*3.2. Applications*

The first data to be considered is related to the soil fertility influence and the characterization of the biologic fixation of N2 for the *Dimorphandra wilsonii* Rizz growth. It was originally studied by [24] and it also figures in the work of [25]. For 128 plants, the phosphorus concentration in the leaves was quantified. Here are the numbers: 0.22, 0.17, 0.11, 0.10, 0.15, 0.06, 0.05, 0.07, 0.12, 0.09, 0.23, 0.25, 0.23, 0.24, 0.20, 0.08, 0.11, 0.12, 0.10, 0.06, 0.20, 0.17, 0.20, 0.11, 0.16, 0.09, 0.10, 0.12, 0.12, 0.10, 0.09, 0.17, 0.19, 0.21, 0.18, 0.26, 0.19, 0.17, 0.18, 0.20, 0.24, 0.19, 0.21, 0.22, 0.17, 0.08, 0.08, 0.06, 0.09, 0.22, 0.23, 0.22, 0.19, 0.27, 0.16, 0.28, 0.11, 0.10, 0.20, 0.12, 0.15, 0.08, 0.12, 0.09, 0.14, 0.07, 0.09, 0.05, 0.06, 0.11, 0.16, 0.20, 0.25, 0.16, 0.13, 0.11, 0.11, 0.11, 0.08, 0.22, 0.11, 0.13, 0.12, 0.15, 0.12, 0.11, 0.11, 0.15, 0.10, 0.15, 0.17, 0.14, 0.12, 0.18, 0.14, 0.18, 0.13, 0.12, 0.14, 0.09, 0.10, 0.13, 0.09, 0.11, 0.11, 0.14, 0.07, 0.07, 0.19, 0.17, 0.18, 0.16, 0.19, 0.15, 0.07, 0.09, 0.17, 0.10, 0.08, 0.15, 0.21, 0.16, 0.08, 0.10, 0.06, 0.08, 0.12, 0.13. Table 3 brings some descriptive statistics.

**Table 3.** Descriptive statistics for soil fertility dataset.


We fitted the Normal-Weibull distribution (NW) (7) to the soil fertility dataset and compared it to the fits of Weibull (W), Exponentiated Weibull (ExpW) [1], Marshall-Olkin Extended Weibull (MOEW) [26], Kumaraswamy-Weibull (KwW) [9], Beta-Weibull (BW) [8] and McDonald-Weibull (McW) [7]. The function goodness.fit of the R package AdequacyModel provides, besides the MLEs and the standard errors (SE), some criteria for model selection (AIC, CAIC, BIC and HQIC); they are shown in Table 4.


**Table 4.** Fitted distributions to the soil fertility dataset (estimates and information criteria).

Information criteria may be used as relative goodness-of-fit measures, such that the lowest values will characterize the best fitted models. In this sense, the Normal-Weibull distribution outperforms the other ones.

Figure 5 shows the histogram of soil fertility data and the fitted densities with the three lowest values of AIC among the distributions in the first column of Table 4. Although the Normal-Weibull and Exponentiated Weibull curves appear to be very close, the blue one (NW) seems to be closer to the histogram.

**Figure 5.** Histogram of soil fertility dataset and fitted densities.

The modified versions of Anderson-Darling (A∗) and Cramér-von Mises (W∗) statistics (more details in [27]) are typically used to investigate the quality of fit of probabilistic models. Table 5 brings these statistics concerning the fitted models to soil fertility data.


**Table 5.** Goodness-of-fit test statistics.

The measures portrayed in Table 5 represent the difference between the empirical distribution function and the real underlying cdf; hence we will consider that the models with lower values of A∗ and W∗ fit the data better. Therefore, once again the Normal-Weibull distribution beats the competing models.

The second application concerns to a dataset representing waiting times (in seconds) between 65 successive eruptions of water through a hole in the cliff at the coastal town of Kiama (New South Wales, Australia), known as the Blowhole. These data can be obtained in [17,28]. Here are they: 83, 51, 87, 60, 28, 95, 8, 27, 15, 10, 18, 16, 29, 54, 91, 8, 17, 55, 10, 35,47, 77, 36, 17, 21, 36, 18, 40, 10, 7, 34, 27, 28, 56, 8, 25, 68, 146, 89, 18, 73, 69, 9, 37, 10, 82, 29, 8, 60, 61, 61, 18, 169, 25, 8, 26, 11, 83, 11, 42, 17, 14, 9, 12. Table 6 provides descriptive statistics.

**Table 6.** Descriptive statistics for eruption dataset.


We fitted the Normal-log-logistic distribution (NLL) (9) to the eruption dataset and compared it to the fits of Log-logistic (LL), Exponentiated Log-logistic (ExpLL), Beta-log-logistic (BLL), Kumaraswamy-log-logistic (KwLL) and Gompertz-log-logistic (GoLL); the four latter along the lines of [1,8,9,11] respectively. Table 7 brings the MLEs, SEs and information criteria.


**Table 7.** Fitted distributions to the eruption dataset (estimates and information criteria).

Since the Normal-log-logistic fitted model presents the smallest values of AIC, CAIC, BIC and HQIC compared to the fits of the other distributions, selecting it rather than the others is a reasonable decision in this case.

In Figure 6 the histogram of eruption data and the fitted densities with the three lowest values of AIC among the distributions in the first column of Table 7 are depicted. By a visual comparison, the three curves are apparently good approximations to the histogram, but the Normal-log-logistic's seems to explain the behavior of the data more accurately.

**Figure 6.** Histogram of eruption dataset and fitted densities.

Table 8 provides the values of A∗ and W∗ of the distributions in the first column of Table 7. These statistics sugges<sup>t</sup> that GoLL and NLL models fit the eruption dataset very closely. Nonetheless, in order to pick a more parsimonious model, one should prefer the NLL, since it has fewer parameters than GoLL.


**Table 8.** Goodness-of-fit tests.

It is worth mentioning that [17] proposed the new class Exponentiated Kumaraswamy-*G* and fitted one of its submodels (with Weibull as baseline) to the same eruption dataset. It presented A∗ = 0.7594 and W∗ = 0.1037, whereas NLL presented lower values of these statistics as one can check in Table 8.
