Statistical Methods and Modeling of Seismogenesis. Eleftheria Papadimitriou

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 {M_i}, 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,

and the other symbols are the same as previously.

       – 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

only has one mode for h≥hcrit and more than one mode for h<hcrit. Similarly for bumps. The critical bandwidths are denoted by

for modes and bumps, respectively.

       – Step 5. Estimating The estimator [1.3] is differentiable:

[1.23]

[1.24]

For

only has one zero. For

only has one zero. The search for

begins from small values of h, for which the derivatives [1.23] and [1.24] have more than one zero, respectively. One increases h gradually until the number of zeros of the respective derivatives becomes one. Acceptable critical bandwidth estimates are obtained when the step change of h is equal to 0.001. The estimates are, in this case, attained with precision 10^-3. However, to search the critical bandwidths, more complex numerical methods may also be applied.

       – 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:

[1.25]

      where

comes from n-times uniform selection with replacement from the original data

(standard bootstrap), and εi are the standard normal random numbers.

      Silverman (1986) and Efron and Tibshirani (1993) indicated that the samples

have greater variance than the original data,

and that this leads to an artificial increase of the significance of the null hypothesis. To remedy this problem, Efron and Tibshirani (1993) recommended using:

[1.26]

where

are the mean and the sample variance of

rather than

because

has approximately the same variance as

However, when testing magnitudes with the use of samples [1.26], we were occasionally receiving distinctly erroneous results; therefore, we suggest carrying out the tests independently twice, using samples [1.25] and [1.26], respectively. The tests results should be comparable.

      When testing H02 we use

in [1.25] and [1.26].

       – Step 7. Calculating the test p-value. To do this we count how many samples of R, drawn in step 6, result in the kernel