An omnibus test for departures from normality is an idea developed by E.S. Pearson (letter to Bowman); he thought a test including skewness b 1 and kurtosis b 2, both of which scale and location free, would give more information than the test using lower moments only. For the normal distribution, the population skewness is \(\sqrt{{ \beta }_{1}} = 0\) and the population kurtosis is β2 = 3.
D’Agostino (1973) introduced a goodness-of-fit test for departures from normality using sample skewness \(\sqrt{{b}_{1}}\) and sample kurtosis b 2, thus
where \(\sqrt{{b}_{1}} = {m}_{3}/{m}_{2}^{3/2}\), b 2 = m 4 ∕ m 2 2, and \({m}_{s} ={ \sum \nolimits }_{1}^{n}{({x}_{j} -\bar{ x})}^{s}/n,j = 1,2,\cdots \,,n\), n sample size. D’Agostino and Pearson considered the Johnson’s (1965) S U and S B transformed distribution for \(\sqrt{{b}_{1}}\) and b 2. Johnson’s system of distributions has the advantage that it...
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Bowman KO, Shenton LR (1975a) Tables of moments of the skewness and kurtosis statistics in non-normal sampling. Union Carbides Nuclear Division report UCCND-CSD-8
Bowman KO, Shenton LR (1975b) Omnibus test contours for departures from normality based on \(\sqrt{\mathrm{b} 1}\) and b2. Biometrika 62: 243–250
Bowman KO, Shenton LR (1986) Moment (\(\sqrt{\mathrm{b} 1}\); b2) techniques, Goodness-of-fit techniques, Chapter 7. Marcel Dekker, New York
D’Agostino RB, Pearson ES (1973) Tests for departure from normality. Empirical results for the distributions of b2 and \(\sqrt{\mathrm{b} 1}\). Biometrika 60(3):613–622
Johnson NL (1965) Tables to facilitate fitting SU frequency curves. Biometrika 52:547–548
Patrinos AAN, Bowman KO (1980) Weather modification from cooling tower: a test based on the distributional properties of rainfall. J Appl Meteorol 19(3):290–297
Pearson ES, D’Agostine RB, Bowman KO (1977) Test of departure from normality: comparison of powers. Biometrika 64:231–246
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Bowman, K.O., Shenton, L.R. (2011). Omnibus Test for Departures from Normality. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_426
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