Augmenting Shapiro-Wilk Test for Normality

Abstract

The expected value of the i-th order statistic is considered to be γ 0 + γ 1 c i + γ 2(c 2 i − λ) + γ 3(c 3 i − μ c i ) + ..., where c i is the expected value of the i-th order statistic from N(0, 1). If the underlying distribution is normal, then γ 1 = σ, γ 2 = 0, γ 3 = 0, .... Shapiro-Wilk test is designed to test the hypothesis γ 1 = σ. The possibility of testing the multiple hypothesis γ 1 = σ, γ 2 = 0, γ 3 = 0, … , is considered in the present paper. Some improvement in efficiency over the Shapiro-Wilk test is demonstrated.