Vedlegg 7 - Anderson-Darling test for normalitet

Jfr. Anderson and Darling (1954), Stephens (1979), Jäntschi and Bolboacă (2018) og Zaiontz (2020)

Fordeling	AD.verdi
Generisk/uniform	\[A=-n-\frac{1}{n}\sum_{i=1}^{n}(2_i-1)[lnF(X_i)+ln(1-F(X_{n-i+1}))]\]
Normal/lognormal	\[A^2=A(1+\frac{75}{n}+\frac{2.25}{n^2})\]
Gamma \(k=1\)	\[A(1+\frac{6}{n})\]
Gamma \(k\ge1\)	\[A+(\frac{0.2+\frac{0.3}{k}}{n})\]
Eksponensiell	\[A(1+\frac{6}{\sqrt{n}})\]
Weilbull/Gumbel	\[A(1+\frac{0.2}{\sqrt{n}})\]
Logistisk	\[A(1+\frac{0.25}{\sqrt{n}})\]

For gammafordeling er k parameteret (“shape parameter”) definert som \(\alpha\).

Kritiske verdier for Anderson-Darling test. Merk: hvis man søker på nett finner man ulike framstillinger med noen avvikende verdier – vi har basert verdiene for normalfordeling på D’Agostino and Stephens (1986, 123, tabell 4.7). Øvrige verdier er basert på Marsaglia and Marsaglia (2004) og Stephens (1974, 1976, 1977, 1978, 1979).

Du kan laste ned tabellen i en Excel-fil her: Download AD-alpha.xlsx

Tabell over \(\alpha\) verdier

Anderson, T. W., and D. A. Darling. 1954. “A Test of Goodness of Fit.” Journal of the American Statistical Association 49 (268): 765. https://doi.org/10.2307/2281537.

D’Agostino, R. B., and M. A. Stephens, eds. 1986. Goodness-of Fit Techniques. Marcel Dekker.

Jäntschi, Lorentz, and Sorana D. Bolboacă. 2018. “Computation of Probability Associated with Anderson Statistic.” Mathematics 6 (6): 88. https://doi.org/10.3390/math6060088.

Marsaglia, George, and John Marsaglia. 2004. “Evaluating the Anderson-Darling Distribution.” Journal of Statistical Software 9 (2). https://doi.org/10.18637/jss.v009.i02.

Stephens, M. A. 1974. “EDF Statistics for Goodness of Fit and Some Comparisons.” Journal of the American Statistical Association 69 (347): 730–37. https://doi.org/10.1080/01621459.1974.10480196.

———. 1976. “Asymptotic Results for Goodness-of-Fit Statistics with Unknown Parameters.” The Annals of Statistics 4 (2): 357–69.

———. 1977. “Goodness of Fit for the Extreme Value Distribution.” Biometrika 64 (3): 583–88. https://doi.org/10.1093/biomet/64.3.583.

———. 1978. “Goodness of Fit with Special Reference to Tests for Exponentiality.” Technical Report No. 262.

———. 1979. “The Anderson-Darling Statistic.” In Technical Report No. 39. Stanford: Stanford University.

———. 1979. “Tests of Fit for the Logistic Distribution Based on the Empirical Distribution Function.” Biometrika 66 (3): 591–95. https://doi.org/10.1093/biomet/66.3.591.

Zaiontz, Charles. 2020. “One-Sample Anderson-Darling Test.” Real Statistics Using Excel.