THE ANDERSON-DARLING STATISTIC BY MICHAEL …

1 Mmm OF mwnb> mmmmwmm SU FGRB, THE ANDERSON-DARLING STATISTIC BY MICHAEL A. STEPHENS TECHNICAL REPORT NO. 39 OCTOBER 31, 1979 PREPARED UNDER GRANT DAAG29-77-G-0031 FOR THE ARMY RESEARCH OFFICE Reproduction in Whole or in Part is Permitted for any purpose of the United States Government Approved for public release; distribution unlimited. DEPARTMENT OF STATISTICS STANFORD UNIVERSITY STANFORD, CALIFORNIA THE ANDERSON-DARLING STATISTIC By MICHAEL A. Stephens TECHNICAL REPORT NO. 39 OCTOBER 31, 1979 Prepared under Grant DAAG29-77-G-0O31 For the Army Research Office Herbert Solomon, Project Director Approved for public releasej distribution unlimited.

2 DEPARTMENT OF STATISTICS STANFORD UNIVERSITY STANFORD, CALIFORNIA Partially supported under Office of Naval Research Contract NOOOli*-76-C-01+75 (NR-0U2-267) and issued as Technical Report No. 278. The findings in this report are not to be construed as an official Department of the Army position, unless so designated by other authorized documents. THE AM3 ERS0N- darling STATISTIC 1. Introduction. The Anderson darling STATISTIC is a member of the group of Goodness-of-Fit statistics which has come to be known as EDF statistics (Stephens, 1974) because they are based on a comparison of the empirical distribution function of a given sample with the theoretical distribution to be tested.

3 It is designed to test that random variable X has a continuous cumulative distribution F(x;6); 8 is a vector of one or more parameters entering into the distribution function. Thus for the normal distribution, the vector 9 = (u^cr ). The empirical distribution function (EDF) is defined as F ( ) number of sample values <, x n^ ' n ' where the n values x,, xp, .., x are assumed to be a random sample of X. From the x. let x/n \, X/ s, .. x, N be the order statistics, l (iy \2y (ja.) in ascending order. F (x) is then defined by F (x) = 0 , x < x n (1) F (x) = i/n , x.)

4 5 x < x , i = l,..,(n-l) n (i) (i+l) F (x) = 1 , x < X . n (n) Since F (x) gives the proportion of a random sample < x , one might expect n it to give a good estimate of F(x; ), which is the probability of X less than x, and Fn(x) is in fact a consistent estimator. It is therefore natural to test whether the sample appears to come from F(xje) by using a STATISTIC based on the discrepancy between F (x) and F(x$e). Many statistics of this type have been proposed, the most famous, and one of the oldest, being the Kolmogorov STATISTIC D. This STATISTIC is based on the largest vertical discrepancy between the two functions.

5 An alternative measure is the Cramer-von Mises family, based on the squared integral of the difference between the EDF and the distribution tested: W = J {PM(x)-P(xj0)) \|r(x) dx 5 (l) J-c n the function i|r(x) gives a weighting to the squared difference. One member of W is the Cramer-von Mises STATISTIC itself, W with \|r(x) = 1. The Anderson darling STATISTIC , the subject of this article, is W with \|r(x) = [{F(xje)}{l-F(x)e)}]_1 This weight function counteracts the fact that the discrepancy between Fn(x) and F(xj0) is necessarily becoming smaller in the tails, since both approach 0 and 1 at the extremes.}

6 The weight function given weights the discrepancy by a factor inversely proportional to its variance, and has the effect of giving greater importance to observations in the tail than do most of the EDF statistics. Since tests of fit are often needed implicitly or explicitly to guard against wayward observations in the tails, the STATISTIC is a recommended one, with, as we shall see, generally good power properties over a wide range of alternative distributions when F(x;0) is not the true distribution. 2. Computing Formula, For practical purposes, the definition of the ANDERSON-DARLING STATISTIC given above needs to be turned to a computational formula.

7 This is done in the following sequence of steps: (a) Calculate z. = F(x,. \',B), i = l,..,n . (b) The ANDERSON-DARLING STATISTIC is given by p n = -{ (2i-l)[ln z1 + ln(l-zn+1_i)]}/n-n . (2) i=l Note that since the x,. v are in ascending order, the z. will also be in ascending order, though the usual notation of order statistics has been omitted. 3 Goodness-of-fit test for a completely specified continuous distribution. The formula for z. above assumes that the tested distribution F(x;0) is completely specified, , the parameters in 9 must be known. When 2 this is the case we describe the situation as Case 0.

8 The STATISTIC A was introduced by Anderson and darling (1952, 195*0* and for Case 0 they gave the asymptotic distribution and tables of percentage points. For 2 testxng purposes the upper tail of A will be used; large discrepancies between the EDF and the tested distribution will indicate a bad fit. Later, Lewis (1961) demonstrated that the distribution of A for a finite sample approaches the asymptotic distribution extremely quickly, so that for practical purposes only the asymyptotic distribution is required for sample sizes greater than 5 A table of percentage points is given in 2 Table 1.)

9 To make the goodness of fit test, A is calculated as in Equation (2) above, and compared with these percentage points; the null hypotheses that random variable X has the distribution F(x;0) 2 is rejected at level a if A exceeds the appropriate percentage point at this level. k. Asymptotic theory of the ANDERSON-DARLING STATISTIC . 2 The distribution of A for Case 0 is the same for all distributions tested. This is because the probability integral transformation is made at step (a) and the values of z. are ordered values from a uniform distri- 2 bution with limits 0 and 1. A is therefore a function of ordered uniform random variables.

10 The asymptotic distribution theory for this special case can be found from the asymptotic theory of the EDF, or more specifically of the function yn(z) =./n(Fn(z)-z) , where F (z) is the EDF of n uniform random variables as above. For a modern treatment of the empirical process given by y (z) aee Durbin (1973a, 1973b). When 0 contains unknown components, the z. given by the transfor- A nation (a) above, when an estimate 0 replaces 0, will not be ordered uniform 2 random variables and the distribution theory of A , as for all other EDF statistics, becomes substantially more difficult.