# Anderson-Darling test

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Anderson-Darling test

The Anderson-Darling test, named after Theodore Wilbur Anderson, Jr. (1918&ndash;?) and Donald A. Darling (1915&ndash;?), who invented it in 1952 [cite journal | first = T. W. | last = Anderson | author link = Theodore W. Anderson, Jr.
coauthors = Darling, D. A.
year = 1952 | month =
title = Asymptotic theory of certain "goodness-of-fit" criteria based on stochastic processes
journal = Annals of Mathematical Statistics
volume = 23 | issue = | pages = 193–212
id = | url =
doi = 10.1214/aoms/1177729437
] , is a form of minimum distance estimation, and one of the most powerful statistics for detecting most departures from normality. It may be used with small sample sizes "n" &le; 25. Very large sample sizes may reject the assumption of normality with only slight imperfections, but industrial data with sample sizes of 200 and more have passed the Anderson-Darling test. Fact|date=February 2007

The Anderson-Darling test assesses whether a sample comes from a specified distribution. The formula for the test statistic $A$ to assess if data

: $A^2 = -N-S$

where

: $S=sum_\left\{k=1\right\}^N frac\left\{2k-1\right\}\left\{N\right\}left \left[ln F\left(Y_k\right) + lnleft\left(1-F\left(Y_\left\{N+1-k\right\}\right) ight\right) ight\right] .$

The test statistic can then be compared against the critical values of the theoretical distribution (dependent on which $F$ is used) to determine the P-value.

The Anderson-Darling test for normality is a distance or empirical distribution function (EDF) test. It is based upon the concept that when given a hypothesized underlying distribution, the data can be transformed to a uniform distribution. The transformed sample data can be then tested for uniformity with a distance test (Shapiro 1980).

In comparisons of power, Stephens (1974) found $A^2$ to be one of the best EDF statistics for detecting most departures from normality. [cite journal
first = M. A. | last = Stephens | authorlink = | coauthors =
year = 1974 | month =
title = EDF Statistics for Goodness of Fit and Some Comparisons
journal = Journal of the American Statistical Association
volume = 69 | issue = | pages = 730–737 | id = | url =
doi = 10.2307/2286009
] The only statistic close was the $W^2$ Cramér-von Mises test statistic.

Procedure

(If testing for normal distribution of the variable "X")

1) The data $X_i$, for $i=1,ldots n$, of the variable $X$ that should be tested is sorted from low to high.

2) The mean and standard deviation $s$ are calculated from the sample of $X$.

3) The values $X_i$ are standardized as

::

4) With the standard normal CDF $Phi$, $A^2$ is calculated using ::$A^2 = -n -frac\left\{1\right\}\left\{n\right\} sum_\left\{i=1\right\}^n \left(2i-1\right)\left(ln Phi\left(Y_i\right)+ ln\left(1-Phi\left(Y_\left\{n+1-i\right\}\right)\right)\right)$

or without repeating indices as

::$A^2 = -n -frac\left\{1\right\}\left\{n\right\} sum_\left\{i=1\right\}^nleft \left[\left(2i-1\right)lnPhi\left(Y_i\right)+\left(2\left(n-i\right)+1\right)ln\left(1-Phi\left(Y_i\right)\right) ight\right] .$

5) $A^\left\{*2\right\}$, an approximate adjustment for sample size, is calculated using

::$A^\left\{*2\right\}=A^2left\left(1+frac\left\{0.75\right\}\left\{n\right\}+frac\left\{2.25\right\}\left\{n^2\right\} ight\right)$

6) If $A^\left\{*2\right\}$ exceeds 0.752 then the hypothesis of normality is rejected for a 5% level test.

Note:

1. If "s" = 0 or any $Phi\left(Y_i\right)=$(0 or 1) then $A^2$ cannot be calculated and is undefined.

2. Above, it was assumed that the variable $X_i$ was being tested for normal distribution. Any other theoretical distribution can be assumed by using its CDF. Each theoretical distribution has its own critical values, and some examples are: lognormal, exponential, Weibull, extreme value type I and logistic distribution.

3. Null hypothesis follows the true distribution (in this case, N(0, 1)).

ee also

*Kolmogorov-Smirnov test
*Shapiro-Wilk test
*Smirnov-Cramér-von-Mises test
*Jarque-Bera test

* [http://www.itl.nist.gov/div898/handbook/eda/section3/eda35e.htm US NIST Handbook of Statistics]
* [http://www.analyse-it.com/blog/2008/8/testing-the-assumption-of-normality.aspx Testing the assumption of normality] .

References

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