Anderson-Darling goodness-of-fit test

This method deals with the modelling of a probability distribution of a random vector \vect{X} = \left( X^1,\ldots,X^{n_X} \right). It seeks to verify the compatibility between a sample of data \left\{ \vect{x}_1,\vect{x}_2,\ldots,\vect{x}_N \right\} and a candidate probability distribution previous chosen. The Anderson-Darling Goodness-of-Fit test allows to answer this question in the one dimensional case n_X =1, and with a continuous distribution. The current version is limited to the case of the Normal distribution.

Let us limit the case to n_X = 1. Thus we denote \vect{X} = X^1 = X. This goodness-of-fit test is based on the distance between the cumulative distribution function \widehat{F}_N of the sample \left\{ x_1,x_2,\ldots,x_N \right\} and that of the candidate distribution, denoted F. This distance is a quadratic type, as in the Cramer-Von Mises test, but gives more weight to deviations of extreme values:

\begin{aligned} D = \int^{\infty}_{-\infty} \frac{\displaystyle \left[F\left(x\right) - \widehat{F}_N\left(x\right)\right]^2 }{\displaystyle F(x) \left( 1-F(x) \right) } \, dF(x) \end{aligned}

With a sample \left\{ x_1,x_2,\ldots,x_N \right\}, the distance is estimated by:

\begin{aligned} \widehat{D}_N = -N-\sum^{N}_{i=1} \frac{2i-1}{N} \left[\ln F(x_{(i)})+\ln\left(1-F(x_{(N+1-i)})\right)\right] \end{aligned}

where \left\{x_{(1)}, \ldots, x_{(N)}\right\} describes the sample placed in increasing order.

The probability distribution of the distance \widehat{D}_N is asymptotically known (i.e. as the size of the sample tends to infinity). If N is sufficiently large, this means that for a probability \alpha and a candidate distribution type, one can calculate the threshold / critical value d_\alpha such that:

  • if \widehat{D}_N>d_{\alpha}, we reject the candidate distribution with a risk of error \alpha,

  • if \widehat{D}_N \leq d_{\alpha}, the candidate distribution is considered acceptable.

Note that d_\alpha depends on the candidate distribution F being tested; the current version is limited to the case of the Normal distribution.

An important notion is the so-called “p-value” of the test. This quantity is equal to the limit error probability \alpha_\textrm{lim} under which the candidate distribution is rejected. Thus, the candidate distribution will be accepted if and only if \alpha_\textrm{lim} is greater than the value \alpha desired by the user. Note that the higher \alpha_\textrm{lim} - \alpha, the more robust the decision.

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