Estimation methods for the discrete Poisson-Lindley and discrete Lindley distributions with actuarial measures and applications in medicine

2.1 VaR measure

The VaR of any random variable X is the $p^{\mathit{th}}$ quantile of its CDF as shown in Artzner (1999), and it is defined, for a probability level $p\in (0,1)$ , by ${\mathit{Var}}_p=F_X^{-1}\left(p\right)$ .

The VaR of the DPL distribution with PMF (1) is derived as $${\mathit{VaR}}_p=-3-\alpha -\frac{1}{\alpha }-\frac{1}{\log (\alpha +1)}W\left[\frac{{(\alpha +1)}^{-\frac{{\alpha }^2+1}{\alpha }}(p-1)\log (\alpha +1)}{\alpha }\right]\text{.}$$ The VaR of the DL model with PMF (5) takes the form $${\mathit{VaR}}_p=-2-\frac{1}{\beta }-\frac{1}{\beta }W\left[e^{-\beta -1}(\beta +1)(p-1)\right]\text{.}$$

2.2 TVaR Measure

The TVaR is an important actuarial measure, and it is used to determine the expected value of the loss given that an event outside a given probability level has occurred. The TVaR is defined in Klugman et al. (2012) by the following equation $$\begin{array}{cc}\hfill {\mathit{TVaR}}_p=& E\left(X\right|X>x_p)=x_p+\frac{{\displaystyle \sum _{x=x_p}^{\infty }}(x-x_p)}{p\left(x\right)}\hfill \\ \hfill =& x_p+\frac{E\left(X\right)-x_p+{\displaystyle \sum _{x=0}^{x_p-1}}(x_p-x)p\left(x\right)}{1-F\left(x_p\right)}\text{.}\hfill \end{array}$$ Using Eqs. (3) and (4), the TVaR of the DPL distribution is derived as $${\mathit{TVaR}}_p=x_p+\frac{(\alpha +1)\left[\frac{1}{\alpha (\alpha +3)+\alpha x_p+1}+1\right]}{\alpha }\text{.}$$ Similarly, based on Eqs. (7) and (8), the TVaR of the DL distribution follows as $${\mathit{TVaR}}_p=x_p+\frac{e^{\beta }\left[-\beta +e^{\beta }(2\beta +1)+\left(e^{\beta }-1\right)\beta x_p-1\right]}{{\left(e^{\beta }-1\right)}^2\left(2\beta +\beta x_p+1\right)}\text{.}$$

2.3 Simulations for risk measures

In this sub-section, we present some numerical computations for the VaR and TVaR measures of the DPL and DL distributions for different parametric values. The results can be obtained as follows.

1. A random sample of size $n=100$ is generated from the DPL and DL distributions and their parameters are estimated by the maximum likelihood method.

2. The VaR and TVaR of the two distributions are calculated from 2,000 repetitions.

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Fig. 1

Plots of the VaR (top panel) and TVaR (bottom panel) of the DPL and DL distributions using the values in Tables 1 and 2.

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The results in Tables 1 and 2 and the plots in Fig. 1 reveal that the VaR and TVaR measures are increasing functions in the parameter $\alpha$ , for the DPL distribution, and $\beta$ , for the DL distribution.

3.1 Maximum likelihood estimation

Let $x_1,x_2,\dots ,x_n$ be a random sample of size n from the DPL model with PMF (2), then the log-likelihood function reduces to

Let $x_1,x_2,\dots ,x_n$ be a random sample of size n from the DL model with PMF (5), then the log-likelihood function is given by

3.2 Least-squares and weighted least-squares estimation

Let $x_{1:n},x_{2:n},\dots ,x_{n:n}$ be the order statistics from the DPL distribution. By minimizing the following equation, we have LSE (Swain et al., 1988) of the parameter $\alpha$ of the DPL distribution $${\mathit{LS}}_{\mathit{DPL}}={\displaystyle \sum _{i=1}^n}{\left[F\left(x_{i:n}\right)-\frac{i}{n+1}\right]}^2={\displaystyle \sum _{i=1}^n}{\left\{1-{(\alpha +1)}^{-x_{i:n}-3}[1+\alpha (\alpha +x_{i:n}+3\left)\right]-\frac{i}{n+1}\right\}}^2,$$ it can be also obtained by solving the following non-linear equation $${\displaystyle \sum _{i=1}^n}\left\{1-{(\alpha +1)}^{-x_{i:n}-3}[1+\alpha (\alpha +x_{i:n}+3\left)\right]-\frac{i}{n+1}\right\}\mathrm{\Delta }\left(x_{i:n}\right)=0,$$ where

Let $x_{1:n},x_{2:n},\dots ,x_{n:n}$ be the order statistics of a random sample from the DL distribution. The LSE of the DL parameter $\beta$ follows by minimizing the following equation with respect to $\beta$ $${\mathit{LS}}_{\mathit{DL}}={\displaystyle \sum _{i=1}^n}{\left[F\left(x_{i:n}\right)-\frac{i}{n+1}\right]}^2={\displaystyle \sum _{i=1}^n}{\left\{1-\frac{\exp [-\beta (x_{i:n}+1\left)\right]}{1+\beta }[1+\beta (2+x_{i:n}\left)\right]-\frac{i}{n+1}\right\}}^2\text{.}$$ The LSE of $\beta$ can also be obtained by solving the following non-linear equation $${\displaystyle \sum _{i=1}^n}\left\{1-\frac{\exp [-\beta (x_{i:n}+1\left)\right]}{1+\beta }[1+\beta (2+x_{i:n}\left)\right]-\frac{i}{n+1}\right\}\mathrm{\Phi }\left(x_{i:n}\right)=0,$$ where

3.3 Cramér-von Mises Estimation

The CVME of the parameter $\alpha$ of the DPL distribution is obtained by minimizing the following equation $$\begin{array}{cc}\hfill {\mathit{CV}}_{\mathit{DPL}}=& \frac{1}{12n}+{\displaystyle \sum _{i=1}^n}{\left[F\left(x_{i:n}\right)-\frac{2i-1}{2n}\right]}^2\hfill \\ \hfill =& \frac{1}{12n}+{\displaystyle \sum _{i=1}^n}{\left\{1-{(\alpha +1)}^{-x_{i:n}-3}[1+\alpha (\alpha +x_{i:n}+3\left)\right]-\frac{2i-1}{2n}\right\}}^2,\hfill \end{array}$$ or by solving the following non-linear equation $${\displaystyle \sum _{i=1}^n}\left\{1-{(\alpha +1)}^{-x_{i:n}-3}[1+\alpha (\alpha +x_{i:n}+3\left)\right]-\frac{2i-1}{2n}\right\}\mathrm{\Delta }\left(x_{i:n}\right)=0,$$ where $\mathrm{\Delta }\left(x_i\right)$ is defined in Eq. (13). Further details about the CVME can be explored in Macdonald (1971) and Luceño (2006). The CVME of the DL parameter $\beta$ can be calculated by minimizing the following equation $${\mathit{CV}}_{\mathit{DL}}=\frac{1}{12n}+{\displaystyle \sum _{i=1}^n}{\left[F\left(x_{i:n}\right)-\frac{2i-1}{2n}\right]}^2=\frac{1}{12n}+{\displaystyle \sum _{i=1}^n}{\left\{1-\frac{\exp [-\beta (x_{i:n}+1\left)\right]}{1+\beta }[1+\beta (2+x_{i:n}\left)\right]-\frac{2i-1}{2n}\right\}}^2,$$ or by solving the following non-linear equation $${\displaystyle \sum _{i=1}^n}\left\{1-\frac{\exp [-\beta (x_{i:n}+1\left)\right]}{1+\beta }[1+\beta (2+x_{i:n}\left)\right]-\frac{2i-1}{2n}\right\}=0,$$ where $\mathrm{\Phi }\left(x_i\right)$ is given in Eq. (14).

3.4 Percentile Estimation

This estimation method was introduced by Kao (1958, 1959). Let $p_i=\frac{i}{n+1}$ be an estimate of $F\left(x_i\right)$ , then the PCE of DPL parameter $\alpha$ follows by minimizing $${\mathit{PC}}_{\mathit{DPL}}={\displaystyle \sum _{i=1}^n}{\left[x_{i:n}-Q\left(p_i\right)\right]}^2={\displaystyle \sum _{i=1}^n}{\left\{x_{i:n}+3+\alpha +\frac{1}{\alpha }+\frac{W\left(z\right)}{\log (\alpha +1)}\right\}}^2,$$ where $$z=\frac{{(\alpha +1)}^{-\frac{{\alpha }^2+1}{\alpha }}(p_i-1)\log (\alpha +1)}{\alpha }\text{.}$$ We can also obtain the PCE of the parameter $\alpha$ by solving the following non-linear equation $${\displaystyle \sum _{i=1}^n}\left\{x_{i:n}+3+\alpha +\frac{1}{\alpha }+\frac{W\left(z\right)}{\log (\alpha +1)}\right\}\mathrm{\Psi }\left(x_{i:n}\right)=0,$$ where $$\mathrm{\Psi }\left(x_{i:n}\right)=\frac{\partial }{\partial \alpha }Q\left(p_i\right)=\frac{\alpha W\left(z\right)\left[\left({\alpha }^2+\alpha +2\right)\log (\alpha +1)+\alpha W\left(z\right)\right]-(\alpha -1){(\alpha +1)}^2{\log }^2(\alpha +1)}{{\alpha }^2(\alpha +1){\log }^2(\alpha +1)\left(W\right(z)+1)}\text{.}$$ Let $p_i=\frac{i}{n+1}$ is an estimate of $F\left(x_i\right)$ , then the PCE of the DL parameter $\beta$ follows by minimizing $${\mathit{PC}}_{\mathit{DL}}={\displaystyle \sum _{i=1}^n}{[x_{i:n}-Q(p_i\left)\right]}^2={\displaystyle \sum _{i=1}^n}{\left\{x_{i:n}+\frac{2\beta +W\left[e^{-\beta -1}(\beta +1)(p_i-1)\right]+1}{\beta }\right\}}^2\text{.}$$ The PCE of the parameter $\beta$ can also be determined by solving the non-linear equation $${\displaystyle \sum _{i=1}^n}\left\{x_{i:n}+\frac{2\beta +W\left[e^{-\beta -1}(\beta +1)(p_i-1)\right]+1}{\beta }\right\}\mathrm{\Omega }\left(x_{i:n}\right)=0,$$ where $$\mathrm{\Omega }\left(x_{i:n}\right)=\frac{1+W\left[e^{-\beta -1}(\beta +1)(p-1)\right]}{{\beta }^2}+{\left\{\beta +\frac{\beta +1}{W\left[e^{-\beta -1}(\beta +1)(p-1)\right]}+1\right\}}^{-1}\text{.}$$