A Comprehensive Study on Power of Tests for Normality

“JSTA-17-4-7_print” — 2018/12/21 — 12:53 — page 647 — #1 Journal of Statistical Theory and Applications Vol. 17(4), December 2018, pp. 647–660 DOI: 10.2991/jsta.2018.17.4.7; ISSN 1538-7887 https://www.atlantis-press.com/journals/jsta A Comprehensive Study on Power of Tests for Normality Hadi Alizadeh Noughabi1,* 1 Department of Statistics, University of Birjand, Birjand, Iran ARTICLE INFO ABSTRACT Article History Many statistical procedures assume that the underling distribution is normal. In this paper, we consider the popular and powerful tests for normality and investigate the power values of these tests to detect deviations from normality. The family of fourparameter generalized lambda distributions (FMKL) for its high flexibility is considered as alternative distributions. We then compare the power values of normality tests against these alternatives and for different sample sizes. The considered tests are Kolmogorov-Smirnov, Anderson-Darling, Kuiper, Jarque-Bera, Cramer von Mises, Shapiro-Wilk and Vasicek. These tests are popular tests which are commonly used in practice and statistical software. The tests are described and then power values of the tests are compared against FMKL family by Monte Carlo simulation. The results are discussed and interpreted. Finally, we apply some real data examples to show the behavior of the tests in practice. Received 14 Oct 2017 Accepted 17 May 2018 Keywords Test of normality Monte Carlo simulation Power of test The generalized lambda distribution 2000 Mathematics Subject Classification 62G10, 62P20, 62P30 © 2018 The Authors. Published by Atlantis Press SARL. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/). 1. INTRODUCTION Ramberg and Schmeiser [1] introduced the four-parameter generalized lambda distribution (GLD) as Q (u) = 𝜆1 + u𝜆3 − (1 − u)𝜆4 , 𝜆2 where Q(u) is the quantile function, 0 ≤ u ≤, 𝜆1 , 𝜆2 are the location and scale parameters, and 𝜆3 , 𝜆4 are shape parameters jointly related to the strengths of the lower and upper tails, respectively. For its high flexibility it is used in many fields such as modeling financial data. Because of the limitations of the Ramberg and Schmeiser (RS) parameterizations, Freimer et al. [2] proposed a new parameterization called FMKL as Q (u) = 𝜆1 + 1 u𝜆3 − 1 (1 − u)𝜆4 − 1 − , ) 𝜆2 ( 𝜆3 𝜆4 where 0 ≤ u ≤ 1, 𝜆1 , 𝜆2 are the location and scale parameters, respectively. Also 𝜆3 and 𝜆4 determine the shape characteristics and for a symmetric distribution 𝜆3 = 𝜆4 . The five different shapes of the FMKL are: unimodal, U-shaped, J-shaped, S-shaped, and monotone, which may be symmetric and asymmetric with smooth, abrupt, truncated, long, medium or short tails. In many situations, a goodness of fit test about the distribution of the population using observations is necessary. Since the normal distribution is widely used in many statistical procedures and also is the most fundamental distribution, test for the normal hypothesis is indispensable. Moreover, testing normality is one of the most areas of statistical research. For example in statistical modeling the normal assumption of the underlying error distribution must be checked. Therefore, many tests for normality are proposed by authors. A fair of normality tests can be found in the statistical literature. In many situations, a goodness of fit test about the distribution of the population is necessary. Since the normal distribution is the most basic distribution and use widely in statistics, test for the normal hypothesis has been studied by many authors. See for example, D’Agostino and Stephens [3], Huber-Carol et al. [4], Thode [5]. Recently, testing normality has been considered by Alizadeh Noughabi and Arghami [6], Harri and Coble [7], Sanqui et al. [8], Zamanzade and Arghami [9], Marmolejo-Ramos and González-Burgos [10], Joenssen and Vogel [11] and Wang [12]. *Corresponding author. Email: Pdf_Folio:61 “JSTA-17-4-7_print” — 2018/12/21 — 12:53 — page 648 — #2 648 Hadi Alizadeh Noughabi / Journal of Statistical Theory and Applications 17(4) 647–660 In this article, we consider seven popular (like Kolmogorov-Smirnov, Anderson-Darling) and powerfulness (like Shapiro-Wilk, Vasicek) normality tests and compare power values of these tests against the GLD (FMKL) with different parameters. We show that no single test procedure is uniformly more powerful than others. However, the powerful tests can be determined based on type of alternatives. Thus, tests for normality based on type of alternatives are classified. The methodologies of the considered tests are presented in Section 2. Power values of the tests are compared with each other against the FMKL family by Monte Carlo simulation in Section 3. In Section 4, the applicability of the tests in practice is shown by real data. Finally, some conclusions are given in Section 5. 2. TESTS FOR NORMALITY Given a random sample X1 , ..., Xn from a continuous probability distribution F with a density f(x), over the real line and with mean 𝜇 and variance 𝜎2 < ∞, the hypothesis of interest is x−𝜇 2 1 , for some (𝜇, 𝜎) ∈ Θ , x ∈ ℝ H0 ∶ f (x) = f0 (x; 𝜇, 𝜎) = exp {− 12 ( 𝜎 )} √2𝜋𝜎 where 𝜇 and 𝜎 are unspecified and Θ = ℝ × ℝ+ . The alternative to H0 is H1 ∶ f (x) ≠ f0 (x; 𝜇, 𝜎) for any (𝜇, 𝜎) ∈ Θ . In this section, we consider seven popular tests for the above hypothesis. The considered tests are Cramer von Mises [13], KolmogorovSmirnov [14], Anderson-Darling [15], Kuiper [16], Shapiro-Wilk [17], Vasicek [18] and Jarque-Bera [19]. These tests are commonly used in practice and software. For example, the Shapiro-Wilks test is used in SAS software for testing normality. The description of each normality tests is presented in Table 1. From the aforementioned tests, Vasicek’s test, Shapiro-Wilk and Jarque-Bera test are specific in the sense that the null hypothesis is normal, while the rest are suitable for any null family of distributions. For further study about this tests, see D’Agostino and Stephens [3] and references there in. 3. SIMULATION STUDY In this section, Type I error of the tests are obtained and then power values of the tests against flexible FMKL family are computed through a Monte Carlo simulation. Table 1 Tests of normality. Test of normality Test statistic Notations n 1- Cramer von Mises 2- Kolmogorov-Smirnov CH = D+ = max 2 1 2i − 1 + − Zi ; i = 1, ..., n. ( 2n ) 12n ∑ i=1 i i−1 − Zi , D− = max Zi − , i = 1, ..., n {n } { n } D = max(D+ , D− ) V = D+ + D− 3- Kuiper X(i) − X : where Φ is the ( SX ) cdf of standard normal distribution. X(i) − X Zi = Φ : where Φ is the ( SX ) cdf of standard normal distribution. D+ and D− are as above. Zi = Φ [n/2] (∑ i=1 W= 4- Shapiro-Wilk a(n−i+1) (X(n−i+1) − X(i) )2 ) n 2 (X(i) − X) ∑ The coefficients ai are tabulated in Pearson and Hartley [20] i−1 n (2i − 1) {ln(Zi ) + ln(1 − Zn−i+1 )} ∑ 5- Anderson-Darling 6- Vasicek i=1 2 A = −n − KLmn = n exp(H(n, (...truncated)