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Many statistical procedures are based on the assumption of normality. Classical tests such as Jarque-Bera (JB), Anderson-Darling (AD), Kolmogorov-Smirnov (KS), Shapiro-Wilk (SW), and D’Agostino-Pearson K2 (DP) are widely used, but in small samples are based on asymptotic approximations and can be unbalanced in terms of sensitivity to skewness versus tail departures. The paper presents a Modified Skewness-Kurtosis (MSK) test which unites non-linear weighted skewness and kurtosis calculation with bootstrap techniques for critical value determination. When parameters α=0.5,β=1,C1=1.5 along with C2=2.5 are used in the MSK test it upholds the 5% Type I error rate and boosts power for detecting non-normal distribution patterns in both heavy-tailed and skewed alternative distributions. The MSK test achieves notable success compared to the Jarque-Bera test in simulation studies that generated 20,000 Monte Carlo replications alongside 1,000 bootstrap samples per replication. t demonstrates comparable performance to Shapiro–Wilk, Anderson–Darling, Kolmogorov–Smirnov, and D’Agostino–Pearson K2 tests. The evaluations performed under diverse distributions including N(0,1),t(3),t(5), lognormal, loglogistic, logistic, Gamma, Beta, Weibull, Gompertz, Uniform, Exponential, Cauchy and χ2. We have determined that the MSK test functions as an established tool that provides reliable results for checking normality distribution. This study utilizes modern advancements in machine learning methods along with kernel-based methods to offer supporting background for developing new testing procedures.
Bokhari et al. (Wed,) studied this question.