Anderson-Darling test


Anderson-Darling test

The Anderson-Darling test, named after Theodore Wilbur Anderson, Jr. (1918–?) and Donald A. Darling (1915–?), 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" ≤ 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 {Y_1 (note that the data must be put in order) comes from a distribution with cumulative distribution function (CDF) F is

: A^2 = -N-S

where

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

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 ar{X} and standard deviation s are calculated from the sample of X.

3) The values X_i are standardized as

::Y_i=frac{X_i-ar{X{s}

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

or without repeating indices as

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

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

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

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

Note:

1. If "s" = 0 or any Phi(Y_i)=(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

External links

* [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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