**1. Introduction**

The last few years in applied sciences have been marked by the need and volume of data to be analyzed. To meet this need, new models have been proposed, and their improvement is a hot topic. These require, among other things, the underlying development of new (statistical or probabilistic) distributions. In this regard, one idea is to modify existing distributions in order to make the corresponding models more flexible and adaptable to several kinds of data. Hence, several modifications based on mathematical techniques have been proposed, generating distributions classified under "families of distributions". The readers are referred to [1] for a bird's-eye view. In recent times, the families described by "trigonometric transformations" have gained a lot of interest because of their applicability and working capability in a variety of situations. Related to this topic, Refs. [2–4] were among the first to study the sinusoidal transformation that leads to the sine generated (S-G) family. For this family, the cumulative distribution function (cdf) and probability density function (pdf) are given by

$$F\_{\mathcal{S}}(\mathbf{x};\boldsymbol{\eta}) = \sin\left[\frac{\boldsymbol{\pi}}{2}G(\mathbf{x};\boldsymbol{\eta})\right], \mathbf{x} \in \mathbb{R},\tag{1}$$

$$f\_S(\mathbf{x}; \boldsymbol{\eta}) = \frac{\pi}{2} \mathbf{g}(\mathbf{x}; \boldsymbol{\eta}) \cos \left[ \frac{\pi}{2} G(\mathbf{x}; \boldsymbol{\eta}) \right], \mathbf{x} \in \mathbb{R}, \tag{2}$$

respectively, where *G*(*x*; *η*) and *g*(*x*; *η*) represent the cdf and pdf of a certain continuous distribution with a parameter vector denoted by *η*. Thus, the functions *FS*(*x*; *η*) and *fS*(*x*; *η*) are linked to a baseline or parent distribution determined beforehand, relying on

**Citation:** Tomy, L.; G, V.; Chesneau, C. The Sine Modified Lindley Distribution. *Math. Comput. Appl.* **2021**, *26*, 81. https://doi.org/ 10.3390/mca26040081

Academic Editors: Nicholas Fantuzzi and Paweł Olejnik

Received: 30 September 2021 Accepted: 13 December 2021 Published: 16 December 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

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the purpose of study. It is worth noting that the baseline cdf has not been supplemented with any additional parameters. The S-G family was developed as a viable substitute for the parent distribution; we can see it from the following first-order stochastic ordering (FOSO) property:

$$G(\mathbf{x}; \boldsymbol{\eta}) \le F\_S(\mathbf{x}; \boldsymbol{\eta}) \tag{3}$$

for all *<sup>x</sup>* <sup>∈</sup> <sup>R</sup>, as well as the possibility of creating versatile statistical distributions that can accept a wide range of data. To make the statement clearer, the exponential distribution is used as a parent distribution by [2] to define the sine exponential distribution. The inverse Weibull (IW) distribution proposed by [5] is used as the reference distribution by [4], thus creating the sine IW (SIW) distribution. The sine power Lomax distribution investigated by [6] is one of the most recent works highlighting the importance of the S-G family. It enhances the parental power Lomax distribution on several functional aspects. Among the trigonometric families of distributions, a few of them, including the C-S family by [7], SKum-G family by [8], STL-G family by [9], and T-G family by [10], were influenced by these efforts.

In this research, we contribute to the developments of the S-G family by linking it to a particular one-parameter distribution introduced by [11]: the modified Lindley (ML) distribution. The sine ML (S-ML) distribution is thus introduced. In order to comprehend the outlined approach, a review of the ML distribution is essential. As a first comment, the ML distribution presented by [11] is achieved by implementing the tuning exponential function *e*−*θx*, with *θ* > 0, to the Lindley distribution, with the motive of modifying its capabilities for new modeling perspectives. On the mathematical side, the cdf and pdf of the ML distribution are defined by

$$G\_{ML}(\mathbf{x};\boldsymbol{\theta}) = \begin{cases} 1 - \left[1 + e^{-\theta \mathbf{x}} \frac{\theta \mathbf{x}}{1 + \theta} \right] e^{-\theta \mathbf{x}}, & \text{if } \mathbf{x} > \mathbf{0} \\\ 0, & \text{if } \mathbf{x} \le \mathbf{0} \end{cases} \tag{4}$$

and

$$g\_{ML}(\mathbf{x};\theta) = \begin{cases} \frac{\theta}{1+\theta} e^{-2\theta \mathbf{x}} \left[ (1+\theta)e^{\theta \mathbf{x}} + 2\theta \mathbf{x} - 1 \right], & \text{if } \mathbf{x} > \mathbf{0} \\\ 0, & \text{if } \mathbf{x} \le \mathbf{0} \end{cases},\tag{5}$$

respectively. Basically, the ML distribution satisfies the following FOSO property:

$$G\_L(\mathbf{x}; \theta) \le G\_{ML}(\mathbf{x}; \theta) \le G\_E(\mathbf{x}; \theta) \tag{6}$$

for all *<sup>x</sup>* <sup>∈</sup> <sup>R</sup>, where *GL*(*x*; *<sup>θ</sup>*) and *GE*(*x*; *<sup>θ</sup>*) represent the cdfs of the Lindley and exponential distributions, respectively. In this sense, the ML distribution constitutes a real alternative to these two classical distributions. The ML distribution is also identified as a linear combination of the exponential distribution with parameter *θ* and the gamma distribution with parameters (2, 2*θ*), and it has an "increasing-reverse bathtub-constant" hazard rate function (hrf). The real benefit is quite noteworthy; the ML model is superior to the Lindley and exponential models for the three data sets seen in [11]. A few inspired distributions enhancing or generalising the ML distribution were proposed for the purpose of optimality. These include the Poisson ML distribution by [12], wrapped ML distribution by [13], and discrete ML distribution by [14].

The immediate aim of the S-ML distribution is to use the S-G technique to enhance the effectiveness of the ML distribution on diverse data sets. In particular, thanks to the FOSO properties in Equations (3) and (6), it is a real and attractive alternative to the Lindley and ML distributions. Further exploration in the following research will reveal deeper motives. To summarise, the S-ML model's utility and adaptability make it particularly appealing to fit data from various fields. Remarkably, the characterized pdf shows a variety of curve shapes, some of which have only one mode, are decreasing, and are asymmetrical to the right. In comparison to the pdf of the ML distribution, when it is unimodal, the pdf of

the S-ML distribution has a more rounded peak, meaning that it is better adapted to fit a data histogram presenting a high kurtosis level. Furthermore, the S-ML distribution exhibits a non-monotonic hrf which is "increasing-reverse bathtub-constant" shaped. The hrf of the ML distribution has this feature as well. As with other competent models, the accuracy of the fits is persistent in the case of the S-ML model due to their characteristics. The claim is demonstrated by examining two published real-world data sets, primarily from engineering and climate data, against twelve competent models.

We prepare the rest of the paper in the following manner. The concept, quality, and key aspects of the S-ML distribution are covered in Section 2. A moment analysis is conducted in Section 3. The maximum likelihood estimation of the parameter *θ* is explained in Section 4. A simulation study is presented in Section 5. Section 6 assesses the proposed model's applicability to real-world data. Finally, in Section 7, the conclusions are provided.

### **2. The S-ML Distribution**

The mathematical foundation for the S-ML distribution is first presented.

#### *2.1. Functional Analysis*

To begin, we perform a functional analysis of the S-ML distribution. By substituting Equations (4) and (5) in Equations (1) and (2), respectively, we derive the major functions of the S-ML distribution; the cdf and pdf are given as follows

$$F\_{S-ML}(\mathbf{x};\theta) = \begin{cases} \cos\left[\frac{\pi}{2}\left(1 + e^{-\theta\mathbf{x}}\frac{\mathbf{x}\theta}{1+\theta}\right)e^{-\theta\mathbf{x}}\right], & \text{if } \mathbf{x} > \mathbf{0} \\\ 0, & \text{if } \mathbf{x} \le \mathbf{0} \end{cases}$$

and

$$f\_{S-ML}(\mathbf{x};\boldsymbol{\theta}) = \begin{cases} \frac{\pi}{2} \frac{\theta}{1+\theta} e^{-2\theta\mathbf{x}} \Big[ (1+\theta)\boldsymbol{\epsilon}^{\theta\mathbf{x}} + 2\mathbf{x}\boldsymbol{\theta} - 1 \Big] \sin\left[\frac{\pi}{2} \left(1 + \boldsymbol{\epsilon}^{-\theta\mathbf{x}} \frac{\mathbf{x}\boldsymbol{\theta}}{1+\theta}\right) \boldsymbol{\epsilon}^{-\theta\mathbf{x}} \right], & \text{if } \mathbf{x} > \mathbf{0} \\\ 0, & \text{if } \mathbf{x} \le \mathbf{0} \end{cases},\tag{7}$$

with *θ* > 0. As a primary result mentioned in the introduction section, the following FOSO property holds: *GML*(*x*; *<sup>θ</sup>*) <sup>≤</sup> *FS*−*ML*(*x*; *<sup>θ</sup>*) for any *<sup>x</sup>* <sup>∈</sup> <sup>R</sup>, making an immediate difference between the ML and S-ML modeling from the cdf viewpoint. Differences can also be observed on the respective pdfs, as discussed below. Naturally, variant forms of *fS*−*ML*(*x*; *θ*) can be obtained by changing the value of *θ*. Due to the relative complexity of this function in the analytical sense, we propose a graphical study for shape analysis. The more representative shapes of this pdf are shown in Figure 1.

**Figure 1.** Plots of (**a**) unimodal shapes and (**b**) decreasing shapes for *fS*−*ML*(*x*; *θ*).

We can observe from Figure 1, that, for smaller values of *θ*, the plot of *fS*−*ML*(*x*; *θ*) is unimodal, and for larger values of *θ*, the plot of *fS*−*ML*(*x*; *θ*) is decreasing. As a result, the S-ML distribution is suitable for modeling a vast majority of lifetime phenomena. Compared to the parent ML distribution, the following observations are made: When it is unimodal, we observe that the pdf of the S-ML distribution has a more rounded peak, meaning that it is better adapted to fit a data histogram presenting a high kurtosis level. In other words, the S-ML model is more able to analyze data of a leptokurtic nature.

#### *2.2. Reliability Analysis*

We complete the previous functional analysis by studying the complementary reliability functions, such as the survival function (sf), hrf (for hazard rate function), reversed hrf (rhrf), second rate of failure (srf), and the cumulative hrf (chrf) of the S-ML distribution. In a broader sense, the sf measures the probability that the life of an item will survive beyond any specified time. Mathematically, the sf of the S-ML distribution is given by

$$S\_{S-ML}(\mathbf{x};\boldsymbol{\theta}) = 1 - F\_{\mathbb{S}-ML}(\mathbf{x};\boldsymbol{\theta}) = \begin{cases} 1 - \cos\left[\frac{\pi}{2} \left( 1 + e^{-\theta \mathbf{x}} \frac{\mathbf{x}\boldsymbol{\theta}}{1 + \theta} \right) e^{-\theta \mathbf{x}} \right], & \text{if } \mathbf{x} > \mathbf{0} \\\ 1, & \text{if } \mathbf{x} \le \mathbf{0} \end{cases}.$$

The hrf measures the likelihood of an item deteriorating or expiring depending on its lifetime. As a direct consequence, it is critical in the classification of survival distributions. The hrf of the S-ML distribution is specified by

$$\begin{split} h\_{S-ML}(\mathbf{x};\boldsymbol{\theta}) &= \frac{f\_{S-ML}(\mathbf{x};\boldsymbol{\theta})}{S\_{S-ML}(\mathbf{x};\boldsymbol{\theta})} \\ &= \begin{cases} \frac{\pi}{2} \frac{\theta}{1+\theta} e^{-2\theta x} \Big[ (1+\theta)e^{\theta x} + 2x\theta - 1 \Big] \cot \left[ \frac{\pi}{4} \left( 1 + e^{-\theta x} \frac{\mathbf{x}\theta}{1+\theta} \right) e^{-\theta x} \right], & \text{if } x > 0 \\\ 0, & \text{if } x \le 0 \end{cases} \end{split}$$

.

.

.

Further, Figure 2 displays the shapes of this hrf for various values of *θ*.

**Figure 2.** Plots of *hS*−*ML*(*x*; *θ*) with selected values of *θ*.

Figure 2 emphasizes that the hrf of the S-ML distribution has "increasing-reverse bathtub-constant" shapes, which is also possessed by the hrf of the ML distribution. This makes a solid difference between the Lindley and exponential distributions. It is also a desirable property for modelling purposes.

The rhrf is the ratio between the pdf to its cdf and it plays a role in analyzing censored data. Analytically, it corresponds to

$$r\_{S-ML}(\mathbf{x};\theta) = \begin{cases} \frac{\pi}{2} \frac{\theta}{1+\theta} e^{-2\theta \mathbf{x}} \left[ (1+\theta)e^{\theta \mathbf{x}} + 2\mathbf{x}\theta - 1 \right] \tan \left[ \frac{\pi}{2} \left( 1 + e^{-\theta \mathbf{x}} \frac{\mathbf{x}\theta}{1+\theta} \right) e^{-\theta \mathbf{x}} \right], & \text{if } \mathbf{x} > 0\\\ 0, & \text{if } \mathbf{x} \le 0 \end{cases}$$

The srf is the logarithmic ratio of the sf at time *x* and *x* + 1, and it is given by

$$r\_{S-ML}^{\*}(\mathbf{x};\boldsymbol{\theta}) = \begin{cases} \ln\left(\frac{1-\cos\left[\left(\pi/2\right)\left(1+e^{-\theta\mathbf{x}}\right)\left(1+\theta\right)\right]e^{-\theta\mathbf{x}}}{1-\cos\left[\left(\pi/2\right)\left(1+e^{-\theta\left(\mathbf{x}+1\right)}\left(\mathbf{x}+1\right)\theta/\left(1+\theta\right)\right)e^{-\theta\left(\mathbf{x}+1\right)}\right]}\right), & \text{if } \mathbf{x} > 0\\\ 0, & \text{if } \mathbf{x} \le 0 \end{cases}$$

The chrf is the negative logarithm of sf and is given by

$$H\_{S-ML}(\mathbf{x};\theta) = \begin{cases} -\ln\left(1 - \cos\left[\frac{\pi}{2}\left(1 + e^{-\theta x}\frac{\mathbf{x}\theta}{1+\theta}\right)e^{-\theta x}\right]\right), & \text{if } \mathbf{x} > 0\\ 0, & \text{if } \mathbf{x} \le 0 \end{cases}$$

.

With these functions, we conclude different reliability analysis in regard with the S-ML distribution.

#### **3. Moment Analysis**

For any lifetime distribution, a moment analysis is necessary to handle numerically its modeling capacities, identifying the behavior of various central and dispersion moment parameters, as well as moment skewness and kurtosis coefficients.

As a first notion, for any positive integer *r* ≥ 1, and a random variable *X* with the S-ML distribution, the *r*-th moment of *X* exists. It can be expressed as

$$\begin{split} \text{norm}(r) &= \mathbb{E}(X^{r}) = \int\_{0}^{+\infty} \mathbf{x}^{r} f\_{S-ML}(\mathbf{x};\boldsymbol{\theta}) d\mathbf{x} \\ &= \frac{\pi}{2} \frac{\theta}{1+\theta} \int\_{0}^{+\infty} \mathbf{x}^{r} e^{-2\theta x} \Big[ (1+\theta)e^{\theta x} + 2x\theta - 1 \Big] \sin \Big[ \frac{\pi}{2} \left( 1 + e^{-\theta x} \frac{\mathbf{x}\theta}{1+\theta} \right) e^{-\theta x} \Big] d\mathbf{x} . \end{split} \tag{8}$$

Integral developments in the classical sense are limited. Computer software, on the other hand, can be used to quantitatively evaluate it for a given *θ*.

We propose a series development of mom(*r*) in the next result, which can be used for computational purposes in a less opaq method than a "ready to use but black box" computer program.

**Proposition 1.** *The r-th moment of X can be expanded as*

$$\text{mom}(r) = \frac{r}{\theta^r} \sum\_{k=1}^{+\infty} \sum\_{\ell=0}^{2k} \binom{2k}{\ell} \frac{(-1)^{k+1}}{(2k)!} \left(\frac{\pi}{2}\right)^{2k} (1+\theta)^{-\ell} \frac{(\ell+r-1)!}{(\ell+2k)^{r+\ell}}.$$

**Proof.** For the proof, we do not directly use the integral expression of mom(*r*) as described in (8). An integration by part gives

$$\text{mom}(r) = \int\_0^{+\infty} \mathbf{x}^r f\_{S-ML}(\mathbf{x}; \theta) d\mathbf{x} = r \int\_0^{+\infty} \mathbf{x}^{r-1} S\_{S-ML}(\mathbf{x}; \theta) d\mathbf{x} \dots$$

Now, by utilizing the series expansion of the cosine function and the classical binomial formula, we obtain

$$\begin{split} S\_{S-ML}(\mathbf{x};\boldsymbol{\theta}) &= 1 - \sum\_{k=0}^{+\infty} \frac{(-1)^{k}}{(2k)!} \left(\frac{\pi}{2}\right)^{2k} \left(1 + e^{-\theta \mathbf{x}} \frac{\mathbf{x}\boldsymbol{\theta}}{1+\theta}\right)^{2k} e^{-2k\theta \mathbf{x}} \\ &= \sum\_{k=1}^{+\infty} \frac{(-1)^{k+1}}{(2k)!} \left(\frac{\pi}{2}\right)^{2k} \left(1 + e^{-\theta \mathbf{x}} \frac{\mathbf{x}\boldsymbol{\theta}}{1+\theta}\right)^{2k} e^{-2k\theta \mathbf{x}} \\ &= \sum\_{k=1}^{+\infty} \sum\_{\ell=0}^{2k} \binom{2k}{\ell} \frac{(-1)^{k+1}}{(2k)!} \left(\frac{\pi}{2}\right)^{2k} \left(\frac{\theta}{1+\theta}\right)^{\ell} \mathbf{x}^{\ell} e^{-(\ell+2k)\theta \mathbf{x}}. \end{split} \tag{9}$$

Hence, after some developments including the change of variable *y* = (- + 2*k*)*θx* (so that *dx* = [1/((-+ 2*k*)*θ*)]*dy*), and the calculus of gamma-type integral, we get

$$\begin{split} \operatorname{Hom}(r) &= r \int\_0^{+\infty} \mathbf{x}^{r-1} \left[ \sum\_{k=1}^{+\infty} \sum\_{\ell=0}^{2k} \binom{2k}{\ell} \frac{(-1)^{k+1}}{(2k)!} \left( \frac{\pi}{2} \right)^{2k} \left( \frac{\theta}{1+\theta} \right)^{\ell} \mathbf{x}^{\ell} e^{-(\ell+2k)\theta \mathbf{x}} \right] d\mathbf{x} \\ &= r \sum\_{k=1}^{+\infty} \sum\_{\ell=0}^{2k} \binom{2k}{\ell} \frac{(-1)^{k+1}}{(2k)!} \left( \frac{\pi}{2} \right)^{2k} \left( \frac{\theta}{1+\theta} \right)^{\ell} \int\_0^{+\infty} \mathbf{x}^{r+\ell-1} e^{-(\ell+2k)\theta \mathbf{x}} d\mathbf{x} \\ &= \frac{r}{\theta^r} \sum\_{k=1}^{+\infty} \sum\_{\ell=0}^{2k} \binom{2k}{\ell} \frac{(-1)^{k+1}}{(2k)!} \left( \frac{\pi}{2} \right)^{2k} (1+\theta)^{-\ell} \frac{(\ell+r-1)!}{(\ell+2k)^{r+\ell}}. \end{split}$$

Proposition 1 is proved.

Then, based on Proposition 1, the following finite sum approximation remains acceptable:

$$\text{mom}(r) \approx \frac{r}{\theta^r} \sum\_{k=1}^{l\ell} \sum\_{\ell=0}^{2k} \binom{2k}{\ell} \frac{(-1)^{k+1}}{(2k)!} \left(\frac{\pi}{2}\right)^{2k} (1+\theta)^{-\ell} \frac{(\ell+r-1)!}{(\ell+2k)^{r+\ell}}.$$

where *U* represents any large integer.

From the above moment formulas, we can easily derive the mean, variance, moment skewness coefficient and moment kurtosis coefficient; the mean is given by mom(1), the variance is obtained as V(*X*) = E (*<sup>X</sup>* <sup>−</sup> mom(1))<sup>2</sup> , the moment skewness coefficient can be derived as MS = E (*<sup>X</sup>* <sup>−</sup> mom(1))<sup>3</sup> /V(*X*) 3/2 and the moment kurtosis coefficient can be derived as MK = E (*<sup>X</sup>* <sup>−</sup> mom(1))<sup>4</sup> /V(*X*) 2 .

Table 1 gives a glimpse of these values for different values of *θ*.

**Table 1.** Values of various moment measures of the S-ML distribution.


From Table 1, we can observe that, as the value of the parameter *θ* of the S-ML distribution increases, all the considered measures increase. Furthermore, since MS > 0, it is clear that the S-ML distribution is mainly right-skewed, and since MK > 3, it is mainly leptokurtic.

We can complete the previous moment results by investigating the incomplete moments. To begin, let *r* ≥ 1 be an integer, *t* ≥ 0, and *X* be a random variable with the S-ML distribution. Based on this variable, we define its incomplete version by *Y*(*t*) = *X* if *X* ≤ *t* and *Y*(*t*) = 0 if *X* > *t*. Then, the *r*-th incomplete moment of *X* given at *t* exists, and it is defined by

$$\text{mom}(r, t) = \mathbb{E}\left(\mathcal{Y}(t)^r\right) = \int\_0^t \mathbf{x}^r f\_{\mathbb{S}-ML}(\mathbf{x}; \boldsymbol{\theta}) d\mathbf{x} \dots$$

It is involved in developments of important probabilistic objects, such as mean deviations, income curves, etc. More basically, it can be viewed as a truncated version of the standard *r*-moment. We may refer to [15] in this regard.

In the next results, we present a series expansion of mom(*r*, *t*), which can be used for approximation purposes.

**Proposition 2.** *The r-th incomplete moment of X given at t exists and can be expanded as*

$$\begin{split} \text{norm}(r,t) &= -t^{\ell} \left\{ 1 - \cos \left[ \frac{\pi}{2} \left( 1 + e^{-\theta t} \frac{t\theta}{1+\theta} \right) e^{-\theta t} \right] \right\} \\ &+ \frac{r}{\theta^{r}} \sum\_{k=1}^{+\infty} \sum\_{\ell=0}^{2k} \binom{2k}{\ell} \frac{(-1)^{k+1}}{(2k)!} \left( \frac{\pi}{2} \right)^{2k} (1+\theta)^{-\ell} \frac{1}{(\ell+2k)^{r+\ell}} \gamma(r+\ell, (\ell+2k)\theta t), \end{split}$$

*where γ*(*a*, *t*) *denotes the incomplete gamma function defined by γ*(*a*, *t*) = *<sup>t</sup>* <sup>0</sup> *<sup>x</sup>a*−1*e*<sup>−</sup>*xdx, where a* > 0 *and t* ≥ 0*.*

**Proof.** The proof follows the lines of the one of Proposition 1. An integration by part gives

$$\text{norm}(r, t) = \int\_0^t \mathbf{x}^r f\_{\mathcal{S}-ML}(\mathbf{x}; \boldsymbol{\theta}) d\mathbf{x} = -t^r \mathcal{S}\_{\mathcal{S}-ML}(t; \boldsymbol{\theta}) + r \int\_0^t \mathbf{x}^{r-1} \mathcal{S}\_{\mathcal{S}-ML}(\mathbf{x}; \boldsymbol{\theta}) d\mathbf{x} \dots$$

It follows from the series expansion in Equation (9) and the change of variable *y* = (-+ 2*k*)*θx* that

$$\begin{split} \int\_{0}^{t} \mathbf{x}^{r-1} \mathbf{S}\_{S-\text{ML}}(\mathbf{x};\boldsymbol{\theta}) d\mathbf{x} &= \sum\_{k=1}^{+\infty} \sum\_{\ell=0}^{2k} \binom{2k}{\ell} \frac{(-1)^{k+1}}{(2k)!} \left(\frac{\pi}{2}\right)^{2k} \left(\frac{\theta}{1+\theta}\right)^{\ell} \int\_{0}^{t} \mathbf{x}^{r+\ell-1} e^{-(\ell+2k)\theta \mathbf{x}} d\mathbf{x} \\ &= \frac{1}{\theta^{\ell}} \sum\_{k=1}^{+\infty} \sum\_{\ell=0}^{2k} \binom{2k}{\ell} \frac{(-1)^{k+1}}{(2k)!} \left(\frac{\pi}{2}\right)^{2k} (1+\theta)^{-\ell} \frac{1}{(\ell+2k)^{\ell+\ell}} \gamma(r+\ell,(\ell+2k)\theta t). \end{split}$$

Therefore

$$\begin{split} \text{mom}(r,t) &= -t^{\mathbb{P}} \left\{ 1 - \cos \left[ \frac{\pi}{2} \left( 1 + e^{-\theta t} \frac{t\theta}{1+\theta} \right) e^{-\theta t} \right] \right\} \\ &+ \frac{r}{\theta^r} \sum\_{k=1}^{+\infty} \sum\_{\ell=0}^{2k} \binom{2k}{\ell} \frac{(-1)^{k+1}}{(2k)!} \left( \frac{\pi}{2} \right)^{2k} (1+\theta)^{-\ell} \frac{1}{\left( \ell+2k \right)^{r+\ell}} \gamma (r+\ell, (\ell+2k)\theta t). \end{split}$$

This concludes the proof of Proposition 2.

In some sense, Proposition 2 generalizes Proposition 1; by taking *t* → +∞, Proposition 2 becomes Proposition 1.

The rest of the study is devoted to the applicability of the S-ML model, illustrated with concrete examples of data analysis.
