ArticlePDF Available

Abstract

In this paper, two new statistics based on comparison of the theoretical and empirical distribution functions are proposed to test exponentiality. Critical values are determined by means of Monte Carlo simulations for various sample sizes and different significance levels. Through an extensive simulation study, 50 selected exponentiality tests are studied for a wide collection of alternative distributions. From the empirical power study, it is concluded that, firstly, one of our proposals is preferable for IFR (increasing failure rate) and UFR (unimodal failure rate) alternatives, whereas the other one is preferable for DFR (decreasing failure rate) and BFR (bathtub failure rate) alternatives and, secondly, the new tests can be considered serious and powerful competitors to other existing proposals, since they have the same (or higher) level of performance than the best tests in the statistical literature.
Full Terms & Conditions of access and use can be found at
http://www.tandfonline.com/action/journalInformation?journalCode=gscs20
Journal of Statistical Computation and Simulation
ISSN: 0094-9655 (Print) 1563-5163 (Online) Journal homepage: http://www.tandfonline.com/loi/gscs20
A wide review on exponentiality tests and two
competitive proposals with application on
reliability
Hamzeh Torabi, Narges H. Montazeri & Aurea Grané
To cite this article: Hamzeh Torabi, Narges H. Montazeri & Aurea Grané (2018) A wide review
on exponentiality tests and two competitive proposals with application on reliability, Journal of
Statistical Computation and Simulation, 88:1, 108-139, DOI: 10.1080/00949655.2017.1379522
To link to this article: https://doi.org/10.1080/00949655.2017.1379522
Published online: 21 Sep 2017.
Submit your article to this journal
Article views: 70
View Crossmark data
JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION, 2018
VOL. 88, NO. 1, 108–139
https://doi.org/10.1080/00949655.2017.1379522
A wide review on exponentiality tests and two competitive
proposals with application on reliability
Hamzeh Torabia, Narges H. Montazeriaand Aurea Grané b
aDepartment of Statistics, Yazd University, Yazd, Iran; bStatistics Department, Universidad Carlos III de
Madrid, Madrid, Spain
ABSTRACT
In this paper, two new statistics based on comparison of the theoret-
ical and empirical distribution functions are proposed to test expo-
nentiality. Critical values are determined by means of Monte Carlo
simulations for various sample sizes and different significance levels.
Through an extensive simulation study, 50 selected exponentiality
tests are studied for a wide collection of alternative distributions.
From the empirical power study, it is concluded that, firstly, one of our
proposals is preferable for IFR (increasing failure rate) and UFR (uni-
modal failure rate) alternatives, whereas the other one is preferable
for DFR (decreasing failure rate) and BFR (bathtub failure rate) alter-
natives and, secondly, the new tests can be considered serious and
powerful competitors to other existing proposals, since they have
the same (or higher) level of performance than the best tests in the
statistical literature.
ARTICLE HISTORY
Received 1 June 2017
Accepted 11 September 2017
KEYWORDS
Empirical distribution
function; entropy estimator;
exponentiality test;
goodness-of-fit tests; Monte
Carlo simulation
MSC (2010)
62F03; 62F10
1. Introduction
The exponential distribution is probably the one most used in statistical work after the nor-
mal distribution. Usually it appears in problems dealing with reliability theory, life testing
and the theory of stochastic processes. In those contexts, it is commonly assumed that the
data involved in the experiment follow the exponential distribution. One of the objectives
of this paper is to propose goodness-of-t statistics based on a new discrepancy measure
between two distribution functions: the empirical one obtained from the sample under
studyandthehypothesized.Manypaperscanbefoundintheliteraturewithproposals
of goodness-of-t tests for exponentiality. However, we found scarce works devoted to
compare their performances. A second objective of this paper is to shed some light on
this matter. Far from being exhaustive, our intention is to compare a wide representative
selection of classical and recent statistics. In particular, among the most recent tests, we
emphasize on tests based on the following new methods:
Tests based on correcting moments of entropy estimators from Zamanzade and
Arghami [1] and Alizadeh Noughabi and Arghami [2]
CONTACT A. Grané aurea.grane@uc3m.es Statistics Department, Universidad Carlos III de Madrid, C/ Madrid
126, 28903 Getafe, Madrid, Spain
© 2017 Informa UK Limited, trading as Taylor & FrancisGroup
JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 109
Tests based on characterizations of the exponential distribution from Alizadeh and
Arghami [3,4], Volkova [5] and Volkova and Nikitin [6]
Tests based on cumulative residual entropy from Baratpour and Habibi Rad [7]
Tests based on Phi-divergence from Alizadeh and Balakrishnan [8]
Tests based on correlation and covariance from Fortiana and Grané [9]andMontazeri
and Torabi [10].
We start by presenting the goodness-of-t problem. Let X1,...,Xnbe nindependent
and identically distributed (iid) random variables with continuous cumulative distribution
function (cdf) F(.). Along the paper, we will denote the order statistic by (X(1),...,X(n)).
Based on the observed sample x1,...,xn, we are interested in the following goodness-of-t
test for a location-scale family:
H0:FF
H1:F/F,(1)
where F={F0(.; θ)=F0((xμ)/σ ) |θ=,σ) },=R×(0, )and μand σ
are unspecied parameters. The family Fis called location-scale family, where F0(.)=
F0(.; θ)is the standard case for θ=(0, 1).
The goodness-of-t problem for a location-scale family described in Equation (1) has
been discussed by many authors. Recently, Torabi et al. [11]proposedanewdiscrepancy
measure between two continuous cdfs, as follows.
Given Xand Ytwo absolutely continuous random variables with cdfs F0and F,
respectively, these authors dene the following discrepancy measure:
D(F0,F)=
−∞
h1+F0(x;θ)
1+F(x)dF(x)=EFh1+F0(X;θ)
1+F(X),(2)
where EF[.] is the expectation under Fand h:(0, )R+isacontinuousfunction,
decreasing on (0, 1)and increasing on (1, )with an absolute minimum at x=1such
that h(1)=0. This measure satises that D(F0,F)0 and equality holds if and only if
F0=F, almost everywhere. Torabi et al. [11] proposed to use it as a measure of goodness-
of-t of an iid sample x1,...,xn, with empirical distribution function (EDF) Fn,toagiven
distribution F0. In particular, for the goodness-of-t problem for a location-scale family
describedinEquation(1),theyproposethefollowingstatistic:
Hn:=D(F0,Fn)=1
n
n
i=1
h1+F0(x(i);ˆμ,ˆσ)
1+Fn(x(i))=1
n
n
i=1
h1+F0(z(i))
1+i/n,
where ˆμand ˆσare the invariant estimators of μand σ, resp. under location-scale trans-
formations and zi=(xi−ˆμ)/ ˆσ,fori=1, ...,n.Notethatinthisfamily,F0(xi,ˆμ,ˆσ) =
F0(zi).ThestatisticH
nis expected to take values close to zero when H0is true. Hence, the
null hypothesis is rejected for large values of Hn.
TheaimofthispaperistoconsiderF0as the cdf of a scale exponential family and pro-
pose goodness-of-t statistics based on Hnfor testing exponentiality. The organization
of the paper is as follows. In Section 2,wedevelopthestatisticsbasedonH
nfor testing
110 H.TORABIETAL.
exponentiality and nd their critical values for several sample sizes and dierent signi-
cance levels. In Section 3, we review 50 statistics to test exponentiality and in Section 4,
we compare their performances to those of our proposals through a wide set of alterna-
tivedistributionscommonlyusedinreliabilitystudies.Asaglobalsummary,weprovide
a classication of the top ve best statistics for dierent types of alternative distributions
according to the shape of their failure rate function. Finally, we analyse a real data set to
illustrate the test.
2. An exponentiality test based on Hn
Let X1,...,Xnbe niid random variables with cdf F0(x;θ) =1ex,whereθ>0isan
unknown parameter. The problem of interest is
H0:F(x)=F0(x,θ)
H1:F(x)= F0(x,θ).(3)
The goodness-of-t problem in Equation (3) is a particular case of Equation (1). As it has
been said, the assumption of exponentiality in Equation (3) is frequently used in many
modelling situations, particularly in life testing and reliability.
Given an iid sample, x1,...,xn, we start by estimating θby their maximum likelihood
estimator, i.e. ˆ
θ=n1n
i=1Xi=¯
Xnandnextweconsiderthestandardizedsampleyi=
xi/ˆ
θ,fori=1, ...,n. Our proposal is based on the ratio of the cdf of the standardized
sample under H0and the EDF based on the xi’s . I n p art i cu l ar :
H(k)
n:=1
n
n
i=1
hk1+F0(x(i);¯
x)
1+Fn(x(i))=1
n
n
i=1
hk1+F0(y(i))
1+i/n,k=1, 2, (4)
where x(i)and y(i),i=1, ...,n, are the order statistics of xiand yi,resp.,and
h1(x)=(ex1x)I[0,1](x)+3
|x31|I[1,)(x)(5a)
and
h2(x)=(ex1x)I[0,1](x)+(x1)2
(x+1)2I[1,)(x). (5b)
Note that hk:[0,)Risanon-negativefunctionwiththeabsoluteminimumat
x=1, such that hk(1)=0, k=1,2. Under H0,weexpectthatFn(y)F0(y).Hencehk((1+
F0(y))/(1+Fn(y))) 0, since hk(1)=0. Thus H(k)
nis expected to be near zero when H0
is true. Therefore, it is justiable to reject H0for large values of H(k)
n.
Figure 1shows the graphs of the components in Equations (5a) and (5b). Functions hk
were selected based on the simulation study in order to achieve a global better performance
of the statistics.
Proposition 2.1: The support of statistics H(k)
n,k=1,2, are given by
supp(H(1)
n)=[0, 1.9129], supp(H(2)
n)=[0, 0.1111].
JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 111
0.0 0.5 1.0 1.5 2.0
0.0 0.5 1.0 1.5 2.0
x
h(x)
3x31
exp(x1)x
(x1)2(x+1)2
Figure 1. Plot of the three components in h1and h2.
Proof: From Proposition 2.3 of Torabi et al. [11], we have that for all yR
0h11+F0(y)
1+Fn(y)max(h1(1/2),h1(2)) =1.9129
and
0h21+F0(y)
1+Fn(y)max(h2(1/2),h2(2)) =0.1111.
Finally, since H(k)
nis the mean of hk(.)over the transformed data, the result is obtained.
Remember that Hnis expected to take values close to zero when H0is true. Hence,
H0versus H1will be rejected for large values of H(k)
n.Moreover,H
nis consistent under
assumption H1(the proof is straightforward from Proposition 2.5 in Torabi et al. [11]).
Table s 3and 4show the consistency of the test statistics H(1)
nand H(2)
nby simulation.
Moreover, since the statistic H(k)
nconsidered for the goodness-of-t test (3) is a function
of the standardized sample yior their transformed values, it is scale invariant. As a conse-
quence, the null distribution of H(k)
ndoes not depend on the parameter θ(see Proposition
2.4 in Torabi et al. [11]). Table 1contains the critical values for the tests based on H(1)
nand
H(2)
n, which were obtained by Monte Carlo from 100,000 simulated samples for several
sample sizes nand signicance levels α=0.01, 0.05, 0.1.
112 H.TORABIETAL.
Tab le 1. Critical values for H(1)
nand H(2)
nfor α=0.01, 0.05, 0.1.
H(1)
nH(2)
nH(1)
nH(2)
n
n0.01 0.05 0.1 0.01 0.05 0.1 n0.01 0.05 0.1 0.01 0.05 0.1
5 0.4190 0.3502 0.2955 0.0153 0.0097 0.0074 25 0.3894 0.3309 0.2928 0.0032 0.0019 0.0015
6 0.4139 0.3537 0.3057 0.0127 0.0082 0.0061 30 0.3834 0.3262 0.2881 0.0027 0.0016 0.0012
7 0.4156 0.3531 0.3086 0.0111 0.0071 0.0053 35 0.3760 0.3215 0.2852 0.0022 0.0014 0.0010
8 0.4158 0.3519 0.3070 0.0098 0.0061 0.0046 40 0.3704 0.3167 0.2825 0.0020 0.0012 0.0009
9 0.4157 0.3500 0.3054 0.0086 0.0054 0.0041 45 0.3668 0.3129 0.2785 0.0018 0.0010 0.0007
10 0.4154 0.3510 0.3066 0.0078 0.0050 0.0037 50 0.3613 0.3087 0.2746 0.0016 0.0009 0.0007
11 0.4148 0.3499 0.3064 0.0071 0.0045 0.0034 55 0.3592 0.3058 0.2721 0.0014 0.0009 0.0007
12 0.4119 0.3490 0.3051 0.0066 0.0041 0.0031 60 0.3523 0.3019 0.2691 0.0013 0.0007 0.0006
13 0.4100 0.3471 0.3049 0.0060 0.0038 0.0028 65 0.3492 0.2995 0.2668 0.0012 0.0007 0.0006
15 0.4064 0.3438 0.3026 0.0053 0.0033 0.0025 70 0.3445 0.2955 0.2641 0.0011 0.0006 0.0005
16 0.4045 0.3430 0.3016 0.0050 0.0031 0.0023 75 0.3432 0.2943 0.2629 0.0010 0.0006 0.0005
17 0.4033 0.3409 0.3017 0.0045 0.0029 0.0022 80 0.3394 0.2914 0.2609 0.0010 0.0006 0.0004
18 0.4010 0.3406 0.3006 0.0044 0.0027 0.0021 85 0.3383 0.2898 0.2595 0.0009 0.0005 0.0004
19 0.3989 0.3389 0.2989 0.0041 0.0026 0.0020 90 0.3349 0.2873 0.2574 0.0009 0.0005 0.0004
20 0.3971 0.3365 0.2985 0.0040 0.0024 0.0018 100 0.3309 0.2846 0.2536 0.0008 0.0005 0.0004
At this point, we prefer to illustrate the usefulness of the new proposals in detecting
some patterns of alternative distributions. This is the reason why we need to borrow some
results from Section 4,thatwillbediscussedlater.
In Section 4, we drive a simulation study to evaluate the performance of the two pro-
posed statistics. We compare their performances with those of a huge selection of statistics
developed for testing exponentiality, through a wide set of alternative distributions that
are commonly used in reliability studies (or life testing). In fact, in both contexts, it is of
interest the shape of the failure rate (or hazard rate) function. In this sense, distributions
can be classied according to their failure rate in:
Decreasing Failure Rate (DFR)
Increasing Failure Rate (IFR)
Unimodal (increasing–decreasing) Failure Rate (UFR)
Bathtub (decreasing–increasing) Failure Rate (BFR).
To identify the shape of the failure rate function, the empirical-scaled TTT (Total Time
on Test) transform of Aarset [12] can be used. The scaled TTT transform is convex (con-
cave) if the failure rate is decreasing (increasing). For BFR functions, the scaled TTT
transform is rst convex (concave) and then concave (convex).
From the simulation study results (see Tables 57), we may conclude that statistic H(1)
n
is preferable for IFR and UFR alternatives, whereas H(2)
nis preferable for DFR and BFR
alternatives.ThisdiscussionissummarizedinTable2.
Tab le 2. Shapes of the TTT plot for different types of failure rate function.
Shape of the failure rate function TTT plot Best stat.
IFR Concave H(1)
n
DFR Convex H(2)
n
UFR Concave–convex H(1)
n
BFR Convex–concave H(2)
n
JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 113
3. Exponentiality tests under evaluation
As it has been mentioned, the assumption of exponentiality in Equation (3) is frequently
used in many modelling situations, particularly in life testing and reliability. So, many tests
have been developed to determine whether a given data set ts the exponential distribution
or not. However, there are few studies that compare the performances of the proposals that
can be found in the literature. Among them, we can nd the following reviews: Epstein
[13,14] in 1960 described 12 dierent analytical and graphical procedures for determining
if the exponential distribution held. In a similar manner, Gnedenko et al. [15] in 1969 listed
eight tests in this area. Ascher [16] in 1990 discussed and compared 15 tests for exponen-
tiality. A wide selection of classical and recent tests for exponentiality (about 17 tests) were
discussed and compared by Henze and Meintanis [17] in 2005. Alizadeh and Arghami [18]
in 2011 studied ve entropy tests of exponentiality using ve statistics based on dierent
entropy estimates.
The second objective of this work is to help the applied researcher to decide which of
the so many existing proposals could better t his/her experimental situation. Therefore,
we consider a representative selection of 50 tests for exponentiality and investigate their
nite sample performance in an extensive simulation study. The tests have been divided
into 12 categories:
1. Tests based on entropy estimators
2. Tests based on correcting moments of entropy estimators
3. Tests based on Phi-divergence
4. Tests based on characterizations of the exponential distribution
5. Tests based on spacings
6. Tests based on Lorenz curve and Gini index
7. Tests based on the empirical distribution function
8. Tests based on the residual mean function
9. Tests based on the empirical Laplace transformation
10. Tests based on the empirical characteristic function
11. Tests based on correlation and covariance
12. Other tests.
3.1. Tests based on entropy estimators
The statistics considered in this section are based on the asymmetric Kullback–Leibler dis-
tance between two probability density functions (pdfs), where the concept of entropy of a
continuous random variable plays a fundamental role. Therefore, we start by introducing
this concept and several statistics commonly used to estimate it.
Let Xbe a random variable with cdf Fand continuous pdf f.Shannon[19]denedthe
entropy of Xas
H(f)=−
f(x)log f(x)dx.
Inordertoestimatetheentropy,VanEs[20], Correa [21], Vasicek [22], Ebrahimi et al.
[23], Zamanzade and Arghami [1], Alizadeh and Arghami [24]andArghami[25]proposed
114 H.TORABIETAL.
the following estimators:
HVEmn =1
nm
nm
i=1log n+1
m(X(i+m)X(i))
+
n
k=m
1
k+log(m)log(n+1),
HCmn =1
n
n
i=1
log
i+m
j=im(X(j)˜
X(i))(ji)
ni+m
j=imX(j)˜
X(i)2
,
HVmn =1
n
n
i=1
log n
2m(X(i+m)X(im)),
HEmn =1
n
n
i=1
log n
cim(X(i+m)X(im)),
HZmn =1
n
n
i=1
log n
zim(X(i+m)X(im)),
HA1mn =−1
n
n
i=1
log ˆ
f(X(i+m))+ˆ
f(X(im))
2,
HA2mn =1
n
n
i=1
log n
aim(X(i+m)X(im)),
where the ‘window size’ mis a positive integer such that m<n/2, X(i)=X(1)for i<1,
X(i)=X(n)for i>nand ˜
X(i)=i+m
j=imX(j)/(2m+1). Concerning the coecients,
ci=1+i1
mI[1,m](i)+2I[m+1,nm](i)+1+ni
mI[nm+1,n](i),
zi=i
mI[1,m](i)+2I[m+1,nm](i)+ni+1
mI[nm+1,n](i),
ai=I[1,m](i)+2I[m+1,nm](i)+I[nm+1,n](i),
where IA(i)=1wheniAand IA(i)=0 otherwise. Finally, ˆ
fis the kernel density
estimator
ˆ
f(Xi)=1
nh
n
j=1
kXiXj
h,(6)
where the kernel function kis chosen to be the standard normal pdf and bandwidth h=
1.06 sn1.5 is chosen to be the normal optimal smoothing formula, with sbeing the sample
standard deviation.
Let us go back to the goodness-of-t problem described in Equation (3) and assume that
the pdf f(x)has a non-negative support. The asymmetric Kullback–Leibler distance of fto
JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 115
agivenpdff0is given by
D(f,f0)=f(x,θ)log f(x;θ)
f0(x,θ) dx=−H(f)f(x,θ) log f0(x;θ)dx.(7)
The quantity D(f,f0)0 and equality hold i the null hypothesis H0is true, where
D(f,f0)=−H(f)log θ+θEf(X)=−H(f)+log θ+1.
Substituting θby its maximum likelihood estimator ˆ
θand applying a monotone trans-
formation on D(f,f0),wehavethat
exp(D(f,f0)) =exp(H(f)log ¯
x1)=exp(H(f))
exp(log ¯
x+1),
where H(f)can be estimated by the entropy estimators described before, giving rise to
dierent proposals to test exponentiality. For instance, Choi et al. [26] introduced the
following statistics based on Van Es [20]andCorea[21] entropy estimators, resp.:
TVEmn =exp{HVEmn}
exp(log(¯
X)+1),TC
mn =exp{HCmn}
exp(log(¯
X)+1).
Ebrahimi et al. [27]usedVasicek[22] entropy estimator and proposed the statistic:
TV1mn =exp{HVmn}
exp(log(¯
X)+1).
Based on Ebrahimi et al. [23]andAlizadeh[24] entropy estimators, Alizadeh and
Arghami [18]proposedthestatistics:
TE1mn =exp{HEmn}
exp(log(¯
X)+1),TA1
mn =exp{HA1mn}
exp(log(¯
X)+1).
Ebrahimi et al. [23] showed a linear relationship between their estimator and Vasicek’s.
Thus for a xed sample size nand xed m,thetestsbasedonTV1
mn and TE1mn have the
same power.
In our power study, we follow the recommendations of Ebrahimi et al. [27]andconsider
m=3forn=10, m=4forn=20 and m=6forn=50. The null hypothesis H0is rejected
for small values of TV1mn,TE1
mn and TA1mn.
Baratpour and Habibi Rad [7] introduced a new distance between two distributions,
similar to the Kullback–Leibler divergence described in Equation (7). Their proposal is
based on the cumulative residual entropy:
CKLn=n1
i=1((ni)/n)log((ni)/n)(X(i+1)X(i))+(n
i=1X2
i/n
i=1Xi)
n
i=1X2
i/n
i=1Xi
.
The null hypothesis H0is rejected for large values of CKLn.
116 H.TORABIETAL.
3.2. Tests based on correcting moments of entropy estimators
Park and Park [28] derived the non-parametric distribution function of Vasicek HVmn and
Ebrahimi HEmn estimators:
gv(x)=
0x
1or x
n+1
2m
n(x(i+m)x(im))ξi<xξi+1i=1, ...,n,
and
ge(x)=
0x
1or x
n+1
1
ni+1ηi)ηi<xηi+1i=1, ...,n,
resp., where ξi=(x(im)+···+x(i+m1))/2m,and
ηi=
ξm+11
m
m
k=i
(x(m+k)x(1))1im
1
2mx(im)+···+x(i+m1)m+1inm+1
ξnm+1+1
n+mk+1
i
k=nm+2
(x(n)x(km1))nm+2in+1.
In the light of Park and Park [28], Alizadeh and Arghami [2] and Zamanzade and
Arghami [1] derived the non-parametric distribution function of their estimators, resp.:
ga(x)=
0x
1or x
n+1
1
ni+1ηi)ηi<xηi+1i=1, ...,n,
where
ηi=
ξm+11
m
m
k=i
(x(m+k)x(1))1im
1
2mx(im)+···+x(i+m1)m+1inm+1
ξnm+1+1
m
i
k=nm+2
(x(n)x(km1))nm+2in+1
and
gz(x)=
0x
1or x
n+1
1
ni+1ηi)ηi<xηi+1i=1, ...,n,
JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 117
where
ηi=
ξm+1
m
k=i
1
k(x(m+k)x(1))1im
1
2m(x(im)+···+x(i+m1))m+1inm+1
ξnm+1+
i
k=nm+2
1
nk+2(x(n)x(km1))nm+2in+1.
Going back to the testing problem described in Equation (3), we have that
D(f,f0)=−H(f)+log θ+1,
where now H(f)can be estimated by Vasicek [22], Ebrahimi et al. [23], Alizadeh and
Arghami [25] and Zamanzade and Arghami [1] entropy estimators, giving rise to the
following proposals:
TV2mn =−HVmn +log ˆ
θv+1,
TE2mn =−HEmn +log ˆ
θe+1,
TA2mn =−HA2mn +log ˆ
θa+1,
TZmn =−HZmn +log ˆ
θz+1,
where ˆ
θv=Egv(X),ˆ
θe=Ege(X),ˆ
θa=Ega(X),ˆ
θz=Egz(X)and m=[n1], where [x]
means the integer part of x.Inparticular,m=2forn=10, m=3forn=20 and m=6for
n=50. The null hypothesis H0is rejected for large values of TV2mn,TE2
mn,TA2
mn and
TZmn.
3.3. Tests based on Phi-divergence
Alizadeh and Balakrishnan [8] introduced the following statistics based on Phi-divergence:
ˆ
Dφ=1
n
n
i=1
f0(xi,ˆ
θ)
ˆ
f(xi)
φˆ
f(xi)
f0(xi,ˆ
θ),
where ˆ
fwasdenedinEquation(6).Somesuitablechoicesofφare:
Kullback–Leibler (KL) divergence: φ(t)=tlog(t)
Hellinger distance: φ(t)=1
2(t1)2
Jereys distance: φ(t)=(t1)log(t)
Total variation distance: φ(t)=|t1|
χ2-divergence: φ(t)=(t1)2.
Recently, Alizadeh and Balakrishnan [8]usedthetermsTKL,TH,TJ,TTandT
χto
denote the test statistics based on KL, Hellinger, Jereys, total variation and Chi-squared
distance, resp. Again, the null hypothesis H0is rejected for large values of TKL, TH, TJ, TT
and Tχ.
118 H.TORABIETAL.
3.4. Tests based on characterizations of the exponential distribution
Alizadeh and Arghami [3] presented characterizations of the exponential distribu-
tion and then introduced three tests for exponentiality. They estimated the quantity
f(x,θ)log f0(x;θ)dxin Equation (7) by n1n
i=1log(f0(xi)). Therefore, the test statistic
based on Vasicek [22]entropyestimatorisgivenby
T=−HVmn 1
n
n
i=1
log(f0(xi)).
These authors showed that if X1and X2are two independent random variables with the
same distribution as Xwith continuous cdf F,then
(1) W=X1/(X1+X2)is (0, 1)-uniformly distributed i Xis exponential.
(2) Y=X1/X2is Fisher’s F(2,2)distributed i Xis exponential.
(3) Z=(X1X2)/(X1+X2)is (1, 1)-uniformly distributed i Xis exponential.
Using the previous results, they introduced the following statistics based on transfor-
mations of the data:
T1=−1
n
n
i=1
log n
2m(W(i+m)W(im),
T2=−1
n
n
i=1
log n
2m(Y(i+m)Y(im)+2
n
n
i=1
log(1+Yi),
T3=−1
n
n
i=1
log n
2m(Z(i+m)Z(im)+log(2),
where n=n(n1). In our power study, we consider m=9forn=10(n=90),m=15
for n=20(n=380)and m=35 for n=50(n=2450).Still,thesameauthorsin[4]
proposed the statistic
T4=−1
n
n
i=1|W(i)ˆ
f(W(i))F0(W(i))|,
where n=n(n1),ˆ
f(.)wasdenedinEquation(6)andF0(.)is the cdf of the (0, 1)-
uniform distribution. The null hypothesis H0is rejected for large values of T1,T
2,T
3
and T4.
Volk o va [5] constructed an integral type test of exponentiality based on Riedl and
Rossberg [29] characterization of the exponential law. The necessary and sucient con-
dition for Fto be exponential involves the identical distribution of the random variables
(X(i+s)X(i))and Xs,{1,2,...,nj}for some s1andn,whereXs,{1,2,...,ni}is sth-order
statistic of the sample X1,X2,...,Xni. The statistic according to Riedl and Rossbergs
JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 119
characterization is
I1=
0
(Hn(t)Gn(t)) dFn(t),
where
Hn(t)=n
31n
1i<jn
I[X2,{i,j,k}X1,{i,j,k}<t]fort0,
Gn(t)=n
21n
1i<jn
I[min(Xi,Xj)<t]fort0,
where Xs,{i,j,k}s=1,2 are sth-order statistic of the samples Xi,Xj,Xk.Thenullhypothesis
H0is rejected for small or large values of I1.
Volk o va and Ni kit i n [6] constructed an integral type test of exponentiality based on
Ahsanullah [30] characterization of the exponential law. In this case, the necessary and
sucient condition for Fto be exponential involves the identical distribution of the ran-
dom variables (ni)(X(i+1)X(i))and (nj)(X(j+1)X(j))for some i,jand n,1i<
jn. The statistic according to Ahsanullah’s characterization is
I2=
0
(Hn(t)Gn(t)) dFn(t),
where
Hn(t)=1
n2
n
i,j=1
I[|XiXj|<t]fort0,
Gn(t)=1
n2
n
i,j=1
I[2 min(Xi,Xj)<t]fort0.
The null hypothesis H0is rejected for small or large values of I2.
3.5. Tests based on spacings
Kochar [31] introduced the following statistic:
Kn=1
n¯
X
n
i=1
J(i/(n+1))X(i),
where J(i)=2(1i)(1log(1i)) 1. Statistic Kncanbealsoexpressedbasedon
normalized spacings, Di=(n+1i)(X(i)X(i1)),intheform
Kn=n
i=1e(i)Di
n
i=1Di
,
where e(i)=(ni+1)1n
j=iJ(j/(n+1)) and X(0)=0. The H0in Equation (3) is
rejected for large values of Kn.
120 H.TORABIETAL.
3.6. Tests based on Lorenz curve and Gini index
Gail and Gastwirth [32,33] proposed the following statistics based on the Lorenz curve and
Gini statistic:
Ln(p)=[np]
i=1X(i)
n¯
X,
Gn=n
j,k=1|XjXk|
2n(n1)¯
X,
where 0 <p<1and[np] is the largest integer less than or equal to np.Inourpower
study, we take p=0.5. The null hypothesis H0is rejected for small or large values of
Ln(p)and Gn. Additionally, D’Agostino and Stephens [34]proposedstatisticSn=2n
(2/n¯
X)n
i=1iX(i)=(n1)(1Gn), which is equivalent to Gini’s.
3.7. Tests based on the EDF
The statistics of this subsection try to measure the distance/discrepancy between two cdfs:
the theoretical or hypothesized, given by the exponential distribution and the EDF of the
scaled sample, Y1,...,Yn.
In particular, Kolmogorov–Smirnov, Kuiper, Finkelstein and Schafers [35], Cramér–von
Mises and Anderson–Darling statistics are given by (see [34]):
KS =max max
1ini
nF0(y(i)),max
1inF0(y(i))i1
n,
V=max
1ini
nF0(y(i))+max
1inF0(y(i))i1
n,
S=
n
i=1
max
F0(y(i))i
n
,
F0(y(i))i1
n,
W2=1
12n+
n
i=1F0(y(i))2i1
2n2
,
A2=−n1
n
n
i=1
(2i1)(log(F0(y(i))) +log(1F0(y(ni+1)))).
Lilliefors [36] computed the estimated critical points for the Kolmogorov–Smirnov ’s statis-
tic for testing exponentiality when the mean was estimated. The null hypothesis H0is
rejected for large values of KS, V, S,W
2and A2.
3.8. Tests based on the mean residual life function
The distribution of a random variable Xis exponential i E(Xt|X>t)=θfor
each t>0. Since this condition is equivalent to E(min(X,t)) =F(t)/θ ,foreacht>0,
JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 121
Baringhaus and Henze [37] proposed the following Cramér–von Mises and Kol-
mogorov–Smirnov type statistics:
CMn=1
n+
n
j,k=1
[2 3emin(Yj,Yk)2min(Yj,Yk)(eYj+eYk)+2emax(Yj+Yk)],
KSn=nmax(KS+
n,KS
n),
where
KS+
n=max
j=0,...,n11
n(Y(1)+···+Y(j))+Y(j+1)1j
nj
n,
KS
n=max
j=0,...,n1j
n1
n(Y(1)+···+Y(j))Y(j)1j
n.
The null hypothesis H0is rejected for large values of CMnand KSn.
Aboukhamseen and Aly [38] introduced the following statistics based on the mean
residual life function
T1,n=max
1kn|Tn(k/n,1)|,
T2,n=1
n
n
k=1
T2
n(k/n,1),
T1,n(γ ) =max
1kn|Tn(k/n,γ)|,
T2,n(γ ) =1
n
n
k=1|Tn(k/n,γ)|,
where
Tn(u,γ) =n(1/n)n
i=n[nu]+1Dγ
i
(1/n)n
i=1Diγ[nu]
n[1 +γ],0<γ <1,
Di=(n+1i)(X(i)X(i1))and X(0)=0 are normalized spacings. In our power study,
we take γ=0.5. The null hypothesis H0is rejected for large values of T1,n,T
2,n,T
1,n(γ )
and T2,n(γ ).
3.9. Tests based on the empirical Laplace transformation
In these tests, the Laplace transform of the standard exponential distribution is estimated
by its empirical counterpart ψn(t)=n1exp(tYi).
Baringhaus and Henze [39]andHenze[40]proposedthefollowingstatistics:
BHn=1
n
n
j,k=1(1Yj)(1Yk)
Yj+Yk+a+2YjYk(Yj+Yk)
(Yj+Yk+a)2+2YjYk
(Yj+Yk+a)3,
122 H.TORABIETAL.
HEn=1
n
n
j,k=1
1
Yj+Yk+a2
n
j=1
exp(Yj+a)E1(Yj+a)
+n(1aexp(a)E1(a)),
where E1=
zexp(t)/tdtistheexponentialintegral.Inourpowerstudy,wetake
a=2.5. The null hypothesis H0is rejected for large values of BHnand HEn.
3.10. Tests based on the empirical characteristic function
In these tests, the characteristic function of a random variable Xis estimated by its empiri-
cal counterpart φn(t)=n1n
i=1exp(itXi). The following statistics were proposed, resp.,
by Epps and Pulley [41] and Henze and Meintanis [17]:
EPn=48n1
n
n
i=1
exp(Yi)1
2,
T(1)
n,a=a
n
n
j,k=11
a2+Y2
jk+1
a2+Y2
jk+
2a
n2
n
j,k=1
n
l=11
a2+[YjkYl]2+1
a2+[Yjk+Yl]2
+a
n3
n
j,k=1
n
l,m=11
a2+[YjkYlm]2+1
a2+[Yjk+Ylm]2,
T(2)
n,a=1
2nπ
a
n
j,k=1exp Y2
jk
4a+exp Y2
jk+
4a
1
n2π
a
n
j,k=1
n
l=1exp [YjkYl]2
4a+exp [Yjk+Yl]2
4a
+1
2n3π
a
n
j,k=1
n
l,m=1exp [YjkYlm]2
4a+exp [Yjk+Ylm]2
4a,
where Yjk=YjYkand Yjk+=Yj+Yk.FollowingtherecommendationofHenzeand
Meintanis [17], we take a=2.5 in our power study. The null hypothesis H0is rejected for
large values of |EPn|,T
(1)
n,aand T(2)
n,a.
3.11. Tests based on correlation and covariance
Shapiro–Wilk statistic [42] and the Stephens’ modication of the Shapiro–Wilk statistic
[43]aregivenby
W=n(¯
xx(1))2
(n1)n
i=1(xi¯
x)2,
JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 123
Ws=(n
i=1xi)2
n(n+1)n
i=1x2
in(n
i=1xi)2.
The null hypothesis H0is rejected for small or large values of W and Ws.
Fortiana and Grané [9]proposedthefollowingstatisticbasedontheHoedings
maximum correlation, ρ+,
Qn=sn
¯
xn
ρ+(Fn,F0)=n
i=1lix(i)
n
i=1x(i)
,
where ¯
xnand s2
nare the sample mean and variance and li=(ni)log(ni)(n
i+1)log(ni+1)+log(n),i=1, ...,n,with0log0=0. The null hypothesis H0is
rejected for small or large values of the Qnstatistic. As explained in Grané and Fortiana
[44], the right tail of Qnis signicant for DFR alternatives, whereas the left tail is signicant
for IFR alternatives.
Similarly, Montazeri and Torabi [10] implemented the previous idea based on the sam-
ple covariance between Fnand a given cdf F0in the form of a specic test. Their aim
wastoobtainaverysimpleandpowerfultestforexponentiality.Sety=(y(1),...,y(n)),
F0(y):=(F0(y(1)),...,F0(y(n))) and Fn(y):=(Fn(y(1)),...,Fn(y(n))).
COV : =COV(F0(y),Fn(y))
=1
n
n
i=1
(F0(y(i))i/n)1
n
n
i=1
F0(y(i))1
n
n
i=1
i/n
=1
n2
n
i=1
iF0(y(i))n+1
2n2
n
i=1
F0(y(i))=1
2n2
n
i=1
(2in1)F0(y(i)).
As happens with the previous statistic, the null hypothesis H0is rejected for small of COV.
The COV statistic can be seen as a generalization of Fortiana and Grané [45]statis-
tic.Theseauthorsfoundtheexactandasymptoticdistributionoftheteststatisticin
case the Probability Integral Transformed Theorem can be applied, i.e. F0(Y)d
=X,where
XU(0, 1).
3.12. Other tests
Cox and Oakes [46] introduced the statistic
COn=n+
n
i=1
(1Yi)log Yi.
The null hypothesis H0is rejected for small or large values of COn.
124 H.TORABIETAL.
Tchi r in a [ 47]proposedthestatistic
Mn=
1
n
n
i=1
log Xi
¯
X+γ
,
together with its one-sided counterparts
M+
n=1
n
n
i=1
log Xi
¯
X+γ,
M
n=−1
n
n
i=1
log Xi
¯
X+γ,
that are usually called the Moran statistics (see Moran [48]andShorack[49]) and γ=
0.577215 ...is the Euler constant. Tchirina [47]showedthatM
+
nis consistent in the IFR-
class, M
nis consistent in the DFR-class and Mnis consistent in the class of distributions
with monotonic failure rate. The null hypothesis H0is rejected for large values of Mn.
Mimoto and Zitikis [50] introduced the Atkinson statistic for exponentiality, given by
Mn(p)=n
(1/nn
i=1Xp
i)1/p
¯
X((1+p))1/p
,1<p<1,
where (.)is the gamma function. Following the recommendation of Mimoto and Zitikis
[50], we take p=0andp=0.99 in our power study. Note that when p=0, the statistic is
given by the following equation:
Mn(0)=n X1/n
i
¯
Xeγ
,
which is a conversion from the Moran statistic Mnfor exponentiality. The null hypothesis
H0is rejected for large values of Mn(p).
Hollander and Proschan [51] introduced the ‘new better than used’ test for exponen-
tiality based on the statistic
J=2
n(n1)(n2)
i=j,k;j<k
I[xi>xj+xk].
The null hypothesis H0is rejected for small values of J.
Deshpande [52] proposed the test statistic
Jb=1
n(n1)
i=j
I[xi>bxj], 0 <b<1.
In our power study, we take b=0.5. The null hypothesis H0is rejected for large values
of Jb.
JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 125
Kimber [53] proposed a statistic to test for exponentiality based on the Stabilized
Probability Plot:
Dsp =max
i|risi|,
where ri=2arcsin(1exp(xi/¯
x)) and si=2arcsin((i0.5)/n).Thisstatistic
was rst introduced by Michael [54] to test for uniformity and Normality. The null
hypothesis H0is rejected for large values of Dsp.
4. Simulation study
In this section, we study the performance of the exponentiality tests based on H(k)
nand
on the statistics introduced in Section 3through a wide set of alternative distributions
proposed by Henze and Meintains [17]. These families of probability distributions are com-
monly used as alternatives to the exponential model and include pdf’s fwith DFR, IFR as
well as models with UFR functions and BFR functions. In Figure 2,wedepicttheshapeof
their failure rate function for several values of the parameter θ.
1. Weibull distribution, W ),withdensity
f(x)=θxθ1exθ,x>0.
2. Gamma distribution, (θ),withdensity
f(x)=xθ1ex
(θ) ,x>0.
3. Log-Normal distribution, LN(θ),withdensity
f(x)=1
x2πθ2exp (log x)2
2θ2,x>0.
4. Half-Normal distribution, HN, with density
f(x)=2exp{−x2/2},x>0.
5. Uniform distribution, U, with density 1, 0 x1.
6. Chen distribution [55], CH),withcdfas
F(x)=1exp{2(1exθ)},x>0.
7. Linear Increasing Failure Rate law, LF(θ),withdensity
f(x)=(1+θx)exp(xθx2/2),x>0.
8. Modied Extreme Value distribution, EV),withcdfas
F(x)=1exp{θ1(1ex)},x>0.
126 H.TORABIETAL.
Increasing Failure Rate (IFR)
W(1.4)
G(2)
LF(4)
LF(2)
HN
U(0,1)
CH(1.5)
CH(1)
EV(.5)
EV(1.5)
Decreasing Failure Rate (DFR)
W(.8)
G(.4)
Bathtub Failure Rate (BFR)
CH(.5)
Unimodal Failure Rate (UFR)
LN(.8)
LN(1.5)
DL(1.5)
DL(.5)
Figure 2. Shape of the failure rate function for the alternatives considered.
9. Dhillondistribution[56], DL ),withcdfas
F(x)=1exp{−(log(x+1))θ+1},x>0.
Table s 3and 4contain the estimated value of H(1)
nand H(2)
nfor each of the alterna-
tives considered, computed as the average value from 10,000 simulated samples of sizes
n=10,20,50,100,1000. With the aim of showing the consistency of the test statistics based
on H(k)
n,inthelastrowofTables3and 4we give the estimated values of H(k)
nfor n=∞,
that is the value of D(F0,F1)computed by the command integrate in Rsoftware, where
θis replaced by the expectation of F1distribution.
Power against an alternative distribution has been estimated by the relative frequency
of values of the corresponding statistic in the critical region for 100,000 simulated sam-
ples of size n=10, 20, 50. We use Rsoftware to compute the estimated power of the new
tests. We also use the package (exptest)fromRsoftware. The results of this Monte
JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 127
Tab le 3. Estimated value of H(1)
nunder hypothesis H1based on 10,000 simulations for several values of n.
IFR UFR DFR BFR
n. G(1) W(1.4) G(2) HN U(0,1) CH(1) CH(1.5) LF(2) LF(4) EV(0.5) EV(1.5) LN(0.8) LN(1.5) DL(1) DL(1.5) W(0.8) G(0.4) CH(0.5)
10 0.1528 2728 0.2983 0.2378 0.3289 0.2211 0.3543 0.2558 0.2826 0.2203 0.2901 0.2610 0.0661 0.2419 0.3271 0.0960 0.0663 0.0700
20 0.1578 0.3241 0.3499 0.2805 0.3893 0.2593 0.4143 0.3061 0.3372 0.2585 0.3452 0.2903 0.0517 0.2745 0.3765 0.0863 0.0646 0.0685
50 0.1532 0.3709 0.3896 0.3269 0.4395 0.2983 0.4561 0.3538 0.3866 0.2990 0.3953 0.3076 0.0396 0.3017 0.4085 0.0673 0.0679 0.0692
100 0.1473 0.3899 0.4026 0.3530 0.4583 0.3243 0.4692 0.3766 0.4058 0.3253 0.4158 0.3121 0.0369 0.3124 0.4189 0.0569 0.0701 0.0721
1000 0.0109 0.4053 0.4143 0.3814 0.4758 0.3612 0.4804 0.3998 0.4238 0.3613 0.4350 0.3132 0.0339 0.3185 0.4291 0.0520 0.0730 0.0751
0 0.4071 0.4154 0.3847 0.4778 0.3657 0.4816 0.4024 0.4258 0.3657 0.4372 0.3133 0.0334 0.3192 0.4302 0.0519 0.0732 0.0755
Tab le 4. Estimated value of H(2)
nunder hypothesis H1based on 10,000 simulations for several values of n.
IFR UFR DFR BFR
n. G(1) W(1.4) G(2) HN U(0,1) CH(1) CH(1.5) LF(2) LF(4) EV(0.5) EV(1.5) LN(0.8) LN(1.5) DL(1) DL(1.5) W(0.8) G(0.4) CH(0.5)
10 0.0018 0.0015 0.0017 0.0014 0.0020 0.0015 0.0021 0.0015 0.0016 0.0015 0.0016 0.0018 0.0064 0.0016 0.0019 0.0033 0.0077 0.0063
20 0.0009 0.0011 0.0013 0.0009 0.0016 0.0008 0.0019 0.0010 0.0012 0.0008 0.0012 0.0013 0.0057 0.0010 0.0017 0.0021 0.0065 0.0050
50 0.0004 0.0009 0.0012 0.0006 0.0015 0.0004 0.0019 0.0007 0.0010 0.0005 0.0010 0.0010 0.0053 0.0007 0.0016 0.0014 0.0056 0.0042
100 0.0002 0.0008 0.0012 0.0005 0.0014 0.0003 0.0019 0.0006 0.0009 0.0004 0.0009 0.0009 0.0052 0.0006 0.0016 0.0011 0.0053 0.0039
1000 0.00002 0.0008 0.0011 0.0004 0.0014 0.0003 0.0019 0.0006 0.0009 0.0003 0.0009 0.0008 0.0051 0.0005 0.0016 0.0009 0.0050 0.0036
0 0.0008 0.0011 0.0004 0.0014 0.0003 0.0019 0.0006 0.0009 0.0003 0.0009 0.0008 0.0051 0.0005 0.0016 0.0008 0.0050 0.0036
128 H.TORABIETAL.
Tab le 5. Percentage empirical power of different exponentiality tests, α=0.05, n=10.
IFR UFR DFR BFR
G(1) W(1.4) G(2) HN U(0,1) CH(1) CH(1.5) LF(2) LF(4) EV(0.5) EV(1.5) LN(0.8) LN(1.5) DL(1) DL(1.5) W(0.8) G(0.4) CH(0.5)
1 T VE 5 17 23 10 28 8 36 13 18 8 18 24 19 17 33 4 11 7
2TC527 34 18 53 14 59 22 29 14 31 25 4 20 46 2 3 1
3TV15 26 33 18 52 14 56 21 28 14 30 26 6 20 45 2 5 3
4TA1 5 29 36 20 51 14 61 24 31 14 33 26 1 21 50 10 0
5CKL
n5232718 60 14 56 22 29 14 33 17 15 14 34 3 7 5
6 TV2 5 21 27 14 44 11 46 17 22 11 23 23 14 17 36 3 17 11
7 TE2 5 19 25 13 41 11 43 16 20 10 21 23 20 17 35 4 21 14
8 TA2 5 17 23 11 37 9 39 14 18 9 19 23 24 15 33 5 25 18
9 TZ 5 16 22 10 33 8 36 13 16 8 17 23 29 16 31 7 29 21
10 TKL 5 27 34 18 50 14 60 23 31 14 33 24 1 19 46 1 0 0
11 TH 5 1 0 1 5 1 2 1 1 1 2 1 47 1023 66 55
12 TJ 5 1 0 1 5 1 2 1 1 1 2 1 47 1023 66 55
13 TT 5 0 0 1 1 1 0 0 0 1 1 1 48 1023 66 54
14 Tχ5222142 8232 4 347 2021 59 47
15 T152433153212 49192511 24 28 520 47 751 36
16 T2627 36 16 35 13 52 20 26 13 26 30 322 50 437 25
17 T352433153212 49192511 24 28 520 47 750 36
18 T4527 35 18 40 13 54 22 29 13 29 29 121 48 10 0
19 I15 13 20 8 15 6 28 10 14 6 12 21 6 13 31 10 46 35
20 I25 16 22 10 24 7 36 12 17 7 16 19 9 13 33 11 50 38
21 Kn526 32 19 56 15 61 25 32 15 34 20 1 16 41 1 0 0
22 Ln(0.5)5 17 22 11 31 8 43 14 20 9 21 14 32 12 31 14 55 43
23 Gn5 16 21 11 37 9 47 14 20 9 23 14 39 13 33 14 49 37
24 KSn5 16 21 11 28 8 36 14 19 8 19 16 32 13 30 10 41 31
25 V 5 15 20 11 35 8 38 14 19 8 19 17 25 12 29 9 35 25
26 S51926134010 48172310 24 19 351436 1246 34
27 W25 18 24 12 36 9 44 16 22 9 22 18 35 14 35 11 45 34
28 A25 13 18 8 28 7 36 12 17 7 16 13 37 10 27 17 67 54
29 KSn5 21 25 15 42 11 46 19 25 12 27 18 26 15 35 6 28 20
(continued).
JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 129
Tab le 5. Continued.
IFR UFR DFR BFR
G(1) W(1.4) G(2) HN U(0,1) CH(1) CH(1.5) LF(2) LF(4) EV(0.5) EV(1.5) LN(0.8) LN(1.5) DL(1) DL(1.5) W(0.8) G(0.4) CH(0.5)
30 CMn5 17 22 11 36 8 44 17 22 8 22 16 38 13 32 12 45 34
31 T1,n5 13 17 10 30 7 35 12 17 7 17 12 38 9 24 14 46 35
32 T2,n5 16 21 11 35 8 42 14 20 8 21 14 40 10 29 15 50 40
33 T1,n(0.5)51619124010 39152010 21 14 241127 935 25
34 T2,n(0.5)5 16 20 12 43 9 43 15 21 9 22 13 10 11 27 8 18 16
35 BH2.5 5 17 24 11 33 8 45 14 21 8 21 17 40 13 34 14 47 37
36 HE2.5 5 17 24 11 33 8 46 14 21 8 21 17 40 13 35 14 47 37
37 EPn5 16 22 11 35 9 47 13 19 9 22 15 40 14 35 14 47 37
38 T(1)
n,2.5 526 31 18 52 15 60 23 31 15 35 19 51741 2 2 0
39 T(2)
n,2.5 5253017 51 14 59 23 30 14 34 19 19 16 40 5 21 14
40 COn5 16 25 10 30 9 45 14 19 8 19 16 34 15 37 17 68 53
41 Mn5 9 14 6 17 5 29 8 11 5 12 10 31 7 22 21 74 62
42 Mn(0)52434 14 32 11 51 18 25 11 25 26 19 19 46 11 62 46
43 Mn(0.99)5 16 23 10 31 8 44 13 19 8 20 16 40 12 32 15 50 38
44 J 5 25 33 17 47 13 57 22 29 13 31 23 1 18 44 1 0 0
45 J0.5 5 21 28 14 34 10 47 18 24 11 23 23 1 17 40 1 0 0
46 Dsp 5 19 27 12 24 9 39 15 20 9 18 26 27 17 39 10 51 36
47 Wn5 10 10 9 32 7 24 11 14 7 17 8 36 7 10 11 26 20
48 Ws5 16 22 10 37 8 46 14 20 8 22 15 38 11 30 13 35 27
49 Qn5 15 19 11 42 8 44 14 19 8 22 12 36 10 25 12 34 27
50 COV 5 26 36 16 35 12 53 21 28 12 26 31 222 49 11 0
H(1)
n527 36 18 47 16 60 23 31 14 32 24 1 20 46 1 0 0
H(2)
n511121 3111 1 247 1221 65 54
Note. The maximum powers attained (from first to third place) are indicated in bold.
130 H.TORABIETAL.
Tab le 6. Percentage empirical power of different exponentiality tests, α=0.05, n=20.
IFR UFR DFR BFR
G(1) W(1.4) G(2) HN U(0,1) CH(1) CH(1.5) LF(2) LF(4) EV(0.5) EV(1.5) LN(0.8) LN(1.5) DL(1) DL(1.5) W(0.8) G(0.4) CH(0.5)
1TVE 5 27421452 10 62 1827 9 26 52 45 30 61 6 31 18
2 TC 5 37 50 24 88 18 83 30 42 18 48 40 19 26 65 2 24 14
3TV15 37512387 18 82 30 42 18 48 43 26 27 67 3 32 19
4TA1 5 47 63 30 80 20 87 36 52 20 52 49 0 35 81 10 0
5CKL
n533382892 21 86 34 48 21 59 18 43 16 48 8 25 16
6 TV2 5 31 45 19 83 15 76 25 35 15 40 43 42 25 62 6 51 33
7 TE2 5 29 42 17 79 13 73 22 31 13 35 45 50 25 60 9 56 39
8 TA2 5 24 35 13 69 9 64 17 25 9 28 45 59 24 55 13 63 46
9 TZ 5 17 27 7 54 6 50 11 5 21 45 42 64 22 45 17 66 50
10 TKL 5 49 63 31 84 23 91 41 56 22 59 41 0 33 79 10 0
11 TH 5 1 1 2 15 1 50 2 3 1 4 1 70 1134 89 79
12 TJ 5 1 1 2 15 1 5 2 3 1 4 1 70 1133 89 79
13 TT 5 0 0 1 6 1 1 1 1 1 1 1 71 1034 89 80
14 Tχ5111221 9121 4 470 3 2 28 77 64
15 T153555185213 72233413 34 51 103174 1686 73
16 T25 36 55 18 55 14 74 25 36 14 35 50 7 31 75 13 82 67
17 T35 35 55 17 51 13 72 24 35 13 34 50 9 31 74 16 85 70
18 T4546 62 28 69 19 83 35 50 19 49 53 036 80 10 0
19 I15 27 46 12 28 9 60 17 26 9 22 48 11 29 67 18 80 68
20 I252947143910 67203411 28 45 162868 2082 70
21 Kn5445333 88 25 91 42 57 25 64 25 0 23 66 1 0 0
22 Ln(0.5)53446215714 78274014 40 27 552264 2382 70
23 Gn53546217015 84294215 46 24 67 19 62 24 76 63
24 KSn52840185213 67243513 35 30 582056 1771 56
25 V 5 26 37 16 66 12 68 22 32 12 35 32 44 19 54 13 62 47
26 S53549227016 82294315 46 33 612066 2075 61
27 W253447216614 79284214 43 33 622365 2076 61
28 A253045176212 76243612 37 34 622163 2689 78
29 KSn53546247218 79324418 48 28 552262 1462 47
(continued).
JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 131
Tab le 6. Continued.
IFR UFR DFR BFR
G(1) W(1.4) G(2) HN U(0,1) CH(1) CH(1.5) LF(2) LF(4) EV(0.5) EV(1.5) LN(0.8) LN(1.5) DL(1) DL(1.5) W(0.8) G(0.4) CH(0.5)
30 CMn53547227016 83304316 47 27 662163 2275 61
31 T1,n52736186313 70243613 38 23 632052 2173 59
32 T2,n53244217114 80284214 45 25 652160 2477 64
33 T1,n(0.5)52836217416 70263815 42 22 472350 1561 47
34 T2,n(0.5)52938217915 78274115 46 19 242252 1442 36
35 BH2.5 53648216514 84284214 44 26 67 21 64 24 77 63
36 HE2.5 53748216214 84284215 43 29 662064 2479 63
37 EPn53648216615 84284215 45 25 67 20 64 24 76 63
38 T(1)
n,2.5 5455533 86 25 91 42 56 25 63 27 18 25 67 4 33 23
39 T(2)
n,2.5 54250318523 91 39 54 23 61 22 38 21 61 10 50 37
40 COn5375419501381253713 37 33 602572 2891 80
41 Mn5 29 46 13 37 9 70 19 29 9 26 31 52 21 67 31 94 85
42 Mn(0)54464 22 50 16 81 29 41 16 39 47 42 34 81 20 90 77
43 Mn(0.99)53447196313 83274013 43 26 672063 2577 64
44 J 5 47 62 29 78 22 89 39 53 22 55 46 0 35 79 10 0
45 J0.5 54360266519 84354819 49 50 035 77 1 0 0
46 Dsp 53351175112 66233313 31 56 46 34 71 14 79 62
47 Wn52120197213 63253413 43 12 601023 1545 34
48 Ws53240227514 84284214 49 20 611653 1853 42
49 Qn53238238617 85304217 54 18 611550 1956 43
50 COV 5 47 66 25 56 18 83 33 44 18 44 61 3 43 84 00 0
H(1)
n549 60 31 78 24 91 40 55 23 58 33 0 30 74 5 0 0
H(2)
n6 6 10 2 18 2 29 4 7 2 8 8 71 420 34 89 79
Note. The maximum powers attained (from first to third place) are indicated in bold.
132 H.TORABIETAL.
Tab le 7. Percentage empirical power of different exponentiality tests, α=0.05, n=50.
IFR UFR DFR BFR
G(1) W(1.4) G(2) HN U(0,1) CH(1) CH(1.5) LF(2) LF(4) EV(0.5) EV(1.5) LN(0.8) LN(1.5) DL(1) DL(1.5) W(0.8) G(0.4) CH(0.5)
1TVE 5 5582239113 973454 14 53 93 85 65 96 11 78 55
2 TC 5 64 81 42 100 33 100 54 73 33 85 73 69 44 93 9 85 66
3TV15 658443100 32 100 55 74 33 86 79 74 48 95 11 89 72
4TA1 5 8095 48 99 32 100 64 83 33 83 84 0 70 99 00 0
5CKL
n5626654100 44 100 68 85 44 96 24 85 21 76 19 68 49
6TV25 648538100 28 100 51 71 29 82 91 89 57 96 20 95 84
7 TE2 5 58 82 31 100 21 99 43 64 22 75 92 92 58 95 27 96 87
8TA2 5 32611198 6 93 18 34 6 42 85 94 43 86 34 97 88
9TZ 5 517162 0 46 14 3 5 55 93 21 42 33 95 83
10 TKL 5 87 97 61 100 45 100 76 91 46 93 76 0 65 100 00 0
11 TH 5 2 1 4 58 3 22 5 9 3 17 1 94 0156 100 98
12 TJ 5 2 1 4 58 3 22 5 9 3 17 1 93 115599 98
13 TT 5 1 0 3 45 2 12 3 5 2 12 1 94 0057 100 100
14 Tχ5 0 0 0 53 0 16 1 2 0 5 8 92 5 1 39 94 84
15 T156792298518 98 42 62 19 59 94 27 68 99 34 100 98
16 T256992318820 98 45 64 21 62 94 22 68 99 31 100 97
17 T356792298518 98 42 62 19 59 94 27 68 99 34 100 98
18 T458195 52 95 35 100 64 84 35 84 92 5 73 100 386 67
19 I156590256215 96396015 51 93 25 68 98 37 99 97
20 I256791287217 98437517 57 92 29 6899 38 99 97
21 Kn5859071 100 57 100 83 95 57 98 38 0 39 95 0 0 0
22 Ln(0.5)57691469330100 62 81 30 81 60 90 49 98 48 99 97
23 Gn579905499 38 100 69 87 38 90 47 95 39 97 48 98 94
24 KSn56483399326 98 54 73 26 75 71 91 46 95 38 98 92
25 V 5 59 79 35 99 23 99 50 71 24 76 75 84 46 94 28 96 87
26 S576904999 34 100 66 84 34 88 73 93 52 98 43 99 95
27 W2575904898 32 100 64 83 32 86 76 94 52 98 44 99 95
28 A2574924599 29 100 61 81 30 85 86 93 58 99 52 100 99
29 KSn571865099 36 100 65 82 36 88 62 92 43 96 35 97 90
(continued).
JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 133
Tab le 7. Continued.
IFR UFR DFR BFR
G(1) W(1.4) G(2) HN U(0,1) CH(1) CH(1.5) LF(2) LF(4) EV(0.5) EV(1.5) LN(0.8) LN(1.5) DL(1) DL(1.5) W(0.8) G(0.4) CH(0.5)
30 CMn577905399 37 100 69 87 37 90 60 95 44 97 46 99 94
31 T1,n565814399 30 99 58 78 30 83 55 93 38 94 40 98 92
32 T2,n575885099 36 100 68 86 36 90 56 95 42 97 46 99 95
33 T1,n(0.5)5607443100 31 99 56 75 31 83 51 85 35 88 29 93 83
34 T2,n(0.5)5688048100 35 100 63 82 35 88 39 58 32 91 32 86 77
35 BH2.5 579915398 37 100 69 87 37 89 47 95 40 97 48 99 95
36 HE2.5 579915398 37 100 69 87 37 89 46 95 40 97 48 99 94
37 EPn580915498 38 100 69 87 38 90 45 95 39 97 48 99 94
38 T(1)
n,2.5 5819064 100 48 100 77 91 48 96 49 74 43 96 25 92 83
39 T(2)
n,2.5 5808564 100 49 100 78 92 49 95 32 83 32 93 31 93 84
40 COn58296 45 91 30 100 60 80 30 78 66 92 55 99 56 100 99
41 Mn57595 32 77 21 99 48 67 21 63 77 86 60 99 56 100 99
42 Mn(0)58397 41 82 29 100 56 74 28 71 84 82 70 100 48 100 99
43 Mn(0.99)579905298 36 100 68 87 36 89 44 95 39 96 49 99 95
44 J 5 86 96 57 99 41 100 74 89 41 91 86 0 69 99 000
45 J0.5 58496 53 97 38 100 70 86 38 87 91 0 72 99 000
46 Dsp 5699135100 23 99 48 69 24 73 97 83 71 99 31 99 95
47 Wn5 61 57 55 100 41 100 69 86 42 93 16 93 17 61 31 84 70
48 Ws5 74 80 57 100 42 100 72 89 43 94 29 93 28 88 34 88 75
49 Qn5 73 79 59 100 47 100 74 89 47 96 26 93 25 86 35 89 77
50 COV 5 82 96 43 87 30 99 60 78 29 74 96 580 100 000
H(1)
n588 94 65 99 50 100 79 92 49 94 51 0 50 98 000
H(2)
n5 37 62 13 78 7 94 24 44 6 46 47 95 23 87 59 100 98
Note. The maximum powers attained (from first to third place) are indicated in bold.
134 H.TORABIETAL.
Tab le 8. Ranking from first to the fifth of the average powers computed from the values in Tables 57.
Rank IFR UFR DFR–BFR
1H
(1)
n,K
n,TKL COV M
n
2T
(1)
n,2.5,T
(2)
n,2.5 T4T1COn
3J,J
0.5 Dsp H(2)
n
4COVM
n(0),J
0.5 TH, TJ, TT
5TA1,CKL
nTA1 , H(1)
nA2
Empirical and theoretical dens.
Data
Density
01234567
0.0 0.2 0.4 0.6
empirical
theoretical
0123456
0123456
Q−Q plot
Theoretical quantiles
Empirical quantiles
01234567
0.0 0.2 0.4 0.6 0.8 1.0
Empirical and theoretical CDFs
Data
CDF
0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.81.0
P−P plot
Theoretical probabilities
Empirical probabilities
Figure 3. Empirical and theoretical density and cumulative distribution with the P–P and Q–Q plots for
secondary reactor pumps.
Carlo experiment are shown in Tables 58.Inparticular,Tables57report the power,
at a 5% signicance level, of 50 exponentiality tests based on the statistics considered in
Section 3.Foreachcolumn,weindicateinboldthebestthreepowerresults.Finally,Table8
reports a ranking of average powers, from the rst to the fth, computed from the values
in Tables 57.
JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 135
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
Total Time on Test (TTT) plot
r/n
TTT
Figure 4. TTT plot for secondary reactor pumps.
Tab le 9. Results of the exponentiality test for secondary reactor pumps data.
Stat. KSnVKS
nA2W2CMnH(2)
n
p-Value 0.122 0.102 0.099 0.049 0.041 0.036 0.017
4.1. Results and recommendations
Based on these comparisons, the following recommendations can be formulated for the
applicationoftheevaluatedstatisticsfortestingexponentialityinpractice.Asaglobalsum-
mary, in Table 8we give the top ve statistics according to their average power performance
for each type of alternative (IFR, UFR, DFR, BFR). In particular, from Tables 57we nd
out:
1. For IFR alternatives, the tests based on TKL, Kn,H
(1)
n,TA1, T4,J,TC,TV1,T
(1)
n,2.5,
T(2)
n,2.5,COVandM
n(0)are the most powerful, whereas the tests based on Q
n(5),TZ,
TH,TJ,TT,T
χand E are the least powerful. The dierence between the powers of
H(1)
nand TKL is not substantial, and both statistics have the best performance for IFR
alternatives.
2. For UFR alternatives, the tests based on COV, H(1)
n,T
1,T
2,T
3,T
4,J,J
0.5,BH
2.5,HE
2.5
and Dsp are the most powerful, whereas those based on Q
n(5),TH,TJ,TT,T
χand E
136 H.TORABIETAL.
are the least powerful. Note that although LN(1.5)isUFRitisverysimilartoaDFR
alternative (see Figure 2).
3. For DFR and UFR alternatives, the most powerful tests are those based on H(2)
n,TH,
TJ, TT, COn,M
n,M
n(0),T
1,T
2,T
3and T4. On the other hand, tests based on TKL,
J0.5 and J are the least powerful.
4. As expected [57], no single test can be considered the best one for testing exponen-
tiality against all alternatives. Moreover, we observe that the tests based on TVE, Tχ,
E, Q
n(5)are never in the top ve for any alternative considered.
5. Among the EDF tests, the tests based on A2and W2aremorepowerfulthanthose
basedonKSandV.Inparticular,thetestsbasedonA
2and W2statistics are powerful
against DFR and IFR alternatives, resp. However, for small sample size, n=10,20, no
classicaltest(KS,V,S
,W
2,A
2) reaches the top ve for any alternative considered.
6. We observe that TKL statistic performs very well as compared to other statistics, with
the exception of DFR and UFR alternatives. For DFR alternatives, TH, TJ and TT
perform well as compared to other competitors. Against IFR and UFR alternatives,
ourrstproposalH
(1)
nis as good as TKL and our second proposal H(2)
nis as good as
TH, TJ, TT against DFR and BFR alternatives. Therefore, according to Tables 57,we
can conclude that H(1)
nis preferable for IFR and UFR alternatives, and H(2)
nfor DFR
and BFR alternatives.
In general, we can conclude that the statistics H(1)
nand H(2)
nhave good performances.
Therefore, they can be used in practice to test exponentiality.
Finally, based on our simulations, we can recommend the following simple rule to test
the goodness-of-t of a sample to the exponential distribution: rst, identify the shape
of the failure rate function by means of the TTT plot and, second, use one of the top ve
statistics contained in Table 8to test exponentiality. We illustrate this rule with a numerical
example.
4.2. A numerical example
Classical statistics such as Kolmogorov–Smirnov, Kuiper, Cramér–von Mises or Ander-
son–Darling are widely used among practitioners, mainly because they are available in
many statistical softwares. However, one of their drawbacks is the low power against sev-
eral alternatives. We would like to emphasize this fact through a numerical example taken
from Suprawhardana and Sangadji [58].
4.2.1. Data set
This data set consists of 23 measures corresponding to time between failures (in thousands
of hours) of secondary reactor pumps [58]: 2.160, 0.746, 0.402, 0.954, 0.491, 6.560, 4.992,
0.347, 0.150, 0.358, 0.101, 1.359, 3.465, 1.060, 0.614, 1.921, 4.082, 0.199, 0.605, 0.273, 0.070,
0.062, 5.320.
In Figure 3, we plot the empirical and theoretical density and cumulative distribution
functions, as well as P–P and Q–Q plots for this data set. The plots suggest that the expo-
nentialdistributiondoesnotgiveagoodtforthisdataset.Infact,theexibleWeibull
distribution provides a more reasonable t [59].
JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 137
Figure 4contains the TTT plot of the secondary reactor pumps data, where we observe a
convex form, pointing out a DFR function. Therefore, according to Table 2,H
(2)
nwould be
appropriate for testing exponentiality for this data set. Results are summarized in Table