to for Model 1 and Model 2, respectively. For the same set of model parameters (/31, /32, rh and r/2), we have, from eqns (11) and (16) >to2,

when/32 < 1

to1 =to2, w h e n / 3 2 = 1 < toz, w h e n 1 3 2 > l .

4.2 Model 2 In this model, over the interval [0, to), F(t) is given by a two parameter Weibull distribution F~(t) (with shape parameter /31 and scale parameter 771) and, over the interval [to, oc), it is given by a different two parameter Weibull distribution (with shape parameter /32 and scale parameter "qz) scaled by a factor k so that R(t) and f(t) are continuous at to. As a result, the reliability function (R(t)), the density function (f(t)) and the failure rate function (h(t)) for the model are given by

R(t) = [exp[

-

(l/lh)fll],

tkexp[ - (t -/-q2)t~2],

O<_t<_to to
(13)

[(/3,/rh)(t/rl,) ~, ' e x p [ - (t/rh)°,], O<_t<_to f(t) = ik(/32/rlz)(t/r12)~< ,exp[ - t/r12)¢21, t o < t < ~. (14)

Note that the shapes of the plots (WPP, pdf and fr) for Model 1 and Model 2 are similar. This raises the question: which one is more appropriate for a given data set? The answer can be found in Ref. 2.

5 PARAMETRIC CHARACTERIZATION OF DENSITY FUNCTION FOR MODEL 1

The shape o f f ( t ) is given by fl(t) over [0, to) and f2(t) over [to, ~). Since both fl(t) and )~(t) can have two shapes depending on the scale parameters, one can expect f(t) to have one of four possible shapes (Types A - D ) shown in Fig. 1. We consider the two cases /31 >/32 and/31
Parametric study of sectional models 5.1 Case (i): fix > flz . When /32 1 and /32 <- 1, fl(t) is unimodal and )~(t) is decreasing. In this case, f(t) is unimodal and the pdf shape is Type B. . When /31>/32>1, both )~(t) and rE(t) are unimodal. The shape of f(t) depends on the relationship between tpl, tp2 and to where tpi ( i = 1,2) is the point where the peak of f,(t) (i = 1,2) occurs. From eqns (4) and (5), we have

tpl = r/l(1 - 1//31) 1'~',.

(18)

tp2 = "o2(1 - 1//32) 1/°2 + %

(19)

We need to consider the following four cases: (a) to <- min(tpl, tp2), (b) to -> max(tpl, tp2), (c) tp2 < to < tpx and (d) tpl < to < tp2. In the first three cases ((a)-(c)), the pdf shape is always Type B. We show in the Appendix that case (d) is not possible. As a result, the pdf shape can be only either Type A or Type B. The shape is determined by the two shape parameters and the two scale parameters having no influence. Figure 3 shows the parametric characterization of the pdf in the/31-/32 plane.

5.2 Case (ii): fll ( /32 . When /31
2. When 131 - 1 tp2 or to < tp2. In the /31-/32 parameter plane, let F denote the demarcation curve between these two shapes. F is given by the relationship to = tp2 where to is given by eqn (5) and tp2 by eqn (19). As a result, F is given by "0 =/3(1 - 1//32)°~-1)%

(20)

where 7? and/3 =/32//31. Figure 4 shows F, for a range of r/values, in the/31-/32 plane for the region 0 <-/3t < 1 <-/32. For a given 7/, the pdf shape is Type C above F and Type A below. As can be seen, F moves up with r/increasing. 3. When 1
172/'t~1

7? =/3(1 - 1//31)(~-1)/~.

(21)

r2 is obtained from the relationship to = tp2 and is given by eqn (20). Figure 5 shows these two curves in the/31-/32 plane for a given r/. In the region between rl and r:, the pdf shape is Type D and elsewhere in the region (/32 >/31,

~

4

¢q

155

rl-5.0

i

~1-3.0

3

/

B . ..... "

0

.-

•

i

0.2

0.4

i

i

i

i

L

0.6

0.8

1

1~

1.4

1.6

pt

Fig. 3. [Model 1] Parametric characterization of pdf for /31 >/32.

Fig. 4. [Model 1] Parametric characterization of pdf for a range of r/and/31
D. N. P. Murthy, R. Jiang

156 t0

for a range of rI values. As can be seen, both the curves move up as 77 increases.

B

B

6 P A R A M E T R I C C H A R A C T E R I Z A T I O N OF FAILURE RATE FUNCTION FOR M O D E L 1

:./

e |

s:

Note that h(t) = hi(t) for 0 -< t < to and h(t) = h2(t) for to<-t< ~. Since both hi(t) and h2(t) can have one of three different shapes, the shape of h(t) can be one of //

2

....A - 7 I

0

4

5

I

7

8

10

pt

Fig. 5. [Model 1] Parametric characterization of pdf for a given rl.

eight different shapes shown in Fig. 2. They can be divided into four groups: non-increasing (Type A, A1 and A2), non-decreasing (Type B, B1 and B2), bath-tub (Type C) and unimodal (Type D). Since the scale parameters 071 and ~72) have no influence on hi(t) and h2(t), we carry out the parametric characterization of the failure rate in the /31-/32 plane. We consider the two cases /31
separately.

6.1 Case (i): fit < fll 1
n-.3

10 [

~-2

~-1.5

.L

1. When / 3 2 < / 3 1 < 1 , both hi(t) and h:(t) are decreasing and as a result the fr shape is Type A. 2. When/32
/ / / / /

6.2 Case (ii): fll < f12

/ / / /

By following an approach similar to Case (i), the complete parametric characterization of the failure rate in the/31-/32 plane for this case is as shown in the upper triangle of Fig. 7. Note that in this case we have an additional shape C.

/ / / / /

a[-~. .-/'" i

/

. /

/ o

]1// 0

I I

I 2

I

I

5

e

I 7

Fig. 6. [Model 1] Parametric characterization of pdf for a range of rl and 1
7 PARAMETRIC CHARACTERIZATION DENSITY FUNCTION FOR M O D E L 2

OF

As in Model 1, the shape of the density function for this model can be one of four different shapes shown

Parametric study of sectional models

157

The density function cannot have a Type D shape. The proof of this is similar to that for Model 1 and hence omitted. 7.2 Case (ii): fll < f12 As in Model 1, when /3~
k2

/

tp2 = 7/2(1 - 1//32) t/e2.

(22)

As a result, F is given by 7/= [/3(1 - 1//32)1-1/~]1/~1.

~ " " ~ " A1 pl

Fig. 7. [Model 1] Parametric characterization of fr. in Fig. 1. As before, we consider the two cases/31 f12 The parametric characterization of the density function in the /31--/32 plane is identical to that for Model 1. The two possible shapes are Types A and B.

(23)

Figure 8 shows F in the/31-/32 plane for a range of 77. For a given rl, the pdf shape is Type A below F and Type C above. As can be seen, F moves up as 7/ increases. When 1
(24)

For 77> 1.15, there is no/31 and/32 (1
3.S ~-2

2.5

p~ 2

1.$

8 PARAMETRIC CHARACTERIZATION F A I L U R E R A T E FOR M O D E L 2 1"1-0.3

OF

~-0.6

0.5

0 0.2

0.4

t 0.6

I 0.8

I 1

I ~ .2

II 1.4

Fig. 8. [Model 2] Parametric characterization of pdf for a range of r/and fl~ < f12.

The failure rate is given by eqn (15). Note that it is similar to eqn (10) except that h2(t) in eqn (10) has the shift parameter 3,. As indicated in Section 2, 3' has no influence on shape type of the failure rate. As a result, the parametric characterization of the failure rate for Model 2 is identical to that for Model 1.

158

D. N. P. Murthy, R. Jiang

(-) 10 9 8 7 6 5 4 3 2 1 0

1"2 8

q=0.5

I.

/

o09'1 [ //// /

'! 7 o

i

2

1

i

i

,

,

3

4

5

6

4

3:-

131

I I Y IAI

.1'1=k°5

1

(b) 10 9 8 7 6 5 4 3 2 1 o

"q=l.

I I

1"!

131 Fig. 10. [Model 2] Parametric characterization of pdf for a range of r/and 1
7 0

1

J

i

i

i

i

2

3

4

5

6

1. The density function shape can be one of four different types. 2. The failure rate shape can be one of eight different types.

131 These make the sectional models attractive for modelling a range of failure data. In Model 2, the density function for the model is given by f ( t ) = f a ( t ) for 0 - < t < t 0 and =kf2(t) for to-< t < oo where f~(t) is the density function associated with the distribution function F~(t), i = 1 , 2. A different sectional model (Model 2a) is as follows:

(c) 10 9 8 7 6 5 4 3 2 1 0

1"]

2

f ( t ) = kfl(t) for 0-
_ fl (t)

0

1

2

3

4

5

6

131 Fig. 9. [Model 2] Parametric characterization of pdf for a range of rl.

v

9 CONCLUSIONS AND FURTHER RESEARCH In this paper we have given a complete parametric characterization of the density and failure rate functions for two sectional models involving two Weibull distributions. The main results of the paper are as follows:

/

"? I ",

tpl

5i I

i'"

tO

tp2

Fig. 11. Pdf plot for bi-modal shape [Model 1].

Parametric study of sectional models In this case, continuity of the reliability and density functions t = to implies that kF1 (to) = F2(t0) and kfl(to) = )~(t0). In this case, the density function shapes are similar to that for Model 2. However, the failure rate shapes are different because the failure rate is given by r ( t ) = f l ( t ) / [ 1 - k F l ( t ) ] for 0 < - t < t 0 and =r2(t) for to-< t < oo. The characterization of r(t) for t-< to is more complex but for t->t0, it can assume one of three shapes depending on /32. This topic is currently under investigation by the authors. The extension to sectional models with more than two Weibull distributions is a still more difficult problem as it involves study in a higher dimensional parameter space.

9. 10. 11. 12.

159

for the exponential distribution with changing parameters. Technometrics, 1996, 8, 217-227. Shooman, M., Probabilistic Reliability: An Engineering Approach, McGraw-Hill, 1968. Mann, N. R., Schafer, R. E. and Singpurwalla, N. D., Methods for Statistical Analysis of Reliability and Life Data, John Wiley and Sons, New York, 1974. Colvert, R. E. and Boardman, T. J. Estimation in the piece-wise constant hazard rate model. Communications in Statistics A (Theory & Methods), 1976, 1, 1013-1029. Elandt-Johnson, R. C. and Johnson, N. L., Survival Model and Data Analysis, John Wiley and Sons, New York, 1980.

APPENDIX

We prove by contradiction that for Model 1 the density function cannot have a bimodal shape when

t31
1. Jiang, R. and Murthy, D. N. P., Modeling failure data by mixture of two Weibull distributions: a graphical approach. IEEE Transactions on Reliability, 1995, 44, 477-488. 2. Jiang, R. and Murthy, D. N. P., Reliability modeling involving two Weibull distributions. Reliability Engineering and System Safety, 1995, 47, 187-198. 3. Jiang, R., Failure m~xlels involving two Weibuli distributions. Doctoral Thesis, The University of Queesland, Australia, 1996. 4. Jiang, R. and Murthy, D. N. P., Mixture of Weibull distributions--parametric characterization of failure rate function. Applied Stochastic Models and Data Analysis (accepted). 5. Jiang, R. and Murthy, D. N. P., Parametric study of competing risk model involving two Weibull distributions. Reliability, Qtudity and Safety Engineering (accepted). 6. Jiang, R. and Murthy, D. N. P., Parametric study of multiplicative model involving two Weibull distributions. Reliability Engineering and System Safety (accepted). 7. Kao, J. H. K. A graphical estimation of mixed Weibull parameters in life-testing of electron tubes. Technometrics,1995, 1, 389-407. 8. Aroian, L. A. and Robsion, D. E. Sequential life tests

Suppose that the density function has a bimodal shape. It must be of the shape shown in Fig. 11. This implies tpl < to < tp2. Let

Sl = Rl(tpl),S2 = Rz(tp2) and s3 = F(tp2) - F(tpl). Since the area under the curve must equal one, we have 1 = (1 - S l ) + S 3 + S 2 . Note that s3 > 0 for the shape to be bimodal. This implies 1 - s l + s 2 < 1.

(A1)

Using eqns (4) and (5) in eqn (8), we have

s I = gl(tpl )

=

exp( - 1 + 1//31)

and

s2 = Rz(tp2) = exp( - 1 + 1//32). Since/31 >/32, this implies that sa < st, or 1 - sl + s2 > 1.

(A2)

Equation (A2) contradicts eqn (A1) and hence the result.