View source for Anderson-Darling statistic

<!--
a1106001.png
$#A+1 = 21 n = 1
$#C+1 = 21 : ~/encyclopedia/old_files/data/A110/A.1100600 Anderson\ANDDarling statistic
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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]]  $  X $
belongs to a given set of distribution functions. In the simplest case this set consists of one completely specified (continuous) distribution function  $  F _ {0} $, 
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  $  F _ {0} $. 
The empirical distribution function  $  F _ {n} $
is a non-parametric [[Statistical estimator|statistical estimator]] of the true distribution function based on a sample  $  X _ {1} \dots X _ {n} $. 
The weighted Cramér–von Mises statistics form a subclass of the EDF statistics. They are defined by

$$ 
\int\limits {\{ F _ {n} ( x ) - F _ {0} ( x ) \}  ^ {2} w ( F _ {0} ( x ) ) }  {d F _ {0} ( x ) } ,
$$

where  $  w $
is a non-negative weight function. The weight function is often used to put extra weight in the tails of the distribution, since  $  F _ {n} ( x ) - F _ {0} ( x ) $
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

$$ 
w ( x ) = [ F _ {0} ( x ) \{ 1 - F _ {0} ( x ) \} ] ^ {- 1 } .
$$

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

$$ 
- n - n ^ {- 1 } \sum _ { i } \{ ( 2i - 1 ) { \mathop{\rm ln} } z _ {i} + ( 2n + 1 - 2i ) { \mathop{\rm ln} } ( 1 - z _ {i} ) \} ,
$$

with  $  z _ {i} = F _ {0} ( X _ {( i ) }  ) $
and  $  X _ {( 1 ) }  < \dots < X _ {( n ) }  $
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  $  0.96 $, 
and hence the test is close to optimal.

In practice, it is of more interest to test whether the distribution function of  $  X $
belongs to a class of distribution functions  $  \{ F ( x, \theta ) \} $
indexed by a nuisance parameter  $  \theta $, 
as, for instance, the class of normal, exponential, or logistic distributions. The Anderson–Darling statistic is now obtained by replacing  $  F _ {0} ( X _ {( i ) }  ) $
by  $  F ( X _ {( i ) }  , {\widehat \theta   } ) $
in calculating  $  z _ {i} $, 
where  $  {\widehat \theta   } $
is an estimator of  $  \theta $. 
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 $P - P$ 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>