**7. Conclusions**

By unifying the Lindley and Lomax distributions, we establish a three-parameter distribution called the minimum Lindley Lomax (minLLx). The quantile function, ordinary and incomplete moments, moment generating function, Lorenz and Bonferroni curves, order statistics, Rényi entropy, stress−strength model, and stochastic ordering are all considered as defining attributes of the new model. The envisaged model's characterizations are evaluated. The model parameters are determined using the optimum likelihood criterion, and these projections are assessed using numerical simulations. Two real-world applications exemplify the utility of the new model.

**Supplementary Materials:** Partial codes used in Section 5 are available online at https://www.mdpi. com/article/10.3390/mca27010016/s1.

**Author Contributions:** Conceptualization, S.K. and G.G.H.; methodology, H.M.R.; software, S.O.; validation, F.J., S.K. and H.M.R.; formal analysis, F.J. and H.M.R.; investigation, S.S.; resources, S.O.; data curation, S.S.; writing—original draft preparation, F.J. and G.G.H.; writing—review and editing S.K. and F.J.; visualization, S.K. and S.S.; supervision, S.O.; project administration, F.J. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

**Theorem A1.** *Let* (Ω , F, *P*) *be a given probability space and let H* = [*a*, *b*] *be an interval for some d* < *b* (*a* = −∞, *b* = ∞ *might as well be allowed*). *Let X* : Ω → *H be a continuous av with the distribution function F and let q*<sup>1</sup> *and q*<sup>2</sup> *be two real functions defined on H, such that*

$$E[q\_2(X)|X \ge x] = E[q\_1(X)|X \ge x]\psi(x), \quad x \in H\_2$$

*is defined with some real function <sup>η</sup>*. *Assume that <sup>q</sup>*1, *<sup>q</sup>*<sup>2</sup> ∈ *<sup>C</sup>*−1(*H*), *<sup>ψ</sup>* ∈ *<sup>C</sup>*2(*H*) *and <sup>F</sup> is a twice continuously differentiable and strictly monotone function on the set H*. *Finally, assume that the equation ψ q*<sup>1</sup> = *q*<sup>2</sup> *has no real solution in the interior of H*. *Then F is uniquely determined by the functions q*1, *q*2*, and ψ*, *particularly*

$$F(\mathfrak{x}) = \int\_{a}^{\mathfrak{x}} \mathbb{C} \left| \frac{\psi'(\mathfrak{u})}{\psi(\mathfrak{u}) \, q\_1(\mathfrak{u}) - q\_2(\mathfrak{u})} \right| \exp(-s(\mathfrak{u})) \, d\mathfrak{u} \, \mathfrak{y}$$

*where function s is a solution of the differential equation s* = *<sup>ψ</sup> <sup>q</sup>*<sup>1</sup> *<sup>ψ</sup> <sup>q</sup>*1−*q*<sup>2</sup> *and <sup>C</sup> is the normalization constant, such that <sup>H</sup> dF* = 1.

We like to mention that this kind of characterization based on the ratio of truncated moments is stable in the sense of weak convergence (see Glanzel [12]), in particular, let us assume that there is a sequence {*Xn*} of avs with a distribution function {*Fn*}, such that the functions *q*1*n*, *q*2*n*, and *ψ<sup>n</sup>* (*n* ∈ *N*) satisfy the conditions of Theorem 1, and let *q*1*<sup>n</sup>* → *q*1, *q*2*<sup>n</sup>* → *q*<sup>2</sup> for some continuously differentiable real functions *q*<sup>1</sup> and *q*2. Finally, let *X* be a chance variable with distribution *F*. Under the condition that *q*1*n*(*X*) and *q*2*n*(*X*) are uniformly integrable and the family {*Fn*} is relatively compact, the sequence *Xn* converges to *X* in distribution if and only if *ψ<sup>n</sup>* converges to *ψ*, where

$$
\psi(x) = \frac{E[q\_2(X) | X \ge x]}{E[q\_1(X) | X \ge x]}.
$$

This stabilization theorem ensures that the precision of the distribution function is duplicated in the subsequent convergence of functions *q*1, *q*2, and *ψn*. It ensures, e.g., that the characterization on the Wald distribution coincides with that on the Levy-Smirnov distribution if *α* → ∞. The application of this theorem over certain challenges in analytical techniques, such as the estimation of the parameters of discrete distributions, is yet another corollary of Theorem 1 s stability condition. The functions *q*1, *q*2, and in particular, *ψ* should be as straightforward and feasible for this reason. Although the function quartet is not distinctive, it is frequently possible to choose *ψ* as a linear combination. As a direct consequence, it is worth considering a few specific instances in order to develop innovative characterizations that capture the link between individual continuous univariate distributions and are relevant in other disciplines of science.

#### **Appendix B**

The components of the observed information matrix are the following

$$\frac{\partial^2 \ell}{\partial \theta^2} = \frac{-n}{\left(1+\theta\right)^2} + \sum\_{i=1}^n \left\{ \frac{2\left(1+\mathbf{x}\_i\right)\left(1+\lambda\mathbf{x}\_i\right)\left[\lambda\beta\left(1+\theta+\theta\mathbf{x}\_i\right) + \theta^2\left(1+\lambda\mathbf{x}\_i\right)\left(1+\lambda\mathbf{x}\_i\right)\right]}{-\left[\lambda\beta\left(1+\mathbf{x}\_i\right) + 2\theta\left(1+\mathbf{x}\_i\right)\left(1+\lambda\mathbf{x}\_i\right)\right]^2} \right\},$$

$$\left[\lambda\beta(1+\theta+\theta\mathbf{x}\_i) + \theta^2(1+\lambda\mathbf{x}\_i)\left(1+\lambda\mathbf{x}\_i\right)\right]^2$$

*∂*2*l ∂θ ∂λ* <sup>=</sup> *n* ∑ *i*=1 ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ (1 + *xi*) (*β* + 2*θxi*) *λβ*(1 + *θ* + *θxi*) + *θ*2(1 + *xi*)(1 + *λxi*) <sup>−</sup>[*λβ* <sup>+</sup> <sup>2</sup>*θ*(<sup>1</sup> <sup>+</sup> *<sup>λ</sup>xi*)] *β*(1 + *θ* + *θxi*) + *θ*<sup>2</sup>*xi*(1 + *xi*) [*λβ*(<sup>1</sup> + *<sup>θ</sup>* + *<sup>θ</sup>xi*) + *<sup>θ</sup>*2(<sup>1</sup> + *<sup>λ</sup>xi*)(<sup>1</sup> + *<sup>λ</sup>xi*)]<sup>2</sup> ⎫ ⎪⎪⎪⎬ ⎪⎪⎪⎭ , *∂*2*l ∂θ ∂β* <sup>=</sup> *<sup>λ</sup> n* ∑ *i*=1 ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ (1 + *xi*) *λβ*(1 + *θ* + *θxi*) + *θ*2(1 + *xi*)(1 + *λxi*) −(1 + *θ* + *θxi*)[*λβ*(1 + *xi*) + 2*θ*(1 + *xi*)(1 + *λxi*)] [*λβ*(<sup>1</sup> + *<sup>θ</sup>* + *<sup>θ</sup>xi*) + *<sup>θ</sup>*2(<sup>1</sup> + *<sup>λ</sup>xi*)(<sup>1</sup> + *<sup>λ</sup>xi*)]<sup>2</sup> ⎫ ⎪⎪⎪⎬ ⎪⎪⎪⎭ , *∂*2*l ∂λ*<sup>2</sup> <sup>=</sup> <sup>−</sup> *n* ∑ *i*=1 *β*(1 + *θ* + *θxi*) + *θ*<sup>2</sup>*xi*(1 + *xi*) 2 [*λβ*(<sup>1</sup> + *<sup>θ</sup>* + *<sup>θ</sup>xi*) + *<sup>θ</sup>*2(<sup>1</sup> + *<sup>λ</sup>xi*)(<sup>1</sup> + *<sup>λ</sup>xi*)]<sup>2</sup> , *∂*2*l ∂λ ∂β* <sup>=</sup> *n* ∑ *i*=1 *xi* 1 + *λxi* + *n* ∑ *i*=1 *θ*2(1 + *xi*)(1 + *θ* + *θxi*) [*λβ*(<sup>1</sup> + *<sup>θ</sup>* + *<sup>θ</sup>xi*) + *<sup>θ</sup>*2(<sup>1</sup> + *<sup>λ</sup>xi*)(<sup>1</sup> + *<sup>λ</sup>xi*)]<sup>2</sup> , *∂*2*l ∂β*<sup>2</sup> <sup>=</sup> <sup>−</sup>*λ*<sup>2</sup> *<sup>n</sup>* ∑ *i*=1 (1 + *θ* + *θxi*) 2 [*λβ*(<sup>1</sup> + *<sup>θ</sup>* + *<sup>θ</sup>xi*) + *<sup>θ</sup>*2(<sup>1</sup> + *<sup>λ</sup>xi*)(<sup>1</sup> + *<sup>λ</sup>xi*)]<sup>2</sup> .

#### **References**

