The Anderson-Darling test (Anderson and Darling, 1952) is used to test if a sample of data comes from a specific distribution. It is a modification of the Kolmogorov-Smirnov (K-S) test and gives more weight to the tails than the K-S test. The K-S test is distribution free in the sense that the critical values do not depend on the specific distribution being tested.
The Anderson-Darling test makes use of the specific distribution in calculating critical values. This has the advantage of allowing a more sensitive test and the disadvantage that critical values must be calculated for each distribution.
The Anderson-Darling test is only available for a few specific distributions. The test is calculated as:
AD2 = integral{[F_o(x)-F_t(x)]^2/[F_t(x)(1-F_t(x)0]}dF_t(x)
AD2a = AD2*a
Note that for a given distribution, the Anderson-Darling statistic may be multiplied by a constant, a (which usually depends on the sample size, n). These constants are given in the various papers by Stephens (1974, 1977a, 1977b, 1979, 1986). This is what should be compared against the critical values. Also, be aware that different constants (and therefore critical values) have been published. You just need to be aware of what constant was used for a given set of critical values (the needed constant is typically
given with the critical values).
The critical values for the Anderson-Darling test are dependent on the specific distribution that is being tested. Tabulated values and formulas have been published for a few specific distributions (normal, lognormal, exponential, Weibull, logistic, extreme value type 1). The test is a one-sided test and the hypothesis that the distribution is of a specific form is rejected if the test statistic, AD2a, is greater than the critical value.
Here, we develop the m-file for detecting departure from normality. It is one of the most powerful statistics for test this.
Input:
x - data vector
alpha - significance level (default = 0.05)
Output:
- Complete Anderson-Darling normality test |