The Anderson Darling test assesses whether a sample comes from a specified distribution. It makes use of the fact that, when given a hypothesized underlying distribution and assuming the data does arise from this distribution, the cumulative distribution function (CDF) of the data can be assumed to follow a uniform distribution. Let us consider $X_1,X_2,\cdots ,X_n$ be a random sample. Anderson-Darling statistic $A^2$ (here we denote as TAD) is given by Anderson and Darling [7] as follows:

$TAD=-n-\frac{1}{n}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\left(2i-1\right)\left[\ln \left(F\left(Y_i\right)\right)+\ln \left(1-F\left(Y_{n+1-i}\right)\right)\right],$ (2)

where $Y_1,Y_2,\cdots ,Y_n$ be the ordered measurements and $F$ is the CDF (Cumulative distribution function) of (1). Zhang and Wu [8] proposed Likelihood-ratio Anderson-Darling test for exponentiality test as follows:

$LAD=-\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\left\{\frac{\log F\left(Y_i\right)}{n-i+0.5}+\frac{\log \left[1-F\left(Y_i\right)\right]}{i-0.5}\right\}$ (3)

Extensive research has been conducted on the asymptotic distributions of these statistics. But here we are proposing simulation distribution under the null hypothesis to obtain the upper tail p-value for the tests (2 & 3).

Kolmogorov-Smirnov test (Kolmogorov [9] and Smirnov [10] ) is a nonparametric test of the equality of continuous or discontinuous, one-dimensional probability distributions that can be used to test whether a sample came from a given reference probability distribution.

The Kolmogorov-Smirnov statistic quantifies a distance between the empirical distribution function of the sample and the cumulative distribution function of the reference distribution. The empirical distribution function $F_n$ for $n$ independent and identically distributed (i.i.d.) ordered observations $X_i$ is defined as

$TKS=\underset{x}{\sup }\left|F_n\left(x\right)-F\left(x\right)\right|$ (4)

where $F\left(x\right)$ is the CDF of the null hypothesis distribution. Zhang and Wu [8] proposed Likelihood-ratio Kolmogorov-Smirnov test for exponentiality test as follows:

$LKS=\underset{i\in \left\{1,2,\cdots ,n\right\}}{\max }\left\{\left(i-0.5\right)\log \frac{i-0.5}{nF\left(Y_i\right)}+\left(n-i+0.5\right)\log \frac{n-i+0.5}{n\left[1-F\left(Y_i\right)\right]}\right\},$ (5)

A wide range of research is done in obtaining asymptotic distributions of this statistic. But here we are proposing simulation distribution under the null hypothesis to obtain the upper tail p-value.

The Shapiro-Wilk test is a statistical test for the normality of a population, based on sample data. It was introduced by Shapiro and Wilk [11] in testing for normality. Here, we are proposing to implement the test for testing shifted exponential distribution as follows: Let $X_{\left(i\right)}$ be the $i^{th}$ ordered values from a sample size $n$.

$TSW=\frac{{{\displaystyle \sum }}_{i=1}^n{\left(a_i{X}_{\left(i\right)}\right)}^2}{{{\displaystyle \sum }}_{i=1}^n{\left(X_{\left(i\right)}-\bar{X}\right)}^2}$ (6)

where $\bar{X}$ is the mean of the sample,

$\left(a_1,a_2,\cdots ,a_n\right)=\frac{m^{\text{T}}{V}^{-1}}{C},$

where $m_i={\displaystyle {\sum }_{r=1}^{i}1/\left(n-r+1\right)}$, $1\le i\le n$, $V_{ii}={\displaystyle {\sum }_{r=1}^{i}1/{\left(n-r+1\right)}^2}$, $1\le i\le n$, $V_{ij}={\displaystyle {\sum }_{r=1}^{i}1/{\left(n-r+1\right)}^2}$, $1\le i<j\le n$ Balakrishnan and Basu [3] , and $C=\Vert V^{-1}m\Vert ={\left(m^{\text{T}}{V}^{-1}{V}^{-1}m\right)}^{1/2}$.

Note that this is a left tailed test.

The Shapiro-Francia test is a statistical test for the normality of a population, based on sample data. It was introduced by S. S. Shapiro and R. S. Francia in 1972 as a simplification of the Shapiro-Wilk test [12] .

$TSF=\frac{{{\displaystyle \sum }}_{i=1}^n\left(X_{\left(i\right)}-\bar{X}\right)\left(m_i-\bar{m}\right)}{\sqrt{{{\displaystyle \sum }}_{i=1}^n{\left(X_{\left(i\right)}-\bar{X}\right)}^2{{\displaystyle \sum }}_{i=1}^n{\left(m_i-\bar{m}\right)}^2}}$ (1.4)

where $\bar{X}$ is the mean of the sample and $\bar{m}$ is the mean of $m_i$'s, given in section 1.3. Note that this is a left tailed test.

The test statistic is as follows:

$TLC=\frac{1}{12n}+\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left[\frac{2i-1}{2n}-F\left(x_i\right)\right]}^2$ (1.5)

Note that this is a right tailed test.

Standard Chi-Square Goodness-of-fit test is computed as

${\chi }^2=\underset{k=1}{\overset{g}{{\displaystyle \sum }}}{\left(\frac{O_k-E_k}{E_k}\right)}^2$ (1.6)

where $g$ stands for the number of groups, $O_k$ stands for the observed counts in the $k^{th}$ group, and $E_k$ stands for the expected counts under $H_0$ in the $k^{th}$ group. Note that ${\chi }^2$ will follow approximate Chi-square distribution with $g-2-1$ degrees of freedom as both the parameters in the Beta distribution are assumed to be unknown.

We demonstrate the four different parameter estimation procedures given above using real-life data. The data given in Table 3 below is obtained from Bain and Engelhardt [13] and represents the times between successive failures. It is assumed that the times are exponentially distributed while successive failures are assumed to be from a Poission process.

The respective p-values for TKS is 0.251, for TAD is 0.367, for TSW is 0.229, for TSF is 0.234, for LKS is 0.631, for TLC is 0.649, for LAD is 0.541, for TCS is 0.449 and for SCS is 0.765.

Likelihood-ratio Anderson-Darling test has higher power irrespective of alternative distribution. Cramer-von Mises test is the next best test. Between Shapiro-Wilk and Shapiro-Francia tests, the Shapiro-Francia test has higher power.

Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk tests have poor performances as they have very low powers irrespective of alternative distributions.