Some New Goodness-of-fit Tests for Rayleigh Distribution

This article deals with goodness-of-fit test for Rayleigh distribution. Three new tests based on Jeffreys, Lin-Wong and Rényi divergence measures are proposed, and shown to be consistent. Monte Carlo simulations are performed for various alternatives and sample sizes in order to compare the proposed tests with other goodnessof-fit tests for Rayleigh distribution in the literature. Simulation results showed that in comparison with the existing tests, our proposed tests have shown better performances in terms of power and among them Jeffreys test is better than other. Finally, illustrative example for use of some considered more powerful tests are presented and analyzed.

As a matter of fact, divergence measures are generally used to find appropriate distance or difference between two probability distributions. These measures have been applied in several fields like probability distributions, signal processing, pattern recognition, finance, economics, etc.
While GOF tests for exponentiality have a large literature, GOF tests for the Rayleigh distribution (testing Rayleighity) have only recently been considered. The GOF tests for the Rayleigh distribution were proposed in Meintanis and Iliopoulos (2003), Morris  , although ̂ is a biased estimator.
The article is organized as follows. In section 2, some new tests based on the Jeffreys, Lin-Wong and Rényi divergence measures are developed and some theorems about their properties are presented. In Section 3, using p-value method for checking Rayleighity is described. In Section 4, the GOF tests of Rayleighity in literature are reviewed. In Section 5, the power and type I error rate estimation techniques and alternative distributions have been described. Then, the power of the proposed tests against competitors' tests are compared. In section 6, the performance of considered tests for a real data is evaluated. Finally, in section 7 we appraise the ability of considered tests and choose better tests for each category of alternative distributions.

Proposed Tests for the Rayleigh distribution
One of the important problems in many applications is finding a suitable measure of distance (or divergence or difference or discrimination) between two probability distributions.
A measure of divergence is used as a way to appraise the distance (divergence) between any two populations or functions. Let 1 and 2 be two probability density functions which may or may not depend on an unknown parameter of fixed finite dimension. As mentioned, the most well-known measure of divergence is the KL-divergence. Actually, a divergence is a measure of difference or closeness between two distributions 1 and 2 . Here, the focus is on continuous distributions with densities 1 ( ) and 2 ( ), for which the real-valued functional is a convex function of both 1 and 2 . Note that ( 1 , 2 ) is not a metric distance in general (symmetry and the triangle inequality are not required).
In fact, some measures proposed for determining GOF do not possess all properties of a metric function. Therefore, they are rather called as divergence measures. For example, KL-divergence is nonnegative but is not symmetric. Also, it is easy to see that it does not satisfy the triangular inequality (Garrido, 2009) thus it is not a metric function. Hence, it must be interpreted as a pseudo-metric measure only. So we propose some new GOF tests for the Rayleigh distribution which are based on some divergence measures.
Accordingly, in this section we propose using some popular divergence measures such as Jeffreys, Lin-Wong and Rényi divergence for checking the Rayleigh distribution and we guess using these divergence measures leads to a better performance in terms of power against other proposed tests in the statistical literature.

Jeffreys Test
As mentioned earlier, KL-divergence is not symmetric, so in order to obtain a symmetric measure Jeffreys (1946) proposed a symmetric variant of the KL-divergence which is called the Jeffreys' divergence (or sometimes Jdivergence) in the literature. This divergence has several practical uses in statistics such as detecting influential data in regression analysis and model comparisons (see Arellano-Valle et al., 2000). The Jeffreys' divergence is defined as Clearly, ( 1 , 2 ) and ( 1 , 2 ) divergences share most of their properties. As pointed out in Ullah (1996), Jeffreys measure does not satisfy the triangular inequality of metric and hence it is also a pseudo-metric measure. Let 1 , . . . , be nonnegative; independent and identically distributed (iid) random variables from a continuous distribution function with order statistics (1) ≤. . . ≤ ( ) . Let 0 ( ) denote a Rayleigh distribution, where is the unknown parameter. The 0 and 1 hypotheses are as follows 0 : ( ) = 0 ( , ), 1 : ( ) ≠ 0 ( , ).
To discriminate between two hypotheses 0 and 1 , we propose using the Jeffreys' divergence measure of two density functions ( ) and 0 ( ) as (1)

Remark 1
The Jeffreys' statistic for testing Rayleighity can be defined as below where ( ) = (1) for < 1 and ( ) = ( ) for > and ̂ is the ML estimator of . Proof: we can write (1) easily as Now, by replacing 0 the Jeffreys' divergence will be Under the null hypothesis ( , 0 ) = 0 and we expect large values of ( , 0 ) under 1 . Based on Vasicek's method (1976) and supposing ( ) = we have Now, the proposed statistic in Remark 1 will be easily derived by using the relation below (2) Generally, the Null Hypothesis for large values of will be rejected, that is, 0 is rejected if ≥ * for some critical values * . Although, we will reject the null hypothesis when the p-value is less than the level of significance.

Theorem 1
The statistic of is scale-free and invariant with respect to scale transformation. Proof: As is a scale parameter of the Rayleigh distribution we consider a scale transformation group as = { : ( ) = , > 0} . Since ( ( )) = ( ) ∀ , ∀ ∈ , therefore, is a scale-free statistic and invariant under .
, where is the empirical distribution function. Since ̂ is a consistent estimator of as → ∞ and based upon strong law of large numbers we have Meanwhile, Safavinejad et al. (2015) showed that the ML estimator of in the Rayleigh distribution is equivariant. Therefore, the test is a reasonable test for the Rayleigh distribution which has some good properties such as to be scale-free, invariancy and consistency.

Lin-Wong Test
In this subsection, for constructing the test statistic we estimate the Lin-Wong distance similar to Vasicek's (1976) method for estimating the Shannon entropy. To discriminate between the two hypotheses 0 and 1 , we propose using the Lin and Wong (1990) divergence measure of two density functions ( ) and 0 ( ) as Since Lin-Wong divergence belongs to Csiszer family, we have ( , 0 ) ≥ 0 and the equality holds if and only if ( ) = 0 ( ) (See Kapur and Kesavan, 1992). So, it motivates us to use Lin-Wong divergence as a test statistic for Rayleighity. Remark 2 The Lin-Wong statistic for testing Rayleighity can be defined as below where ( ) = (1) for < 1 and ( ) = ( ) for > and ̂ is the ML estimator of . Proof: Similar to proof of Remark 1 by replacing 0 in (3) we have Based on Vasicek's method (1976) and supposing ( ) = we have Finally, the proposed statistic in Remark 2 will be derived by using (2)  is a reasonable test for the Rayleigh distribution which has some good properties such as to be scale-free, invariancy and consistency.

Rényi Test
In this subsection, for constructing the test statistic we estimated the Rényi divergence similar to Vasicek's (1976) method for estimating the Shannon entropy.

Remark 3
The Rényi statistic for testing Rayleighity can be defined as below where ( ) = (1) for < 1 and ( ) = ( ) for > and ̂ is the ML estimator of .
To discriminate between the two hypotheses 0 and 1 , we propose using the Rényi (1961) divergence of two density functions ( ) and 0 ( ) as where > 0 and ≠ 1 .
Finally, the proposed statistic in Remark 3 will be derived by using (2)  is a reasonable test for the Rayleigh distribution which has some good properties such as to be scale-free, invariancy and consistency.

Alizadeh et al. (2012) proposed a GOF test based on KL-divergence for checking Rayleighity. Their proposed statistic is
where ̂ is the ML estimate of and is the Vasicek's sample entropy estimator which is where the window size is a positive integer smaller than /2, (

Cumulative Residual Entropy-Based Test Baratpour and Khodadadi (2012) based on cumulative Kullback-Leibler defined another GOF test for Rayleighity which is
where 0 is rejected at the significance level in favor of 1  , where = 2√2 , ̂= /̂ and ̂ denotes the consistent estimator of . The null hypothesis, 0 , is rejected for large values of .

Empirical Distribution Function Tests
Empirical distribution is a very important estimator in statistics. Many statistical procedures depend on its performance. Consider a random variable with the distribution function ( ). A random sample of size , 1 , ⋯ , , is given from ( ) and let (1) < (2) < ⋯ < ( ) be the order statistics. We assume this distribution to be continuous. The empirical distribution function, ( ), is defined as where has been substituted by ̂ which is the ML estimator of . The null hypothesis, 0 , is rejected for large values of 2 , 2 , 2 and * .

Power Study
In this section, the power of considered tests are estimated against several alternatives. The method is that of Monte Carlo simulation of the distribution of considered tests under alternative distributions. For each alternative, 10000 samples of size were generated. Then, the power was calculated by proportion of rejecting null hypothesis for simulated data from alternative distribution in some iterations. In other words, the power can be estimated by where p-value ( ( ) ) is the probability value of at j-th iteration and 1 is the alternative distribution. This form of definition of power ensures that the approximate power is a number strictly between 0 and 1. The p-value can be easily calculated using Monte Carlo simulation proposed by North et al. (2003) through simulating samples from the distribution specified in the null hypothesis and computing the test statistic, , for each of these samples. Then the p-value will be where ( 1 , … , ) is the statistic from the simulated Rayleigh data ( 1 , … , ) at k-th iteration and is the observed statistic. It is important to mention here that as we checked the power values which were calculated by the method mentioned in this paper and the classical method which is based on using critical values to estimate powers, the calculated powers based on these methods are very close together and the difference between them are usually in thousandths decimal which are negligible. The continuous alternative distributions are classified in the three classes below: • Monotonic decreasing hazard (Dec. Hazard) rate: Chisquare(1), Weibull(0.5,1), • Monotonic increasing hazard (Inc. Hazard) rate: Chisquare(3), Beta(3,1), • Non-Monotonic hazard (Non-Mon. Hazard) rate: Beta(1,0.5), Exponential (2). The powers of the considered tests are compared to that of some other tests for Rayleighity against the same alternatives which are tabulated in Tables 2 and 3.

Determining Optimum values of window size ( )
The GOF test based on entropy involves choosing the best integer parameter . Unfortunately, there is no choice criterion of , and in general it depends on . Ebrahimi et al. (1992) tabulated the values of which maximize the power of the test. Meanwhile, similar tables are given by Abbasnejad (2011), Alizadeh Noughabi and Balakrishnan (2014).
Meanwhile, the optimum value of is a value of which leads to the smallest value of bias and mean square error (MSE). Therefore, we determined the optimum values of based on 10000 iterations which their results are presented in Table 1.
To determine optimum values of for any value of , we compute bias and MSE of for different values of , 1 to /2, by where ( ) = 1 + ( /√2) + /2 is the entropy of the Rayleigh distribution, is the Euler-Mascheroni constant which is already 0.57721 and is the number of iterations. From Table 1 it is evident that with increasing , the optimal choice of also increases.   . Also, we can offer three most powerful tests in overall as , 2 , . Although among non-monotone hazard rate distributions choosing the best test is difficult, three most powerful tests in overall and in entropy type tests are respectively , , . Another important property of a test is type I error rate which is calculated as proportion of rejected null hypothesis for the Rayleigh distribution. As we checked, we conclude that type I error rates of all considered tests are about the supposed significance level. So, they have acceptable performance in this field.

Application to average wind speed data
In this section, we consider one average wind speed data analysis reported in Best et al. (2010). The following data represent 30 average daily wind speeds (in km/h) for the month of November 2007 recorded at Elanora Heights, a northeastern suburb of Sydney, Australia: 2.7, 3.2, 2.1, 4.8, 7.6, 4.7, 4.2, 4.0, 2.9, 2.9, 4.6, 4.8, 4.3, 4.6, 3.7, 2.4, 4.9, 4.0, 7.7, 10.0, 5.2, 2.6, 4.2, 3.6, 2.5, 3.3, 3.1, 3.7, 2.8, 4.0. The data were analyzed initially by Best et al. and Alizadeh et al. (2012), who fitted the Rayleigh distribution successfully. The probability values of testing Rayleighity for some of more powerful tests in this paper are presented in Table 4. The probability values of all tests are greater than 0.05, so they are not significant at 5 percent significance level. Thus, we cannot reject the null hypothesis which means the wind speed data follows the Rayleigh distribution.

Conclusion
The aim of this paper is to evaluate the performance of the proposed test. So, we considered and compared thirteen different GOF tests for evaluating Rayleighity. In order to evaluate the ability of mentioned tests in identifying the Rayleigh distribution, by a simulation study the powers and type I error of the proposed tests were computed under several alternatives and different sample sizes in three different class of hazard rates. The results were tabulated in Tables 2 and 3 which are the power estimates of considered tests at significance level of 0.05. Finally, we concluded that test which is a symmetric version of KL test and belongs to entropy based tests performed better than the other tests. Since for real data we do not know type of hazard rate distribution, the results obtained from this study encourage us using test in all three classes.
The future of this research is to work on proposing GOF tests for censored data such as Type II censored data which is very well-known and very applicable to applied researches.