Faculty of Automotive and Mechanical Engineering
Changsha University of Science and Technology, Changsha, Hunan 410114, China
(jiang@csust.edu.cn)
Abstract This paper gives main characteristics of the failure-rate-based bathtub curve, defines two change points of the bathtub curve and develops nonparametric and parametric methods to estimate the change points. Simple relations for estimating the interval failure rate in the infant mortality and random failure phases are derived. The proposed methods are useful for model selection and can be used to evaluate the appropriateness of any bathtub curve model. These are illustrated through a real-world example.
Key words Failure rate, bathtub curve, change point, nonparametric method, parametric method
I. I NTRODUCTION
The bathtub curve has been widely used to explain the failure behavior of the population of nonrepairable components. It is characterized by three phases (i.e., early use phase, normal use phase and wear-out phase) and associated with different failure mechanisms (i.e., infant mortality, random failure and wear-out failure).
For repairable systems, the failure is characterized by rate of occurrence of failure (or intensity function) rather than the failure rate. The plot of the intensity function versus time can be bathtub-shaped (e.g., see [1] and [2]) and in this case it is also called the bathtub curve. Two kinds of the bathtub curve have different engineering significance. To differentiate, Jiang and Murthy [3] call them the component-level and system-level bathtub curves, respectively. In this paper, we focus the attention on the component-level bathtub curve and let ( ) denote the failure rate function.
There are many approaches to specify the partition points (termed as the change points) between two adjacent phases. For example, Bebbington et al [4] define the change points based on the curvature of the bathtub curve.
The basic idea is that at the change points the curvature has maximal changes. When the estimation is based on the fitted model, the estimated change points depend on the model, which itself needs to be validated; when the estimation is based on a nonparametric approach, this curvature-based approach is probably not robust since it needs to evaluate the curvature, which is a function of
'( ) and r t .
In this paper we develop the methods to specify the change points from an engineering-oriented viewpoint.
The paper is organized as follows. We present some the bathtub curve models in Section II. The methods to specify the change points are presented in Section III, and illustrated in Section IV. The paper is concluded with a brief summary in section V.
II. B ATHTUB C URVE M ODELS
The bathtub curve models are developed for modeling data that display the bathtub-shaped failure rate. There are many such models in the literature, e.g., see [5] and [6].
According to the support, the bathtub curve models roughly fall into the following two broad categories:

Models with infinite support, and

Models with finite support.
A.
Models with infinite support
This category of models is defined in ( 0,

) and given in the form of distribution (or reliability) function, failure rate or cumulative hazard function. Two typical models defined in terms of the distribution (or reliability) function are the Weibull competing risk model, e.g., given by (see [7]):
R t
 
 t
1
)

1

 t
2

( t


2


) ], (1) with 0
 
1
   
2
,

0 and the exponential Weibull distribution given by (e.g., see [8])
( )
   t
   
1

. (2)
A typical example defined in terms of failure rate function is the model given by (see [9]):
( )
 at
 
1 e
 t ,
 
(0,1) . (3)
Two typical models defined in terms of cumulative hazard rate function are given by (see [10]):
( )
 at e t
, 0

1,
 
0 , (4) and given by (see [11]):
( )
 a t
   a
  
0 . (5)

B.
Models with finite support
A decreasing function defined in ( 0,

) can become bathtub shaped when making a variable transformation to change the support to a finite range. The following (see
[12]) is such an example:
H x

(
1 x

/

 x
 
(0,1) , x

. (6)
Here, random variable Y

X / (1
 
X ) follows the
Weibull distribution with the support ( 0,

), and random variable X has a finite support ( 0,1/

). The failure rate is given by ：
 
1 x


1
  x
) . (7)
III. S PECIFICATION OF C HANGE POINTS
A.
Definition of bathtub curve
Usually, the bathtub curve is ambiguously defined as
“first decreasing and then increasing”. A stricter definition is as follows:
(a) There exists a point t
1
before which the dominant failure mode is infant failure and the failure rate is roughly decreasing; t
1
is relatively small and r (0) is usually finite.
(b) There exists a point t after which the dominant
2 failure mode is wear-out and the failure rate is increasing; and t
2
is relatively large.
(c) Between t
1
and t
2
, the dominant failure mode is random failure, the failure rate is roughly constant and t
2
 t
1
is relatively large.
B.
Change points of a bathtub curve
Let r t ,
1
( ) r t
2
and r t
3
denote the failure rate functions associated with the infant, random and wear-out failure modes and, t and
1 t denote the change points,
2 respectively. The total failure rate is their superposition and given by ： r t
 r t
 r t
 r t . (8)
Let t
0
denote the time where r t achieves its minimum. At t
0
we have: r t
 r t
0

0 . (9)
Let

denote r t
0
, i.e., the minimal failure rate. We call t
0
the minimum point.
In the interval ( 0, t
0
), the total failure rate can be characterized by a decreasing function given by r (1) ( ) , which can be viewed as superposition of ( ) and ( ) without the effect of
In the interval ( r t .
3
( ) t
0
,

), the total failure rate can be characterized by an increasing function given by r
(3)
( ) , which can be viewed as superposition of ( ) and ( ) without the effect of ( ) .
The interval ( t t
2
) can be divided into two sub-intervals: ( t t
0
) and ( t t
2
). In the former subinterval
( ) has a weak effect and hence the failure rate can be mildly decreasing; in the latter subinterval ( ) has a weak effect and hence the failure rate can be mildly increasing. As a result, a two-order polynomial can be appropriate for approximating their superposition, which is denoted as r
(2)
( ) .
As such, the left change point is defined as r (1) ( )
1
 r (2) ( )
1
, and the right change point is defined as r (2) ( )
2
 r (3) ( )
2
.
(10)
(11)
C.
Characteristics and significance of the bathtub curve
In the literature the bathtub curve is usually characterized by two characteristic points. The first characteristic point is actually or somehow similar to t
0 defined above and given different terms such as critical point (e.g., see [12]) or turning point (e.g., see [13]); and the second characteristic point is somehow similar to t
2 defined above and termed the instability point by [10].
Here we characterize the bathtub curve by four parameters: t
1
, t
0
,

and t
2
. The magnitude of t
1 represents the manufacturing quality and provides the information about burn-in.
The magnitude of t reflects the type of
0 manufacturing defects. According to Jiang and Murthy
[14], there are two types of basic manufacturing defects: assembly defects and nonconforming components. t
0 can be relatively small for the former and relatively large for the latter. Also, it can reflect the beginning time of ageing. A large [small] t implies that the aging begins
0 lately [early].

reflects the reliability associated with the random failure mode.

Finally, t
2
provides the information about the intervention time of preventive maintenance and t
2
 t
1 reflects the useful life.
D.
Non-parametric estimation of failure rate
Suppose that a dataset is given by x n
1 1
 x n
2
(
2
)
  x n m
( m
) ， where x i
,1
  m
(12)
, is a failure time (i.e., it is not a censored observation) and n i
is the number of failures at that time. We view interval ( x i
as a representative point of x i

1
2
 x i , x i

2 x i

1 ). At the beginning of this interval, the number of surviving items is given by
N i
.
.
For example, for a complete dataset it is given by:
N i
 m  n j
. (13) at
As such, a nonparametric estimate of the failure rate x i
is given by r i
 i
2 n i i

1
 x i

1
)
,1 i m , (14) where x
0

0 and
Discussion: x m

1
is defined as 2 x m
 x m

1
.
1.
For the first failure observation x
1
, we may view it as the representative point of interval ( 0, x
1
 x
2
2
) rather than ( x
1 , x
1

2 2 x
2 ). As such, (14) is revised as r
1

2.
For group data, let
2 n
1
1
(
1
 x
2
)
. n x i

1 x i
(15)
denote the number of failures in the interval ( x i

1
, x i
) and number of surviving items at representative point is x i

1 x i

( x i

1
 x i
. The interval
) / 2 , and the failure rate is estimated as: r x i

( n x x i
 i

1 x i

1 x
) i
N i
,1 i m . (16)
3.
The empirical intensity function can be defined in a similar way (e.g., see [1]). Here,
N denote the i
N i
usually or almost is unchanged.
E.
Nonparametric approach to specify the change points
If the plot of r i
versus x i
is bathtub shaped, we propose the following multi-step procedure to specify the characteristic points of a bathtub curve.
The first step is to get the initial estimates of the change points through examining the empirical failure rate curve obtained from the above approach. Let

and
1

denote the initial estimates of
2 t
1
and t
2
, respectively.
The second step is to estimate the minimum point.
This is done by fitting the data points in (
 
) to the
1
,
2 following two-order polynomial given by r (2) ( ) (
 t
0
) 2 . (17)
The third step is to estimate the left change point.
This is done by fitting the data points in ( 0,

) to the
1 following negative exponential function given by r
(1)
( )
 e a
1
 b t
. (18)
As such, t
1
is given by e a
1
 b t
2
(
1
 t
0
) 2 . (19)
If the data in ( 0, t
1
) are different from the data in ( 0,

1
), this step is repeated until the data in these two intervals are the same or t
1
 
.
1
The fourth step is to estimate the right change point.
This is done by fitting the data points in (
 
) to the
2
, following exponential function given by r (3) ( )
 e a
3

(

0
)
. (20)
As such, t
2
is given by e a
3

3
(
2
 t
0
)
2
(
2
 t
0
) 2
. (21)
If the data in ( t

) are different from the data in (
2
,
 
),
2
, this step is repeated until the data in these two intervals are the same or t

2

2
.
The interval mean failure rate in the early use period is given by

1
 t
1
1 
0 t
1
( )
 e a
1
(1
 e

)
. (22)
The interval mean failure rate in the normal use phase is given by


2

1 t
2
 t
1 t
2  t
1
( ) b
2
( t
2
 t
0
)
2 
( t
2
 t
0
)( t
1
 t
0
 t
1
 t
0
)
2
3
. (23)
F.
Parametric approach
The change points need to be differently defined when a parametric model is fitted to the data with bathtub shaped failure rate. This is because the three parts of the failure rate in three phases are superposed together and the total failure rate is given by a single function.
We start with the minimum point, which can be obtained from the fitted model. According to (9), we have r t
 r t
 
. (24)
When t
 t
1
, the dominant failure mode is the infant failure so that we have r t
 r t ; when t
 t
1
, the dominant failure mode is the random failure so that we have r t
 r t . When t = t
1
, the effects of the two modes are indifferent so that we have r t
 r t
1
 
or r t

2

. (25)
As such, the left change point is defined by (25). In a similar argument, the right change point is defined by r t
2

2

. (26)
IV. I LLUSTRATION
We illustrate the method using the data shown in
Table I, which come from [15].
T ABLE I
L IFE T IME OF 50 D EVICES
0.1 0.2
7 11
36
67
84
40
67
84
1 1 1 1 1 2 3 6
12 18 18 18 18 18 21 32
45 46 47 50 55 60 63 63
67 67 72 75 79 82 82 83
84 85 85 85 85 85 86 86
A Non-parametric estimates
Using the non-parametric approach outlined in Part D of Section III, we obtained the empirical failure rate shown in Fig. 1 As seen, the data have a bathtub-shaped failure rate and the initial estimates of change points are about (

,
1

2
) = (2, 80).
Fitting the empirical failure rate to (17) yields the estimate of the minimum point, which is shown in the second row of Table II.
Fitting the empirical failure rate with t
 
1
to (18) yields the estimate of the left change point, from which we have t
1

3.36
 
1
. This implies that the observation x

3 should be included when fitting (18). In this case, we have t
1

2.94
, which is close to the previous estimate and hence the iterative process terminates.
Fitting the empirical failure rate with t
 
2
to (20) yields the estimate of the right change point, from which we have t
2

79.65
 
2
and hence a new iteration is not needed.
The final estimates are shown in the second row of
Table II and Fig. 1. As seen from the figure, the estimates look reasonable.
 
2
Using the estimated parameters to (23) we have
0.01597
. On the other hand, the average of the empirical failure rates over ( t t
1 2
) equals 0.01569, which is very close to the estimate of

from (23). This
2 confirms the appropriateness of the nonparametric method.
0.2
0.15
0.1
0.05
0
0 10 20 30 40 t
50 60 70 80 90
Fig. 1. Non-parametric estimates of failure rate
Model
T ABLE II
E MPIRICAL AND F ITTED F AILURE R ATES t
0 t
2

,

10
2
Non-parametric
(1)
(4)
2.94 21.54 79.65
-205.391 6.74 74.41 80.03
-227.155 1.29 10.33 44.09
1.2532
1.2360
1.0484
(6) -205.146 4.84 28.10 62.60 0.7783
B Estimates derived from a fitted parametric model
Consider the models given by (1), (4) and (6). They are fitted to the data using the maximum likelihood method. The characteristic points derived from the fitted models are shown in Table II, along with the log-likelihood values. Fig. 2 shows the failure rate functions of the fitted models.
In terms of the log-likelihood, the best model is the model given by (6). However, compared with the empirical failure rate curve, the best model is the model given by (1). The characteristics obtained from (4) are considerably different from those obtained from the non-parametric method, implying that it is inappropriate for modeling the data.

The above analysis shows that the non-parametric method can provide plausible estimation for the characteristics of the bathtub curve. However, a bathtub curve model may give misleading results.
0.1
0.08
0.06
0.04
0.02
0
0 20 40
(4)
60
(6)
(1)
80 t
Fig.2. Parametric estimates of characteristic points
V. C ONCLUSIONS
In this paper, we have strictly defined the failure-rate-based bathtub curve and characterized the bathtub curve using three characteristic points. We have proposed a non-parametric method to estimate the empirical failure rate and its characteristic points. The interval failure rates associated with the early and normal use phases have been derived. These have been illustrated by an example. Two main findings have been:

In the normal use phase, the failure rate can be mildly impacted by both infant failure and wear-out modes so that it can be convex, decreasing or increasing as shown in Fig. 2, and

A bathtub curve model may give misleading results unless it is appropriately validated.
The proposed nonparametric estimates for the failure rate and its characteristic points can be used to evaluate the appropriateness of a bathtub curve model and to select an adequate model for modeling a given dataset. The results presented in this paper can be easily extended to the case of intensity-function-based bathtub curve.
A CKNOWLEDGEMENT
The research was supported by the National Natural
Science Foundation (No. 71071026).
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