*2.2. Identifiability*

A set of unknown parameters of a model is stated to be identifiable if different sets of parameters give different distributions for a given *x*. Here, the identifiability property of the DPsL distribution is proved. Let *P*DPsL(*x*; *λ*1) and *P*DPsL(*x*; *λ*2) be different pmfs of the DPsL distribution indexed by *λ*<sup>1</sup> = (*θ*1, *β*1) and *λ*<sup>2</sup> = (*θ*2, *β*2), respectively. Then, the likelihood ratio is given by:

$$\begin{array}{rcl} \mathbf{U} &=& \frac{P\_{\mathrm{DPsL}}(\mathbf{x};\lambda\_{1})}{P\_{\mathrm{DPsL}}(\mathbf{x};\lambda\_{2})} \\ &=& \frac{\frac{(\beta\_{1}+\theta\_{1}\mathbf{x})e^{-\theta\_{1}\mathbf{x}}-(\beta\_{1}+\theta\_{1}(\mathbf{x}+1))e^{-\theta\_{1}(\mathbf{x}+1)}}{\beta\_{1}} \\ &=& \frac{\frac{(\beta\_{2}+\theta\_{2}\mathbf{x})e^{-\theta\_{2}\mathbf{x}}-(\beta\_{2}+\theta\_{2}(\mathbf{x}+1))e^{-\theta\_{2}(\mathbf{x}+1)}}{\beta\_{2}} \\ &=& \frac{\frac{\beta\_{2}}{\beta\_{1}}}{\beta\_{1}}\frac{(\beta\_{1}+\theta\_{1}\mathbf{x})e^{-\theta\_{1}\mathbf{x}}-(\beta\_{1}+\theta\_{1}(\mathbf{x}+1))e^{-\theta\_{1}(\mathbf{x}+1)}}{(\beta\_{2}+\theta\_{2}\mathbf{x})e^{-\theta\_{2}\mathbf{x}}-(\beta\_{2}+\theta\_{2}(\mathbf{x}+1))e^{-\theta\_{2}(\mathbf{x}+1)}}. \end{array} \tag{5}$$

Taking logarithm of this ratio, we obtained:

$$\begin{split} \log \mathcal{U}\_{-} &= \quad \log \left( \frac{\beta\_{2}}{\beta\_{1}} \right) + \log \left( (\beta\_{1} + \theta\_{1} \mathbf{x}) e^{-\theta\_{1} \mathbf{x}} - (\beta\_{1} + \theta\_{1} (\mathbf{x} + 1)) e^{-\theta\_{1} (\mathbf{x} + 1)} \right) \\ &- \log \left( (\beta\_{2} + \theta\_{2} \mathbf{x}) e^{-\theta\_{2} \mathbf{x}} - (\beta\_{2} + \theta\_{2} (\mathbf{x} + 1)) e^{-\theta\_{2} (\mathbf{x} + 1)} \right). \end{split}$$

Now, by considering *x* as a continuous variable and taking the partial derivative of log U with respect to *x* and equating it to 0, we obtained:

$$\frac{\theta\_1 \left(\theta\_1 + \theta\_1 - 1 + \theta\_1 \mathbf{x} - \epsilon^{\theta\_1} (\theta\_1 \mathbf{x} + \theta\_1 - 1)\right)}{(\theta\_1 + \theta\_1 \mathbf{x})e^{-\theta\_1 \mathbf{x}} - (\theta\_1 + \theta\_1 (\mathbf{x} + 1))e^{-\theta\_1 (\mathbf{x} + 1)}} = \frac{\theta\_2 (\theta\_2 + \theta\_2 - 1 + \theta\_2 \mathbf{x} - \epsilon^{\theta\_2} (\theta\_2 \mathbf{x} + \theta\_2 - 1))}{(\theta\_2 + \theta\_2 \mathbf{x})e^{-\theta\_2 \mathbf{x}} - (\theta\_2 + \theta\_2 (\mathbf{x} + 1))e^{-\theta\_2 (\mathbf{x} + 1)}},$$

which implies that:

$$e^{-\left(\theta\_{2}-\theta\_{1}\right)\mathbf{x}}\frac{\left(\beta\_{2}+\theta\_{2}\mathbf{x}\right)-\left(\beta\_{2}+\theta\_{2}\left(\mathbf{x}+1\right)\right)\mathbf{e}^{-\theta\_{2}}}{\left(\beta\_{1}+\theta\_{1}\mathbf{x}\right)-\left(\beta\_{1}+\theta\_{1}\left(\mathbf{x}+1\right)\right)\mathbf{e}^{-\theta\_{1}}}=\frac{\theta\_{2}\left(\theta\_{2}+\beta\_{2}-1+\theta\_{2}\mathbf{x}-\mathbf{e}^{\theta\_{2}}\left(\theta\_{2}\mathbf{x}+\beta\_{2}-1\right)\right)}{\theta\_{1}\left(\theta\_{1}+\beta\_{1}-1+\theta\_{1}\mathbf{x}-\mathbf{e}^{\theta\_{1}}\left(\theta\_{1}\mathbf{x}+\beta\_{1}-1\right)\right)}.$$

By performing *<sup>x</sup>* <sup>→</sup> <sup>+</sup>∞, we obtained 0 <sup>=</sup> *<sup>θ</sup>*<sup>2</sup> <sup>2</sup> (1−*eθ*<sup>2</sup> ) *θ*2 <sup>1</sup> (1−*eθ*<sup>1</sup> ) or <sup>+</sup><sup>∞</sup> <sup>=</sup> *<sup>θ</sup>*<sup>2</sup> <sup>2</sup> (1−*eθ*<sup>2</sup> ) *θ*2 <sup>1</sup> (1−*eθ*<sup>1</sup> ) according to *θ*<sup>2</sup> > *θ*<sup>1</sup> or *θ*<sup>2</sup> < *θ*1, respectively, which is impossible since *θ*<sup>1</sup> > 0 and *θ*<sup>2</sup> > 0. Therefore, *θ*<sup>1</sup> = *θ*2. By taking into account this equality, by taking *x* = 0 in (5), we obtained *<sup>β</sup>*1−(*β*1+*θ*1)*e*−*θ*<sup>1</sup> *<sup>β</sup>*2−(*β*2+*θ*1)*e*−*θ*<sup>1</sup> <sup>=</sup> *<sup>β</sup>*<sup>1</sup> *β*2 , which is possible if, and only if, *β*<sup>1</sup> = *β*2. Therefore, we concluded that the DPsL model is identifiable and that the parameters uniquely determine the distribution, that is, *P*DPsL(*x*; *λ*1) = *P*DPsL(*x*; *λ*2) ⇐⇒ *λ*<sup>1</sup> = *λ*2.

#### *2.3. Moments, Skewness and Kurtosis*

In the rest of the study, *X* denotes a random variable that follows the DPsL distribution. Then, the probability generating function (pgf) of *X* can be derived as:

$$\begin{aligned} G(\mathbf{s}) &= \quad \mathcal{E}(\mathbf{s}^X) = \sum\_{\mathbf{x}=\mathbf{0}}^{\infty} \mathbf{s}^x \mathcal{P}\_{\mathrm{DFsL}}(\mathbf{x}; \boldsymbol{\theta}, \boldsymbol{\beta}) \\ &= \quad \frac{\mathfrak{e}^{2\mathfrak{g}}\boldsymbol{\beta} - \mathfrak{e}^{\mathfrak{g}}(\boldsymbol{\beta} + \mathfrak{s}\boldsymbol{\beta} + \boldsymbol{\theta} - \mathfrak{e}\boldsymbol{\mathfrak{s}}) + \mathfrak{s}\boldsymbol{\beta}}{(\mathfrak{e}^{\mathfrak{g}} - \mathfrak{s})^2 \boldsymbol{\beta}}, \quad |\mathbf{s}| < \mathfrak{e}^{\mathfrak{g}}. \end{aligned}$$

When *s* in pgf is substituted by *e<sup>t</sup>* , the moment generating function (mgf) follows as:

$$M(t) = \mathcal{E}(\varepsilon^{tX}) = \frac{\varepsilon^{2\theta}\beta - \varepsilon^{\theta}(\beta + \varepsilon^{t}\beta + \theta - \theta\varepsilon^{t}) + \varepsilon^{t}\beta}{(\varepsilon^{\theta} - \varepsilon^{t})^{2}\beta}, \ t < \theta.$$

By using the well-known relationship between *M*(*t*) and the (standard) moments of *X*, the first four moments of the DPsL distribution are:

$$\mathcal{E}(X) = \frac{\epsilon^{\theta}(\beta + \theta) - \beta}{(\epsilon^{\theta} - 1)^{2}\beta},\tag{6}$$

$$\mathcal{E}(X^{2}) = \frac{\epsilon^{2\theta}\beta + 3\epsilon^{\theta}\theta + \epsilon^{2\theta}\theta - \beta}{(\epsilon^{\theta} - 1)^{3}\beta},$$

$$\mathcal{E}(X^{3}) = \frac{-\beta - 3\epsilon^{\theta}\beta + 3\epsilon^{2\theta}\theta + \epsilon^{2\theta}\beta + 7\epsilon^{\theta}\theta + 10\epsilon^{2\theta}\theta + \epsilon^{3\theta}\theta}{(\epsilon^{\theta} - 1)^{4}\beta}$$

and

$$\operatorname{E}(X^{4}) = \frac{-\beta - 10e^{\theta}\beta + 10e^{3\theta}\beta + e^{4\theta}\beta + 15e^{\theta}\theta + 55e^{2\theta}\theta + 25e^{3\theta}\theta + e^{4\theta}\theta}{\left(e^{\theta} - 1\right)^{5}\beta}.$$

Based on E(*X*) and E(*X*2), the variance of *X* follows from the Koenig–Huygens formula as:

$$\text{Var}(X) = \frac{e^{\theta} [(e^{\theta} - 1)^2 \beta^2 + (e^{2\theta} - 1)\beta \theta - e^{\theta} \theta^2]}{(e^{\theta} - 1)^4 \beta^2}. \tag{7}$$

Expressions for skewness and kurtosis of the DPsL distribution can be derived explicitly by using the following formulas:

$$\text{Skewness}(X) = \frac{\text{E}\{X^3\} - 3\text{E}\{X^2\}\text{E}(X) + 2[\text{E}(X)]^3}{[\text{Var}(X)]^{3/2}}$$

and

$$\text{Kurtosis}(X) = \frac{\text{E}\left(X^4\right) - 4\text{E}\left(X^2\right)\text{E}(X) + 6\text{E}\left(X^2\right)[\text{E}(X)]^2 - 3[\text{E}(X)]^4}{[\text{Var}(X)]^2}.$$

#### *2.4. Coefficient of Variation and Dispersion Index*

The expressions of the coefficient of variation (*CV*) and dispersion index (*DI*) of *X* are given by:

$$\text{CV}(X) \;= \; ^1 \frac{\sqrt{\text{Var}(X)}}{\text{E}(X)} = \frac{\sqrt{(\epsilon^{\theta} - 1)^2 \beta^2 + (\epsilon^{2\theta} - 1)\beta \theta - \epsilon^{\theta} \theta^2}}{\sqrt{\epsilon^{\theta}} ((\beta + \theta) - \beta \epsilon^{-\theta})}$$

and

$$DI(X) \quad = \frac{\text{Var}(X)}{\text{E}(X)} = \frac{(e^{\theta} - 1)^2 \beta^2 + (e^{2\theta} - 1)\theta\theta - e^{\theta}\theta^2}{(e^{\theta} - 1)^2 \beta e^{\theta} (\beta + \theta)},\tag{8}$$

respectively.

In full generality, when the DI is one, the distribution is equi-dispersed, and if DI is greater than (less than) one, the distribution is over-dispersed (under-dispersed). Some numerical values of the mean, variance, DI, skewness and kurtosis for the DPsL distribution for some values of the parameters are presented in Tables 1 and 2.

From the information contained in these tables, it is clear that the DPsL distribution would be an appropriate option for modelling under as well as over-dispersed and positively skewed datasets.


**Table 1.** Values for some moment measures for the DPsL distribution for *β* = 1.5 and different values of *θ*.

**Table 2.** Values for some moment measures for the DPsL distribution for *θ* = 2 and different values of *β*.


#### *2.5. Mean Residual Lifetime and Mean Past Lifetime*

The mean residual lifetime (mrl) and mean past lifetime (mpl) of a component are two widely used measures to study the ageing behaviour of components. Both measures characterize the distribution uniquely. By assuming that the lifetime of a component is modelled by *X*, the mrl of *X* at *i* = 0, 1, 2, . . . is defined as:

$$\begin{aligned} \mathcal{Z}(i) &=& \operatorname{E}(X - i \mid X \ge i) \\ &=& \frac{1}{1 - F\_{\text{DPsL}}(i - 1; \theta\_\prime \beta)} \sum\_{j = i + 1}^{\infty} (1 - F\_{\text{DPsL}}(j - 1; \theta\_\prime \beta)) .\end{aligned}$$

That is:

$$\begin{aligned} \mathcal{L}(i) &= \begin{array}{c} 1 \\ \frac{\varepsilon^{-\theta i}(\beta + \theta i)}{\varepsilon} \end{array} \sum\_{j=i+1}^{\infty} \varepsilon^{-\theta j}(\beta + \theta j) \\ &= \begin{array}{c} \frac{\varepsilon^{i\theta}\left((\varepsilon^{\theta} - 1)\beta - i\theta + \varepsilon^{\theta}(1 + i)\theta\right)}{\varepsilon^{-\theta i}(\beta + \theta i)(\varepsilon^{\theta} - 1)^{2}}. \end{array} \end{aligned}$$

Furthermore, the mpl of *X* is another reliability measure that corresponds to the time elapsed since the failure of *X* given that the system has already failed before some *i*. Thus, the mpl of *X* at *i* = 1, 2, . . . is defined by:

$$\begin{aligned} \mathcal{L}^\*(i) &= \begin{array}{c} \mathrm{E}(i-X \mid X < i) \\ \hline \hline \mathrm{F\_{\mathrm{DPsL}}(i-1; \theta, \beta)} \end{array} \sum\_{m=1}^i F\_{\mathrm{DPsL}}(m-1; \theta, \beta), \end{aligned}$$

where *ζ*∗(0) = 0. That is:

$$\begin{split} \zeta\_{\varepsilon}^{\*}\left(i\right) &= \begin{array}{c} 1 \\ \beta - \varepsilon^{-\theta i}\left(\beta + i\theta\right) \end{array} \sum\_{m=1}^{i} \left(\beta - \varepsilon^{-m\theta}\left(\beta + m\theta\right)\right) \\ &= \frac{\varepsilon^{-i\theta}}{\beta - \varepsilon^{-\theta i}\left(\beta + i\theta\right)\left(\varepsilon^{\theta} - 1\right)^{2}} \times \\ &e^{-i\theta} \left\{ \left[ \left(\varepsilon^{\theta} - 1\right)\left(1 + \varepsilon^{\theta i}(1 + i) - \varepsilon^{\theta i}(1 + i)\right) \right] \beta - \left[\varepsilon^{\theta(1 + i)} + i - \varepsilon^{\theta}(1 + i)\right] \theta \right\}. \end{split}$$

#### *2.6. Stress–Strength Analysis*

Stress–strength reliability has wide applications in almost all fields of engineering and machine learning. Let *Xstress* and *Xstrength* be random variables that model the stress and strength of a system, respectively. Then, the expected reliability can be calculated by the following formula:

$$\operatorname{Re}\_{\mathrm{Stress}-\mathrm{Strength}} = \Pr\left[X\_{\mathrm{Stress}} \le X\_{\mathrm{Strength}}\right] = \sum\_{\mathbf{x}=\mathbf{0}}^{\infty} P\_{\mathbf{X}\_{\mathrm{Stress}}}(\mathbf{x}) S\_{\mathbf{X}\_{\mathrm{Strength}}}(\mathbf{x})\_{\mathbf{x}}$$

where *PX*(*x*) and *SX*(*x*) denote the pmf and sf, respectively, of a random variable *X*. Suppose that *Xstress* and *Xstrength* are two independent random variables following the DPsL (*θ*1, *β*1) and DPsL (*θ*2, *β*2) distributions, respectively. Then, from (3) and (4), the expected reliability is obtained in closed form as:

$$\begin{split} & \operatorname{Re}\_{\mathrm{Stress}-\mathrm{Strength}} = \\ & \frac{1}{\beta\_1 \beta\_2 (\epsilon^{\theta\_1 + \theta\_2} - 1)^3} \Big\{ (\epsilon^{\theta\_1} - 1)(\epsilon^{\theta\_1 + \theta\_2} - 1) \beta\_1 \left[ (\epsilon^{\theta\_1 + \theta\_2} - 1)\beta\_2 + \theta\_2 \epsilon^{\theta\_1 + \theta\_2} \right] \Big| \\ & - \theta\_1 \epsilon^{\theta\_1} \Big[ (\epsilon^{\theta\_2} - 1)(\epsilon^{\theta\_1 + \theta\_2} - 1) \beta\_2 + \epsilon^{\theta\_2} (1 - 2\epsilon^{\theta\_1} + \epsilon^{\theta\_1 \theta\_2}) \theta\_2 \Big] \Big\}. \end{split}$$

Some numerical values for *ReStress*−*Strength* for different values of the parameters are given in Tables 3–5.

From Tables 3 and 4, it is clear that the expected reliability increases (decreases) as *β*<sup>1</sup> → ∞ (*β*<sup>2</sup> → ∞). In addition, from Table 5, the expected reliability (decreases) as *θ*<sup>1</sup> → ∞ (*θ*<sup>2</sup> → ∞).

**Table 3.** Numerical values of *ReStress*−*Strength* associated with the DPsL distribution at *<sup>θ</sup>*<sup>1</sup> = 0.3, *θ*<sup>2</sup> = 0.1 for different values of *β*<sup>1</sup> and *β*2.



**Table 4.** Numerical values of *ReStress*−*Strength* associated with the DPsL distribution at *<sup>θ</sup>*<sup>1</sup> = 0.6, *θ*<sup>2</sup> = 0.01 for different values of *β*<sup>1</sup> and *β*2.

**Table 5.** Numerical values of *ReStress*−*Strength* associated with the DPsL distribution at *<sup>β</sup>*<sup>1</sup> = 1, *β*<sup>2</sup> = 1.5 for different values of *θ*<sup>1</sup> and *θ*2.

