Fractional Entropy-Based Test of Uniformity with Power Comparisons

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.


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) F(x) � P(X < x), the CRE is defined by where F(x) � 1 − F(x) 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 U(0, 1), Rao et al. [1] showed that the value of the CRE is 1/4. e literature abounds with many different results for Shannon's entropy and its modifications. Interested readers may refer to [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][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) e measure CRE q (F) is a nonadditive and nonnegative. Moreover, it is a convex function of the parameter q, CRE 0 (F) � E(X), and CRE 1 (F) � CRE(F). Xiong et al. [16] derived the FCRE for some well-known distributions; for example, FCRE of the CDF U(0, 1) is Γ(q + 1)/2 q+1 .
Misagh et al. [15] proposed a weighted form of CRE, which is shift-dependent.
is information-theoretic uncertainty measure is called the weighted cumulative residual entropy (WCRE), and it is defined by 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 U(0, 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. is fact gives a satisfactory motivation of our study. roughout 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. e rest of this work is systematic as follows. In Section 2, 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. en, 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 (⟶

Theoretical Aspects and Test Statistic
To establish our test of the null hypothesis H 0 , we need the following theorem, which shows that, for a CDF with support [0, 1], one always has 0 ≤ CRE q (F) ≤ e − q , and for the distribution U(0, 1), we have FCRE � Γ(q + 1)/2 q+1 , and this value is uniquely attained by the uniform distribution, whenever q is fixed. Theorem 1. Let X be a nonnegative RV with an absolutely continuous CDF F with a support [0, 1]. From (2), it holds 0 ≤ CRE q (F) ≤ e − q , and CRE q (F) � Γ(q + 1)/2 q+1 is uniquely acquired by the distribution U(0, 1).
On the other hand, using the strict convexity of f(x) � x(− ln x) q , it is easy to see that FCRE is a concave function of distribution (with support [0, 1]). is shows that CRE q (F) � Γ(q + 1)/2 q+1 is uniquely acquired by the distribution U(0, 1). is completes the proof.
Let X 1 , X 2 , . . . , X n be a random sample with a continuous CDF F, with support [0, 1]. Furthermore, let X (1) ≤ X (2) ≤ · · · ≤ X (n) be the corresponding order statistics X 1 , X 2 , . . . , X n . According to (2), we can obtain the empirical FCRE as an estimator of FCRE(F) by where F n (x) � 1 − F n (x) and F n (x) is the empirical CDF, which is defined by To perform a consistent test of the hypothesis of uniformity, we suggest the consistent statistic test where Moreover, under the null hypothesis H 0 , we get R q n ⟶ p n Γ (q + 1)/2 q+1 . On the other hand, under the alternative hypothesis (that F is any continuous CDF with support [0, where r is a smaller or larger number than e test based on the sample estimate R q n is consistent.
Proof. From Glivenko-Cantelli theorem (see Tucker [19] On the other hand, eorem 3 in Xiong et al. [16] asserts that CRE q (F n ) ⟶ a.s. n CRE q (F), which proves the theorem. □ Theorem 3. Suppose that the random sample X 1 , X 2 , . . . , X n has been drawn from an unknown continuous CDF F defined on [0, 1]. en, from (6), Proof. Since the function f(p) � p(− ln p) q , 0 < p < 1, has a maximum value at e − q , 0 ≤ q ≤ 1; therefore, 2 Journal of Mathematics is completes the proof of the theorem. □ Theorem 4. Under H 0 , from (6), the mean and the variance of R q n are, respectively, Proof. e proof directly follows by noting that, for any e critical region, which describes the test procedure, is given by the following two inequalities: where α is the desired level of significance, and CRE * q α is the α− quantile of the asymptotic, or approximated, CDF of the test statistic CRE q (F n ), under H 0 . In the next section, we derive the asymptotic and approximated CDF of the test statistic CRE q (F n ). ese quantiles are computed by using the Monte Carlo method.

Percentage Points of the Test Statistic
In this section, we obtain the asymptotic distribution of R q n n). us, we can see that T i 's have the following probability density function (PDF): e mean and variance of T i are, respectively, According to Lyapunov central limit theorem (see where N is the standard normal RV (in the sequel, the standard normal distribution will be denoted by N(0, 1)). erefore, under H 0 , the percentage point (α− quantile) CRE * q α is approximated according to the asymptotic normality of R q n for large n by where Z α corresponds to the quantile (α × 100) of the CDF N(0, 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 R q n for finite n as follows: where the RV Y has Beta(a, b) distribution, and A i � (1 − (i/n))(− ln(1 − (i/n))) q , 0 ≤ q ≤ 1, i � 1, 2, . . . , n − 1. According to (14), the mean and variance of R q n are, respectively, Now, by using this approximation of R q n , the quantiles of order α/2 and 1 − (α/2) of the approximated CDF of the test statistic CRE q (F n ) under H 0 are, respectively, Journal of Mathematics where F − 1 (.) is the quantile function of the CDF F, F is the Beta(a, b) distribution, and a and b are defined in (15).

Empirical Weighted Cumulative Residual Entropy.
From (3), Misagh et al. [15] proposed the empirical WCRE by where A We suggest the following statistic of a consistent test based on (18): Theorem 5. e test based on the sample estimate T w n is consistent.

Proof.
Since the function f(p) � − p ln p, 0 < p < 1, has a maximum value at 1/e; therefore, is completes the proof.   U(0, 1), respectively. Consequently, for R q n , we present the percentage points of the Monte Carlo method, asymptotic normality, and beta approximation by using (10), (13), and (17), respectively. e 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 R q n , the accuracy of the Monte Carlo method is more than the other two methods. Figures 1-4 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.

Power Analysis
In this section, we study the power test of Monte Carlo study under alternative distributions. e power of R q n is estimated by the proportion of the generated samples falling into the critical region. Under seven alternative distributions, the power of the test statistic R q n is calculated by the Monte Carlo study of generating 50,000 samples each of size n, where n � 20, 30, 50. e alternative CDFs proposed by Stephens [18] in power study of uniformity tests are as follows: In Table 2, based on the Monte Carlo study, we recorded the power values of the proposed test statistics R q n , T w n , Kolmogorov-Smirnov (K-S), Kuiper (V), Cramer-von Mises (W 2 ), Watson (U 2 ), and Anderson-Darling (A 2 ), for n � 10, 20, 30 and α � 0.05. From Table 2, we can conclude the following: (1) If q increases and tends to 1 (q ⟶ 1), the power of CRE q test, for alternative A l (B l )(C l ), decreases (increases) (increases), and vice versa, if q decreases and tends to 0 (q ⟶ 0). (2) If q ⟶ 1, the CRE q test, for alternative A l (B l ), gives the worst (best) performance compared with the other tests. (3) To compare the performance between CRE q and CRE w tests, we observe that: (a) For the alternative A l , q ⟶ 1, CRE w performs better than CRE q and vice versa if q ⟶ 0, n increases. (b) For the alternative B l , q ⟶ 1, CRE q performs better than CRE w , and vice versa, if q ⟶ 0, n increases. (c) For the alternative C l , q ⟶ 0, CRE w performs better than CRE q , and vice versa, if q ⟶ 1.
Stephens [18] noted that V and U 2 tests will reveal a change at variance. erefore, we observe the following: (1) For alternative A l , q ⟶ 0, CRE q performs better than V and U 2 , and vice versa, if q ⟶ 1.

Journal of Mathematics 5
(2) For the alternative B l , q ⟶ 1, CRE q performs better than V and U 2 , and vice versa, if q ⟶ 0, n increases. (3) For the alternative C l , q ⟶ 0, V and U 2 performs better than CRE q . (4) CRE w performs better than V and U 2 against the alternative A l . (5) CRE w performs better than V and U 2 against the alternative B l , n increases. But, V and U 2 perform better than CRE w against the alternative C l .
Consequently, based on alternatives with a change toward a smaller variance, the tests CRE w and CRE q , q ⟶ 1, are the best. Meanwhile, under alternatives with a change toward a larger variance, the tests CRE w and CRE q , q ⟶ 0, are weaker.

Conclusion
For the CDFs with support [0, 1], we exhibited that the values of CRE q and CRE w 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 CRE q and CRE w . Besides, for CRE q , 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 CRE w 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.

Data Availability
e simulated data used to support the findings of this study are included within the article.