Mathematical and Statistical Methods for Actuarial Sciences and Finance

96 M. Corazza, A. Ellero and A. Zorzi

4.1 Overall analysis

Here we compare the overall probability distributions of the first significant digit of
the considered prices and returns against Benford’s law and the uniform probability
distribution (see Fig. 1) by means of the chi-square goodness-of-fit test. Uniform
probability distribution is used as the (intuitive) benchmark alternative to the (coun-
terintuitive) Benford’s law.
At a visual inspection, both the empirical probability distributions seem to be
rather Benford-like (in particular, the one associated to returns). Nevertheless, in
both the comparisons the null is rejected. In Table 1 we report the values of the
associated chi-square goodness-of-fit tests with 8 degrees of freedom (we recall that
χ 82 , 0. 95 = 15 .51).
From a qualitative point of view, our results are analogous to the ones obtained
by Ley (see [8]). In particular, that author observed that, despite the fact that the
chi-square goodness-of-fit tests on DJIA and S&P Indexes suggest rejection of the
null, this was due to the large number of observations considered. In fact, the same
kind of analysis performed only on 1983–1993 data suggestedacceptance of the
null. Moreover, the rejection with respect to the uniform probability distribution is
stronger and stronger than the rejection with respect to Benford’s law. In other words,

7002 ,rebotcO 71 – 5991 ,tsuguA 41 :ruoivaheb llarevo tigid tnacifingis tsrif snruter dna secirP

00.0

50.0

01.0

51.0

02.0

52.0

03.0

53.0

1 2 3 4 5 6 7 8 9

tigid tnacifingis tsriF

Probability

drofneB

mrofinU

secirP

snruteR

Fig. 1.Overall empirical probability distributions

Ta b le 1 .Overall calculated chi-square

Reference probability distribution χ^2 w.r.t. prices χ^2 w.r.t. returns

Benford 151527.74 7664.84

Uniform 780562.24 673479.62