the randomized values in every interval [M-0.5δM, M+0.5δM] repeat exponential distribution of the whole dataset. We can object that the best fitted exponential distribution is used first to randomize the data, and next, this fit is tested. Indeed, this randomization slightly amplifies the probability that H0 will not be rejected. However, other randomizations, e.g., the one assuming normal distribution in [M−0.5δM, M+0.5δM], is logically incorrect and strongly acts against H0.
Within the second approach, two independent null hypotheses are checked: H01 (the magnitude distribution is at most unimodal) and H02 (the number of bumps in the magnitude density ≤ 1). A bump is an interval [a,b] such that the probability density is concave over [a,b] but not over any larger interval. PDFs with two modes and two bumps, respectively, are illustrated in Figure 1.3.
Figure 1.3. Left – PDF with two modes. Right – PDF with two bumps. Reprinted from Lasocki and Papadimitriou (2006, Figure 1)
This approach is suitable to test the applicability of the exponential model, as well as the above-mentioned alternatives, because all of these models have at most, one mode.
These two null hypotheses, H01 and H02, are studied using the smooth bootstrap test for multimodality presented by Silverman (1986) and Efron and Tibshirani (1993), with some adaptations to the magnitude distribution problem provided by Lasocki and Papadimitriou (2006). Given the magnitude data sample {Mi}, i = 1,.., N, the procedure consists of the following steps:
– Step 1. Estimating the catalog (sample) completeness level, MC. We can do this either visually, selecting MC as the smallest magnitude of the linear part of the semi-logarithmic magnitude-frequency graph, or using more sophisticated methods presented, e.g., in Mignan and Woessner (2012), Leptokaropoulos et al. (2013) and the references therein.
– Step 2. Reducing the sample to its complete part {Mi|Mi ≥ Mc}, i = 1,.., n .
– Step 3. Estimating the exponential model of magnitude distribution [1.19]. The maximum likelihood estimator of β is (Aki 1965; Bender 1983):
[1.22]
where 〈M〉 stands for the mean value of the reduced (complete) sample,
– Step 4. Randomizing {Mi}, i = 1,.., n according to equation [1.21], in which F(•) is CDF of the exponential distribution with parameter . The result is
The next steps begin from the kernel density estimate of magnitude, equation [1.3]. As it is shown in Figure 1.2, the number of modes and bumps of a kernel estimate depends on the bandwidth value, h. The greater h is, the fewer modes/bumps occur. Thus, there exists a critical value of bandwidth, hcrit, such that
– Step 5. Estimating The estimator [1.3] is differentiable:
For
– Step 6. Drawing R n-element samples from (equation [1.3]) when testing H01 (modes), and from when testing H02 (bumps). The sampling is done through the smoothed bootstrap technique. For testing H01, the drawn samples are:
where
Silverman (1986) and Efron and Tibshirani (1993) indicated that the samples
where
