**4. Inferential Analysis**

The inference of the S-ML model is covered in this section. The parameter *θ* is supposed to be unknown. In order to estimate it, the maximum likelihood estimation method is employed. We adopt the methodology as described in a broader context, as seen in [16].

Thus, the next is a mathematical representation of this methodology in the setting of the S-ML distribution. First, let *n* be a positive integer and *x*1, *x*2, ... , *xn* be observations drawn from a random variable *X* following the S-ML distribution. Then, the corresponding likelihood function and log-likelihood function are as follows

$$\begin{split} L &= \prod\_{i=1}^{n} f\_{\mathbb{S}-ML}(\mathbf{x}\_{i}; \boldsymbol{\theta}) = \left(\frac{\pi}{2}\right)^{n} \left(\frac{\theta}{1+\theta}\right)^{n} e^{-2\theta \sum\_{i=1}^{n} \mathbf{x}\_{i}^{\prime}} \prod\_{i=1}^{n} \left[ (1+\theta)e^{\theta \mathbf{x}\_{i}} + 2\mathbf{x}\_{i}\theta - 1 \right] \\ &\times \prod\_{i=1}^{n} \sin\left[\frac{\pi}{2} \left(1 + e^{-\theta \mathbf{x}\_{i}} \frac{\mathbf{x}\_{i}\theta}{1+\theta}\right) e^{-\theta \mathbf{x}\_{i}}\right] \end{split}$$

and

$$\begin{aligned} \ln L &= n \ln \pi - n \ln 2 + n \ln \theta - n \ln(1 + \theta) - 2\theta \sum\_{i=1}^{n} \mathbf{x}\_i \\ &+ \sum\_{i=1}^{n} \ln \left[ (1 + \theta)e^{\theta \mathbf{x}\_i} + 2\mathbf{x}\_i \theta - 1 \right] + \sum\_{i=1}^{n} \ln \left\{ \sin \left[ \frac{\pi}{2} \left( 1 + e^{-\theta \mathbf{x}\_i} \frac{\mathbf{x}\_i \theta}{1 + \theta} \right) e^{-\theta \mathbf{x}\_i} \right] \right\}, \end{aligned}$$

respectively. The maximum likelihood estimate (MLE) of *θ* can be defined via the following argmax definition:

$$\theta = \operatorname{argmax} \ln\_{\theta \ge 0} L. \tag{10}$$

This estimate can be formalized through the solution of the non-linear equations expressed as *d* ln *L*/*dθ* = 0, where

$$\begin{split} \frac{d}{d\theta} \ln L &= \frac{n}{\theta} - \frac{n}{1+\theta} - 2\sum\_{i=1}^{n} \mathbf{x}\_{i} + \sum\_{i=1}^{n} \left[ \frac{e^{\theta \mathbf{x}\_{i}} (\theta \mathbf{x}\_{i} + \mathbf{x}\_{i} + 1) + 2\mathbf{x}\_{i}}{(1+\theta)e^{\theta \mathbf{x}\_{i}} + 2\mathbf{x}\_{i}\theta - 1} \right] + \\ &\sum\_{i=1}^{n} \left[ \frac{\pi}{2} e^{-\theta \mathbf{x}\_{i}} \left( -\frac{\theta \mathbf{x}\_{i}^{2} e^{-\theta \mathbf{x}\_{i}}}{\theta+1} + \frac{\mathbf{x}\_{i}e^{-\theta \mathbf{x}\_{i}}}{\theta+1} - \frac{\theta \mathbf{x}\_{i}e^{-\theta \mathbf{x}\_{i}}}{(\theta+1)^{2}} \right) - \frac{\pi}{2} \mathbf{x}\_{i} e^{-\theta \mathbf{x}\_{i}} \left( 1 + \frac{\theta \mathbf{x}\_{i}e^{-\theta \mathbf{x}\_{i}}}{\theta+1} \right) \right] \\ &\times \cot \left[ \frac{\pi}{2} \left( 1 + e^{-\theta \mathbf{x}\_{i}} \frac{\mathbf{x}\_{i}\theta}{1+\theta} \right) e^{-\theta \mathbf{x}\_{i}} \right]. \end{split}$$

There is no analytical solution for this equation, but ˆ *θ* can be determined at least numerically with any statistical software such as the R software (see [17]). Based on ˆ *θ*, the estimated pdf (epdf) of the S-ML model is given by *fS*−*ML x*; ˆ *θ* and the estimated cdf (ecdf) of the S-ML model is given by *FS*−*ML x*; ˆ *θ* .

Let *I*(*θ*) = −*E <sup>d</sup>*<sup>2</sup> ln[ *fS*−*ML*(*X*; *<sup>θ</sup>*)]/*dθ*<sup>2</sup> be the expected Fisher information matrix. Then, the estimated standard error (SE) of *θ* is achieved by considering the value of the diagonal component of *I* ˆ *θ* −<sup>1</sup> raised to half.

#### **5. Simulation Study**

In the framework of the S-ML model, a simulation study is carried out to study the performance of ˆ *θ* given as Equation (10) in terms of their bias (bias) and mean squared error (MSE). The simulated procedure can be described as follows:

We generate samples of sizes *n* = 20, 50, 100, 200, 500, 1000 from the S-ML distribution with *θ* = (1.25, 1.50, 2.00, 2.50). For each sample, the MLE ˆ *θ* is calculated. Here, 1000 such repetitions are made to calculate the standard mean MLE (MMLE), bias and MSE of these estimates using the formula:

$$\text{MMLE}(\hat{\theta}) = \frac{1}{1000} \sum\_{i=1}^{1000} \hat{\theta}\_{i\prime} \\ \text{Bias}\_{\theta}(\hat{\theta}) = \frac{1}{1000} \sum\_{i=1}^{1000} \left(\hat{\theta}\_{i} - \theta\right)^{2}$$

and

$$\text{MSE}\_{\theta} \left( \hat{\theta} \right) = \frac{1}{1000} \sum\_{i=1}^{1000} \left( \hat{\theta}\_{i} - \theta \right)^{2} \lambda$$

respectively, where ˆ *θ<sup>i</sup>* is the estimate of *θ* for each iteration in the simulation study; *i* is from 1 to 1000. The results of the study are reported in Table 2.


**Table 2.** Outcome of the simulation study.

From Table 2, it is observed that as sample size *n* increases,


#### **6. Applications of the S-ML Model**

We use the S-ML model on two data sets based on the maximum likelihood method as introduced previously. The data differ in size, traits, and background, but they are all of current interest in their areas.

## *6.1. Method*

We proceed as follows for each data set:


The adequacy measures that are used for model fitting are provided here. Suppose *x*1, *x*2, ... , *xn* represent the data and *x*(1), *x*(2), ... , *x*(*n*) be their ordered values. As an initial step, we consider the Cramér von-Mises, Anderson Darling, and Kolmogorov–Smirnov statistics defined by

$$A^\* = -n - \sum\_{i=1}^n \frac{2i - 1}{n} \left[ \ln \left( F\_{\mathcal{S} - ML} \left( \mathbf{x}\_{(i)}; \theta \right) \right) + \ln \left( S\_{\mathcal{S} - ML} \left( \mathbf{x}\_{(n+1-i)}; \theta \right) \right) \right],$$

$$W^\* = \frac{1}{12n} + \sum\_{i=1}^n \left( F\_{\mathcal{S} - ML} \left( \mathbf{x}\_{(i)}; \theta \right) - \frac{2i - 1}{2n} \right)^2$$

and

$$D\_n = \max\_{i=1,2,\dots,n} \left( F\_{\mathbb{S}-ML} \left( \mathfrak{x}\_{(i)}; \theta \right) - \frac{i-1}{n}, \frac{i}{n} - F\_{\mathbb{S}-ML} \left( \mathfrak{x}\_{(i)}; \theta \right) \right),$$

respectively. The *p*-value of the Kolmogorov–Smirnov test linked to *Dn* is also examined. Of course, the above definitions can be adapted to any other model than the S-ML model. The measures of adequacy are extensively employed to determine which model is best in terms of fitting the data set under study. The model having the least value for the *W*∗ or *A*∗, and the highest *p*-value, is considered to give the best fit that is in correspondence with the data.

Furthermore, we consider the following goodness-of-fit measures: Akaike information criterion (AIC) and Bayesian information criterion (BIC), given as follows

$$\text{AIC} = 2k - 2\text{LL}, \text{BIC} = -2\text{LL} + k\ln(n)\_\prime$$

respectively, where LL is the value of the log-likelihood function taken at ˆ *θ* and *k*, being the number of parameters of the model, here *k* = 1 for the S-ML model. As it is widely understood, the model with the lowest value for AIC or BIC is selected as the greatest player of models that fits the data set compared to the other models. For more information on the usage and the underlying meaning of the measures *W*∗, *A*∗, *Dn*, AIC and BIC, we refer to [18].

In order to study the best fit of the S-ML model, we aim to compare it with some useful and competent models, which include the ML, Lindley, sine exponential and sine Lindley models listed in Table 3. It is worth noting that models with three parameters are also considered. The aim is to prove that our model can be efficient enough to outperform more complex models in the literature.


**Table 3.** Competent models with the S-ML model.
