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.
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) with a cumulative distribution function (CDF) , the CRE is defined bywhere 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 , Rao et al. [1] showed that the value of the CRE is . 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 , the FCRE for the RV is defined by
The measure is a nonadditive and nonnegative. Moreover, it is a convex function of the parameter , , and . Xiong et al. [16] derived the FCRE for some well-known distributions; for example, FCRE of the CDF is .
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
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 is .
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 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. 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 (), () and () stand for convergence in probability, convergence in distribution, and almost surely, as .
2. Theoretical Aspects and Test Statistic
To establish our test of the null hypothesis , we need the following theorem, which shows that, for a CDF with support [0, 1], one always has , and for the distribution , we have , and this value is uniquely attained by the uniform distribution, whenever is fixed.
Theorem 1. Let be a nonnegative RV with an absolutely continuous CDF with a support . From (2), it holds , and is uniquely acquired by the distribution .
Proof. Since , and the function has a maximum at , , we get . On the other hand, using the strict convexity of , it is easy to see that FCRE is a concave function of distribution (with support [0, 1]). This shows that is uniquely acquired by the distribution . This completes the proof.
Let be a random sample with a continuous CDF , with support [0, 1]. Furthermore, let be the corresponding order statistics . According to (2), we can obtain the empirical FCRE as an estimator of bywhere and is the empirical CDF, which is defined bywhere is the indicator function, i.e., ; .
To perform a consistent test of the hypothesis of uniformity, we suggest the consistent statistic testwhere and , , .
Xiong et al. [16] proved that . Moreover, under the null hypothesis , we get . On the other hand, under the alternative hypothesis (that is any continuous CDF with support [0, 1], which is not the uniform), we have , where is a smaller or larger number than .
Theorem 2. The test based on the sample estimate is consistent.
Proof. From Glivenko–Cantelli theorem (see Tucker [19]), we have . On the other hand, Theorem 3 in Xiong et al. [16] asserts that which proves the theorem.
Theorem 3. Suppose that the random sample has been drawn from an unknown continuous CDF defined on . Then, from (6), we have .
Proof. Since the function , , has a maximum value at , ; therefore,This completes the proof of the theorem.
Theorem 4. Under , from (6), the mean and the variance of are, respectively,
Proof. The proof directly follows by noting that, for any , the RV , based on the CDF , has beta distribution, i.e., (cf. [20]). This completes the proof.
Remark 1. Under , from (6), (8), and (9), we have and , where is the FCRE of the CDF .
The critical region, which describes the test procedure, is given by the following two inequalities:where is the desired level of significance, and is the quantile of the asymptotic, or approximated, CDF of the test statistic , under . In the next section, we derive the asymptotic and approximated CDF of the test statistic . 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 of under . From (6), we can write , where , , and . Thus, we can see that ’s have the following probability density function (PDF):
The mean and variance of are, respectively,
According to Lyapunov central limit theorem (see Billingsley [21]), we have , where is the standard normal RV (in the sequel, the standard normal distribution will be denoted by ). Therefore, under , the percentage point (quantile) is approximated according to the asymptotic normality of for large bywhere corresponds to the quantile of the CDF .
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 . By adopting the same procedure of Noughabi [14], we can obtain an approximation of for finite as follows:where the RV has distribution,and , , . According to (14), the mean and variance of are, respectively,
Now, by using this approximation of , the quantiles of order and of the approximated CDF of the test statistic under are, respectively,where is the quantile function of the CDF , is the distribution, and and are defined in (15).
3.1. Empirical Weighted Cumulative Residual Entropy
From (3), Misagh et al. [15] proposed the empirical WCRE bywhere , , .
We suggest the following statistic of a consistent test based on (18):
Theorem 5. The test based on the sample estimate is consistent.
Proof. From Mirali et al. [12] and by using Glivenko–Cantelli theorem, (see Tucker [19]), we have which proves the theorem.
Theorem 6. Let be a random sample drawn from an unknown continuous CDF defined on . Then, from (18), we get .
Proof. Since the function , , has a maximum value at ; therefore,This completes the proof.
3.2. Percentage Points
We generate samples of size , where , from . Using (6), the test statistic is estimated by the empirical for each sample and the same for . Moreover, we can see that , , and , where and are the FCRE and WCRE of the CDF , respectively. Consequently, for , 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 increases. Besides, for , 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 . When 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 .
4. Power Analysis
In this section, we study the power test of Monte Carlo study under alternative distributions. The power of is estimated by the proportion of the generated samples falling into the critical region. Under seven alternative distributions, the power of the test statistic is calculated by the Monte Carlo study of generating 50,000 samples each of size , where . The 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 , , Kolmogorov–Smirnov (K-S), Kuiper (V), Cramer-von Mises (), Watson (), and Anderson-Darling (), for and . From Table 2, we can conclude the following:(1)If increases and tends to 1 (), the power of test, for alternative , decreases (increases) (increases), and vice versa, if decreases and tends to 0 .(2)If , the test, for alternative , gives the worst (best) performance compared with the other tests.(3)To compare the performance between and tests, we observe that:(a)For the alternative , performs better than and vice versa if increases.(b)For the alternative , performs better than , and vice versa, if increases.(c)For the alternative , performs better than , and vice versa, if . Stephens [18] noted that and tests will reveal a change at variance. Therefore, we observe the following:(1)For alternative , performs better than and , and vice versa, if .(2)For the alternative , performs better than and , and vice versa, if increases.(3)For the alternative , V and performs better than .(4) performs better than and against the alternative .(5) performs better than and against the alternative increases. But, and perform better than against the alternative .
Consequently, based on alternatives with a change toward a smaller variance, the tests and , , are the best. Meanwhile, under alternatives with a change toward a larger variance, the tests and , , are weaker.
5. Conclusion
For the CDFs with support , we exhibited that the values of and are within and , respectively. Moreover, the test of uniformity was proposed by calculating the percentage points and power analysis of and . Besides, for , 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 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 , we indicated when the test has higher and lower power compared with the other tests.
Data Availability
The simulated data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest concerning the publication of this article.