Anderson-Darling Test for Normality

Performs the Anderson-Darling test for normality by computing the \(A^2\) test statistic:

$$A^2 = n\int_{-\infty}^{\infty}\frac{(F_n(x)-F(x))^2F'(x)}{F(x)(1-F(x))}dx$$

The Anderson-Darling test evaluates whether a sample comes from a normal distribution by computing an integral over the empirical cumulative distribution function (ECDF) and comparing it against the normal distribution's CDF.

Interpretation:

• When \(A^2/n\) approaches zero as sample size increases, the normality hypothesis is supported

• When \(A^2/n\) converges to a positive finite value, the normality hypothesis lacks support

Important: The input data vector x must be sorted in ascending order. Unsorted data will trigger an error.

Usage

anderson_darling_normality_statistic(x, mu = 0, sd = 1)

Arguments

x

A numeric vector of sample data (must be sorted in ascending order).

mu

The mean of the normal distribution to test against. Default is 0.

sd

The standard deviation of the normal distribution to test against. Default is 1.

Value

The Anderson-Darling \(A^2\) test statistic.

See also

Boost Documentation for more details on the mathematical background.

Examples

# Anderson-Darling test for normality with sorted data
x <- sort(rnorm(100))
anderson_darling_normality_statistic(x, 0, 1)
#> [1] 1.071972