Anderson–Darling test

The Anderson–Darling and Cramér–von Mises statistics belong to the class of

quadratic EDF statistics (tests based on the empirical distribution function). If the hypothesized distribution is $F$, and empirical (sample) cumulative distribution function is $F\_n$, then the quadratic EDF statistics measure the distance between $F$ and $F\_n$ by

:$$

n \int_{-\infty}^\infty (F_n(x) - F(x))^2\,w(x)\,dF(x),

where $n$ is the number of elements in the sample, and $w(x)$ is a weighting function. When the weighting function is $w(x)=1$, the statistic

is the Cramér–von Mises statistic. The Anderson–Darling (1954) test

{{Cite journal|author1=Anderson, T.W. |author2=Darling, D.A.|title = A Test of Goodness-of-Fit| journal = Journal of the American Statistical Association| year = 1954| volume = 49|issue=268 |pages = 765–769| doi=10.2307/2281537|jstor=2281537 }} is based on the distance

:$$

A^2 = n \int_{-\infty}^\infty \frac{(F_n(x) - F(x))^2}{F(x)\; (1-F(x))} \, dF(x),

which is obtained when the weight function is $w(x)=[F(x)\backslash ;\; (1-F(x))]^\{-1\}$. Thus, compared with the Cramér–von Mises distance, the Anderson–Darling distance places more weight on observations in the tails of the distribution.

The Anderson–Darling test assesses whether a sample comes from a specified distribution. It makes use of the fact that, when given a hypothesized underlying distribution and assuming the data does arise from this distribution, the cumulative distribution function (CDF) of the data can be transformed to what should follow a uniform distribution. The data can be then tested for uniformity with a distance test (Shapiro 1980). The formula for the test statistic $A$ to assess if data

: $A^2\; =\; -n-S\; \backslash ,,$

where

: $S=\backslash sum\_\{i=1\}^n\; \backslash frac\{2i-1\}\{n\}\backslash left[\backslash ln(\; F(Y\_i))\; +\; \backslash ln\backslash left(1-F(Y\_\{n+1-i\})\backslash right)\backslash right].$

The test statistic can then be compared against the critical values of the theoretical distribution. In this case, no parameters are estimated in relation to the cumulative distribution function $F$.

Essentially the same test statistic can be used in the test of fit of a family of distributions, but then it must be compared against the critical values appropriate to that family of theoretical distributions and dependent also on the method used for parameter estimation.

Empirical testing has found{{cite journal|last=Razali |first=Nornadiah |author2=Wah, Yap Bee |title=Power comparisons of Shapiro–Wilk, Kolmogorov–Smirnov, Lilliefors and Anderson–Darling tests |journal=Journal of Statistical Modeling and Analytics |year=2011 |volume=2 |issue=1 |pages=21–33}} that the Anderson–Darling test is not quite as good as the Shapiro–Wilk test, but is better than other tests. Stephens found $A^2$ to be one of the best empirical distribution function statistics for detecting most departures from normality.

The computation differs based on what is known about the distribution:{{Cite book|editor1=D'Agostino, R.B. |editor2=Stephens, M.A. |

year = 1986|

title = Goodness-of-Fit Techniques|

chapter = Tests for the Normal Distribution|

author = Ralph B. D'Agostino|

publisher = Marcel Dekker|

location = New York|

isbn = 0-8247-7487-6

}}

• Case 0: The mean $\backslash mu$ and the variance $\backslash sigma^2$ are both known.
• Case 1: The variance $\backslash sigma^2$ is known, but the mean $\backslash mu$ is unknown.
• Case 2: The mean $\backslash mu$ is known, but the variance $\backslash sigma^2$ is unknown.
• Case 3: Both the mean $\backslash mu$ and the variance $\backslash sigma^2$ are unknown.

The n observations, $X\_i$, for $i=1,\backslash ldots\; n$, of the variable $X$ must be sorted such that $X\_1\backslash leq\; X\_2\backslash leq\; ...\; \backslash leq\; X\_n$ and the notation in the following assumes that X_i represent the ordered observations. Let

:$$

\hat{\mu} =

\begin{cases}

\mu, & \text{if the mean is known.} \\

\bar{X} = \frac{1}{n} \sum_{i = 1}^n X_i, & \text{otherwise.}

\end{cases}

:$$

\hat{\sigma}^2 =

\begin{cases}

\sigma^2, & \text{if the variance is known.} \\

\frac{1}{n} \sum_{i = 1}^n (X_i - \mu)^2, & \text{if the variance is not known, but the mean is.} \\

\frac{1}{n - 1} \sum_{i = 1}^n (X_i - \bar{X})^2, & \text{otherwise.}

\end{cases}

The values $X\_i$ are standardized to create new values $Y\_i$, given by

:$Y\_i=\backslash frac\{X\_i-\backslash hat\{\backslash mu\}\}\{\backslash hat\{\backslash sigma\}\}.$

With the standard normal CDF $\backslash Phi$, $A^2$ is calculated using

:$A^2\; =\; -n\; -\backslash frac\{1\}\{n\}\; \backslash sum\_\{i=1\}^n\; (2i-1)(\backslash ln\; \backslash Phi(Y\_i)+\; \backslash ln(1-\backslash Phi(Y\_\{n+1-i\}))).$

An alternative expression in which only a single observation is dealt with at each step of the summation is:

:$A^2\; =\; -n\; -\backslash frac\{1\}\{n\}\; \backslash sum\_\{i=1\}^n\backslash left[(2i-1)\backslash ln\backslash Phi(Y\_i)+(2(n-i)+1)\backslash ln(1-\backslash Phi(Y\_i))\backslash right].$

A modified statistic can be calculated using

:$$

A^{*2} =

\begin{cases}

A^2\left(1+\frac{4}{n}-\frac{25}{n^2}\right), & \text{if the variance and the mean are both unknown.} \\

A^2, & \text{otherwise.}

\end{cases}

If $A^\{2\}$ or $A^\{*2\}$ exceeds a given critical value, then the hypothesis of normality is rejected with

some significance level. The critical values are given in the table below for values of $A^\{*2\}$.

{{Cite journal | first = G.| last = Marsaglia|year = 2004 | title = Evaluating the Anderson-Darling Distribution

| journal = Journal of Statistical Software

| volume = 9 | issue = 2 | pages = 730–737 | doi = 10.18637/jss.v009.i02| doi-access = free| citeseerx = 10.1.1.686.1363}}

Note 1: If $\backslash hat\{\backslash sigma\}$ = 0 or any $\backslash Phi(Y\_i)=$(0 or 1) then $A^2$ cannot be calculated and is undefined.

Note 2: The above adjustment formula is taken from Shorack & Wellner (1986, p239). Care is required in comparisons across different sources as often the specific adjustment formula is not stated.

Note 3: Stephens notes that the test becomes better when the parameters are computed from the data, even if they are known.

Note 4: Marsaglia & Marsaglia provide a more accurate result for Case 0 at 85% and 99%.

Alternatively, for case 3 above (both mean and variance unknown), D'Agostino (1986) in Table 4.7 on p. 123 and on pages 372–373 gives the adjusted statistic:

:$A^\{*2\}=A^2\backslash left(1+\backslash frac\{0.75\}\{n\}+\backslash frac\{2.25\}\{n^2\}\backslash right)\; .$

and normality is rejected if $A^\{*2\}$ exceeds 0.631, 0.754, 0.884, 1.047, or 1.159 at 10%, 5%, 2.5%, 1%, and 0.5% significance levels, respectively; the procedure is valid for sample size at least n=8. The formulas for computing the p-values for other values of $A^\{*2\}$ are given in Table 4.9 on p. 127 in the same book.

Above, it was assumed that the variable $X\_i$ was being tested for normal distribution. Any other family of distributions can be tested but the test for each family is implemented by using a different modification of the basic test statistic and this is referred to critical values specific to that family of distributions. The modifications of the statistic and tables of critical values are given by Stephens (1986) for the exponential, extreme-value, Weibull, gamma, logistic, Cauchy, and von Mises distributions. Tests for the (two-parameter) log-normal distribution can be implemented by transforming the data using a logarithm and using the above test for normality. Details for the required modifications to the test statistic and for the critical values for the normal distribution and the exponential distribution have been published by Pearson & Hartley (1972, Table 54). Details for these distributions, with the addition of the Gumbel distribution, are also given by Shorack & Wellner (1986, p239). Details for the logistic distribution are given by Stephens (1979). A test for the (two parameter) Weibull distribution can be obtained by making use of the fact that the logarithm of a Weibull variate has a Gumbel distribution.

Fritz Scholz and Michael A. Stephens (1987) discuss a test, based on the Anderson–Darling measure of agreement between distributions, for whether a number of random samples with possibly different sample sizes may have arisen from the same distribution, where this distribution is unspecified.{{cite journal |last1=Scholz |first1=F. W. |last2=Stephens |first2=M. A. |year=1987 |title=K-sample Anderson–Darling Tests |journal=Journal of the American Statistical Association |volume=82 |issue=399 |pages=918–924 |doi=10.1080/01621459.1987.10478517 }} The R package kSamples and the Python package Scipy implements this rank test for comparing k samples among several other such rank tests.{{cite web |title=kSamples: K-Sample Rank Tests and their Combinations |work=R Project |date=7 October 2023 |url=https://cran.r-project.org/web/packages/kSamples/index.html }}{{Cite web |title=The Anderson-Darling test for k-samples. Scipy package. |url=https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.anderson_ksamp.html}}

For $k$ samples the statistic can be computed as follows under the assumption that the distribution function $F\_i$ of $i$-th sample is continuous

:$$

A^2_{kN} = \frac{1}{N} \sum_{i=1}^k \frac{1}{n_i} \sum_{j=1}^{N-1} \frac{(NM_{ij} - jn_i)^2}{j(N-j)}

where

• $n\_i$ is the number of observations in the $i$-th sample
• $N$ is the total number of observations in all samples
• is the pooled ordered sample
• $M\_\{ij\}$ is the number of observations in the $i$-th sample that are not greater than $Z\_j$.

• Kolmogorov–Smirnov test
• Kuiper's test
• Shapiro–Wilk test
• Jarque–Bera test
• Goodness of fit

{{Reflist}}

• Corder, G.W., Foreman, D.I. (2009).Nonparametric Statistics for Non-Statisticians: A Step-by-Step Approach Wiley, {{ISBN|978-0-470-45461-9}}
• Mehta, S. (2014) Statistics Topics {{ISBN|978-1499273533}}
• Pearson E.S., Hartley, H.O. (Editors) (1972) Biometrika Tables for Statisticians, Volume II. CUP. {{ISBN|0-521-06937-8}}.
• Shapiro, S.S. (1980) How to test normality and other distributional assumptions. In: The ASQC basic references in quality control: statistical techniques 3, pp. 1–78.
• Shorack, G.R., Wellner, J.A. (1986) Empirical Processes with Applications to Statistics, Wiley. {{ISBN|0-471-86725-X}}.
• Stephens, M.A. (1979) Test of fit for the logistic distribution based on the empirical distribution function, Biometrika, 66(3), 591–5.

• [http://www.itl.nist.gov/div898/handbook/eda/section3/eda35e.htm US NIST Handbook of Statistics]

{{DEFAULTSORT:Anderson-Darling test}}

Category:Nonparametric statistics

Category:Normality tests