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 $A^2$ is given by Anderson and Darling [3] as follows:

$A^2=-n-\frac{1}{n}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\left(2i-1\right)\left[\ln \left(F\left(Y_i\right)\right)+\ln \left(1F\left(Y_{n+1-i}\right)\right)\right],$ (1.2)

where $Y_1,Y_2,\cdots ,Y_n$ be the ordered measurements and $F$ is the CDF (Cumulative distribution function) of (1.1). Extensive research has been conducted on the asymptotic distributions of this statistic. But here, we are proposing simulation distribution under the null hypothesis to obtain the upper tail p-value.

Kolmogorov-Smirnov test (K-S test or KS test) (Kolmogorov [4] and Smirnov [5] ) 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 the 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

$D_n=\underset{x}{\sup }\left|F_n\left(x\right)-F\left(x\right)\right|$ (1.3)

where $F\left(x\right)$ is the CDF of the null hypothesis distribution. In literature, a wide range of research has been done to obtain 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-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 [6] . Let $X_{\left(i\right)}$ be the $i^{th}$ ordered values from a sample size $n$. Let $m_{i:n}$ be the mean of the $i^{th}$ order statistic when making $n$ independent draws from a normal distribution.

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

where $\bar{X}$ is the mean of the sample and $\bar{m}$ is the mean of $m_i$’s. Note that this is a left tailed test. Rahman and Pearson [7] showed that practical computation of $m_{i:n}$’s can be done using exclusive Monte–Carlo simulation. In implementing this test for testing Beta distribution, Uniform $U\left(0,1\right)$ can be used as the Standard Normal $N\left(0,1\right)$ is used in case of test for Normal distribution. In case of

$U\left(0,1\right)$, $m_i=\frac{i}{n+1}$. To show the accuracy of such choice, we also implement Monte-Carlo simulation as was done by Rahman and Pearson [7] in case of Normal distribution.

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

${\chi }^2=\underset{k=1}{\overset{g}{{\displaystyle \sum }}}{\left(\frac{O_k-E_k}{E_k}\right)}^2$ (1.5)

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.

When we applied the above mentioned goodmess-of-fit procedures in the Relative humidity data of air in May2007 from the Haarweg Wageningen weather station [8] .

The respective p-values for AD is 0.510, for KS is 0.000, for WE is 0.000, for WS is 0.000, and for CS is 0.160 ( Table 2 ).

Performance of Kolmogorov-Smirnov test is more consistent than other tests considered. Shapiro-Francia test has adequate performance and simulation means for order statistics are not needed as standard uniform distribution means have similar results as for the simulated means. The Chi-Square test performed poorly in terms of powers. Anderson-Darling test is not reliable for various alternatives.

It is also demonstrated that in testing for Beta Distribution using Shapiro-Francia test, there is no need to use simulated means of order statistics, it is suffices to use $m_i=\frac{i}{n+1}$.