*5.1. Failure Times*

The data of failure times for a sample of 15 electronic components in an acceleration life test (see [22]) were considered here. These data were based on the discretization concept. Adopting a data analysis setting, we compared the DPsL, discrete three-parameter Lindley (DTPL) (see [9]), discrete log-logistic (DLL) (see [23]), discrete inverse Weibull (DIW) (see [2]), discrete Burr–Hutke (DBH) (see [6]), discrete Pareto (DP) (see [3]), Poisson (P) and geometric (G) distributions. The MLEs with standard errors (SEs) and confidence intervals (CIs) for the parameter(s), estimated −log Likelihood (−L), Akaike information criterion (AIC), Bayesian information criterion (BIC) and goodness of fit statistic (Kolmogorov statistic (K-S) and *p*-value) of these distributions for this dataset are given in Table 9.

**Table 9.** The MLEs, CIs, −L, AIC, BIC, K-S and *p*-values of all the fitted distributions for the failure times data.


From Table 9, it is evident that, besides the DPsL distribution, the DTPL, G and DLL distributions also performed quite well, but it is clear that the DPsL distribution was the best among them, since it had the lowest K-S, AIC and BIC, with a higher *p*-value. In order to illustrate this claim, Figure 3 provides the probability–probability (P–P) plots, and Figure 4 displays the estimated cdfs of the fitted distributions.

**Figure 3.** The P–P plots for the fitted distributions using the failure times data.

**Figure 4.** Estimated cdfs of the fitted distributions using the failure times data.

From the above figures, we could infer that the DPsL distribution yielded a better fit among other fitted distributions. Table 10 completes these results by presenting some descriptive measures of the fitted DPsL distribution. Hence, it is evident that the fitted DPsL distribution was over dispersed, moderately right skewed and leptokurtic.

**Table 10.** Values of some descriptive statistics of the DPsL distribution for the failure times data.


## *5.2. Numbers of Borers*

The second dataset was the biological experiment data, which represented the number of European corn borer (No. ECB) larvae Pyrausta in the field (see [24]). It was an experiment conducted randomly on eight hills in 15 replications, and the experimenter counted the number of borers per hill of corn. The fits of the DPsL distribution were compared together with some competitive distributions which were the new Poisson weighted exponential (NPWE) (see [16]), DIW, discrete Burr-XII (DBXII) (see [23]), discrete Bilal (DBl) (see [8]), DP, DBH and Poisson (P) distributions. The MLEs with their corresponding SEs, CIs under the form (lower bound of the CI (LCI), upper bound of the CI (UCI)) for the parameter(s) and goodness of fit test for the numbers of borers dataset are reported in Table 11.

**Table 11.** The MLE, LCI, UCI, <sup>−</sup>L, AIC, BIC, *<sup>χ</sup>*<sup>2</sup> and *<sup>p</sup>*-values for the one parameter distributions considered using the number of borers dataset.


From the above table, it is evident that, besides the DPsL distribution, the NPWE distribution also performed quite well, but it is clear that the DPsL distribution was the best among them, since it had the lowest −L, AIC, BIC and *<sup>χ</sup>*<sup>2</sup> value with the highest *<sup>p</sup>*-value. From Figure 5, we could infer that the DPsL distribution yielded a better fit among other fitted distributions. To complete this, Table 12 contains some descriptive measures of

the fitted DPsL distribution. Hence, here also, it is evident that the fitted DPsL distribution was over-dispersed, moderately right skewed and leptokurtic.

**Figure 5.** The estimated pmfs of the fitted distributions for the number of borers dataset.

**Table 12.** Values of some descriptive statistics of the DPsL distribution for the number of borers dataset.

