Modified Lilliefors goodness-of-fit test for normality

Abstract

The first aim of the paper is to put into practice the $\left(\alpha , \beta \right)$-corrected Lilliefors goodness-of-fit test for normality (LF). This modification consists in varying a formula of calculating the empirical distribution function (EDF). Values of constants $\alpha , \beta$ in the formula depend on values of sample skewness and excess kurtosis, which is recommended in order to increase the power of the LF test. Critical values are obtained with the Monte Carlo method for sample sizes $n=10, 14, 20$ and at a significance level $\alpha =0.05.$ The power of several normality tests for a wide collection of alternative distributions is calculated. Alternative distributions are divided into 12 groups according to their skewness and excess kurtosis. The second aim is to propose a similarity measure between the normal distribution and an alternative distribution. The third aim is to propose two new alternative distributions created in order to obtain the desired values of skewness and excess kurtosis. The fourth aim is to calculate the power of tests for new alternative distributions. The paper shows that values of constants $\alpha , \beta$ in the formula of calculating the EDF influences the power of the LF test. The performance of the new proposal through the analysis of real data sets is illustrated.

Keywords:

• Empirical distribution function
• Goodness-of-fit test
• Lilliefors test
• Power of test

Mathematics Subject Classification:

• 62G10
• 62G30

Acknowledgments

The author is grateful to the unknown referees for their valuable comments that contributed to the improvement of the original version of the paper.

Notes

1 Note: in mathematical environments such as Mathcad, SciLab and R is 0⁰ = 1.