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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.

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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:F∈F

H1:F/∈F,(1)

where F={F0(.; θ)=F0((x−μ)/σ ) |θ=(μ,σ) ∈},=R×(0, ∞)and μand σ

are unspecied 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 dene 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 satises 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 dierent 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 classication of the top ve best statistics for dierent 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) =1−e−x/θ ,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=n−1n

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)=(ex−1−x)I[0,1](x)+3

|x3−1|I[1,∞)(x)(5a)

and

h2(x)=(ex−1−x)I[0,1](x)+(x−1)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 justiable 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)

3x3−1

exp(x−1)−x

(x−1)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 y∈R

0≤h11+F0(y)

1+Fn(y)≤max(h1(1/2),h1(2)) =1.9129

and

0≤h21+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 signicance 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 classied 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 5–7), 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 dierent 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 dierent

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

n−m

n−m

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=i−m(X(j)−˜

X(i))(j−i)

ni+m

j=i−mX(j)−˜

X(i)2⎫

⎬

⎭

,

HVmn =1

n

n

i=1

log n

2m(X(i+m)−X(i−m)),

HEmn =1

n

n

i=1

log n

cim(X(i+m)−X(i−m)),

HZmn =1

n

n

i=1

log n

zim(X(i+m)−X(i−m)),

HA1mn =−1

n

n

i=1

log ˆ

f(X(i+m))+ˆ

f(X(i−m))

2,

HA2mn =1

n

n

i=1

log n

aim(X(i+m)−X(i−m)),

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=i−mX(j)/(2m+1). Concerning the coecients,

ci=1+i−1

mI[1,m](i)+2I[m+1,n−m](i)+1+n−i

mI[n−m+1,n](i),

zi=i

mI[1,m](i)+2I[m+1,n−m](i)+n−i+1

mI[n−m+1,n](i),

ai=I[1,m](i)+2I[m+1,n−m](i)+I[n−m+1,n](i),

where IA(i)=1wheni∈Aand IA(i)=0 otherwise. Finally, ˆ

fis the kernel density

estimator

ˆ

f(Xi)=1

nh

n

j=1

kXi−Xj

h,(6)

where the kernel function kis chosen to be the standard normal pdf and bandwidth h=

1.06 sn−1.5 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 ¯

x−1)=exp(H(f))

exp(log ¯

x+1),

where H(f)can be estimated by the entropy estimators described before, giving rise to

dierent 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=n−1

i=1((n−i)/n)log((n−i)/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<ξ

1or x>ξ

n+1

2m

n(x(i+m)−x(i−m))ξi<x≤ξi+1i=1, ...,n,

and

ge(x)=⎧

⎪

⎨

⎪

⎩

0x<η

1or x>η

n+1

1

n(ηi+1−ηi)ηi<x≤ηi+1i=1, ...,n,

resp., where ξi=(x(i−m)+···+x(i+m−1))/2m,and

ηi=

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

ξm+1−1

m

m

k=i

(x(m+k)−x(1))1≤i≤m

1

2mx(i−m)+···+x(i+m−1)m+1≤i≤n−m+1

ξn−m+1+1

n+m−k+1

i

k=n−m+2

(x(n)−x(k−m−1))n−m+2≤i≤n+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<η

1or x>η

n+1

1

n(ηi+1−ηi)ηi<x≤ηi+1i=1, ...,n,

where

ηi=

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

ξm+1−1

m

m

k=i

(x(m+k)−x(1))1≤i≤m

1

2mx(i−m)+···+x(i+m−1)m+1≤i≤n−m+1

ξn−m+1+1

m

i

k=n−m+2

(x(n)−x(k−m−1))n−m+2≤i≤n+1

and

gz(x)=⎧

⎪

⎨

⎪

⎩

0x<η

1or x>η

n+1

1

n(ηi+1−ηi)ηi<x≤ηi+1i=1, ...,n,

JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 117

where

ηi=

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

ξm+1−

m

k=i

1

k(x(m+k)−x(1))1≤i≤m

1

2m(x(i−m)+···+x(i+m−1))m+1≤i≤n−m+1

ξn−m+1+

i

k=n−m+2

1

n−k+2(x(n)−x(k−m−1))n−m+2≤i≤n+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=[√n−1], 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(√t−1)2

•Jereys distance: φ(t)=(t−1)log(t)

•Total variation distance: φ(t)=|t−1|

•χ2-divergence: φ(t)=(t−1)2.

Recently, Alizadeh and Balakrishnan [8]usedthetermsTKL,TH,TJ,TTandT

χto

denote the test statistics based on KL, Hellinger, Jereys, 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 n−1n

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=(X1−X2)/(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(i−m),

T2=−1

n

n

i=1

log n

2m(Y(i+m)−Y(i−m)+2

n

n

i=1

log(1+Yi),

T3=−1

n

n

i=1

log n

2m(Z(i+m)−Z(i−m)+log(2),

where n=n(n−1). 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(n−1),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 sucient con-

dition for Fto be exponential involves the identical distribution of the random variables

(X(i+s)−X(i))and Xs,{1,2,...,n−j}for some s≥1andn,whereXs,{1,2,...,n−i}is sth-order

statistic of the sample X1,X2,...,Xn−i. The statistic according to Riedl and Rossberg’s

JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 119

characterization is

I1=∞

0

(Hn(t)−Gn(t)) dFn(t),

where

Hn(t)=n

3−1n

1≤i<j≤n

I[X2,{i,j,k}−X1,{i,j,k}<t]fort≥0,

Gn(t)=n

2−1n

1≤i<j≤n

I[min(Xi,Xj)<t]fort≥0,

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

sucient condition for Fto be exponential involves the identical distribution of the ran-

dom variables (n−i)(X(i+1)−X(i))and (n−j)(X(j+1)−X(j))for some i,jand n,1≤i<

j≤n. 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[|Xi−Xj|<t]fort≥0,

Gn(t)=1

n2

n

i,j=1

I[2 min(Xi,Xj)<t]fort≥0.

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(1−i)(1−log(1−i)) −1. Statistic Kncanbealsoexpressedbasedon

normalized spacings, Di=(n+1−i)(X(i)−X(i−1)),intheform

Kn=n

i=1e(i)Di

n

i=1Di

,

where e(i)=(n−i+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|Xj−Xk|

2n(n−1)¯

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)=(n−1)(1−Gn), 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

1≤i≤ni

n−F0(y(i)),max

1≤i≤nF0(y(i))−i−1

n,

V=max

1≤i≤ni

n−F0(y(i))+max

1≤i≤nF0(y(i))−i−1

n,

S∗=

n

i=1

max

F0(y(i))−i

n

,

F0(y(i))−i−1

n,

W2=1

12n+

n

i=1F0(y(i))−2i−1

2n2

,

A2=−n−1

n

n

i=1

(2i−1)(log(F0(y(i))) +log(1−F0(y(n−i+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(X−t|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 −3e−min(Yj,Yk)−2min(Yj,Yk)(e−Yj+e−Yk)+2e−max(Yj+Yk)],

KSn=√nmax(KS+

n,KS

−

n),

where

KS+

n=max

j=0,...,n−11

n(Y(1)+···+Y(j))+Y(j+1)1−j

n−j

n,

KS−

n=max

j=0,...,n−1j

n−1

n(Y(1)+···+Y(j))−Y(j)1−j

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

1≤k≤n|Tn(k/n,1)|,

T2,n=1

n

n

k=1

T2

n(k/n,1),

T1,n(γ ) =max

1≤k≤n|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+1−i)(X(i)−X(i−1))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)=n−1exp(−tYi).

Baringhaus and Henze [39]andHenze[40]proposedthefollowingstatistics:

BHn=1

n

n

j,k=1(1−Yj)(1−Yk)

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+a−2

n

j=1

exp(Yj+a)E1(Yj+a)

+n(1−aexp(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)=n−1n

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+[Yjk−−Yl]2+1

a2+[Yjk−+Yl]2

+a

n3

n

j,k=1

n

l,m=11

a2+[Yjk−−Ylm−]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 −[Yjk−−Yl]2

4a+exp −[Yjk−+Yl]2

4a

+1

2n3π

a

n

j,k=1

n

l,m=1exp −[Yjk−−Ylm−]2

4a+exp −[Yjk−+Ylm−]2

4a,

where Yjk−=Yj−Ykand 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’ modication of the Shapiro–Wilk statistic

[43]aregivenby

W=n(¯

x−x(1))2

(n−1)n

i=1(xi−¯

x)2,

JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 123

Ws=(n

i=1xi)2

n(n+1)n

i=1x2

i−n(n

i=1xi)2.

The null hypothesis H0is rejected for small or large values of W and Ws.

Fortiana and Grané [9]proposedthefollowingstatisticbasedontheHoeding’s

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=(n−i)log(n−i)−(n−

i+1)log(n−i+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 signicant for DFR alternatives, whereas the left tail is signicant

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 specic 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

(2i−n−1)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

X∼U(0, 1).

3.12. Other tests

Cox and Oakes [46] introduced the statistic

COn=n+

n

i=1

(1−Yi)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

¯

X−e−γ

,

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(n−1)(n−2)

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(n−1)

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|ri−si|,

where ri=2/π arcsin(1−exp(−xi/¯

x)) and si=2/π arcsin((i−0.5)/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(θ ),withdensity

f(x)=θxθ−1e−xθ,x>0.

2. Gamma distribution, (θ),withdensity

f(x)=xθ−1e−x

(θ) ,x>0.

3. Log-Normal distribution, LN(θ),withdensity

f(x)=1

x√2πθ2exp −(log x)2

2θ2,x>0.

4. Half-Normal distribution, HN, with density

f(x)=2/π exp{−x2/2},x>0.

5. Uniform distribution, U, with density 1, 0 ≤x≤1.

6. Chen distribution [55], CH(θ),withcdfas

F(x)=1−exp{2(1−exθ)},x>0.

7. Linear Increasing Failure Rate law, LF(θ),withdensity

f(x)=(1+θx)exp(−x−θx2/2),x>0.

8. Modied Extreme Value distribution, EV(θ),withcdfas

F(x)=1−exp{θ−1(1−ex)},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(θ ),withcdfas

F(x)=1−exp{−(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 S∗51926134010 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 S∗53549227016 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

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

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 S∗576904999 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

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 5–7.

Rank IFR UFR DFR–BFR

1H

(1)

n,K

n,TKL COV M

n

2T

(1)

n,2.5,T

(2)

n,2.5 T4−T1COn

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 5–8.Inparticular,Tables5–7report the power,

at a 5% signicance 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 5–7.

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 5–7we 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 dierence 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,

ourrstproposalH

(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 5–7,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-

nentialdistributiondoesnotgiveagoodtforthisdataset.Infact,theexibleWeibull

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