Anderson-Darling Goodness-of-Fit Test

The Anderson-Darling (A-D) test is a statistical method designed to evaluate whether a given sample of data follows a specified continuous probability distribution. It is especially sensitive to deviations in the tails of the distribution, making it effective for distributions with extreme values.

Test Procedure

The Anderson-Darling test statistic $A^2$ is calculated using the following formula:

$$A^2=-N-\sum _{i=1}^N\frac{(2i-1)}{N}[\ln (F(x_i))+\ln (1-F(x_{N+1-i}))]$$

where:

• $N$ is the number of observations in the dataset.
• $x_i$ represents the sorted observations in ascending order.
• $F(x_i)$ is the cumulative distribution function (CDF) of the hypothesized distribution evaluated at $x_i$.

Hypothesis Testing

The hypotheses for the Anderson-Darling test are defined as:

• Null hypothesis $H_0$: The data follows the specified probability distribution.
• Alternative hypothesis $H_a$: The data does not follow the specified probability distribution.

To determine the outcome, compare the calculated Anderson-Darling statistic to the critical value or p-value:

1. Critical value approach: If the test statistic $A^2$ exceeds the critical value from the A-D distribution tables at the significance level $\alpha$ (usually 0.05), the null hypothesis is rejected.

2. p-value approach: Reject the null hypothesis if the calculated p-value is less than the chosen significance level.

Interpretation

• Rejected (True): The dataset significantly deviates from the hypothesized distribution; reject the null hypothesis.
• Not Rejected (False): The dataset is consistent with the hypothesized distribution; accept the null hypothesis.

Implementation in Phitter

In the Phitter library, the Anderson-Darling test is implemented through the method:

phi.get_test_anderson_darling(id_distribution: str) -> dict

Returned values:

• test_statistic: The Anderson-Darling test statistic.
• critical_value: The critical value from the Anderson-Darling distribution tables corresponding to the significance level.
• p_value: The p-value associated with the test statistic.
• rejected: Boolean indicating whether the null hypothesis is rejected (True) or not (False).

Example Usage

import phitter

# Define dataset
data = [...]

# Fit distributions
phi = phitter.Phitter(data)
phi.fit()

# Anderson-Darling test for a specific distribution, e.g., "normal"
ad_results = phi.get_test_anderson_darling("normal")
print(ad_results)

This returns a dictionary containing the Anderson-Darling test statistic, critical value, p-value, and whether the null hypothesis is rejected.

Output:

{
    "test_statistic": 4.621,
    "critical_value": 11.070,
    "p_value": 0.795,
    "rejected": False
}

Apply Anderson Darling test to single distribution

Continous case

import phitter

# Define dataset
data = [...]

# Continous measures
continuous_measures = phitter.continuous.ContinuousMeasures(data)

# Define distribution instance
distribution_inst = phitter.continuous.Normal(continuous_measures=continuous_measures)

# Get test result
ad_results = phitter.continuous.evaluate_continuous_test_anderson_darling(distribution_inst, continuous_measures)

Additional Section: Insights from Evaluating the Anderson-Darling Distribution (Marsaglia & Marsaglia)

In their paper Evaluating the Anderson-Darling Distribution, George Marsaglia and John C. W. Marsaglia present methods for accurately determining the distribution of the Anderson-Darling (A–D) test statistic—both in the limiting case as the sample size $n\to \mathrm{\infty }$ and for finite $n$. While the Anderson-Darling test is often introduced in the context of checking for uniformity in $[0,1]$ (i.e., testing whether sorted samples $x_1<x_2<\cdots <x_n$ are drawn from a uniform distribution), their results also underpin more general uses, including tests for other continuous distributions by transforming data appropriately.

Below is a concise overview of the key ideas and the most relevant mathematical expressions from their work.

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1. Limiting Distribution $A_{\mathrm{\infty }}$

The Anderson-Darling statistic $A_n$ for $n$ ordered uniform samples has a limiting distribution as $n\to \mathrm{\infty }$. This limit is often denoted:

$$A_{\mathrm{\infty }}=\underset{n\to \mathrm{\infty }}{lim}{A}_n.$$

Marsaglia & Marsaglia denote the cumulative distribution function (CDF) of $A_{\mathrm{\infty }}$ by

$${\text{AD}}_{\mathrm{\infty }}(z)=\underset{n\to \mathrm{\infty }}{lim}Pr(A_n<z).$$

They show that ${\text{AD}}_{\mathrm{\infty }}(z)$ can be computed by expanding a complicated series directly and then using two-term recursions to achieve very high precision. Concretely, their presentation re-derives the asymptotic form given by Anderson & Darling and provides a practical numerical scheme to evaluate it.

1.1. Asymptotic Formula

A common form for ${\text{AD}}_{\mathrm{\infty }}(z)$ can be written as:

$${\text{AD}}_{\mathrm{\infty }}(z)=\frac{1}{z}\sum _{j=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{-\frac{1}{2}}{j})(4j+1)f(z,j),$$

where $f(z,j)$ involves integrating an exponential series in terms of $(1+w^2{)}^{-1}$. Although the exact expression is somewhat intricate, Marsaglia & Marsaglia derive a pair of practical, piecewise approximations that work extremely well (to about 6–7 digits of accuracy or better):

• For $0<z<2$:

  $${\text{AD}}_{\mathrm{\infty }}(z)\approx z^{-\frac{1}{2}}\exp (-\frac{1.2337141}{z})(2.00012+(0.247105-(0.0649821-(0.0347962-(0.0116720-0.00168691z)z)z)z)z).$$

• For $z\ge 2$:

  $${\text{AD}}_{\mathrm{\infty }}(z)\approx \exp (-\exp (1.0776-(2.30695-(0.43424-(0.082433-(0.008056-0.0003146z)z)z)z)z)).$$

These two pieces meet smoothly at $z=2$ and give a fast, convenient way to approximate the limiting distribution without resorting to full numerical integration.

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2. Finite-$n$ Correction

Although ${\text{AD}}_{\mathrm{\infty }}(z)$ is a valid limiting form, for finite sample sizes $n$ there can be notable differences—especially around the lower percentiles. Marsaglia & Marsaglia address this by introducing an “error-correction” function, $\text{errfix}(n,x)$, which refines the distribution for finite $n$.

If $Z$ is the Anderson-Darling statistic from $n$ observations (ordered as $x_1<\cdots <x_n$), then its exact finite-$n$ CDF is approximated by:

$$Pr(A_n<z)\approx {\text{AD}}_{\mathrm{\infty }}(z)+\text{errfix}(n,{\text{AD}}_{\mathrm{\infty }}(z)),$$

where $0<{\text{AD}}_{\mathrm{\infty }}(z)<1$. Their simulations of size ${10}^{10}$ show that carefully choosing $\text{errfix}(n,x)$ in a piecewise fashion can match the observed finite-$n$ distributions to near-uniform accuracy.

A simplified version of $\text{errfix}(n,x)$ is:

$$\text{errfix}(n,x)=\{\begin{array}{llll}{\displaystyle (0.0037/n^3+0.00078/n^2+0.00006/n)g_1(\frac{x}{c(n)}),}& 0\le x<c(n),\mathrm{\$}\$6pt](0.04213/n+0.01365/n^2)g_2(\frac{x-c(n)}{0.8-c(n)}),& c(n)\le x<0.8,\mathrm{\$}\$6pt]{\displaystyle \frac{g_3(x)}{n},}& 0.8\le x\le 1,\end{array}$$

where $c(n)=0.01265+0.1757/n$, and $g_1,g_2,g_3$ are specific polynomials or piecewise polynomials constructed to fit the simulation data.

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3. Practical Relevance

1. High Accuracy for P-Values
   When using the Anderson-Darling statistic $A_n$ in hypothesis testing, the p-value can be computed as

   $$\text{p-value}={\text{AD}}_{\mathrm{\infty }}(A_n)+\text{errfix}(n,{\text{AD}}_{\mathrm{\infty }}(A_n)).$$

   This formula yields a near-uniform distribution of p-values under the null hypothesis for sample sizes up to at least $n=128$ (and beyond), thereby improving on simply using the large-$n$ tables or the raw limit.

2. Better Finite-$n$ Behavior
   Marsaglia & Marsaglia show that directly applying the limiting distribution ${\text{AD}}_{\mathrm{\infty }}$ for moderate $n$ can introduce systematic bias in the lower to middle quantiles. The correction $\text{errfix}(n,x)$ mitigates this error.

3. Algorithmic Simplicity
   Although the derivations in the paper involve expansions and two-term recursions, the final “shortcut” formulas for ${\text{AD}}_{\mathrm{\infty }}$ and $\text{errfix}$ are straightforward to implement. This makes them suitable for inclusion in codebases like Phitter where computational speed and simplicity are valuable.

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References

• Marsaglia, G. and Marsaglia, J. C. W. (2004). Evaluating the Anderson-Darling Distribution.