View source for Anderson-Darling statistic

In the goodness-of-fit problem (cf. [[Goodness-of-fit test|Goodness-of-fit test]]) one wants to test whether the distribution function of a [[Random variable|random variable]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110600/a1106001.png" /> belongs to a given set of distribution functions. In the simplest case this set consists of one completely specified (continuous) distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110600/a1106002.png" />, say. A well-known class of test statistics for this testing problem is the class of EDF statistics, thus called since they measure the discrepancy between the empirical distribution function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110600/a1106003.png" />. The empirical distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110600/a1106004.png" /> is a non-parametric [[Statistical estimator|statistical estimator]] of the true distribution function based on a sample <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110600/a1106005.png" />. The weighted Cramér–von Mises statistics form a subclass of the EDF statistics. They are defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110600/a1106006.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110600/a1106007.png" /> is a non-negative weight function. The weight function is often used to put extra weight in the tails of the distribution, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110600/a1106008.png" /> is close to zero at the tails and some form of relative error is more attractive.

A particular member of this subclass is the Anderson–Darling statistic, see [[#References|[a1]]], [[#References|[a2]]], obtained by taking

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110600/a1106009.png" /></td> </tr></table>

To calculate the Anderson–Darling statistic one may use the following formula:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110600/a11060010.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110600/a11060011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110600/a11060012.png" /> the ordered sample.

It turns out, cf. [[#References|[a7]]] and references therein, that the Anderson–Darling test is locally asymptotically optimal in the sense of Bahadur under logistic alternatives (cf. [[Bahadur efficiency|Bahadur efficiency]]). Moreover, under normal alternatives its local Bahadur efficiency is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110600/a11060013.png" />, and hence the test is close to optimal.

In practice, it is of more interest to test whether the distribution function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110600/a11060014.png" /> belongs to a class of distribution functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110600/a11060015.png" /> indexed by a nuisance parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110600/a11060016.png" />, as, for instance, the class of normal, exponential, or logistic distributions. The Anderson–Darling statistic is now obtained by replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110600/a11060017.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110600/a11060018.png" /> in calculating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110600/a11060019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110600/a11060020.png" /> is an estimator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110600/a11060021.png" />. Often, the maximum-likelihood estimator (cf. also [[Maximum-likelihood method|Maximum-likelihood method]]) is used, but see [[#References|[a5]]] for a discussion on the use of other estimators.

Simulation results, cf. [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]] and references therein, show that the Anderson–Darling test performs well for testing normality (cf. [[Normal distribution|Normal distribution]]), and is a reasonable test for testing exponentiality and in many other testing problems.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.W. Anderson,   D.A. Darling,   "Asymptotic theory of certain  "goodness-of-fit"  criteria based on stochastic processes"  ''Ann. Math. Stat.'' , '''23'''  (1952)  pp. 193–212</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.W. Anderson,   D.A. Darling,   "A test of goodness-of-fit"  ''J. Amer. Statist. Assoc.'' , '''49'''  (1954)  pp. 765–769</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Baringhaus,   R. Danschke,   N. Henze,   "Recent and classical tests for normality: a comparative study"  ''Comm. Statist. Simulation Comput.'' , '''18'''  (1989)  pp. 363–379</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Baringhaus,   N. Henze,   "An adaptive omnibus test for exponentiality"  ''Comm. Statist. Th. Methods'' , '''21'''  (1992)  pp. 969–978</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  F.C. Drost,   W.C.M. Kallenberg,   J. Oosterhoff,   "The power of EDF tests to fit under non-robust estimation of nuisance parameters"  ''Statistics and Decision'' , '''8'''  (1990)  pp. 167–182</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  F.F. Gan,   K.J. Koehler,   "Goodness-of-fit tests based on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110600/a11060022.png" /> probability plots"  ''Technometrics'' , '''32'''  (1990)  pp. 289–303</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  Ya.Yu. Nikitin,   "Asymptotic efficiency of nonparametric tests" , Cambridge Univ. Press  (1995)</TD></TR></table>