A review of tests for exponentiality with Monte Carlo comparisons

Abstract

In this paper, 91 different tests for exponentiality are reviewed. Some of the tests are universally consistent while others are against some special classes of life distributions. Power performances of 40 of these different tests for exponentiality of datasets are compared through extensive Monte Carlo simulations. The comparisons are conducted for different sample sizes of 10, 25, 50 and 100 for different groups of distributions according to the shape of their hazard functions at 5 percent level of significance. Also, the techniques are applied to two real-world datasets and a measure of power is employed for the comparison of the tests. The results show that some tests which are very good under one group of alternative distributions are not so under another group. Also, some tests maintained relatively high power over all the groups of alternative distributions studied while some others maintained poor power performances over all the groups of alternative distributions. Again, the result obtained from real-world datasets agree completely with those of the simulation studies.

1. Introduction

In applied statistics, it is often misleading to assume that a set of data $x_1,x_2,\dots ,x_n,$ for a certain statistical inference follows a specific distribution without statistically ascertaining the true distribution from where such set of data is drawn. This is because most statistical inferences applied to data analysis are sensitive to underlying distribution of the data set and as a result, the outcome of such statistical inferences is invalidated with wrong distributional assumption, the consequence of which varies with study. For instance, in life sciences and reliability studies where investigations are carried out with high precision, misleading results owing to wrong distributional assumption goes with dare consequences.

One distributional law that underlies a good number of statistical studies in life sciences, especially those that are related to renewal process, birth and death processes, Markov process, queuing theory and every other process characterized with appreciation and/or progressive decay, and reliability studies as well as survival analysis and generally in modeling is the exponential distribution. Goodness-of-fit test for exponentiality of a data set has been discussed extensively in the literature by a good number of authors. These include Gnedenko [51], Harris [54], Deshpande [37], Cox and Oakes [37], Epps and Pulley [41], Gail and Gastwirth [46], Baringhaus and Henze [23,24,25], Henze [58], Henze and Klar [59] and Abbasnejad et al. [1] to mention but a few. In fact, there exist dozens of different tests for assessing the exponentiality of data sets in the literature. The tests are based on certain characterizations of the exponential distribution such as empirical distribution function $(EDF)$, empirical characteristic function $(ECF)$, empirical Laplace transform $(ELT)$, normalized spacing, Shannon’s entropy estimators and mean residual life functions.

Let $x_1,x_2,\dots ,x_n,$ be a set of independent observations from an unknown distribution $F(x)$ with probability function $f(x)$. Let $F_0(x)$ be a specified distribution function of the exponential distribution, $F_0(x)=1-e^{-\theta x}$ with probability function, $f_0(x)=\theta e^{-\theta x};x>0,\theta >0$. The problem of testing for the exponentiality of the parent distribution of the sample involves testing the goodness-of-fit:

$$H_0:F(x)=F_0(x)\text{against}{H}_1:F(x)\ne F_0(x)$$ (1)

The ability of the various tests for exponentiality to reject the null hypothesis in (1) when the true distribution is not exponential, known as the power of the test, varies from one test to another and depends on the nature of the true distribution of the data set [19]. Despite the fact that an amount of effort has been devoted to assessing the quality of these tests with respect to their power, no comprehensive work exists in the literature for comparing all the possible tests for exponentiality at least up to the time of such work.

Prior to Epstein [42] who compared the powers of 12 different statistics for assessing exponentiality of data sets, it appears that no work had been devoted to this subject. Ascher [19] compared the powers of fifteen different tests for exponentiality against three classes of distributions, namely: alternative distributions with increasing hazard rate $(IHR)$, alternative distributions with decreasing hazard rate $(DHR)$ and alternative distributions with non-monotone hazard rate $(NMHR)$.

In a similar way, Henze and Meintanis [62] in their partial review of tests for exponentiality compared thirteen old and newer tests for exponential distribution, discarding less powerful ones according to Ascher [19]. Also, Rogozhnikov and Lemeshko [99] included over eighteen different procedures in their comparative study. While Ascher [19] and Rogozhnikov and Lemeshko [99] comparisons are based on the classes of the alternative distribution in terms of the monotonicity of hazard rate, Henze and Meintanis [62] tend to be compared along classes of competing tests based on the characterizations of the exponential distribution. Allison et al. [10] also compared the powers of 20 different procedures. Rahman and Wu [97], in much broader study, compared the powers of about twenty four different statistics for assessing exponentiality. Some of the statistics, as it is known, have asymptotic approximations in addition to their evaluation through simulations. As a result, they included these tests evaluated through simulations and through asymptotic approximations in their study at different levels of significance ( $\alpha =0.01,0.05,0.10$). In what appears to be the most comprehensive comparison of tests for exponentiality so far, Torabi et al. [105] described and compared the powers of about fifty different statistics for assessing exponentiality of data sets. Their comparison is based on four classes of distributions alternative to the exponential distribution according to failure rates, which they gave as distributions with increasing failure rate (IFR), distributions with decreasing failure rate (DFR), distributions with bathtub failure rate (BFR) and distributions with unimodal failure rate (UFR).

In addition to what would appear to be test procedures which are universally consistent against any class of alternative distribution to the exponential distribution such as the L²-class of tests, there are in the literature quite a number of test procedures for accessing exponentiality of data sets which are consistent against alternatives belonging to different important classes of life distributions. Some of such classes include the IFR, the increasing failure rate average (IFRA), Laplace increasing failure rate average (LIFRA), new better than used (NBU), new better than used complex average (NBUCA), new better than used in expectation (NBUE), harmonic new better than used in expectation (HNBUE), decreasing mean residual life (DMRL) distribution class, decreasing mean time to failure (DMTTF) class of life distributions as well as the class of life distributions based on the Laplace-Stieltjes transform usually known as the L-class. Some of such tests of exponentiality include Belzunce et al. [30] and Mitra and Anis [87] for the IFR class, Aly [11] and El-Bassiouny [39] for the IFRA class, Prajitha and Nadarajah [96] for the LIFRA class, Atallah et al [21] for the NBU class, Al-Gashgari et al. [9] for the NBUCA class, Hollander and Proschan [63] for the NBUE class, Klar [71] and Ghosh and Mitra [49,50] for HNBUE class, Bergman and Klefsjo [32] for the DMRL class, Li and Xu [79] and Asha and Nair [20] for the DMTTF class as well as Chaudhuri [34] and Majumder and Mitra [83] for the L-class.

Also in the literature are test procedures for assessing exponentiality against alternatives belonging to the well-known non-monotonic ageing classes of life distributions such as the increasing then decreasing mean residual life (IDMRL) distributions, new worse then better than used in expectation (NWBUE) class and the bathtub failure rate (BFR) class. Some of the tests include Guess et al. [52], Hawkins et al. [56] and Na and Lee [91] for the IDMRL class, Ghosh and Mitra [48] for the NWBUE class as well as Na et al. [90] for the BFR class.

More recently, other classes of ageing life distributions have continued to be developed in the literature. Some of such classes include renewal new better than renewal used in expectation (RNBRUE), see Abdel-Aziz [2]; new better than used in average at specific interval (NBUASI), see Abdul-Moniem [3]; renewal new better than renewal used (RNBRU), see Elbatal [40]; harmonic new better than renewal used in expectation (HNBRUE), see Al-Zahrani and Stoyanov [8]; overall decreasing life (ODL), see Diab and El-Atfy [38] and new better than renewal used in Laplace transform (NBRUL), see Mahmoud et al. [82]. Tests for exponentiality against some of these new classes have also been proposed in the literature. However, our review in this paper shall not include such tests.

These dozens of test procedures for assessing exponentiality against different classes of monotonic and non-monotonic life distributions have been developed using different characterizations of the life distributions such as the total time on test (TTT), see for instance Klefsjo [74]; the Laplace order dominance, see for instance Majumder and Mitra [83]; U – statistics, see for instance Ahmad [4]; Stochastic ordering of random numbers, see for instance Belzunce et al. [30]; moment inequality, see for instance Mahmoud et al. [82]; excess – wealth function, see for instance Fernandez – Ponce and Rodriguez – Grinolo [44]; measure of deviation, see for instance Anis and Mitra [18] to mention but a few.

In this paper, detailed review of these test procedures for assessing exponentiality is provided. In addition, a Monte Carlo power comparison of some of the procedures is provided. In section 2, all the test procedures are described. Section 3 gives the power comparison of the tests through extensive simulation studies. The real – life applications are also presented in this section. Section 4 summarizes the study with a general conclusion of the study.

2. Description of the exponentiality tests

In this section, all the competing test procedures are classified into seven different families (or groups) according to the theory behind their derivations and according to the monotonicity of the classes of life distributions (for tests against some special classes of life distributions). The classification scheme used here is however a matter of convenience, in line with how the authors perceived the test procedures. They include tests based on spacings, entropy estimators and divergence measures: tests based on empirical functions; tests based on residual mean functions; tests based on other characterizations of the exponential distribution; tests against monotonic classes of life distributions; tests against non-monotonic classes of ageing life distributions and other tests for exponentiality.

In some of the statistics which shall be described in this section, scaled observations $Y_j=X_j/{\hat{\theta }}_n$ are used, or their transformed observations, $Z_j=1-\exp (-Y_j);1\le j\le n$, where ${\hat{\theta }}_n={\overline{X}}_n=\frac{1}{n}\sum _{j=i}^n{X}_j$ is the maximum likelihood estimator of the parameter $\theta$. The order statistics of $X_j,Y_j$ and $Z_j$ shall be denoted by $X_{(j)},Y_{(j)}$ and $Z_{(j)}$ respectively. Also, some of the statistics use normalized spacing defined as: $D_j=(n-j+1)(X_{(j)}-X_{(j-1)});1\le j\le n$ with $X_{(0)}=0$ while $D_{(1)}\le D_{(2)}\le \dots \le D_{(n)}$ are the order statistics of the normalized spacings. The statistics are presented in what follows.

2.1. Tests based on spacing, entropy estimators and divergence measures

The Epstein $E{p}_n$ test. Epstein [43] introduced a test for exponentiality based on normalized spacing as defined earlier. The statistic is $E{p}_n=2n[\ln \left(n^{-1}\sum _{j=1}^r{D}_j\right)-n^{-1}\sum _{j=1}^n\ln (D_j)]/[1+(n+1)/6n]$.

The test rejects the null hypothesis of exponentiality for large values of the statistic and under the assumption of exponentiality, the statistic is asymptotically distributed as chi-square with (n–1) degrees of freedom.

The $G{D}_{n,r}$ test of Gnedenko [51]. This test involves ordering of a set of non negative observations of size n and splitting the ordered set into two subsets of r and n – r ordered observations such that the first subset contains the first r smallest observations. The statistic is given by $G{D}_{n,r}=\left(\sum _{j=1}^r{D}_j/r\right)/\left(\sum _{j=r+1}^n{D}_j/(n-r)\right)$.

The test rejects the null hypothesis of exponentiality for both small and large values of the statistic which follows an F distribution asymptotically with 2r and 2(n – r) degrees of freedom.

The $H{a}_{n,r}$ test of Harris [54]. This is a modification of the Gnedenko [51] test. Harris [54] in order to improve the power of the Gnedenko [51] modified it by splitting the ordered observations into three subsets with the first subset containing the first r smallest observations, the last subset containing the last r largest observations and the remainder of n – 2r forming the second subset. The test rejects the null hypothesis of exponentiality for both small and large values of the statistic: $H{a}_{n,r}=\left[\left(\sum _{j=1}^r{D}_j+\sum _{j=n-r+1}^r{D}_j\right)/2r\right]/\left[\sum _{j=r+1}^{n-r}{D}_j/(n-2r)\right]$. Also, Harris [54] recommended $n/4$ for r.

The Tiko $T{k}_n$ test. Sadeghpour et al. [100] described this test which is based on spacing when comparing their newly developed test for exponentiality with some other existing ones. The statistic is given as $T{k}_n=\left(2\sum _{j=1}^{n-1}(n-j+1)(n-j)(X_{(j+1)}-X_{(j)})\right)/(\phantom{\sum _{j=1}^{n-1}}(n-2)\sum _{j=1}^{n-1}(n-j)(X_{(j+1)}-X_{(j)}))$. The statistic rejects the null hypothesis for high values of $T{k}_n$.

The $K{o}_n$ test of Kochar [77]. Also, Kochar [77] introduced a test for exponentiality based on the statistic:

$$K{o}_n=\sqrt{\frac{108n}{17}}\left(\sum _{j=1}^{n}J(j/(n+1))X_{(j)}\right)/\left(\sum _{j=1}^n{X}_j\right);j=1,2,...,n$$

where $J(j)=2(1-j)(1-\log (1-j))-1$. It is asymptotically standard normally distributed and the null hypothesis is rejected for large values of $K{o}_n$.

The $BH{a}_n$ test of Baratpour and Habibirad [22]. Baratpour and Habibirad [22] obtained an estimator of the cumulative residual entropy $(CRE)$ of a distribution $F$ and gave the cumulative Kullback-Leibler $(CKL)$ divergence between two distributions $F$ and $F_0$ of random variables $X$ and $Y$ respectively. It is well known that $CKL(F:F_0)\ge 0$ but if $F$ and $F_0$ have the same distribution (in this instance exponential with parameter $\lambda$), $F(x)=F_0(x)$ such that $CKL(F:F_0)=0$. By employing a relation between $CRE(F)$ and $CKL(F:F_0)$, and using their estimators, they obtained a statistic given as:

$$BH{a}_n=\frac{\sum _{j=1}^n\frac{n-j}{n}\left[In\left(\frac{n-j}{n}\right)\right](X_{(j+1)}-X_{(j)})+\left(\sum _{j=1}^n{X}_j^2\right)/\left(2\sum _{j=1}^n{X}_j\right)}{\left(\sum _{j=1}^n{X}_j^2\right)/\left(2\sum _{j=1}^n{X}_j\right)};1\le j\le n$$

The null hypothesis is rejected for large values of $BH{a}_n$ and the statistic is said to be consistent against any fixed alternative.

The $Pk{1}_n$ test of Park, Rao and Shin [94]. On a separate study of the estimator of the cumulative residual entropy of a distribution using the Kullback-Leibler information, Park et al. [94] obtained a statistic for testing the exponentiality of a data set. The statistic is given as:

$$\begin{array}{rl}Pk{1}_n& =1-{\left[\overline{X}-\sum _{j=0}^n\left(\frac{n-j}{n}\right)\log \left(\frac{n-j}{n}\right)(X_{(j+1)}-X_{(j)})\right]}^2/\left[2\sum _{j=0}^n{X}_j^2/n\right];\\ & j=0,1,2,...,n\end{array}$$

The statistic is said to be consistent against any fixed alternative and rejects the null hypothesis for large values of $Pk{1}_n$.

The $A{b}_n$ Test of Abbasnejad et al. [1]. Lin and Wong [109] introduced a divergence distance between any two distributions $F$ and $G$ which equals zero if $F$ and $G$ are the same. Abbasnejad et al. [1] applied this divergence distance instead of the CKL divergence distance to obtain a test of exponentiality as: $A{b}_n=-\frac{1}{n}\sum _{j=1}^{n}In\left[\frac{1}{2}+\frac{n}{4m\overline{X}}(X_{(j+m)}-X_{(j-m)})\exp \left(-\frac{X_{(j)}}{\overline{X}}\right)\right]$; where $X_{(j)}=X_{(1)}$ for $j<1$ and $X_{(j)}=X_{(n)}$ for $j>n$. $m$ is the integer parameter of goodness-of-fit tests based on entropy which is estimated as $m=\sqrt{n}+0.5$. The affine invariant and consistent test rejects the null hypothesis for large values of $A{b}_n$.

The $Pk{2}_n$ test of Park, Choi and Jung [93]. Let X be a nonnegative random variable so that $E(X)$ can be expressed as ${\int }_0^{\mathrm{\infty }}\overline{F}(x)dx$. They defined the equilibrium density function as $f^{\ast }(x)=\overline{F}(x)/{\int }_0^{\mathrm{\infty }}\overline{F}(x)dx$ and stated the exponential density function is the only continuous density function that satisfies the condition $f^{\ast }(x)=f(x)$. Now, suppose the distribution functions of X and Y are F and G respectively. They used the concept of the Kullback-Leibler information of the equilibrium density functions $f^{\ast }(x)=\overline{F}(x)/E(X)$ and $g^{\ast }(y)=\overline{G}(y)/E(Y)$ to obtain a statistic for testing exponentiality as $Pk{2}_n={\overline{X}}^{-1}\sum _{j=1}^n\left(\frac{n-j}{n}\right)\log \left(\frac{n-j}{n}\right)(X_{(j+1)}-X_{(j)})+\frac{1}{2n{\overline{X}}^2}\sum _{j=1}^n{X}_j^2$. The statistic is affine invariant, consistent and rejects null hypothesis for large values of $Pk{2}_n$.

The SBH_n,r of Sadeghpour, Baratpour and Habibirad [100]. Renyi [98] obtained a more generalized entropies called the Renyi entropy. Sadeghpour et al. [100] introduced distances based on equilibrium distributions, of which one of them based on the Renyi distance is

$$D_R(f_{X^{\ast }},f_{Y^{\ast }})=\frac{1}{1-r}\log \left[\frac{E^{r-1}(Y)}{E^r(X)}{\int }_0^{\mathrm{\infty }}{\left[\frac{{\overline{F}}_X(x)}{{\overline{F}}_Y(x)}\right]}^{r-1}{\overline{F}}_X(x)dx\right],r\ne 1,r>0.$$

By letting $F_X(x)=F_n(x)$ and $F_Y(x)=F_0(x,\lambda )$, an exponential distribution with an unknown parameter $\lambda$. They obtained a statistic for exponentiality testing given by

$$SB{H}_{n,r}=\sum _{j=0}^{n-1}{\left(\frac{n-j}{n}\right)}^r\left[\exp \left\{\frac{(r-1)X_{(j+1)}}{{\overline{X}}_n}\right\}-\exp \left\{\frac{(r-1)X_{(j)}}{{\overline{X}}_n}\right\}\right].$$

The test is affine invariant and rejects the null hypothesis for large values of the statistic.

The $AH{B}_{\alpha ,n}$ of Ahrari, Habibirad and Baratpour [6]. The alpha-divergence of a continuous random variable was proposed by Chernoff [35] and investigated extensively by Amari [13,14] as well as Amari and Nagaoke [15]. Ahrari et al. [6] gives the alpha-divergence based on equilibrium distributions and used it to propose an affine invariant test for exponentiality which rejects the null hypothesis of exponentiality for large values of the statistic:

$$AH{B}_{\alpha ,n}=\frac{1}{\alpha {(\alpha -1)}^2}\left[\sum _{j=0}^{n-1}{\left(\frac{n-j}{n}\right)}^{\alpha }\left(e^{-\frac{X_{(j+1)}(1-\alpha )}{{\overline{X}}_n}}-e^{-\frac{X_{(j)}(1-\alpha )}{{\overline{X}}_n}}\right)+1-\alpha \right].$$

2.2. Tests based on empirical functions

The empirical functions referred to in this subsection include the distribution function, the Laplace function, the characteristic function and the quantile function.

2.2.1. Empirical distribution function (EDF)

In goodness-of-fit tests generally, procedures in this category are based on the absolute difference between the $EDF$ of the data set $X_1,X_2,\dots ,X_n,$ which is denoted by $F_n(x)$, and the distribution function, $F_0(x)$ of the hypothesized distribution. As a result, tests based on the distance between the $EDF$ of the scaled data $Y_1,Y_2,\dots ,Y_n$ and the distribution function of the unit exponential distribution, $F_0(x)=1-\exp (-x)$ shall be considered. The statistics are presented in what follows.

The Kolmogorov–Smirnov $K{S}_n$ test. This test rejects the null hypothesis of exponentiality for large values of the statistic:

$$\begin{array}{rl}K{S}_n& =max\left(\underset{1\le j\le n}{max}[j/n-Z_{(j)}];\underset{1\le j\le n}{max}[Z_{(j)}-(j-1)/n]\right)\\ & =max(K{S}^{+},K{S}^{-}),\text{where}K{S}^{+}\\ & =\underset{1\le j\le n}{max}[j/n-Z_{(j)}]\text{and}K{S}^{-}=\underset{1\le j\le n}{max}[Z_{(j)}-(j-1)/n].\end{array}$$

The Anderson–Darling $A{D}_n$ test. Like the Kolmogorov–Smirnov $K{S}_n$ test, this goodness-of-fit procedure rejects the null hypothesis of exponentiality for large values of the statistic: $A{D}_n=-n-2\sum _{j=1}^n\left[\frac{2j-1}{2n}In{Z}_j+\left(1+\frac{2j-1}{2n}\right)In(1-Z_j)\right]$.

The Cramer von Mises $C{M}_n$ test. Cramer von Mises test also rejects the null hypothesis of exponentiality for large values of the statistic: $C{M}_n=\sum _{j=1}^n{\left(Z_{(j)}-\frac{2j-1}{2n}\right)}^2+\frac{1}{12n}$.

The Kimber–Michael $K{M}_n$ test. Michael [85] obtained stabilized probability plots and used it to introduce test for uniformity and normality. Kimber [70] adapted the statistic of Michael [85] to propose a test for exponentiality of data sets which measures the absolute differences between arcsine of the square root of the observed and theoretical distribution functions. The test rejects the null hypothesis of exponentiality of the data set for large values of the statistic:

$$K{M}_n=\underset{1\le j\le n}{max}|r_j-s_j|;r_j=\frac{2}{\pi }\arcsin \sqrt{(j-0.5)/n},s_j=\frac{2}{\pi }\arcsin \sqrt{1-\exp (-Y_{(j)})}.$$

The $F{r}_n$ test of Frozini [45]. Frozini [45] proposed a test of goodness-of-fit to exponential distribution based on the absolute differences between the theoretical and empirical distribution functions. The test rejects the null hypothesis of exponentiality for large values of the statistic:

$$F{r}_n=\frac{1}{\sqrt{n}}\sum _{j=1}^n\left|1-\exp \left(\frac{X_{(j)}}{{\overline{X}}_n}\right)-\frac{j-0.5}{n}\right|.$$

The $K{l}_n$ statistic of Klar [72]. The integrated distribution function (IDF) of a non negative continuous random variable X is given by: $\psi (t)=E(X-t{)}^{+}={\int }_t^{\mathrm{\infty }}(1-F(x))dx$. Klar [72] obtained the empirical integrated distribution function (EIDF) of a random variable X as ${\psi }_n(t)={\int }_t^{\mathrm{\infty }}(1-F_n(x))dx=n^{-1}\sum _{j=1}^n(X_j-t)I\{X_j>t\}$. Based on the difference between the estimated IDF of the standardized exponential random variable and the EIDF, the author proposed an affine invariant and consistent test for exponentiality which rejects the null hypothesis for large values with statistic:

$$\begin{array}{rl}K{l}_n& =\frac{1}{n}\sum _{i,j=1}^n\left[{\int }_0^{min(Y_i,Y_j)}(Y_i-u)(Y_j-u)du-{\int }_0^{Y_i}(Y_i-u)e^{-u}du+{\int }_0^{\mathrm{\infty }}{e}^{-2u}du\right]\\ & =\frac{n}{2}-2\sum _{i=1}^n{e}^{-Y_i}-{(3n)}^{-1}\sum _{i=1}^n(n-i-1)Y_{(i)}^3+n^{-1}\sum _{i<j}^{}{Y}_{(i)}^2{Y}_{(j)}.\end{array}$$

The Zhang $Z{K}_n$, $Z{A}_n$ and $Z{C}_n$ tests. Zhang [108] proposed tests for exponentiality based on the distribution function of the standard exponential law, which are respectively alternative to the Kolmogorov–Smirnov, Anderson–Darling and the Cramer von Mises tests given above. In each of the tests, null hypothesis of exponentiality is rejected for large values of the statistics. They are:

$$\begin{array}{rl}Z{K}_n& =\underset{1\le j\le n}{max}\left[(j-0.5)\ln \left(\frac{j-0.5}{n{Z}_{(j)}}\right)+(n-j+0.5)\ln \left(\frac{n-j+0.5}{n\{1-Z_{(j)}\}}\right)\right]\\ Z{A}_n& =-\sum _{j=1}^n\left[\frac{\ln Z_{(j)}}{n-j+0.5}+\frac{\ln (1-Z_{(j)})}{j-0.5}\right]\text{and}\\ Z{C}_n& =\sum _{j=1}^n{\left[\ln \left(\frac{Z_{(j)}^{-1}-1}{(n-0.5)/(j-0.75)-1}\right)\right]}^2.\end{array}$$

The $J{T}_{1,n}$ and $J{T}_{2,n}$ Tests of Jammalamadaka and Taufer [65]. Consider a random sample $X_1,X_2,\dots ,X_n$ from an exponential distribution, $f(x)=1/\theta \exp (-x/\theta );$ $x>0,\theta >0$ with empirical distribution function $F_n(t)$. Let $Y_1,Y_2,\dots ,Y_n$ be a corresponding sample from normalized spacings of the exponential sample observations whose empirical distribution function is $G_n(t)$. Jammalamadaka and Taufer [65] obtained two statistics for exponentiality by measuring the distance between the $ED{F}^{\mathrm{\prime }}s$. The statistics are:

$$J{T}_{1,n}=\sqrt{\frac{n}{2}}\underset{0\le t<\mathrm{\infty }}{sup}|F_n(t)-G_n(t)|\text{and}J{T}_{2,n}=\frac{n}{2\overline{X}}\int {(F_n(t)-G_n(t))}^2\exp \left(-\frac{t}{\overline{X}}\right)dt,$$

where $\overline{X}=n^{-1}\sum _{j=1}^n{X}_j$. The null hypothesis is rejected for large values of each statistic.

The BT_n test of Baringhaus and Taherizadeh [27]. Baringhaus and Taherizadeh [26] proved that the Hankel transform of a standard exponential random variable X is $H_X(t)=\exp (-t);t\ge 0$. Using some uniqueness theorem of the Hankel transform, Baringhaus and Taherizadeh [27] obtained a Kolmogorov–Smirnov type statistic for testing exponentiality as:

$$B{T}_n=\sqrt{n}\underset{0\le t\le 1}{sup}|H_n(t)-\exp (-t)|,$$

where $H_n(t)=n^{-1}\sum _{j=1}^n{J}_0\left(2\sqrt{t{Y}_j}\right);0\le t\le 1$ is the empirical Hankel transform of the empirically standardized variable $Y_j=X_j/{\overline{X}}_n$. It is scale invariant and consistent against fixed alternatives.

2.2.2. Empirical Laplace function (ELF)

The Laplace transform of a random variable $X$ which is unit exponentially distributed is given as $\psi (t)=(1+t{)}^{-1}$ while its empirical counterpart is ${\psi }_n(t)=\frac{1}{n}\sum _{j=1}^n\exp (-t{Y}_j)$ and a measure of distance between $\psi (t)$and ${\psi }_n(t)$ has been used to construct tests for exponentiality. In this study three such tests are considered.

The $BH{e}_{n,a}$ Test of Baringhaus and Henze [23]. With an appropriate choice of a smoothing parameter ‘ $a$’, Baringhaus and Henze [23] proposed a test of exponentiality which rejects the null hypothesis of exponentiality for a large value of the statistic:

$$\begin{array}{rl}BH{e}_{n,a}& =n{\int }_0^{\mathrm{\infty }}{[(1+t){{\psi }_n}^{\mathrm{\prime }}(t)+{\psi }_n(t)]}^2\exp (-at)dt\\ & =\frac{1}{n}\sum _{j,k=1}^n[\frac{(1-Y_j)(1-Y_k)}{Y_j+Y_k+a}-\frac{Y_j+Y_k}{{(Y_j+Y_k+a)}^2}\\ & +\frac{2Y_j{Y}_k}{{(Y_j+Y_k+a)}^2}+\frac{2Y_j{Y}_k}{{(Y_j+Y_k+a)}^3}]\end{array}$$

The test is said to be consistent against any distribution with positive finite mean, $\theta$.

The $H{e}_{n,a}$ Test of Henze [58]. Henze [58] obtained a more direct $L^2$- distance type statistic between ${\psi }_n$ and $\psi$ of a fitted exponential distribution and gives a goodness-of-fit statistic for exponentiality based on this statistic as:

$$\begin{array}{rl}H{e}_{n,a}& =n{\int }_0^{\mathrm{\infty }}({\psi }_n(t)-{(1+t)}^{-1})\exp (-at)dt\\ & =\frac{1}{n}\sum _{j,k=1}^n\frac{1}{(Y_j+Y_k+a)}-2\sum _{j=1}^n\exp (Y_j+a)E_1(Y_j+a)+n[1-9\exp (a)E_1(a)]\end{array}$$

where $E_1(a)={\int }_a^{\mathrm{\infty }}\frac{\exp (-t)}{t}dt$ and $a$ is a positive constant. It is consistent against any fixed alternative and rejects the null hypothesis for large values of $H{e}_{n,a}$.

The $HM{L}_{n,a}$ Test of Henze and Meintanis [60]. Henze and Meintanis [60] generalizes the Henze [58] test with a weight function $w(t)$ which is assumed to satisfy $w(t)=O(t^k)$ as $t\to \mathrm{\infty }$, for some integer k. The generalized statistic is:

$$\begin{array}{rl}HM{L}_{n,a}& =n{\int }_0^{\mathrm{\infty }}{[(1+t){\psi }_n(t)-1]}^{2}w(t)\exp (-at)dt\\ & =\frac{1}{n}\sum _{j,k=1}^n\frac{1+{(Y_j+Y_k+a+1)}^2}{{(Y_j+Y_k+a)}^3}-2\sum _{j=1}^n\frac{1+Y_j+a}{{(Y_j+a)}^2}+\frac{n}{a}\end{array}$$

Notice that if the weight function $w(t)$ is given as $w(t)=(1+t{)}^{-2}$, $HM{L}_{n,a}$ collapses to $H{E}_{n,a}$ statistic of Henze [58].

2.2.3. Empirical characteristic function (ECF)

Let $X_1,X_2,\dots ,X_n$ be a random sample from a specified distribution, the empirical characteristic function $(ECF)$ of the random sample is defined as ${\phi }_n(t)=\frac{1}{n}\sum _{j=1}^n\exp (it{X}_j).$ Several goodness – of – fit tests are based on the $ECF$.

The $E{P}_n$ Test of Epps and Pulley [41]. An exponential random variable with parameter $\theta$ has a characteristic function $\phi (t)=1/(1-i\theta t)$ with a parametric estimator $\hat{\phi }(t)=1/(1-i\overline{X}t)$, where $\overline{X}$ is the sample mean. Comparing this with the $ECF$ of an exponential random sample, Epps and Pulley [41] proposed a statistic:

$$\begin{array}{rl}E{P}_n& =(48n{)}^{1/2}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\phi }_n(t)-\frac{1}{(1-i\overline{X}t)}\right)\frac{{\overline{X}}_n}{2\pi (1+i\overline{X}t)}dt\\ & =(48n{)}^{1/2}\left[\frac{1}{n}\sum _{j=1}^n\exp (-Y_j)-\frac{1}{2}\right].\end{array}$$

The asymptotic distribution of the consistent $E{P}_n$ statistic is standard normal and the null hypothesis of exponentiality is rejected for large values of the statistic.

The $H{M}_n^{(1)}$ and $H{M}_n^{(2)}$ Tests of Henze and Meintanis [61]. Henze and Meintanis [61] proposed two statistics for assessing exponentiality of a data set based on weighted distance between the $ECF$ of the data and the characteristic function $(CF)$ of a unit exponential distribution varying the weight functions. The universally consistent statistics which reject the null distribution of exponentiality for large values are:

$$\begin{array}{rl}H{M}_{n,a}^{(1)}& =\frac{a}{2n}\sum _{j,k=1}^n[\frac{1}{a^2+{(Y_j-Y_k)}^2}-\frac{1}{a^2+{(Y_j+Y_k)}^2}-\frac{4(Y_j+Y_k)}{{[a^2+{(Y_j+Y_k)}^2]}^2}\\ & +\frac{2a^2-6{(Y_j-Y_k)}^2}{{[a^2+{(Y_j-Y_k)}^2]}^3}+\frac{2a^2-6{(Y_j+Y_k)}^2}{{[a^2+{(Y_j+Y_k)}^2]}^3}]\end{array}$$

and

$$\begin{array}{rl}H{M}_{n,a}^{(2)}& =\frac{\sqrt{\pi }}{4n\sqrt{a}}\sum _{j,k=1}^n[\left(1+\frac{2a-{(Y_j-Y_k)}^2}{4a^2}\right)\exp \left(-\frac{{(Y_j-Y_k)}^2}{4a}\right)\\ & +\left(\frac{2a-{(Y_j+Y_k)}^2}{4a^2}-\frac{(Y_j+Y_k)}{a}-1\right)\exp \left(-\frac{{(Y_j+Y_k)}^2}{4a}\right)]\end{array}$$

2.2.4. Empirical quantile function (EQF)

Relatively new in the class of tests for exponentiality based on empirical functions are those based on the quantile function. The quantile function, $F_0^{-1}(\text{x})$, associated with a standard exponential random variable X, with distribution function $F_0(\text{x})=1-\exp (-X)$ is given by: $Q(p)=F_0^{-1}(p)=-\ln (1-p);p\in (0,1)$. A measure of distance between this and an appropriate measure of its empirical form, $F_n^{-1}(p)$ is used to obtain tests for exponentiality.

The $M_n$test of Madukaife [81]. Madukaife [81] gives the pth empirical quantile function (EQF) of a rescaled set of observations $y_1,y_2,...,y_n$ as the jth order statistic $Y_{(j)}$ of the observations, where $p=(j-0.5)/n$. By comparison with the theoretical quantile function, he proposed an affine invariant and consistent test which rejects the null hypothesis of exponentiality for large values of the statistic: $M_n=\sum _{j=1}^n{\left[Y_{(j)}+\ln \left(\frac{n-j+0.5}{n}\right)\right]}^2$.

2.3. Tests based on the mean residual function

It is well known that a random variable $X$ is distributed exponentially under the assumption that $0<\theta <\mathrm{\infty }$ if and only if $E[X-t|X>t]=\theta$ for each $t>0$ and this is equivalent to $E[min(X,t)]=\theta .F(t)$ for each $t>0$.

Basing upon this characterization of the exponential distribution, Baringhaus and Henze have at different points proposed statistics for testing goodness-of-fit for an exponential distribution. They are Baringhaus and Henze [24,25] which are the Kolmogorov–Smirnov and Cramer–von Mises type statistics. Specifically, Baringhaus and Henze [24] proposed a Kolmogorov–Smirnov type statistic, $BH{K}_n$ and a Cramer–von Mises type statistic, $BH{C}_n$. They are given by:

$$\begin{array}{rl}BH{K}_n& =\sqrt{n}\underset{t\ge 0}{sup}\left|\frac{1}{n}\sum _{j=1}^{n}min(U_j,t)-\frac{1}{n}\sum _{j=1}^{n}1(U_j\le t)\right|=\sqrt{n}max[K{S}_n^{+},K{S}_n^{-}],\\ K{S}_n^{+}& =\underset{j\in \{0,1,\dots ,n-1\}}{max}\left[\frac{1}{n}(Y_{(1)}+...+Y_{(j)}+Y_{(j+1)})\left(1-\frac{j}{n}\right)-\frac{j}{n}\right],\\ K{S}_n^{-}& =\underset{j\in \{0,1,\dots ,n-1\}}{max}\left[\frac{j}{n}-\frac{1}{n}(Y_{(1)}+...+Y_{(j)})-Y_{(j)}\left(1-\frac{j}{n}\right)\right]\text{and}\\ BH{C}_n& =n{\int }_0^{\mathrm{\infty }}{\left[\frac{1}{n}\sum _{j=1}^{n}min(U_j,t)-\frac{1}{n}\sum _{j=1}^{n}1(U_j\le t)\right]}^2{e}^{-t}dt\\ & =\frac{1}{n}+\sum _{j=1}^n\sum _{k=1}^n[2-3\exp \{-min(Y_j,Y_k)\}-2min(Y_j,Y_k)(e^{-Y_j}+e^{-Y_k})\\ & +2\exp \{-max(Y_j+Y_k)\}]\end{array}$$

Baringhaus and Henze [25] suggested to generalize the statistic in a natural way by the more general weight function $e^{-at},t>0$, where $a>-1$ is some real smoothing parameter. The test statistic is

$$\begin{array}{rl}BH{C}_{n,a}& =n{\int }_0^{\mathrm{\infty }}{\left[\frac{1}{n}\sum _{j=1}^{n}min(U_j,t)-\frac{1}{n}\sum _{j=1}^{n}1(U_j\le t)\right]}^2{e}^{-at}dt\\ & =\frac{1}{n}\sum _{j,k=1}^n[\frac{2}{a^3}-\left(\frac{1}{a^3}+\frac{1}{a}\right)min(Y_j,Y_k)\{\exp (-a{Y}_j)+\exp (-a{Y}_k)\}\\ & -\left(\frac{2}{a^3}+\frac{1}{a^2}\right)\exp (-amin(Y_j,Y_k))+\left(\frac{1}{a^2}+\frac{1}{a}\right)\exp (-amax(Y_j,Y_k))]\end{array}$$

For $a=0$, $BH{C}_{n,0}=\frac{1}{n}\sum _{j,k=1}^n[\frac{1}{3}min{(Y_j,Y_k)}^3+(min(Y_j,Y_k)-1)(max(Y_j,Y_k)-\frac{1}{2}{Y}_j^2-\frac{1}{2}{Y}_k^2\phantom{\sum _{j,k=1}^n})]$

The null hypothesis of exponentiality is rejected for large values of the statistics and the tests are said to be universally consistent.

2.4. Tests based on other characterizations of the exponential distribution

The Ahsanullah $A{h}_n$ test. Let $X_1,X_2,...,X_n$ be non-negative iid random variables with distribution function $F(x)$. Ahsanullah [7] proved that a necessary and sufficient condition for $F(x)$ to be exponential is that for some j and k, the statistics $(n-j)(X_{(j+1)}-X_{(j)})$ and $(n-k)(X_{(k+1)}-X_{(k)})$ are identically distributed for $1\le j<k<n$. Volkova and Nikitin [106] utilized this characterization to propose a test for exponentiality. The test rejects the null hypothesis of exponentiality for large values of the statistic: $A{h}_n={\int }_0^{\mathrm{\infty }}(H_n(t)-G_n(t))d{F}_n(t),$ where $F_n(t)$is the EDF and $H_n(t)=n^{-2}\sum _{j,k=1}^{n}1\{|X_j-X_k|<t\};t>0,$ $G_n(t)=n^{-2}\sum _{j,k=1}^{n}1\{2min(X_j,X_k)<t\};t>0$.

2.5. Tests against monotonic classes of life distributions

2.5.1. Tests against IFR alternatives

A random variable X with distribution function F and a survival function $\overline{F}=1-F$ is said to have an IFR alternative of ageing life distribution if $\{\overline{F}(x+y)/\overline{F}(x)\}$ is decreasing in $x\ge 0$ for all $y\ge 0$ ([4]). Different characterizations have been used to obtain various tests of exponentiality against alternatives satisfying this condition. They are presented in what follows.

The $K_n^{(1)}$ of Klefsjo [73]. Klefsjo [73] used the concept of total time on test (TTT) transform to obtain a test of exponentiality against IFR alternatives.

Let $D_j=(n-j+1)(X_{(j)}-X_{(j-1)});1\le j\le n$ with $X_{(0)}=0$ be the usual normalized spacing and let $S_{(j)}=\sum _{k=1}^j{D}_{(k)}\text{with}{S}_0=0;1\le j\le n$ denote the TTT at $X_{(j)}$. Also, let $u_{(j)}=S_{(j)}/S_{(n)}$. Recall that the TTT – plot is the plot of $u_{(j)}$ against j/n; j = 1, 2, . . ., n life distributions, which is concave for IFR life distributions. The test statistic obtained based on this principle is given as $K_n^{(1)}=\sum _{j=1}^{n-1}(2u_{(j)}-u_{(j-1)}-u_{(j+1)})$.

The null hypothesis of exponentiality is rejected for positive values of the statistic. It is important to note here that Klefsjo [73] also used the same principle to test for exponentiality against IFRA, NBUE, DMRL as well as their duals.

The Ad_n test of Ahmad [4]. Using the method of U-statistic approach, Ahmad [4] obtained an empirical measure of departure of a nonnegative random variable X from exponential distribution into that of IFR and used the measure to obtain the test statistic as:

$$\begin{array}{rl}A{d}_n& =\sqrt{\frac{27n}{8}}{\overline{X}}^{-2}{\hat{\delta }}_{F_n}^{(1)},{\hat{\delta }}_{F_n}^{(1)}\\ & =[n(n-1){]}^{-1}\sum _{i\ne }\sum _j\{2{[min(X_i{X}_j)]}^2-X_i{X}_j\}\text{and}\overline{X}\\ & =n^{-1}\sum _{j=1}^n{X}_j.\end{array}$$

The statistic is scale invariant and is asymptotically standard normally distributed.

The Mitra–Anis $M{A}_n$ test. Mitra and Anis [87] proposed another test for exponentiality which is consistent against ageing life distributions with IFR. The technique is an L-statistic which is a linear combination of order statistics and null hypothesis of exponentiality is rejected by the test for large values of the statistic: $M{A}_n=\left(n^{-2}\sum _{j=1}^n(2+3n-4j)X_{(j)}\right)/({\overline{X}}_n)$

The statistic is scale invariant and $\text{as}n\to \mathrm{\infty },\sqrt{3n}M{A}_n\to N(0,1)$.

2.5.2. Tests against IFRA and LIFRA alternatives

A continuous life distribution function F with a survival function $\overline{F}=1-F$ is said to be an IFRA if and only if $-\log \overline{F}(bx)\le -b\log \overline{F}(x)$ for all $x>0,0\le b\le 1$ or equivalently if and only if $\overline{F}(bx)\ge \{\overline{F}(x){\}}^b$ [37]. In addition to this equivalent condition for existence of IFRA, Aly [12] obtained two other equivalent inequalities as conditions. They are if and only if $\overline{F}(ax)\le \{\overline{F}(x){\}}^a$ and $\{\overline{F}(ax){\}}^c\le \{\overline{F}(x){\}}^{ac}$ for all $x>0,a>0$ and $c>0$.

The Deshpande [37] $D{e}_{n,b}$ test. Let $x_1,x_2,...,x_n$ be a random sample of size n from a nonnegative continuous random variable X. Deshpande [37] used the first equivalent condition to define a function, $h_b(x_j,x_k)=\{\begin{array}{cc}1& (x_j>b{x}_k)\\ 0& \text{otherwise}\end{array};b\in (0,1)$. Based on the function, the author proposed a test for exponentiality which rejects the null hypothesis for large values of the statistic: $D{e}_{n,b}=\frac{1}{n(n-1)}\sum _{j\ne k}{h}_b(x_j,x_k);b\in (0,1)$.

The JTZ_n test of Jammalamadaka, Tiwari and Zalkikar [66]. Jammalamadaka et al. [66] defined a process based on the equivalent condition for IFRA as: $J_n(b)=\frac{1}{2}+\frac{1}{n(n-1)}\sum _{1\le i<j\le n}I\left(\frac{X_{(i)}}{X_{(j)}}>b\right)$, where $I(.)$ is an indicator function. Based on this process, they proposed U-statistic type test for exponentiality against IFRA which is asymptotic normal under the null hypothesis of exponentiality with a computational form as: $JT{Z}_n=\frac{1}{2}+\frac{1}{n(n-1)}\sum _{i=1}^n\sum _{j=i+1}^n\frac{X_{(i)}}{X_{(j)}}$

The El_n,r test of El-Bassiouny [39]. El-Bassiouny [39] proposed a test for exponentiality against IFRA based on moment inequality approach. The statistic, which is scale-invariant and asymptotically normal under the null hypothesis of exponentiality, is given by:

$$E{l}_{n,r}=\frac{1}{{\overline{X}}^{r+1}}\left\{\frac{2}{n(n-1)}\sum _{i<}\sum _j\left[min(X_i^{r+1},X_j^{r+1})-\frac{X_i^{r+1}}{2^{r+1}}\right]\right\};$$

for all integer $r\ge 0$.

The PN_n,a test of Prajitha and Nadarajah [96]. Let $L(t)$ denote the Laplace transform of a nonnegative continuous life distribution function F. Then, Prajitha and Nadarajah [96] state that F is said to have LIFRA if $[1-L(t)]/L(t)$ is non-decreasing for all $t>0$. Based on a measure of deviation of F from the null hypothesis of exponentiality and by using the empirical Laplace transform, $L_n(t)=n^{-1}\sum _{j=1}^n\exp \{-t{Y}_j\};Y_j={\overline{X}}_n^{-1}{X}_j,j=1,2,...,n$, Prajitha and Nadarajah [96] obtained a consistent and scale-invariant weighted integral as a test against LIFRA. The computational form of the statistic which rejects the null hypothesis for large is given as:

$$P{N}_{n,a}=\frac{1}{n^{3/2}}\sum _{j=1}^n\sum _{k=1}^n\frac{1}{Y_j+Y_k+a}-\frac{1}{n^{1/2}}\sum _{j=1}^n\frac{1}{Y_j+a}+\frac{1}{n^{1/2}}\sum _{j=1}^n\frac{Y_j}{{(Y_j+a)}^2};a>0.$$

2.5.3. Tests against NBU and NBUCA classes of alternatives

A nonnegative continuous random variable X with distribution function F and survival function $\overline{F}$ is said to be NBU if $\overline{F}(x+y)\le \overline{F}(x)\overline{F}(y)$ for all $x,y>0$ [4]. Also, Ahmad et al. [5] state that X is NBUCA if and only if ${\int }_0^{\mathrm{\infty }}{\int }_x^{\mathrm{\infty }}\overline{F}(u+t)dudx\le \overline{F}(t){\int }_0^{\mathrm{\infty }}{\int }_x^{\mathrm{\infty }}\overline{F}(u)dudx$ for all $t>0$.

The $AM{A}_{n,\alpha ,\lambda }$ test of Atallah, Mahmoud and Al-Zahrani [21]. Using the definition of the NBU class of life distributions, Atallah et al [22] proposed a measure of departure from the exponential distribution. Based on the proposed measure, they obtained a statistic for testing exponentiality of a dataset against NBU alternative distributions. The statistic, which is asymptotically normal under the null hypothesis of exponentiality is given as:

$$AM{A}_{n,\alpha ,\lambda }=\frac{1}{\alpha \lambda (\alpha +\lambda )n^2}\sum _{i=1}^n\sum _{j=1}^n[\lambda e^{\lambda X_i}-(\alpha +\lambda )e^{\lambda X_i}{e}^{-\alpha X_j}+\alpha e^{-\alpha X_i}];\alpha ,\lambda >0.$$

The $AS{M}_{n,s}$ test of Al-Gashgari, Shawky and Mahmoud [9]. Based on the method of moment iniqualities, Al-Gashgari et al. [9] proposed a statistic for assessing exponentiality of a dataset against NBUCA alternatives. The statistic is given by:

$$AS{M}_{n,s}=-\frac{1}{n^{2}s}\sum _{i=1}^n\sum _{j=1}^n\left[\frac{2X_i}{s}+\frac{2}{s^2}{e}^{-s{X}_i}-X_i^2{e}^{-s{X}_i}-\frac{2}{s^2}\right];s>0$$

2.5.4. Tests against NBUE class of alternatives

A life distribution $F(x)$ having a finite mean $\mu$ and a survival function $\overline{F}(x)$ is said to possess NBUE if and only if ${\int }_x^{\mathrm{\infty }}\overline{F}(t)dt\le \mu \overline{F}(x)$ for all $t>0$ [68]. Using different properties of this class of distributions in relation to the exponential distribution, a number of test statistics of exponentiality against the class are presented as follows:

The $K_n^{(1)}$ and $K_n^{(2)}$ test of Koul [78].

$$K_n^{(1)}=n^{-1}\sum _{j=1}^{n}J(W_{ni})\text{and}{K}_n^{(2)}=\underset{1\le i\le n}{max}[W_{ni}-i/n];i=1,2,...,n,$$

where $W_{ni}=\left[\sum _{j=1}^i(n-j+1)(X_{(j)}-X_{(j-1)})\right]{\left[\sum _{j=1}^n(n-j+1)(X_{(j)}-X_{(j-1)})\right]}^{-1};i=1,2,...,n$.

The null hypothesis of exponentiality is rejected for large values of the statistics.

The HP_n test of Hollander and Proschan [63].

$$H{P}_n=\frac{1}{{\overline{X}}_n}\sum _{j=1}^n{X}_{(j)}\left[\frac{1}{2n}+{\left(\frac{n-j+1}{n}\right)}^2-{\left(\frac{n-j}{n}\right)}^2\right];j=1,2,...,n.$$

Large values of the scale invariant statistic rejects the null hypothesis of exponentiality.

The $BC{R}_n^{(1)}$ test of Belzunce, Candel and Ruiz [31].

$$BC{R}_n^{(1)}=\frac{1}{n^4{\overline{X}}_n}\sum _{i=0}^{n-2}{(n-i)}^{2}n\left({\lambda }_n(i)+\frac{\sum _{i=1}^n{X}_{(i)}}{2n}\right);X_{(0)}=0;$$

${\lambda }_n(i)=\frac{1}{{(n-i)}^2}\sum _{j=i+1}^n(n-2j+i+1)X_{(j)}$. The null hypothesis is rejected for large values of $BC{R}_n^{(1)}$.

The AM_n,k test of Anis and Mitra [18].

$$A{M}_{n,k}=\frac{1}{{\overline{X}}_n}\sum _{j=1}^n{X}_{(j)}\left[\frac{a_1}{n}+a_2\left\{{\left(\frac{n-j+1}{n}\right)}^{k+1}-{\left(\frac{n-j}{n}\right)}^{k+1}\right\}\right];j=1,2,...,n,$$

where $a_1=[k(k+1){]}^{-1},a_2=k^{-1}$ and k is any positive real number. The test rejects the null hypothesis of exponentiality for large values of the statistic.

The FR_n test of Fernandez-Ponce and Rodriguez-Grinolo [44].

$$F{R}_n=\frac{1}{2}-\left(\sum _{j=2}^n(j-1)D_j\right)/\left(n\sum _{j=1}^n{D}_j\right),$$

where $D_j$ is the usual normalized spacings of the sample observations.

The $K{M}_n^{\ast }$ test of Kattumannil and Mathew [68].

$$K{M}_n^{\ast }=\left(\sum _{i=1}^n{d}_{i,n}{D}_i\right)/\left(\sum _{i=1}^n{D}_i\right);d_{i,n}=\frac{n-2i+1}{2(n-1)},i=1,2,...,n.$$

The null hypothesis of exponentiality is rejected for large values of the statistic.

The $SK{A}_n^{(1)}$ test of Sreelakshmi, Kattumannil and Asha [103].

$$SK{A}_n^{(1)}=\frac{1}{n{\overline{X}}_n}\left\{\sum _{i=1}^n{X}_{(i)}-\frac{1}{n}\sum _{i=1}^{n-1}\frac{2i}{n-i}\sum _{j=i+1}^n(X_{(j)}-X_{(i)})\right\}.$$

The scale invariant test rejects null hypothesis of exponentiality for large values of the statistic.

2.5.5. Tests against DMRL alternatives

Let $F(x)$ be a nonnegative continuous life distribution function with a survival function $\overline{F}(x)=1-F(x),F(0)=0$ and a finite mean $\mu ={\int }_0^{\mathrm{\infty }}\overline{F}(x)dx$. The mean residual life, $e(x)$, of $F(x)$ ([32]) is given by: $e(x)=\{\begin{array}{ll}\frac{1}{\overline{F}(x)}{\int }_x^{\mathrm{\infty }}\overline{F}(t)dt& \text{if}\overline{F}(x)>0\\ 0& \text{if}\overline{F}(x)=0\end{array}$

They went further to state that $F(x)$ is said to have a decreasing (increasing) mean residual life if $e(x)$ is decreasing (increasing). Using different characterizations of the life distributions with DMRL in relation to the exponential distribution, which has a constant MRL, the following statistics for testing exponentiality against DMRL have been developed:

The $B{K}_{n,\alpha ,\beta }$ test of Bergman and Klefsjo [32].

$$\begin{array}{rl}B{K}_{n,\alpha ,\beta }& =\frac{1}{{\overline{X}}_n}\sum _{j=1}^n\left[a_1\left(\frac{n-j+1}{n}\right)+a_2{\left(\frac{n-j+1}{n}\right)}^{\beta +1}+a_3{\left(\frac{n-j+1}{n}\right)}^{\alpha +\beta +2}\right]\\ & \times [X_{(j)}-X_{(j-1)}],\end{array}$$

where $\alpha$ and $\beta$ are any positive integers, $a_1=-[\beta (\beta +1)(\alpha +\beta +1){]}^{-1},a_2=-[\beta (\alpha +1){]}^{-1}$ and $a_3=-[\alpha +\beta +2][\beta (\beta +1)(\alpha +\beta +1){]}^{-1}$. A large positive value of the statistic indicates rejection of the null hypothesis of exponentiality.

The $BC{R}_n^{(2)}$ test of Belzunce, Candel and Ruiz [31].

$$\begin{array}{rl}BC{R}_n^{(2)}& =\frac{1}{n^6{\overline{X}}_n}\sum _{i=0}^{n-2}\sum _{j=i+1}^{n-1}{(n-i)}^2(n-j{)}^2[{\lambda }_n(j)-{\lambda }_n(i)];{\lambda }_n(i)\\ & =\frac{1}{{(n-i)}^2}\sum _{j=i+1}^n(2-2j+i+1)X_{(j)}.\end{array}$$

The test rejects the null hypothesis for large values of the scale invariant statistic.

The $A_n^{\ast }$ of Anis [16].

$$\begin{array}{rl}{A}_n^{\ast }& =\frac{1}{n^2{\overline{X}}_n^3}\sum _{j=0}^{n-1}\left[(X_{(j+1)}-X_{(j)}){(\sum _{i=j+1}^{n-1}{X}_{(i)})}^2\right]\\ & +\frac{2}{3n^2{\overline{X}}_n^3}\sum _{j=0}^{n-1}[{(n-j)}^2(X_{(j+1)}^3-X_{(j)}^3)]\\ & -\frac{2}{n^2{\overline{X}}_n^3}\sum _{j=0}^{n-1}\left[(n-j)(X_{(j+1)}^2-X_{(j)}^2)\sum _{i=j+1}^n{X}_{(i)}\right]-\frac{1}{2n{\overline{X}}_n^2}\sum _{j=1}^n{X}_j^2.\end{array}$$

The statistic is scale invariant and rejects the null hypothesis for large values of the statistic.

The SM_n test of Sankaran and Midhu [101].

$$\begin{array}{rl}S{M}_n& ={\int }_0^1[\hat{M}(p)-\hat{M}(0)]dp,\text{where}\\ \hat{M}(p)& =(1-p{)}^{-1}\sum _{j=k}^n{Q}_n{F}_n[T_{(i)}-T_{(k)}](S_{(i)}-S_{(i-1)}).\end{array}$$

$Q_n$ is the empirical quantile function; $F_n$ is the empirical distribution function; $T_{(i)}$ is the ith order statistic of the right censored sample and $k=[np{]}^{+},$ which is the greatest integer function. Also, $S_i=0\text{if}i=0,F_n{T}_{(i)}\text{if}i=1,2,\dots ,n-1\text{and}1\text{if}i=n$. The test rejects the null hypothesis for large values of the statistic.

2.5.6. Tests against HNBUE alternatives

A life distribution $F(x)$ having finite mean $\mu$ and a survival function $\overline{F}(x)=1-F(x)$ is said to be in HNBUE class if and only if ${\int }_x^{\mathrm{\infty }}\overline{F}(t)dt\le \mu \exp (-t/\mu )$ for all $t>0$. A number of tests for exponentiality against the HNBUE alternatives have been developed using different characterizations of the HNBUE class of life distributions in relation to the exponential distribution. They are:

The $K{l}_n^{(1)}$ and $K{l}_n^{(1)}$ tests of Klefsjo [74].

$$K{l}_n^{(1)}=\sum _{j=1}^n{J}_1(j/n)X_{(j)}/S_n\text{and}K{l}_n^{(2)}=\sum _{j=1}^n{J}_2(j/n)X_{(j)}/S_n,$$

where $J_1(u)=-3^{-1}+3(1-u{)}^2$, $J_2(u)=11(6^{-1})-3u^2$ and $S_n=\sum _{j=1}^n{D}_j$, $D_j$ is the usual normalized spacings of the observations. The statistics are asymptotically normally distributed.

The $K_{n,a}$ test of Klar [71].

$$\begin{array}{rl}{K}_{n,a}& =\frac{1}{n{a}^2}\sum _{j=1}^n{e}^{-a{Y}_j}-\frac{1}{a^2(1+a)}\text{and}\\ K_{n,0}& =\frac{1}{2n}\sum _{j=1}^n{Y}_j^2-1;a>0,Y_j=X_j/{\overline{X}}_n,j=1,2,...,n.\end{array}$$

The test rejects null hypothesis for large negative values of the statistic.

The GM_n,_r test of Ghosh and Mitra [49].

$$G{M}_{n,r}=\frac{1}{n(r+1)(r+2)}\sum _{j=1}^n{Y}_j^{r+2}-\mathrm{\Gamma }(r+1);r>-1,Y_j=X_j/{\overline{X}}_n,j=1,2,...,n.$$

The scale invariant test rejects the null hypothesis for large negative values of the statistic.

The $G{M}_{n,a}^{(1)}$ and $G{M}_{n,a}^{(2)}$ tests of Ghosh and Mitra [50].

$$\begin{array}{rl}G{M}_{n,a}^{(1)}& =\frac{1}{n}\sum _{i=1}^n\left(\frac{1}{a^2}{Y}_i{e}^{-a{Y}_i}+\frac{2}{a^3}{e}^{-a{Y}_i}-\frac{3a+2}{a^3{(1+a)}^2}\right)\text{and}\\ G{M}_{n,a}^{(2)}& =\frac{1}{n}\sum _{i=1}^n\left(\frac{1}{a^2}{Y}_i^2{e}^{-a{Y}_i}+\frac{4}{a^3}{Y}_i{e}^{-a{Y}_i}+\frac{6}{a^4}{e}^{-a{Y}_i}-\frac{2(3+8a+6a^2)}{a^4{(1+a)}^3}\right);\\ & a>0,i=1,2,\dots ,n,\end{array}$$

where $Y_i$ has its usual meaning. The null hypothesis of exponentiality is rejected for large negative values of the statistics.

2.5.7. Tests against the L-class of alternatives

This class of ageing life distributions was introduced by Klefsjo [75]. He defined an absolutely continuous distribution $F(x)$ with finite mean $\mu$ and survival function $\overline{F}(x)=1-F(x)$ as one belonging to the L-class of ageing distributions if for any $s\ge 0$, ${\int }_0^{\mathrm{\infty }}\exp (-st)\overline{F}(t)dt\ge \mu {(1+s\mu )}^{-1}$. Some tests of exponentiality have been proposed in the literature. They are:

The C_n test of Chaudhuri [34].

$$C_n=\frac{n^{1/2}}{{\overline{X}}_n}\underset{1\le j\le m}{max}\left\{\sum _{i=1}^{n-1}\left(1-\frac{i}{n}\right)a_{ij}-\frac{{\overline{X}}_n}{1+(j/m)F^{-1}(1-\epsilon ){\overline{X}}_n}\right\};\epsilon \ge 0,$$

where $m=[n(1-\epsilon )]$ and $a_{ij}=\frac{\exp (-(j/m)F^{-1}(1-\epsilon )X_{(i)})-\exp (-(j/m)F^{-1}(1-\epsilon )X_{(i+1)})}{(j/m)F^{-1}(1-\epsilon )}$.

Large values of the statistic leads to rejection of the null hypothesis of exponentiality.

The $H{K}_{n,a}^{(1)}$ and $H{K}_{n,a}^{(2)}$ tests of Henze and Klar [59].

$$\begin{array}{rl}H{K}_{n,a}^{(1)}& =n^{-1}\sum _{j=1}^n{(Y_j+a)}^{-1}-e^a{E}_1(a)\text{and}\\ H{K}_{n,a}^{(1)}& =n^{-1}\sum _{j=1}^n({(Y_j+a)}^{-1}+{(Y_j+a)}^{-2})-a^{-1};a>0,\end{array}$$

where $e^a{E}_1(a)={\int }_0^{\mathrm{\infty }}\exp (-at)/(1+t)dt$ and the null hypothesis is rejected for large negative values of the statistics.

The $B{M}_{n,r}$ test of Basu and Mitra [29].

$$\begin{array}{rl}B{M}_{n,r}& =\left[\sqrt{n}\left(n^{-1}\sum _{j=1}^n({\overline{X}}_n/X_j{)}^r-\pi /\mathrm{\Gamma }(r)\sin r\pi \right)\right]/\\ & \left[\sqrt{(\pi /\mathrm{\Gamma }(2r)\sin 2r\pi )-{(\pi /\mathrm{\Gamma }(r)\sin r\pi )}^2}\right];0\le r\le 1/2\end{array}$$

The test rejects the null hypothesis of exponentiality for large negative values of $B{M}_{n,r}$ and the statistic is asymptotically standard normally distributed.

The $M{M}_{n,a,b}^{(1)}$ and $M{M}_{n,a,b}^{(2)}$ tests of Majumder and Mitra [83].

$$\begin{array}{rl}M{M}_{n,a,b}^{(1)}& =\frac{\mathrm{\Gamma }(b+1)}{n}\sum _{i=1}^n\frac{1}{{(a+Y_i)}^{b+1}}-e^a\mathrm{\Gamma }(b+1)E_{b+1}(a)and\\ M{M}_{n,a,b}^{(2)}& =-\frac{\mathrm{\Gamma }(b+1)}{a^{b+1}}+\frac{\mathrm{\Gamma }(b+1)}{n}\sum _{i=1}^n\frac{1}{{(a+Y_i)}^{b+1}}+\frac{\mathrm{\Gamma }(b+2)}{n}\sum _{i=1}^n\frac{1}{{(a+Y_i)}^{b+2}},\end{array}$$

where $E_n(a)={\int }_a^{\mathrm{\infty }}{e}^{-t}{t}^{-n}dt$ is an exponential integral, $Y_i$ has the usual meaning and $a>0,b>-1$. The null hypothesis of exponentiality is rejected for large negative values of the statistics.

2.5.8. Tests against DMTTF alternatives

MTTF was introduced by Barlow and Proschan [28] who gave it in an age replacement model as $m(t)=[F(t){]}^{-1}{\int }_0^t\overline{F}(x)dx\text{for}\text{all}t>0$. An ageing life distribution F is said to have DMTTF if and only if $m(t)$ is decreasing for all $t>0$. Some tests for exponentiality against the DMTTF class of alternatives have been proposed. They are:

The KAIA_n test of Kayid, Ahmad, Izadkhah and Abouammoh [69].

$$\begin{array}{rl}KAI{A}_n& =\frac{1}{{\overline{X}}_n}[\frac{1}{n(n-1)...(n-j-2)}\sum _{i_1\ne {\stackrel{`}{i}}_2\ne \stackrel{`}{\phantom{e}}\dots \ne {\stackrel{`}{i}}_{j+2}}(j+2)min(X_{i_1},..,X_{i_{j+2}})\\ & -\frac{1}{n(n-1)...(n-j-1)}\sum _{i_1\ne {\stackrel{`}{i}}_2\ne \stackrel{`}{\phantom{e}}\dots \ne {\stackrel{`}{i}}_{j+1}}(j+1)min(X_{i_1},..,X_{i_{j+1}})],\end{array}$$

where $j\ge 1$ such that $n>(j+2)$. Large values of the statistic leads to rejection of the null hypothesis of exponentiality.

The $BK{M}_n$ test of Bhattacharyya, Khan and Mitra [33].

$$\begin{array}{rl}BK{M}_n& =\frac{1}{n{\overline{X}}_n}\sum _{i=1}^n{t}_{ni}{X}_{(i)},\text{where}{t}_{ni}\\ & =\frac{4}{3}{\left(\frac{i}{n}\right)}^3-{\left(\frac{i}{n}\right)}^2-\left(\frac{i}{n}\right)+\frac{2}{3}+\frac{i}{n^2}-\frac{2i^2}{n^3}+\frac{4i}{3n^3}+\frac{1}{2n}-\frac{1}{3n^2}-\frac{1}{3n^3}.\end{array}$$

The scale invariant test rejects the null hypothesis for large values of the statistic.

2.6. Tests against non-monotonic classes of ageing life distributions

2.6.1. Tests against IDMTTF alternatives

The IM_n test of Izadi and Manesh [64].

$$\begin{array}{rl}I{M}_n& =\underset{0\le k\le n}{max}\frac{\sqrt{n}}{{\overline{X}}_n}\{\phantom{\sum _{i=k+1}^{n-1}}\left[2\left(2-\frac{k}{n}\right)-1\right]\sum _{i=0}^{n-1}\left(1-\frac{i}{n}\right)D_i-2\sum _{i=0}^{n-1}{\left(1-\frac{i}{n}\right)}^2{D}_i\\ & +\sum _{i=k+1}^{n-1}\left(1-\frac{i}{n}\right)\left[4\left(1-\frac{i}{n}\right)-2\left(2-\frac{k}{n}\right)\right]D_i\}\end{array}$$

where $D_i=X_{(i+1)}-X_{(i)},i=0,1,...,n-1$. Rejection of the null hypothesis of exponentiality is prompted by the test for large values of IM_n.

The $K{A}_n^{\ast }$ test of Kattumannil and Anisha [67].

$$K{A}_n^{\ast }=\frac{1}{n(n-1){\overline{X}}_n}\sum _{i=1}^n(3n-4i+1)X_{(i)};i=1,2,...,n.$$

The test rejects null hypothesis of exponentiality for large values of $K{A}_n^{\ast }$.

2.6.2. Tests against NWBUE alternatives

This class of non-monotonic ageing life distributions has been extensively studied in the literature, see for instance Klefsjo [76], Hawkins and Kochar [55] and Mitra and Basu [88]. What follows therefore are tests of exponentiality against the class that has been proposed.

The $A{M}_n^{\ast }$ test of Anis and Mitra [17].

$$A{M}_{n,r}^{\ast }=\frac{\sqrt{n}\left[(1/n){\sum }_{j=1}^n(X_j/{\overline{X}}_n{)}^r-\mathrm{\Gamma }(r+1)\right]}{\sqrt{[\mathrm{\Gamma }(2r+1)-{(\mathrm{\Gamma }(r+1))}^2]}};r>0$$

The null hypothesis of exponentiality is rejected for large negative values of $A{M}_n^{\ast }$ and the statistic is asymptotically standard normal.

The $G{M}_{n,a}$ test of Ghosh and Mitra [48].

$$\begin{array}{rl}G{M}_{n,a}& =\frac{1}{{\overline{X}}_n}[\sum _{i=1}^{k-1}\frac{1}{n}\left\{c_{p,a}-\frac{a+1}{a}{\left(1-\frac{i}{n}\right)}^a\right\}{X}_{(i)}\\ & +\sum _{i=k}^n\frac{1}{n}\left\{c_{p,a}-\frac{2{(1-p)}^a}{a}+\frac{a+1}{a}{\left(1-\frac{i}{n}\right)}^a\right\}{X}_{(i)}]\end{array}$$

where $k=[np{]}^{+}$, the smallest integer greater than or equal to np, p is the proportion of sample contained in the set of order statistics $X_{(1)}\text{to}{X}_{(i)}$, $c_{p,a}=a^{-1}-(a+1{)}^{-1}+2(a+1{)}^{-1}(1-p{)}^{a+1}$ and $a>0$. Large values of the statistic leads to rejection of null hypothesis of exponentiality.

2.6.3. Tests against IDMRL alternatives

The GHP_n test of Guess, Hollander and Proschan [52].

Let F be IMRL up to a point $\tau$ then changes to DMRL, where the change point $\tau$ is known, then

$$\begin{array}{rl}GH{P}_n& =\sum _{i=1}^k{B}_1\left(\frac{n-i+1}{n}\right)(X_{(i)}-X_{(i-1)})+B_1\left(\frac{n-k}{n}\right)(\tau -X_{(k)})\\ & +B_2\left(\frac{n-k}{n}\right)(X_{(k+1)}-\tau )+\sum _{i=k+2}^k{B}_2\left(\frac{n-i+1}{n}\right)(X_{(i)}-X_{(i-1)}),\end{array}$$

where $B_1(u)=\left[\frac{2}{3}-F_n(\tau )+\frac{1}{2}{F}_n^2(\tau )\right]u+\left[-1+F_n(\tau )-\frac{1}{2}{F}_n^2(\tau )\right]u^2+\frac{1}{3}{u}^4$, $B_2(u)=\left[-\frac{1}{6}+\frac{1}{2}{F}_n(\tau )-\frac{1}{2}{F}_n^2(\tau )+\frac{1}{3}{F}_n^3(\tau )\right]u+\left[\frac{1}{2}-F_n(\tau )+\frac{1}{2}{F}_n^2(\tau )\right]u^2-\frac{1}{3}{u}^4$ and k is the position in the order statistics which is less than or equal to the position of the change point. The statistic is asymptotically normal and its large value leads to rejection of null hypothesis of exponentiality.

The $HK{L}_n^{(1)}$ and $HK{L}_n^{(2)}$ tests of Hawkins, Kochar and Loader [56].

$$\begin{array}{rl}HK{L}_n^{(1)}& =\frac{\sqrt{n}}{{\overline{X}}_n}\underset{0\le k\le n\ast }{max}\left\{\frac{n}{n-k}\sum _{j=k}^{n-1}\left(1-\frac{j}{n}\right)D_j-{\overline{X}}_n\right\}\text{and}\\ HK{L}_n^{(2)}& =\frac{\sqrt{n}}{{\overline{X}}_n}\underset{0\le k\le n}{max}\left\{A_n-2\left(1-\frac{k}{n}\right)\sum _{j=k}^{n-1}\left(1-\frac{j}{n}\right)D_j+4\sum _{j=k}^{n-1}{\left(1-\frac{j}{n}\right)}^2{D}_j\right\},\end{array}$$

where $A_n=-X_{(1)}+\sum _{j=1}^{n-1}\left[1-\frac{j}{n}-2{\left(1-\frac{j}{n}\right)}^2\right]D_j$, $D_j=X_{(j+1)}-X_{(j)}$ and $n\ast =[n(1-\epsilon )]$. The null hypothesis is rejected for large values of the statistic.

The $N{L}_{n,a}$ test of Na and Lee [91]

$$N{L}_{n,a}=\frac{1}{{\overline{X}}_n}\left[\underset{0\le k\le n}{max}\left\{\sqrt{n}(2n_a(k)-n_a(0))\right\}\right];k=0,1,...,n,$$

where $n_a(k)=\frac{1}{a+1}\sum _{i=k}^{n-1}\left\{(a+2){\left(\frac{n-2}{n}\right)}^{a+2}-{\left(\frac{n-k}{n}\right)}^{a+1}\left(\frac{n-2}{n}\right)\right\}(X_{(i+1)}-X_{(i)})$ and $a$ is any integer greater than or equal to –1. Large values of the statistic leads to rejection of the null hypothesis.

2.6.4. Tests against BFR alternatives

A non-monotonic ageing life distribution F is said to have a BFR if and only if there exists change points $0<{\tau }_1\le {\tau }_2<\mathrm{\infty }$ such that the failure rate $r(t)$ is strictly decreasing in $[0,{\tau }_1)$, constant in $[{\tau }_1,{\tau }_2)$ and then strictly increasing in $[{\tau }_2,\mathrm{\infty })$, $r(t)=f(t)/\overline{F}(t)$ for all $t\ge 0$ ([90]). Using different characterizations, the following tests have been developed for assessing exponentiality against the BFR alternatives:

The P_n test of Park [92].

$$\begin{array}{rl}{P}_n& =\frac{1}{{\overline{X}}_n}\{\sum _{i=1}^{k-1}{n}^{-2}[4i-(2+p)n]X_{(i)}\\ & +\sum _{i=k}^n{n}^{-2}[-4i+(3+p)n]X_{(i)}-F_n^{-1}(p)(1-p)(1-2p)\},\end{array}$$

where $k=[np{]}^{+},F_n^{-1}(p)=X_{(k)}\text{and}p=k/n$. The null hypothesis of exponentiality is rejected for large values of the scale invariant statistic.

The $NJ{P}_n$ test of Na, Jeon and Park [90].

$$NJ{P}_n={\overline{X}}_n^{-1}\left[\underset{0\le k\le n}{max}\left\{\sqrt{n}(n_1(k)+n_2(0)-2n_2(k))\right\}\right];k=0,1,...,n,$$

where $n_1(k)=\sum _{i=1}^k\left(\frac{n-i+1}{n}\right)(X_{(i)}-X_{(i-1)})$ and $n_2(k)=\sum _{i=k+1}^n\{\left(\frac{n-k}{n}\right)\left(\frac{n-i+1}{n}\right)-2{\left(\frac{n-i+1}{n}\right)}^2\}(X_{(i)}-X_{(i-1)})$.

The null hypothesis of exponentiality is rejected for large values of the statistic.

The $M{M}_{n,j,k}$ test of Majumder and Mitra [84].

$$\begin{array}{rl}M{M}_{n,j,k}& =\frac{1}{{\overline{X}}_n}[\sum _{i=1}^{m-1}\frac{-a_1}{n}{\left(\frac{n-i}{n}\right)}^{k-1}{X}_{(i)}\\ & -\sum _{i=1}^{m-1}\frac{1}{n}\left\{\frac{{(1-p)}^k}{a_1}{\left(\frac{n-i}{n}\right)}^{j-1}+a_2{\left(\frac{n-i}{n}\right)}^{j+k-1}\right\}{X}_{(i)}\\ & +\sum _{i=m}^n\frac{1}{n}\left\{a_2{\left(\frac{n-i}{n}\right)}^{j+k-1}+a_1{(1-p)}^j{\left(\frac{n-i}{n}\right)}^{k-1}\right\}{X}_{(i)}\\ & -\left\{{(1-p)}^j\left(\frac{1}{j}+\frac{1}{k}\right)-\frac{1}{j}\right\}{(1-p)}^k{\hat{F}}_n^{-1}(p)],\end{array}$$

where $m=[np{]}^{+},{\hat{F}}_n^{-1}(p)=X_{(m)},p=m/n,a_1=-k/j,a_2={(j+k)}^2/jk,j>0\text{ and }k>1/2.$ The test rejects the null hypothesis of exponentiality for large values of the statistic.

2.7. Other tests for exponentiality

The $C{O}_n$ test of Cox and Oakes [37]. Cox and Oakes [37] developed a two-sided test of exponentiality whose statistic is: $C{O}_n=n+\sum _{j=1}^n(1-Y_j)In{Y}_j;j=1,2,\dots ,n$

The statistic rejects the null hypothesis of exponentiality for both small and large values of $C{O}_n$. It is also obtained that the asymptotic null distribution of the statistic is such that $(6/n{)}^{1/2}(C{O}_n/\pi )\to N(0,1)$. As a result of the asymptotic null distribution, it is possible to use standard normal critical values when conducting a test of exponentiality via this statistic.

The $G{G}_n$ test of Gail and Gastwirth [47]. Gail and Gastwirth [46] studied a scale-free goodness-of-fit test for exponentiality which is based on the normalized spacing and proposed a rejection of the null hypothesis for large values of the statistic: $G_n=\frac{1}{2n(n-1)}\sum _{j,k=1}^n|Y_j-Y_k|$

They also obtained that the asymptotic distribution of $G^{\ast }=[12(n-1){]}^{1/2}(G_n-1/2)$ is standard normal even for samples as small as $n=10$ and claimed that the statistic has very high power against Weibull, Uniform and Gamma alternatives. Henze and Meintanis [62] have however stated that testing exponentiality based on $G_n$ is equivalent to testing based on a two – sided statistic: $G{G}_n=\sum _{j=1}^{n-1}{U}_j=2n-\frac{2}{n}\sum _{j=1}^{n}j{Y}_{(j)},$ where $U_k=\sum _{j=1}^k{D}_j/\sum _{j=1}^n{X}_j;1\le k\le n-1$.

This relation, according to them, is because $(n-1{)}^{-1}G{G}_n=1-G_n$.

The Hegazy – Green $Hg{1}_n$ and $Hg{2}_n$ tests. Hegazy and Green [57] proposed two tests for exponentiality which are based on order statistics. The statistics are given by:

$$Hg{1}_n=\frac{1}{n}\sum _{j=1}^n\left|X_{(j)}+\ln \left(1-\frac{j}{n+1}\right)\right|\text{and}Hg{2}_n=\frac{1}{n}\sum _{j=1}^n{\left[X_{(j)}+\ln \left(1-\frac{j}{n+1}\right)\right]}^2$$

The tests reject the null hypothesis of exponentiality for large values of the statistics.

The Wong–Wong $W{W}_n$ test. This is another test that is based on the order statistics of a set of non negative observations. Wong and Wong [107] obtained an extremal quotient of a set of order statistics as an appropriate test for exponentiality. The statistic is given by: $W{W}_n=X_{(n)}/X_{(1)}$, where $X_{(1)}$ and $X_{(n)}$ are respectively the first and last order statistics of the data set.

The Hahn–Shapiro $W{e}_n$ test. Hahn and Shapiro [53] proposed a test for exponentiality based on ratio of sum of squares and square of sum. The test rejects the null hypothesis of exponentiality for large values of the statistic: $W{e}_n=\sum _{j=1}^n{(X_j-\overline{X})}^2/{\left(\sum _{j=1}^n{X}_j\right)}^2$.

The Shapiro–Wilk $S{W}_n$ statistic. Similar to the Hahn and Shapiro [53] $W{e}_n$ statistic is the $S{W}_n$. Shapiro and Wilk [102] proposed a test for exponentiality based on correlation coefficient. The statistic is given by: $S{W}_n=[n{(\overline{X}-X_{(1)})}^2]/\left[(n-1)\sum _{j=1}^n{(X_j-\overline{X})}^2\right]$.

They stated that the statistic is both scale and origin invariant with a maximum value of 1 and a minimum value of $(n-1{)}^{-2}$.

The Pietra $P{t}_n$ test. Another test for exponentiality which has been used extensively in the literature is the Pietra statistic, discussed by Gail and Gastwirth [47]. Its statistic is given by:

$$P{t}_n=\sum _{j=1}^n|X_j-\overline{X}|/2n\overline{X}.$$

The Wald $W_n$ test. This test is a general test of hypothesis that is equivalent to the likelihood ratio test and the Rao’s score test. It uses the asymptotic normality of the maximum likelihood estimator of parameter $\theta$, given as $\hat{\theta }$. The case of testing for exponentiality using the Wald statistic has been discussed in Cox and Oakes [37]. The statistic can be given as:

$$W_n=[\hat{\theta }-1]/[SE(\hat{\theta })]$$

where $SE(\hat{\theta })$ is the standard error of the estimate, obtained by the inverse of the Fisher information matrix. The statistic follows a chi – square distribution.

The Patwardhan $P{a}_n$ test. Let $Y_{(1)},Y_{(2)},...,Y_{(n)}$ be the order statistics of a transformed sample from a non negative distribution. Patwardhan [95] obtained a test of exponential distribution based on the probability plot of the order statistics. The statistic is $P{a}_n=(n+1)n^{-1}\{{\mathbf{\text{y}}}^{\mathrm{\prime }}{\mathbf{\Theta }}^{-1}\mathbf{\text{y}}-n\}$, where $\mathbf{\text{y}}$ is an n-component vector given as ${\mathbf{\text{y}}}^{\mathrm{\prime }}=(Y_{(1)},Y_{(2)},...,Y_{(n)})$ and $\mathbf{\Theta }$ is a covariance matrix of the n-component vector of the order statistics. The statistic is scale invariant with maximum value of n(n+1) and minimum value of (n+1). The null hypothesis is rejected for large values of the statistic.

The Moran $M{o}_n$ statistic. Tchirina [104] used the statistic according to Moran [89] to propose a test for exponentiality. The test rejects the null hypothesis for large value of the statistic: $M{o}_n=\left|n^{-1}\sum _{j=1}^n\ln (X_j/\overline{X})+\gamma \right|$; where $\gamma =0.577215\dots$ is the Euler constant.

The statistic is said to be consistent against the class of distributions with monotone failure rate.

The Atkinson $A{t}_n$ test. Mimoto and Zitikis [86] proposed another test for exponentiality based on the Atkinson statistic. The test statistic is $A{t}_{n,p}=\sqrt{n}|\left({(n^{-1}\sum _{j=1}^n{X}_j^p)}^{1/p}/\overline{X}\right)-{(\mathrm{\Gamma }(1+p))}^{1/p}|;p\in (-1,1)$, where $\mathrm{\Gamma }(.)$ is the gamma function. Mimoto and Zitikis [86] through extensive Monte Carlo simulations obtained that the power of the statistic is maximum when p is close to 0 or 0.99. Hence, in our power study, p = 0.99 is used. They equally stated that the statistic is consistent against fixed alternatives.

3. Power study

In this section, the results of the Monte Carlo simulation study which is carried out to investigate the power performance of forty (40) different procedures for testing exponentiality of a set of data which are selected from the tests described in section 2 are presented. Also, the power performance of these techniques is studied when they are applied to real-life datasets.

It is well known that exponential distribution has a constant hazard rate. Consequently, we considered three different categories of the competing distributions, namely: distributions with increasing hazard rate $(IHR)$ as group I, distributions with decreasing hazard rate $(DHR)$ as group II and distributions with non – monotone hazard rate $(NMHR)$ as group III.

Some of the statistics can be transformed to follow, asymptotically, some known distributions. Such statistic is the CO_n which is known to follow standard normal distribution, for instance. Despite this valuable property of such statistics, empirical computation of the powers of the statistics is used throughout the study to ensure uniformity of comparison. Precisely, 10,000 replications of each distribution are simulated at different sample sizes of 10, 25, 50 and 100. Under each sample size in each of the competing test procedures, the reported power performance is the percentage of the 10,000 replications that reject the null hypothesis of exponentiality at 5 percent level of significance. The results as well as well as their discussions are presented in the supplemental material which can be assessed from the journal website.

3.1. Real-life applications

In this sub-section, the power performances of the techniques under review are obtained from two real-life datasets. The datasets include the monthly rainfall (mm) of the Andaman and Nicobar Islands of India from January 2011 through December 2015. The 48 point dataset (n = 48) was accessed from www.kaggle.com/datasets and total monthly precipitation (mm) in the month of June from 1960 through 2016 in Sao Carlos, Brazil. The 56 point dataset was obtained from Louzada et al. [80].

Preliminary graphical investigations carried out on the datasets by the use of empirical and theoretical densities as well as empirical and theoretical CDF’s with Q-Q and P–P plots showed somewhat, a bad fit of the datasets to exponential distribution. Complete application of the competing tests to the datasets is contained in the supplemental material.

4. Conclusion

In this paper, we reviewed 91 different tests for exponentiality. Some of the tests are said to be universally consistent while some others are against some special classes of alternative distributions. Also, some of the tests have smoothing parameters which can be varied for optimal power performance. Such tests include BHe_n, He_n, BHC_n, etc. As a result, use of any of such tests in practice will require prior determination of best value of the parameter since it is evident that the choice of values of the parameters affects the powers of the tests.

The result of our simulation study (see the supplemental material assessed from the journal website) reveals that some tests which are powerful procedures under alternatives with one shape may not be so under alternatives with another shape. Also, some tests which are less powerful at small sample sizes appear more powerful at large sample sizes and vice versa. Finally, the results of the real-life applications are completely in agreement with the results of the simulation studies.

Disclosure statement

No potential conflict of interest was reported by the author(s).