Abstract
An omnibus test for normality with an adjustment for symmetric alternatives is developed using the empirical likelihood ratio technique. We first transform the raw data via a jackknife transformation technique by deleting one observation at a time. The probability integral transformation was then applied on the transformed data, and under the null hypothesis, the transformed data have a limiting uniform distribution, reducing testing for normality to testing for uniformity. Employing the empirical likelihood technique, we show that the test statistic has a chi-square limiting distribution. We also demonstrated that, under the established symmetric settings, the CUSUM-type and Shiryaev–Roberts test statistics gave comparable properties and power. The proposed test has good control of type I error. Monte Carlo simulations revealed that the proposed test outperformed studied classical existing tests under symmetric short-tailed alternatives. Findings from a real data study further revealed the robustness and applicability of the proposed test in practice.
Highlights
Test DevelopmentConsider an unknown continuous distribution with nonordered random variables denoted by X1, X2, . . . , Xn that are assumed to be independent and identically distributed (i.i.d.). e intention is to test whether the observations are consistent with a normal distribution. us, we intended to test whether to accept or reject the following null hypothesis: H0: X1, X2, . . . , Xn ∼ Nμ, σ2,
From the various proposed empirical likelihood (EL) testing procedures as well as in the current statistical practice, it is evident that the problem of testing composite hypotheses of normality is undeniably the most common research focus in GoF testing. e continued growing need for normality tests is attributed to the frequent use and applications of normally distributed data in various areas of pure and applied statistical practices
To test for normality, Dong and Giles [11] proposed an omnibus test statistic by directly utilizing the EL methodology outlined by Owen [17]. ey utilized the first four moment constraints that characterize the normal distribution
Summary
Consider an unknown continuous distribution with nonordered random variables denoted by X1, X2, . . . , Xn that are assumed to be independent and identically distributed (i.i.d.). e intention is to test whether the observations are consistent with a normal distribution. us, we intended to test whether to accept or reject the following null hypothesis: H0: X1, X2, . . . , Xn ∼ Nμ, σ2,. Zn become asymptotically independent while under the null hypothesis they are distributed according to a t distribution with n − 2 df, which as n grows approaches the standard normal. In addition to this transformation, we further adopted the probability integral transformation (see [25] as well as [8]). Yn. at is, under the null hypothesis, the transformed data follow the uniform distribution asymptotically. From the uniformly transformed observations, we proposed to test for the following null hypothesis: H0: Y1, Y2, . Considering the null and alternative hypotheses, the test statistic is given by (−2LLR)k. Bold represents the most superior CS-type statistic and italicized represents the most superior SR statistic
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