Reliability Engineering and System Safety 43 (1994) 37-41
On partial orderings and testing of new better
than renewal used classes
A. M. Abouammoh, S. A. Abdulghani
Department of Statistics and Operations Research, College of Science, King Saud University, PO Box 2455, Riyadh 11451,
Saudi Arabia
&
I. S. Qamber
Department of Electrical Engineering, University of Bahrain, PO Box 33831, ISA Town, State of Bahrain
(Received 13 July 1992; accepted 4 May 1993)
Partial orderings are used to compare ageing behaviour of components or
systems. In this paper a new partial ordering, namely the renewal superadditivity, is introduced. This concept is used to characterise the new better
than renewal used criterion of ageing. The dual criterion the new worse than
renewal used is characterised by the renewal sub-additivity concept. The
relation of these ageing properties with existing criteria of ageing is shown via
the partial orderings. Test statistic of exponentiality versus the new better
(worse) than renewal used as alternatives is established. The percentiles of this
test statistic are tabulated for small samples. These tables can be used by
reliability engineers to check whether a set of life data has the new renewal
better than used property or its dual. The power estimates of the test are
simulated for some commonly used distributions in reliability. Two sets of real
data are used as examples to elucidate the use of the proposed test statistics
for practical reliability analysis.
1 INTRODUCTION
between F and H, ' < ' say is said to be partial ordering
0
if it is antisymmetric, transitive and reflexive, see Ref.
2, p. 106. Partial orderings and the tail heaviness of
distributions are used to characterise different classes
of life distributions. Note that increasing and
decreasing are used for non-decreasing and nonincreasing, respectively. Now we present the following
concepts of ageing.
Classes of life distributions in reliability theory,
inventory and biometry are usually based on different
criteria of ageing. Most of these criteria are
characterised via partial orderings with respect to
exponential distribution. Partial orderings are used to
order components or their systems according to their
ageing behaviour. Vessely ~ has shown interest in
calculating the ageing effects by using the available
probability risk analysis. His methodology is based on
incorporating ageing effects into probability risk
analysis and utilising Taylor expansion to separate
their corresponding models.
Let F(_t) be the life distribution with survival
function F(t) and density function (assumed to exist)
f(t) where F ( 0 - ) - - 0 . Let H be a continuous
distribution of non-negative variable. The failure rate
of F is defined by r(t)=f(t)/F(t). Any ordering
1.1 Definition 1
The random variable T, F(t), F(t), f(t) or r(t) are
said to have the following concepts:
increasing failure rate (IFR) if F(t + x)/[~(t) is
decreasing in t, Vx-> 0 or r(t) is increasing in
t, t->0;
(ii) increasing failure rate average (IFRA) if
t -1 log F(t) is decreasing in t, t->-0;
(iii) new better than used (NBU) if /e(x + t)-< F
(x)F(t) Vx >- O, t >- 0;
(iv) new better than renewal used ( N B R U ) if
l~e(x I t) -</¢(x) Vx -> 0, t >- 0 where We(t) =
(i)
Reliability Engineering and System Safety 0951-8320/94/$06.00
© 1994 Elsevier Science Publishers Ltd, England.
37
A. M. Abouammoh, S. A. Abdulghani, I. S. Qamber
38
#-1 S~l~(u ) du and I'VF(X It) = f f ' e ( x + t)/14'F(t);
and
(v) new better than renewal used failure rate
( N B U R F R ) if r~(t) -> r(0), Vt > 0, where
rw(t) = w(t)/ff'(t).
For definitions and properties of the concepts (i)-(iii)
see Refs 2 and 3. The concepts (iv) and (v) are
introduced in Refs 4 and 5, respectively. The
corresponding dual concepts namely D F R , D F R A ,
N W U , N W R U and N W U R F R are defined by
changing the direction of the monotonicity or the
inequality as appropriate. Here, D and W stand for
decreasing and worse, respectively. Cao and Wang 6
introduced the concept of new better than used in
convex ordering (NBUC) property. In fact N B U C is
equivalent to N B R U if ~t = j'~ F(x) < oo.
Note that F is N B R U means
1 <- F(x)
w(t)
P(o)
This implies that
- lim x - 1 W ( t + x ) - W ( t ) <_ _ lim x - ' F ( 0 + x ) - F ( 0 )
x-o
w(t)
,e(0)
This yields that
rw(t) >- r(0),
ORDERINGS
i.e. F is N B U R F R or
N B R U =)>N B U R F R
2.1 Definition 2
Let F and H be two continuous distributions such that
H is strictly increasing and F(0) = G(0) = 0. Then we
have the partial orderings:
VX -->0,
(1)
(i)
convex ordering, F < H if H-~F(t) is convex in
¢
t on the support of F, Sv say where Sv e R ÷;
Now we have the following:
F<H
(ii) star-shaped orderings,
IFR :ff I F R A f f N B U :ff N B R U ::> N B U R F R
(e)
This sequence of implications can be shown via partial
orderings.
Now consider the problem of testing the null
hypothesis
Ho: F(t) = 1 - exp(-~.t),
2 PARTIAL
It is useful to express the concepts of ageing in the
general partial orderings setup. These partial orderings are used to characterise different ageing concepts
when such ordering is made with respect to the
exponential distribution. Now we present the
following.
14'(x + t)/l/C(t) <--F(x)
or
ff'(x + t)
In section 2 we introduce the partial ordering
concept that characterize the N B R U property and
investigate its relation with other existing partial
orderings. A test statistics for testing exponentiality
versus the N B R U is derived in section 3 and its small
sample percentiles are simulated. In section 4 we
study the power estimates for this test with respect to
some commonly used distributions in reliability. Two
sets of data, namely the lifelengths of leukaemia
patients 3 and the lifelengths of blood cancer patients
(unspecified type of cancer) from a hospital of the
ministry of health in Saudi Arabia, are used to
elucidate the application of the N B R U (NWRU)-ness
property for real data.
t -> 0, Z > 0
if H-~F(t) is
star-shaped or t-~H-~F(t) is increasing in t,
t_>0;
(iii) super-additive ordering, F < H if H - ' F ( x +
SU
t) >--H-IF(x) + H-1F(t) for all x, t -> 0; and
(iv) strongly
renewal
ordering,
F<H
if L =
sr
versus the alternative
111: F(t) = A, is not exponential.
Here, A denotes a class of life distributions with
non-constant failure rate. Several authors have
considered this problem for various classes of life
distributions. For example, Proschan and Pyke 7 and
Klefsjo 8 for A = IFRA, Hollander and Proschan, 9
Koul ~° and Kumazawa H for A = NBU. These testing
problems are often faced by reliability engineers for
different sets of field data. It is very important to
know the deterioration trend of data set for
comparison purposes or for investigating an optimal
design for preventive or repair maintenance policies.
l i m y _ o Y - I H - I F ( y ) exists and (d/dt)H-IW(t) >_
L, for t---0 where W ( t ) = l ~ f S [ ~ ( u ) d u ,
where /UF ----J'~/C(u) du.
It is shown in Ref. 2 (p. 109) that
F<H~F<H~F<H.
C
*
Su
Now we introduce the following new concept of
partial ordering
2.2 Definition 3
For the distribution F and H in Definition 2, we call F
is renewal superadditive ordered with respect to H,
New better than renewal used classes
39
3 TESTING NBRU AND NWRU PROPERTIES
denoted by F < H, if
rsu
H-1W(x +t)>_H-~F(x)+H-~W(t),
where
ft
W(t) = / u ; ' Jo F(u) du.
In this section we study the problem of constructing
empirical test statistic for the exponentiality as null
hypothesis
t>-O,
F(t) =
H0:
1 - exp(-)-(t),
)- > 0,
t -> 0,
Now we present the following characterisations for
different ageing properties.
versus the NBRU and N W R U properties, that is
2.3 Theorem 1.
or
HA2: F(t)
Let G(t) = 1 - exp(-)-t), where t -> 0 and )- > 0 be an
exponential distribution and let F be a continuous life
distribution with F ( 0 ) = 0. Then
(i)
HAt: F(t) is NBRU
is N W R U
respectively. This test is based on an independent
sample X1 . . . . . 3(, of size n from a population with
continuous distribution F. The alternative hypothesis
//1 for the NBRU (NWRU) properties states that
F is IFR is and only if (iff) F < G,
g'(x 1/) -- P(x [ 0),
¢
t -->0,
x -->0,
(6)
i.e. the conditional renewal survival at any point x for
a given survival at t coincides with the survival at x.
Now our proposed test for the alternative
hypothesis HA1 is motivated by the following quantity
(ii) F is I F R A iff F < G,
(iii) F is NBU iff F < G,
su
(iv) F is NBRU iff < G, and
fx~P(y+t)dy-P(x) f f P(y)dy<-O
rsu
(v) F is N B U R F R if F < G.
(7)
The empirical estimate of the left-hand side of eqn
(7) at specified time t is given by
sr
2.4 Proof
oo
Parts (i)-(iii) in section 2.3 are found in Ref. 2 (pp.
105-9).
(iv) Since G(t) = 1 - exp(-)-t), )- > 0, t -> 0,
E(i, j) = ~ F(Ti + Tk)(Tk - Tk-l) -- n-l(n --j)
k=j
× ~ n-l(n -k)(Tk - T,_,)
(8)
k=i
then
G-~(t)=-)--llog(1-t),
0_<t_< 1.
This means that
G-iF(t) =
_)--1 log F(t),
(3)
where j = 1. . . . . n and T~. . . . . Tn are the ordered
statistics of the sample X1 . . . . . Xn and T~ = 0. Taking
the summation of E(i,j) over j = l . . . . . n and
i = 1. . . . . n we obtain the test statistic
and
G-~W(t + x)
= _ ) - 1 log
W(x + t).
(4)
i=|j=l
Now we have that F is NBRU or
X
n-I(n-i-k)(T,-T,_l)-n-'(n-j)
lTV(x+ t) <--F(x)ITv(t).
x~n-'(n-k)(Tk-T,_,)}
This is equivalent to
(9)
k=i
- l o g ff'(x + t) --> - l o g F(x) - log l~'(t).
(5)
By substituting for eqns (3) and (4) in eqn (5) we
obtain that F < G. Part (v) is proved in Ref. 4.
where
(n_i_k)={oT'S>T~+T,
rsu
if T~+ T, < Tn,
if T~+ Tk --> T~
Note that eqns (1) and (2) and Theorem 1 give the
following corollary.
The test statistic of eqn (9) can have the form
2.5 Corollary 1
M~=n-'[~
t-i=1
Let F and H be two continuous distributions such that
H is strictly increasing and F ( 0 ) = H ( 0 ) = 0 . Then
F<H~F<H~F<H.
su
rsu
$r
(n-i-k)(Tk-T,_l)
j:l
k=j
-2-t(n-
1)~
~ ( n - k)(T,-
T,_,)] .
i=l k=j
(10)
40
A. M. Abouammoh, S. A. Abdulghani, L S. Qamber
Table 1. Critical values for the/.].-statistics
n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
25
30
35
40
45
50
Lower percentiles
Table 2. Power estimates for the Mn-statistic
Upper percentiles
0.01
0-05
O-10
O-10
0-05
0.01
-0.593
-1.296
-1.905
-2.162
-3.055
-2.778
-3.619
-2-974
-2-802
-2.775
-2-900
-3.109
-3.102
-3.118
-3.273
-3.157
-3-349
-3-290
-3.720
-3.602
-4-228
-4-208
-4.500
-4-455
-4.968
-0.372
-0.903
-1.209
-1.370
-1-463
-1.576
-1.577
-1-656
-1.741
-1.776
-1-870
-1.897
-1-862
-2-008
-1-984
-2.140
-2-163
-2.197
-2.254
-2.486
-2-886
-2.868
-2.877
-3-044
-3.514
-0.289
-0.610
-0.760
-1.004
-0.996
-1.125
-1-137
-1-190
-1.245
-1.308
-1.388
-1.434
-1-374
-1-508
-1-540
-1.510
-1.601
-1.698
-1.756
-1.867
-2.014
-2.227
-2-118
-2-420
-2.721
0.161
0.242
0-360
0.434
0-520
0.622
0.649
0-730
0.763
0.801
0-882
0.938
0-932
1-035
1.082
1.106
1-138
1-221
1.273
1-423
1-450
1-896
2.009
2.112
2.326
0.217
0-334
0.455
0-570
0.683
0.793
0.834
0.903
1.010
1-075
1.139
1-235
1-207
1.347
1.387
1-395
1.454
1.598
1.561
1-930
2.093
2.449
2-512
2.768
2.817
0.329
0-511
0.674
0.839
0.968
1.169
1-235
1.329
1.360
1.494
1-573
1-903
1.930
1.999
1.921
2.240
2-141
2.310
2-258
2.829
2.870
3.437
3.603
3.879
4.018
In fact Mn and - M n are the test statistics for the
N B R U and the N W R U properties, respectively. It is
difficult to find the exact distribution for the M~
statistic and hence simulation of small samples is
investigated. In practice, simulated percentiles for
small samples are commonly used by applied
statisticians and reliability analysts. In order to m a k e
the test scale invariant we consider
Un = M ~ / T
(11)
where T is the mean of T1. . . . . T~. Based on eqn (11)
the lower and upper percentiles in the 0.01, 0.05 and
0-10 regions for the sample sizes n = 2(1), 20(5), 50
are calculated. Table 1 contains the simulated results.
It is noted that the Mn values do not rapidly diverge as
the sample size n increases. This supports the belief
that the distribution of Mn can be asymptotically
normal with some appropriate transformation. Calculating the test statistic M~ for any set of life data,
where n is the sample size, one can easily test whether
this set of data is N B R U , N W R U or exponential. The
NBRU-ness property can be also c o m p a r e d for two or
more sets of data.
4 THE POWER
ESTIMATES
The power of the test statistic Mn is considered for the
significance level te = 0.05 and for commonly used
Distribution
Parameter, 0
Sample size
5
10
20
F~ (Weibull)
2.0
3-0
4.0
0-196
0-455
0.686
0-460
0.868
0.989
0.878
1.000
1-000
F~ (~)
2-0
3.0
4-0
0.297
0.607
0.813
0.541
0.889
0-992
0-847
0.999
1.000
2.0
3.0
4.0
0-668
0.800
0-882
0.901
0.969
0.995
0.992
0.998
0.999
(Pareto)
distributions in reliability modelling. These distributions are the Weibull, g a m m a and Pareto. Their
respective survival functions are the following:
~ ( t ) = e x p ( - t °),
0 -->0, t -> 0,
P~(t)=[r(o)]-' y°-le-Ydy,
O>O,t>--O,
~ ( t ) = (1 + Ot) -'/°,
0 > O, t >--O,
Note that Fm for 0 -> 1, F2 for 0 -> 1 and ~ are N B R U
distributions whereas F~ for 0 -< 1, F2 for 0 -< 1 and F3
have N W R U properties.
Table 2 contains the power estimates for the Un test
statistic with respect to these distributions. The
estimates are based on 1000 simulated samples of sizes
n = 5 , 10, 20 and significance level ~ = 0 . 0 5 . The
distributions F~ with 0 = 1, F2 with 0 = 1 and F3
with 0 = 0 coincide with the exponential F(t)=
1 - e x p ( - t ) , t -> 0. As 0 increases these distributions
become more N B R U . Similarly, 172 become m o r e
N W R U distribution as 0 decreases.
The power estimates in Table 2 shows clearly the
departure from exponentiality towards N B R U or
N W R U properties as 0 increases. In fact Table 2
shows how reliable our proposed test can be based life
distribution with tractable ageing criteria.
5 APPLICATIONS
In this section we evaluate the Un test statistic for two
sets of real data and test whether these data are
exponential, N B R U or N W R U .
The first set consists of the lifelengths, from the
diagnosis, of 43 patients suffering form granuiocytic
leukaemia from the National Cancer Institute cited in
Ref. 3. The ordered survival times in days are in
Table 3.
Table 4 consists of 43 blood cancer patients from
one of the ministry of Health Hospitals in Saudi
Arabia.
New better than renewal used classes
Table 3. Ordered
lifetimes
(in days) of 43 leukaemia
patients
7
47
58
74
177
232
273
385
317
429
440
445
455
408
495
497
532
571
579
581
650
702
U43( •=) 0-066
Table 4. Ordered lifetimes (in
days) of 43 blood cancer
patients
115
181
255
418
441
461
516
739
743
789
807
865
924
983
1 025
1 062
1 063
1 165
1 191
1 222
1 222
1 251
715
779
881
900
930
968
1 077
1 109
1 314
1 334
1 367
1 534
1 712
1 784
1 877
1 886
2045
2 056
2 260
2429
2 509
Note that evaluating the test statistics for these two
sets of data has been used to c o m p a r e between the
health care efficiency related to the corresponding
groups of patients. Also, it cannot lead to the
conclusion that cancer patients lifelengths have this
type of ageing criterion. These sets of data are used to
show the possibility of applying the derived
methodology on real data situation. These sets are the
data at hand which could affect the lifelengths of units
in engineering system.
It was found that the test statistics for the two sets
of data, by using eqn (11), are
tr(2)
v 4 3 = 0-034
for the US and the Saudi sets of patients, respectively.
It is clear from the c o m p u t e d values of the test
statistics that we accept the null hypothesis which
states that both sets of data have neither N B R U - n e s s
nor NWR-ness properties under the significance level
or = 0.01 or even or = 0.10. This means that both sets
of data follow the exponential distributions.
The difference in values of /3, due to the time unit
might be due to calculating the empirical value of the
survival function F ( y +t), see eqns (8) and (9).
There, the n u m b e r of lifelengths which exceed y + t
may be less for larger units of time. Similarly, one can
apply the test statistic for a set or more sets of life
data to check the N B R U ageing criteria or to compare
41
1 277
1 290
1 357
1 369
1 408
1 455
1 478
1 519
1 578
1 578
1 599
1 603
1 605
1 696
1 735
1 799
1 815
1 852
1 899
1 925
1 965
between these sets for the purpose of ordering them
according to their ageing behaviour.
REFERENCES
1. Vesely, W. E., Incorporating ageing effects into
probabilistic risk analysis a Taylor expansion approach.
Reliability Engng Sys. Safety, 32 (1991) 315-37.
2. Barlow, R. E. & Proschan, F., Statistical Theory of
Reliability and Life Testing. To Begin with, Silver
Spring, MD, USA, 1981.
3. Bryson, M. C. & Siddiqui, M. M., Some criteria of
ageing. J. Am. Statistical Assoc., 64 (1969) 1472-83.
4. Abouammoh, A. M., The new better than renewal used
class of life distributions, Arabian J. Sci. & Eng.
(Submitted for publication).
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failure rate classes of life distributions. Statist. Prob.
Letts, 14 (1992) 211-17.
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life distributions. J. Appl. Prob., 23 (1991) 748-56.
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9. Hollander, M. & Proschan, F., Tests for the mean
residual life. Biometrika, 62 (1975) 585-93.
10. Koul, H., A testing for new better than used. Commun.
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