# Fractional Entropy-Based Test of Uniformity with Power Comparisons

Authors: Mohamed S. Mohamed; Haroon M. Barakat; Salem A. Alyami; Mohamed A. Abd Elgawad
Journal: Journal of Mathematics
 (2021)

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## Abstract

In the present paper, we use the fractional and weighted cumulative residual entropy measures to test the uniformity. The limit distribution and an approximation of the distribution of the test statistic based on the fractional cumulative residual entropy are derived. Moreover, for this test statistic, percentage points and power against seven alternatives are reported. Finally, a simulation study is carried out to compare the power of the proposed tests and other tests of uniformity.


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## Body

## 1. Introduction

Rao et al. [1] suggested a nonnegative measure of uncertainty and called it the cumulative residual entropy (CRE). For any nonnegative continuous random variable (RV) X with a cumulative distribution function (CDF) Fx=PX<x, the CRE is defined by(1)CREF=−∫0∞F¯xlnF¯xdx,where F¯x=1−Fx is the reliability function. Rao et al. [1] revealed many salient features of the CRE. For example, the CRE possesses more general mathematical properties than the Shannon entropy, and it can be easily computed from sample data, and these computations asymptotically converge to the true values. Moreover, the CRE deals with the quantity of information in residual life. For the standard uniform distribution, denoted by U0,1, Rao et al. [1] showed that the value of the CRE is 1/4. The literature abounds with many different results for Shannon’s entropy and its modifications. Interested readers may refer to [1–17].Xiong et al. [16] suggested the fractional cumulative residual entropy (FCRE) to extend the CRE to the case of fractional order. For any 0≤q≤1, the FCRE for the RV X is defined by(2)CREqF=∫0∞F¯x−lnF¯xqdx.The measureCREqF is a nonadditive and nonnegative. Moreover, it is a convex function of the parameter q, CRE0F=EX, and CRE1F=CREF. Xiong et al. [16] derived the FCRE for some well-known distributions; for example, FCRE of the CDF U0,1 is Γq+1/2q+1.Misagh et al. [15] proposed a weighted form of CRE, which is shift-dependent. This information-theoretic uncertainty measure is called the weighted cumulative residual entropy (WCRE), and it is defined by(3)CREwF=−∫0∞xF¯xlnF¯xdx.Later, Mirali et al. [12] and Mirali and Baratpour [13] studied several properties of this measure including its dynamic version. It is easy to observe that the WCRE of the U0,1 is 5/36.Stephens [18] offered a practical guide to goodness-of-fit tests using statistics based on the empirical CDF. Moreover, in [18], the power comparisons of some uniformity tests were carried out. Dudewicz and Van der Meulen [9] investigated the power properties of an entropy-based test when used for testing uniformity. Moreover, via a comparison with other tests of uniformity, Dudewicz and Van der Meulen [9] showed that the entropy-based test possesses good power properties for many alternatives. Noughabi [14] constructed a test for uniformity based on the CRE and studied some of its properties. Moreover, he reported the percentage points and power comparison against seven alternative distributions. As a natural extension of the results obtained by Noughabi [14], we study the FCRE and WCRE for testing the uniformity. A result of a simulation study shows that the test based on FCRE and WCRE is competitive with the test based on CRE in terms of power. This fact gives a satisfactory motivation of our study.Throughout this paper, we obtain the percentage points under the WCRE and FCRE by using the Monte Carlo method via the simulation and the normality asymptotic, as well as the beta approximation, respectively. Moreover, a power comparison is performed between the FCRE and WCRE and other tests. The rest of this work is systematic as follows. In Section2, we introduce the FCRE test statistic for uniformity and discuss some of its properties. In Section 3, we propose the methods of finding the percentage points of FCRE and illustrate the WCRE test statistics for uniformity. In addition, we calculate the percentage points of FCRE and WCRE. Then, in Section 4, we use Monte Carlo simulation to perform the power comparison of uniformity of different tests against seven alternative distributions. Section 5 is devoted to the conclusions. Everywhere in what follows, the symbols (⟶np), (⟶nd) and (⟶na.s.) stand for convergence in probability, convergence in distribution, and almost surely, as n⟶∞.


## 2. Theoretical Aspects and Test Statistic

To establish our test of the null hypothesisH0, we need the following theorem, which shows that, for a CDF with support [0, 1], one always has 0≤CREqF≤e−q, and for the distribution U0,1, we have FCRE=Γq+1/2q+1, and this value is uniquely attained by the uniform distribution, whenever q is fixed.Theorem 1.
LetX be a nonnegative RV with an absolutely continuous CDF F with a support 0,1. From (2), it holds 0≤CREqF≤e−q, and CREqF=Γq+1/2q+1 is uniquely acquired by the distribution U0,1.Proof.
Since0≤F¯x−lnF¯xq≤1, and the function fx=x−lnxq has a maximum at x=e−q, 0<x≤1, we get 0≤CREqF≤e−q. On the other hand, using the strict convexity of fx=x−lnxq, it is easy to see that FCRE is a concave function of distribution (with support [0, 1]). This shows that CREqF=Γq+1/2q+1 is uniquely acquired by the distribution U0,1. This completes the proof.
LetX1,X2,…,Xn be a random sample with a continuous CDF F, with support [0, 1]. Furthermore, let X1≤X2≤⋯≤Xn be the corresponding order statistics X1,X2,…,Xn. According to (2), we can obtain the empirical FCRE as an estimator of FCREF by(4)CREqFn=∫0∞F¯nx−lnF¯nxqdx,where F¯nx=1−Fnx and Fnx is the empirical CDF, which is defined by(5)Fnx=∑i=1n−1inIXi,Xi+1x+IXn,∞x,x∈ℜ,where IAx is the indicator function, i.e., IAx=1,x∈A; IAx=0,x∉A.
To perform a consistent test of the hypothesis of uniformity, we suggest the consistent statistic test(6)Rnq=CREqFn=∑i=1n−11−in−ln1−inqXi+1−Xi=∑i=1n−1AiWi,where Ai=1−i/n−ln1−i/nq and Wi=Xi+1−Xi, i=1,2,…,n−1, 0≤q≤1.
Xiong et al. [16] proved that CREqFn⟶npCREqF. Moreover, under the null hypothesis H0, we get Rnq⟶npΓq+1/2q+1. On the other hand, under the alternative hypothesis (that F is any continuous CDF with support [0, 1], which is not the uniform), we have CREqFn⟶npr, where r is a smaller or larger number than Γq+1/2q+1.Theorem 2.
The test based on the sample estimateRnq is consistent.Proof.
From Glivenko–Cantelli theorem (see Tucker [19]), we have suptFnt−Ft⟶na.s.0. On the other hand, Theorem 3 in Xiong et al. [16] asserts that CREqFn⟶na.s.CREqF, which proves the theorem.Theorem 3.
Suppose that the random sampleX1,X2,…,Xn has been drawn from an unknown continuous CDF F defined on 0,1. Then, from (6), we have 0≤Rnq≤e−q.Proof.
Since the functionfp=p−lnpq, 0<p<1, has a maximum value at e−q, 0≤q≤1; therefore,(7)0≤Rnq=CREqFn=∑i=1n−11−in−ln1−inqXi+1−Xi≤∑i=1n−1e−qXi+1−Xi=e−qXn−X1≤e−q.
This completes the proof of the theorem.Theorem 4.
UnderH0, from (6), the mean and the variance of Rnq are, respectively,(8)ERnq=1n+1∑i=1n−1Ai,(9)VarRnq=nn+12n+2∑i=1n−1Ai2.Proof.
The proof directly follows by noting that, for anyi=1,2,…,n−1, the RV Wi=Xi+1−Xi, based on the CDF U0,1, has beta distribution, i.e., Wi∼Beta1,n (cf. [20]). This completes the proof.Remark 1.
UnderH0, from (6), (8), and (9), we have limn⟶∞ERnq=Γq+1/2q+1=CREqU and limn⟶∞VarRnq=0, where CREqU is the FCRE of the CDF U0,1.
The critical region, which describes the test procedure, is given by the following two inequalities:(10)CREqFn≤CREα/2∗q≔lower orCREqFn≥CRE1−α/2∗q≔upper,where α is the desired level of significance, and CREα∗q is the α−quantile of the asymptotic, or approximated, CDF of the test statistic CREqFn, under H0. In the next section, we derive the asymptotic and approximated CDF of the test statistic CREqFn. These quantiles are computed by using the Monte Carlo method.


## 3. Percentage Points of the Test Statistic

In this section, we obtain the asymptotic distribution ofRnq under H0. From (6), we can write Rnq=∑i=1n−1Ti, where Ti=AiWi, i=1,2,…,n−1, and Wi∼Beta1,n. Thus, we can see that Ti’s have the following probability density function (PDF):(11)fTit=nAi1−tAin−1,i=1,2,…,n−1.The mean and variance ofTi are, respectively,(12)μi=ETi=AiEWi=Ain+1,σi2=VarTi=Ai2VarWi=nAi2n+12n+2.According to Lyapunov central limit theorem (see Billingsley [21]), we have ∑i=1n−1Ti−μi/∑i=1n−1σi2=Rnq−ERnq/VarRnq⟶ndN, where N is the standard normal RV (in the sequel, the standard normal distribution will be denoted by N0,1). Therefore, under H0, the percentage point (α−quantile) CREα∗q is approximated according to the asymptotic normality of Rnq for large n by(13)CREα∗q=ERnq+VarRnqZα,where Zα corresponds to the quantile α×100 of the CDF N0,1.Johannesson and Giri [22] proposed an approximation of the CDF of linear combination of the finite number of beta RVs. Noughabi [14] used this approximation to obtain approximately the percentage points of the CRE for finite n. By adopting the same procedure of Noughabi [14], we can obtain an approximation of Rnq for finite n as follows:(14)Rnq≈∑i=1n−1AiY,where the RV Y has Betaa,b distribution,(15)a=n+2∑i=1n−1Ai2n+1∑i=1n−1Ai2−1n+1,b=nn+1n+2∑i=1n−1Ai2∑i=1n−1Ai2−1,and Ai=1−i/n−ln1−i/nq, 0≤q≤1, i=1,2,…,n−1. According to (14), the mean and variance of Rnq are, respectively,(16)ERnq=∑i=1n−1Aiaa+b,VarRnq=∑i=1n−1Ai2aba+b2a+b+1.Now, by using this approximation ofRnq, the quantiles of order α/2 and 1−α/2 of the approximated CDF of the test statistic CREqFn under H0 are, respectively,(17)lower≔∑i=1n−1AiF−1α2,upper≔∑i=1n−1AiF−11−α2,where F−1. is the quantile function of the CDF F, F is the Betaa,b distribution, and a and b are defined in (15).


### 3.1. Empirical Weighted Cumulative Residual Entropy

From (3), Misagh et al. [15] proposed the empirical WCRE by(18)CREwFn=−∑i=1n−1Xi+12−Xi221−inln1−in=∑i=1n−1A′iUi,where A′i=Xi+12−Xi2/2, Ui=−1−i/nln1−i/n, i=1,2,…,n−1.We suggest the following statistic of a consistent test based on (18):(19)Tnw=CREwFn=∑i=1n−1A′iUi.Theorem 5.
The test based on the sample estimateTnw is consistent.Proof.
From Mirali et al. [12] and by using Glivenko–Cantelli theorem, (see Tucker [19]), we have CREwFn⟶na.s.CREwF, which proves the theorem.Theorem 6.
LetX1,X2,…,Xn be a random sample drawn from an unknown continuous CDF F defined on 0,1. Then, from (18), we get 0≤Tnw≤1/2e.Proof.
Since the functionfp=−plnp, 0<p<1, has a maximum value at 1/e; therefore,(20)0≤Tnw=CREwFn=−∑i=1n−1Xi+12−Xi221−inln1−in≤12∑i=1n−11eXi+12−Xi2=12eXn2−X12≤12e.
This completes the proof.


### 3.2. Percentage Points

We generate50,000 samples of size n, where n=10,20,30,40,50,70,100, from U0,1. Using (6), the test statistic Rnq is estimated by the empirical Rnq for each sample and the same for Tnw. Moreover, we can see that CRE0.1U=0.4438, CRE0.5U=0.3133, CRE0.9U=0.2576 and CREwU=0.1388, where CREqU and CREwU are the FCRE and WCRE of the CDF U0,1, respectively. Consequently, for Rnq, we present the percentage points of the Monte Carlo method, asymptotic normality, and beta approximation by using (10), (13), and (17), respectively. The result of this study is given in Table 1, where we note that the difference between the percentage points decreases when n increases. Besides, for Rnq, the accuracy of the Monte Carlo method is more than the other two methods.Table 1
Percentage points of the proposed test statisticsRnq and Tnw at level α=0.05.
nqRnqTnwMonte Carlo methodNormal approximationBeta approximationUpperLowerUpperLowerUpperLowerUpperLower100.10.51310.22820.61650.12930.64950.16720.15740.06690.50.35220.18180.44810.10650.470030.13150.90.29640.149010.37320.08730.39160.1084200.10.506080.30410.60120.21440.62170.23680.15440.09570.50.34580.23410.42820.16320.44150.17770.90.28920.19010.35490.13350.36610.1458300.10.49950.33570.584070.25560.59870.27150.152750.10680.50.34220.25440.41350.190070.4230.20020.90.28530.20660.342070.15550.350090.1641400.10.49450.35410.57090.28090.58240.29310.15150.11260.50.33930.26460.403090.20640.41040.21420.90.28230.21530.33310.169020.33930.1756500.10.49050.36530.56070.29830.57010.30820.15060.11630.50.33740.27120.39530.21770.40130.22410.90.28080.22070.32650.17830.33160.1837700.10.48540.38010.54610.32130.5530.32850.14920.12060.50.33450.27960.38440.23260.38880.23720.90.27760.22760.31720.19060.321030.19451000.10.47990.39190.53170.34180.53670.34690.148050.12420.50.33170.28590.374040.24590.37720.24920.90.275080.23350.30850.20160.31120.2044Figures1–4 represent the empirical PDF’s of the test statistics using Monte Carlo samples with n=10,20,30,50,100. When n increases, it turned out that the test statistics are nearer to the exact values, which implies that the bias and the variance decrease with increasing n.Figure 1
The estimated PDF’s ofRn0.1 based on U (0, 1).Figure 2
The estimated PDF’s ofRn0.5 based on U (0, 1).Figure 3
The estimated PDF’s ofRn0.9 based on U (0, 1).Figure 4
The estimated PDF’s ofTnw based on U (0, 1).


## 3.1. Empirical Weighted Cumulative Residual Entropy

From (3), Misagh et al. [15] proposed the empirical WCRE by(18)CREwFn=−∑i=1n−1Xi+12−Xi221−inln1−in=∑i=1n−1A′iUi,where A′i=Xi+12−Xi2/2, Ui=−1−i/nln1−i/n, i=1,2,…,n−1.We suggest the following statistic of a consistent test based on (18):(19)Tnw=CREwFn=∑i=1n−1A′iUi.Theorem 5.
The test based on the sample estimateTnw is consistent.Proof.
From Mirali et al. [12] and by using Glivenko–Cantelli theorem, (see Tucker [19]), we have CREwFn⟶na.s.CREwF, which proves the theorem.Theorem 6.
LetX1,X2,…,Xn be a random sample drawn from an unknown continuous CDF F defined on 0,1. Then, from (18), we get 0≤Tnw≤1/2e.Proof.
Since the functionfp=−plnp, 0<p<1, has a maximum value at 1/e; therefore,(20)0≤Tnw=CREwFn=−∑i=1n−1Xi+12−Xi221−inln1−in≤12∑i=1n−11eXi+12−Xi2=12eXn2−X12≤12e.
This completes the proof.


## 3.2. Percentage Points

We generate50,000 samples of size n, where n=10,20,30,40,50,70,100, from U0,1. Using (6), the test statistic Rnq is estimated by the empirical Rnq for each sample and the same for Tnw. Moreover, we can see that CRE0.1U=0.4438, CRE0.5U=0.3133, CRE0.9U=0.2576 and CREwU=0.1388, where CREqU and CREwU are the FCRE and WCRE of the CDF U0,1, respectively. Consequently, for Rnq, we present the percentage points of the Monte Carlo method, asymptotic normality, and beta approximation by using (10), (13), and (17), respectively. The result of this study is given in Table 1, where we note that the difference between the percentage points decreases when n increases. Besides, for Rnq, the accuracy of the Monte Carlo method is more than the other two methods.Table 1
Percentage points of the proposed test statisticsRnq and Tnw at level α=0.05.
nqRnqTnwMonte Carlo methodNormal approximationBeta approximationUpperLowerUpperLowerUpperLowerUpperLower100.10.51310.22820.61650.12930.64950.16720.15740.06690.50.35220.18180.44810.10650.470030.13150.90.29640.149010.37320.08730.39160.1084200.10.506080.30410.60120.21440.62170.23680.15440.09570.50.34580.23410.42820.16320.44150.17770.90.28920.19010.35490.13350.36610.1458300.10.49950.33570.584070.25560.59870.27150.152750.10680.50.34220.25440.41350.190070.4230.20020.90.28530.20660.342070.15550.350090.1641400.10.49450.35410.57090.28090.58240.29310.15150.11260.50.33930.26460.403090.20640.41040.21420.90.28230.21530.33310.169020.33930.1756500.10.49050.36530.56070.29830.57010.30820.15060.11630.50.33740.27120.39530.21770.40130.22410.90.28080.22070.32650.17830.33160.1837700.10.48540.38010.54610.32130.5530.32850.14920.12060.50.33450.27960.38440.23260.38880.23720.90.27760.22760.31720.19060.321030.19451000.10.47990.39190.53170.34180.53670.34690.148050.12420.50.33170.28590.374040.24590.37720.24920.90.275080.23350.30850.20160.31120.2044Figures1–4 represent the empirical PDF’s of the test statistics using Monte Carlo samples with n=10,20,30,50,100. When n increases, it turned out that the test statistics are nearer to the exact values, which implies that the bias and the variance decrease with increasing n.Figure 1
The estimated PDF’s ofRn0.1 based on U (0, 1).Figure 2
The estimated PDF’s ofRn0.5 based on U (0, 1).Figure 3
The estimated PDF’s ofRn0.9 based on U (0, 1).Figure 4
The estimated PDF’s ofTnw based on U (0, 1).


## 4. Power Analysis

In this section, we study the power test of Monte Carlo study under alternative distributions. The power ofRnq is estimated by the proportion of the generated samples falling into the critical region. Under seven alternative distributions, the power of the test statistic Rnq is calculated by the Monte Carlo study of generating 50,000 samples each of size n, where n=20,30,50. The alternative CDFs proposed by Stephens [18] in power study of uniformity tests are as follows:(21)Al:Fy=1−1−yl,0≤y≤1,l=1.5,2,Bl:Fy=2l−1yl,0≤y≤0.5,1−2l−11−yl,0.5≤y≤1,l=1.5,2,3,Cl:Fy=0.5−2l−10.5−yl,0≤y≤0.5,0.5+2l−1y−0.5l,0.5≤y≤1,forl=1.5,2.In Table2, based on the Monte Carlo study, we recorded the power values of the proposed test statistics Rnq, Tnw, Kolmogorov–Smirnov (K-S), Kuiper (V), Cramer-von Mises (W2), Watson (U2), and Anderson-Darling (A2), for n=10,20,30 and α=0.05. From Table 2, we can conclude the following:(1)
Ifq increases and tends to 1 (q⟶1), the power of CREq test, for alternative AlBlCl, decreases (increases) (increases), and vice versa, if q decreases and tends to 0 q⟶0.(2)
Ifq⟶1, the CREq test, for alternative AlBl, gives the worst (best) performance compared with the other tests.(3)
To compare the performance betweenCREq and CREw tests, we observe that:(a)
For the alternativeAl,q⟶1, CREw performs better than CREq and vice versa if q⟶0,n increases.(b)
For the alternativeBl,q⟶1, CREq performs better than CREw, and vice versa, if q⟶0,n increases.(c)
For the alternativeCl,q⟶0, CREw performs better than CREq, and vice versa, if q⟶1.Stephens [18] noted that V and U2 tests will reveal a change at variance. Therefore, we observe the following:(1)
For alternativeAl,q⟶0, CREq performs better than V and U2, and vice versa, if q⟶1.(2)
For the alternativeBl,q⟶1, CREq performs better than V and U2, and vice versa, if q⟶0,n increases.(3)
For the alternativeCl,q⟶0, V and U2 performs better than CREq.(4)
CREw performs better than V and U2 against the alternative Al.(5)
CREw performs better than V and U2 against the alternative Bl,n increases. But, V and U2 perform better than CREw against the alternative Cl.Table 2
Power estimates of the tests at levelα=0.05.
nAlternativeRnqTnwK-SVW2U2A20.10.50.910A1.50.107080.09080.072080.140020.126160.07560.14560.077760.1877A20.27710.23270.151040.34140.302980.16310.35510.163080.4761B1.50.104060.13140.13020.08960.073520.09710.07410.10170.1349B20.24270.33790.33570.214020.11840.23070.11040.24810.3269B30.57630.76620.77230.55160.24240.53940.21540.56990.72308C1.50.08430.1190.12170.09420.03420.09740.02390.10310.0222C20.13540.24780.25430.17230.04020.23330.011140.24750.0092420A1.50.24960.19750.09090.25430.21790.12260.252080.12250.3235A20.66790.56720.26540.6370.56160.34860.62410.33580.7538B1.50.14490.28280.27970.20470.08690.16340.07810.17860.1774B20.386020.72230.72220.55110.18490.46470.1620.50670.52802B30.81040.99230.99310.9540.45880.87110.46150.89780.93998C1.50.09410.19790.21010.15160.05090.16210.024060.17910.0213C20.15090.45510.48330.32960.11620.46330.04620.50480.033830A1.50.40450.32730.11580.36860.31440.180020.3660.17210.4498A20.88560.80610.38540.82850.75220.54470.81050.50710.8973B1.50.17070.44660.44810.33310.10210.24770.08730.26670.2281B20.47940.91480.91730.79850.27060.66950.251080.70760.7002B30.90070.999940.999980.99830.67010.975060.72370.98190.99104C1.50.10210.27590.29380.20840.070.24920.03030.26780.0271C20.16270.61230.65130.47360.20770.67110.12580.71110.1105Consequently, based on alternatives with a change toward a smaller variance, the testsCREw and CREq, q⟶1, are the best. Meanwhile, under alternatives with a change toward a larger variance, the tests CREw and CREq, q⟶0, are weaker.


## 5. Conclusion

For the CDFs with support0,1, we exhibited that the values of CREq and CREw are within 0,e−q and 0,1/2e, respectively. Moreover, the test of uniformity was proposed by calculating the percentage points and power analysis of CREq and CREw. Besides, for CREq, we obtained the percentage points by using the Monte Carlo method via the simulation and the normality asymptotic, as well as the beta approximation. Moreover, for CREw the percentage points were derived by using the Monte Carlo method via the simulation. A power comparison was performed between the FCRE and WCRE and other tests, where, by changing the value of q, we indicated when the test has higher and lower power compared with the other tests.



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