*5.1. Estimation*

For *n* randomly selected units from the NMBIII (*c*, *λ*, *k*) population, where *c, λ,* and *k* are unknown, were tested under middle-censoring scheme. In this setting, there are *n*<sup>1</sup> > 0 uncensored observations and *n*<sup>2</sup> > 0 censored observations. Then, by re-ordering the observed data into the uncensored and censored observations, we therefore have the following data

$$\{ (T\_{1}, \dots, T\_{n\_1}, (L\_{n\_1+1}, R\_{n\_1+1}), \dots, (L\_{n\_1+n\_2}, R\_{n\_1+n\_2}) \}\_{\prime \prime} $$

where *n*<sup>1</sup> + *n*<sup>2</sup> = *n*.

The likelihood function of the observed data is given by:

$$\begin{split} L(\mathbf{c}, \lambda, k | \mathbf{x}) &= \omega(k)^{n\_1} \prod\_{i=1}^{n\_1} (\lambda + \frac{\mathbf{c}}{\mathbf{x}\_i}) \prod\_{i=1}^{n\_1} (\mathbf{x}\_i^{-\mathbf{c}} \mathbf{e}^{-\lambda x\_i}) \prod\_{i=1}^{n\_1} (1 + x\_i^{-\mathbf{c}} e^{-\lambda x\_i})^{-k-1} \\ &\times \prod\_{i=n\_1+1}^{n\_1+n\_1} [(1 + r\_i^{-\mathbf{c}} \mathbf{e}^{-\lambda r\_i})^{-k} - (1 + l\_i^{-\mathbf{c}} e^{-\lambda l\_i})^{-k}], \end{split}$$

where *ω* is a normalizing constant depending on *γ* and *θ*, and the estimation of them is not of interest and this is left as a constant. The log-likelihood function is given by

$$\begin{split} l(\mathbf{c},\lambda,k|\mathbf{x}) &= \log \omega + n\_1 \log k + \sum\_{i=1}^{n\_1} \log(\lambda + \frac{\mathbf{c}}{\mathbf{x}\_i}) + n \sum\_{i=1}^{n\_1} \log(\mathbf{x}\_i^{-\mathbf{c}} \mathbf{e}^{-\lambda \mathbf{x}\_i}) - (k+1) \sum\_{i=1}^{n\_1} \log(1 + \mathbf{x}\_i^{-\mathbf{c}} \mathbf{e}^{-\lambda \mathbf{x}\_i}) \\ &+ \sum\_{i=n\_1+1}^{n\_1+n\_1} \log[(1 + r\_i^{-\mathbf{c}} \mathbf{e}^{-\lambda r\_i})^{-k} - (1 + l\_i^{-\mathbf{c}} \mathbf{e}^{-\lambda l\_i})^{-k}]. \end{split}$$

The maximum-likelihood estimation (MLE) of *c, <sup>λ</sup>,* and *<sup>k</sup>*, denoted by *c <sup>M</sup>*, *<sup>λ</sup> <sup>M</sup>*, and *k <sup>M</sup>*, can be derived by solving the following equations:

$$\begin{split} \frac{\partial l(c,\lambda,k|\mathbf{x})}{\partial \mathbf{c}} &= \sum\_{i=1}^{n\_{1}} (\lambda \mathbf{x}\_{i} + \mathbf{c})^{-1} - \sum\_{i=1}^{n\_{1}} \log \mathbf{x}\_{i} + (k+1) \sum\_{i=1}^{n\_{1}} \frac{(\mathbf{x}\_{i}^{-\mathbf{c}} \mathbf{c}^{-\lambda \mathbf{x}\_{i}}) \log \mathbf{x}\_{i}}{1 + \mathbf{x}\_{i}^{-\mathbf{c}} \mathbf{e}^{-\lambda \mathbf{x}\_{i}}} \\ &+ \sum\_{i=n\_{1}+1}^{n\_{1}+n\_{i}} \frac{k(1 + r\_{i}^{-\mathbf{c}} \mathbf{e}^{-\lambda r\_{i}})^{-k-1} (r\_{i}^{-\mathbf{c}} \mathbf{e}^{-\lambda r\_{i}}) \log(r\_{i}) - k(1 + l\_{i}^{-\mathbf{c}} \mathbf{e}^{-\lambda l\_{i}})^{-k-1} (l\_{i}^{-\mathbf{c}} \mathbf{e}^{-\lambda l\_{i}}) \log(l\_{i})}{[(1 + r\_{i}^{-\mathbf{c}} \mathbf{e}^{-\lambda r\_{i}})^{-k} - (1 + l\_{i}^{-\mathbf{c}} \mathbf{e}^{-\lambda l\_{i}})^{-k}]} \end{split}$$

$$\begin{split} \frac{\partial l(c,\lambda,k|\mathbf{x})}{\partial\lambda} &= \sum\_{i=1}^{n\_1} \frac{1}{\lambda + \frac{\mathbf{c}}{x\_i}} - \sum\_{i=1}^{n\_1} \mathbf{x}\_i - (k+1) \sum\_{i=1}^{n\_1} \frac{\mathbf{x}\_i^{-c+1} \mathbf{e}^{-\lambda x\_i}}{1 + \mathbf{x}\_i^{-c} \mathbf{e}^{-\lambda x\_i}} \\ &- \sum\_{i=n\_1+1}^{n\_1+n\_1} \frac{k(1 + r\_i^{-c} \mathbf{e}^{-\lambda r\_i})^{-k-1} (r\_i^{-c+1} \mathbf{e}^{-\lambda r\_i}) - k(1 + l\_i^{-c} \mathbf{e}^{-\lambda l\_i})^{-k-1} (l\_i^{-c+1} \mathbf{e}^{-\lambda l\_i})}{[(1 + r\_i^{-c} \mathbf{e}^{-\lambda r\_i})^{-k} - (1 + l\_i^{-c} \mathbf{e}^{-\lambda l\_i})^{-k}]} \end{split}$$

and

$$\begin{split} \frac{\partial l(\boldsymbol{\varepsilon},\boldsymbol{\lambda},\boldsymbol{k}|\boldsymbol{x})}{\partial\boldsymbol{k}} &= -\sum\_{i=n\_{1}+1}^{n\_{1}+n\_{i}} \frac{(1+r\_{i}^{-\varepsilon}\mathbf{e}^{-\lambda r\_{i}})^{-k}\log(1+r\_{i}^{-\varepsilon}\mathbf{e}^{-\lambda r\_{i}}) - k(1+l\_{i}^{-\varepsilon}\mathbf{e}^{-\lambda l\_{i}})^{-k}\log(1+l\_{i}^{-\varepsilon}\mathbf{e}^{-\lambda l\_{i}})}{[(1+r\_{i}^{-\varepsilon}\mathbf{e}^{-\lambda r\_{i}})^{-k} - (1+l\_{i}^{-\varepsilon}\mathbf{e}^{-\lambda l\_{i}})^{-k}]} \\ &+ \frac{n\_{1}}{k} - \sum\_{i=1}^{n\_{1}}\log(1+x\_{i}^{-\varepsilon}\mathbf{e}^{-\lambda x\_{i}}). \end{split}$$

It is obvious that the MLE of *c, λ,* and *k* cannot be solved explicitly. Therefore, the solutions can be obtained using Newton–Raphson method or numerically using the solve systems of nonlinear equations "*nleqslv*" package in R.

Since the MLE is asymptotically normal, the approximate confidence intervals for the parameters *c, λ* and *k* can be computed as follows: *c*ˆ*<sup>M</sup>* ± *z <sup>α</sup>* 2 *σ*ˆ <sup>2</sup> *<sup>c</sup>* , *<sup>λ</sup>*<sup>ˆ</sup> *<sup>M</sup>* <sup>±</sup> *<sup>z</sup> <sup>α</sup>* 2 & *σ*ˆ 2 *<sup>λ</sup>* and ˆ *kM* ± *z <sup>α</sup>* 2 & *σ*ˆ 2 *<sup>k</sup>* , where *<sup>σ</sup>*<sup>ˆ</sup> <sup>2</sup> (.) are the variances of the respective parameters *c*, *k*, and *λ*, and *z <sup>α</sup>* 2 is the value of the standard normal curve and *α* is the level of significance.

#### *5.2. Simulation Results*

We conducted Monte Carlo simulation studies to assess the finite sample behaviour of the MLEs of the parameters *c*, *k* and *λ* based on two settings; the first is the random variable generated from the NMBIII distribution, while the other considers the case where the NMBIII lifetime data were middle-censored.

The random samples for both settings were generated from distribution NMBIII(*c*, *k*, *λ*) based on accept-reject approach. Without loss of generality, random samples were used with five different sizes viz *n* = 10, 30, 50, 70, and 100 from NMBIII(*c*, *k*, *λ*) distribution with parameters *c* = 1, *k* = 2, and *λ* = 0.5.

The middle censoring settings considered three combinations of the censoring schemes (*γ*−1, *θ*−1)=(0.25, 0.25), (1, 0.75), and (1.25, 0.5).

The results were obtained from 1000 Monte Carlo replications from simulations carried out using the software R, and the average estimates and the mean squared error (MSE) are obtained and reported in Table 3.

Results in Table 3 show that the ML estimates for both settings behave similarly. In general, there is a decreasing function between the sample size and the mean squared error, which verifies the consistency property of the derived estimators. The average estimates are insignificantly effected by the censoring status.


**Table 3.** Average MLE estimates and the corresponding MSE (within brackets).

#### **6. Applications**

This section provides three applications for complete data sets to show how the NM-BIII distribution can be applied in practice. We compare NMBIII distribution to MBIII, BIII, Weibull (W), Gamma (Ga), Lognormal (LN), Generalized Weibull (EW), and Generalised Extreme value type-II (GEV-II) distributions. In these applications, the model parameters are estimated by the method of maximum likelihood. The Akaike information criterion (AIC), Bayesian information criterion (BIC), A\*(Anderson Darling), and W\*(Cramer–von Mises) are computed to compare the fitted models. In general, the smaller the values of these statistics, the better the fit to the data. Additionally, the asymptotic variance-covariance matrices of the NMBIII parameters are also provided. The plots of the fitted PDFs, CDFs, Probability–Probabibility (PP), and Quantile–Quantile (QQ) of NMBIII are displayed for visual comparison. The required computations are carried out in the R software.

The first data set consists of 119 observations on fracture toughness of Alumina (Al2O3) (in the units of MPa m1/2. These data were studied by [32]. The second data set refers to the material thickness of hole (12 mm) and sheet (3.15 mm), comprising 50 observations, as reported by authors in [33]. The third data set was first analysed by [34] and represents the survival times, in weeks, of 33 patients suffering from Acute Myelogenous Leukaemia.

Tables 4–6 list the MLEs, standard errors, AIC, BIC, A\*, and W\* of the model for the data sets 1–3. The results in Tables 4–6 indicate that the NMBIII model provides the best fit as compared to all the other models. Figures 3–5 also support the results of Tables 4–6.



The variance–covariance matrix of the MLEs of the NMBIII distribution for data set 1 is


#### **Table 5.** Data set 2.


The variance–covariance matrix of the MLEs of the NMBIII distribution for data set 2 is


**Table 6.** Data set 3.

⎛


The variance–covariance matrix of the MLEs of the NMBIII distribution for data set 3 is

⎞

0.014574568 0.075708071 −0.0003995590

**Figure 3.** *Cont*.

**Figure 3.** Estimated density (**top left**), cdf (**top right**), QQ-plot (**bottom left**), and PP-plot (**bottom right**) for data set 1.

**Figure 4.** Estimated density (**top left**), cdf (**top right**), QQ-plot (**bottom left**) and PP-plot (**bottom right**) for data set 2.

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**Figure 5.** Estimated density (**top left**), cdf (**top right**), QQ-plot (**bottom left**), and PP-plot (**bottom right**) for data set 3.
