*6.1. Application 1*

Lyu [13] presents a data set related 104 times with programming in the Centre for Software Reliability (CSR). Some descriptive statistics are: mean = 147.8, variance = 60, 071.7, skewness = 3, and kurtosis = 14.6. The moment estimators for the *MSHN* model were *σM* = 74.085 and *qM* = 2.402, which were used as initial values to compute the *ML* estimator in Table 3.

For each distribution we report the estimated log-likelihood. To compare the competing models, we consider the Akaike information criterion (AIC) (Akaike [14]) and the Bayesian information criterion (BIC) (Schwarz [15]), which are defined as AIC = 2*k* − 2 log lik and BIC = *k* log(*n*) − 2 log lik, respectively, where *k* is the number of parameters in the model, *n* is the sample size and log lik is the maximum value for the log-likelihood function. Table 4 displays the AIC and BIC for each model fitted. Figure 3 presents the histogram of the data fitted with the HN, SHN and MSHN distributions, provided with the ML estimations. The QQ-plot for the MSHN and SHN distributions are presented in Figure 4.

**Table 3.** ML estimations with the corresponding SE for the models fitted. Half-normal (HN).


**Table 4.** The Akaike information criterion (AIC) and the Bayesian information criterion (BIC) for each model fitted.

**Figure 3.** Histogram fitted with the HN, SHN and MSHN distributions provided with the ML estimations.

**Figure 4.** QQ plots: (**a**) MSHN distribution and (**b**) SHN distribution.
