A new one-parameter discrete exponential distribution: Properties, inference, and applications to COVID-19 data

2.1 Survival, hazard rate, and quantile functions

The SF of the DMEx distribution reduces to $$S\left(x\right)=\left(1+\frac{1+x}{\beta }\right)\exp \left(-\frac{1+x}{\beta }\right).$$

The HRF of the DMEx distribution takes the form $$h\left(x\right)=\frac{\left(1+\frac{x}{\beta }\right)-\left(1+\frac{1+x}{\beta }\right)\exp \left(-\frac{1}{\beta }\right)}{\left(1+\frac{1+x}{\beta }\right)\exp \left(-\frac{1}{\beta }\right)}.$$

Fig. 2 shows that the behavior of the HRF of the DMEx model is increasing for different values of $\beta$ .

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Fig. 2

The HRF plots of the DMEx distribution.

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The second rate of failure, say $\phi \left(x\right)$ , of the DMEs model is defined as. $$\phi \left(x\right)=\log \left[\frac{S\left(x\right)}{S\left(x+1\right)}\right]=\log \left[\frac{\left(1+\frac{1+x}{\beta }\right)\exp \left(\frac{1}{\beta }\right)}{\left(1+\frac{2+x}{\beta }\right)}\right].$$

The reversed HRF of the DMEx distribution is $$r^{\text{*}}\left(x\right)=\frac{p\left(x\right)}{F\left(x\right)}=\frac{\exp \left(-\frac{x}{\beta }\right)\left[\left(1+\frac{x}{\beta }\right)-\left(1+\frac{1+x}{\beta }\right)\exp \left(-\frac{1}{\beta }\right)\right]}{1-\left(1+\frac{1+x}{\beta }\right)\exp \left(-\frac{1+x}{\beta }\right)}.$$

The quantile function (QF) of the DMEx model takes the form $$Q\left(p\right)=-1-\beta -\beta W\left(\frac{\left(p-1\right)}{\text{exp}\left(1\right)}\right),0<p<1,$$ where $x$ is the integer part of $x$ and $W\left(.\right)$ is the negative branch of the Lambert $W$ function.

2.2 Probability generating function and moments

The probability generating function of the DMEx distribution follows as $$G_x\left(Z\right)=\sum _{x=0}^{\infty }{Z}^{x}p\left(X\right)=1+\left(Z-1\right)\sum _{x=1}^{\infty }{Z}^{x-1}\left(1+\frac{x}{\beta }\right)\exp \left(-\frac{x}{\beta }\right).$$

Differentiating $G_x\left(Z\right)$ with respect to $Z$ and setting $Z=1$ , we can obtain the first four factorial moments of the DMEx distribution as follows $$G^{\prime }x\left(1\right)=\sum _{x=1}^{\infty }\left(1+\frac{x}{\beta }\right)\exp \left(-\frac{x}{\beta }\right).$$ $$G_x^{\text{''}}\left(1\right)=2\sum _{x=1}^{\infty }\left(x-1\right)\left(1+\frac{x}{\beta }\right)\exp \left(-\frac{x}{\beta }\right).$$ $$G_x^{\text{'''}}\left(1\right)=3\sum _{x=1}^{\infty }\left(x-1\right)\left(x-2\right)\left(1+\frac{x}{\beta }\right)\exp \left(-\frac{x}{\beta }\right).$$ $$G_x^{\text{''''}}\left(1\right)=4\sum _{x=1}^{\infty }\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(1+\frac{x}{\beta }\right)\exp \left(-\frac{x}{\beta }\right).$$

The first four ordinary moments of the DMEx distribution can be calculated using the factorial moments as follows $$\mu =G^{\prime }x\left(1\right)=\sum _{x=1}^{\infty }\left(1+\frac{x}{\beta }\right)\exp \left(-\frac{x}{\beta }\right).$$ $${\mu }_2^{\prime }=G^{″}x\left(1\right)+G^{\prime }x\left(1\right)=\sum _{x=1}^{\infty }\left(2x-1\right)\left(1+\frac{x}{\beta }\right)\exp \left(-\frac{x}{\beta }\right).$$ $${\mu }_3^{\prime }=G^{\prime \prime \prime }x\left(1\right)+3G{}^{\prime }x\left(1\right)+G^{\prime }x\left(1\right)\sum _{x=1}^{\infty }\left(3x^2-3x+1\right)\left(1+\frac{x}{\beta }\right)\exp \left(-\frac{x}{\beta }\right).$$ $${\mu }_4^{\prime }=G^{\prime \prime \prime \prime }x\left(1\right)+6G^{\prime \prime \prime }x\left(1\right)+7G^{\prime \prime }x\left(1\right)+G^{\prime }x\left(1\right)=\sum _{x=1}^{\infty }\left(4x^3-6x^2+4x-1\right)\left(1+\frac{x}{\beta }\right)\exp \left(-\frac{x}{\beta }\right).$$

The variance of the DMEx distribution has the form $${\sigma }^2=\sum _{x=1}^{\infty }\left(2x-1\right)\left(1+\frac{x}{\beta }\right)\exp \left(-\frac{x}{\beta }\right)-{\left[\sum _{x=1}^{\infty }\left(1+\frac{x}{\beta }\right)\exp \left(-\frac{x}{\beta }\right)\right]}^2.$$

Further, the coefficient of skewness and kurtosis of the DMEx distribution can be calculated by the formulae $$\mathit{CS}=\frac{{\mu }_3^{\prime }-3{\mu }_2^{\prime }\mu +2{\mu }^3}{{\left({\sigma }^2\right)}^{\frac{3}{2}}}$$ and $$\mathit{CK}=\frac{{\mu }_4^{\prime }-4{\mu }_3^{\prime }\mu +6{\mu }_2^{\prime }{\mu }^2-3{\mu }^4}{{\left({\sigma }^2\right)}^2}.$$

2.3 Dispersion index

The index of dispersion (DI) is defined as $$\mathit{DI}=\frac{\mathit{Variance}}{\mathit{Mean}}.$$

The DI shows that whether a distribution is suitable for modeling under-dispersed data or over-dispersed data. The distribution is over-dispersed for $\mathit{DI}>1$ and for $\mathit{DI}<1$ it is under-dispersed. Table 1 reports some numerical values for the descriptive measures of the DMEx distribution. Table 1 shows that the DMEs model is useful for both over-dispersed and under-dispersed data sets. The DMEx also exhibits positively skewed which supported by the plots of its PMF in Fig. 1.

3.1 Maximum likelihood (ML)

Suppose $x_1,\dots ,x_n$ be a random sample of size $n$ from the DMEx distribution. Then, the log-likelihood function reduces to $$\log L=-\frac{1}{\beta }\sum _{i=1}^n{x}_i+\sum _{i=1}^n\left[\left(1+\frac{x_i}{\beta }\right)-\left(1+\frac{x_i+1}{\beta }\right)\exp \left(-\frac{1}{\beta }\right)\right].$$

The ML estimator (MLE) of $\beta$ follows by solving $\frac{\partial L}{\partial \beta }=0,$ that is $$\frac{\mathit{dL}}{d\beta }=\frac{1}{{\beta }^2}\sum _{i=1}^n{x}_i+\sum _{i=1}^n\frac{\left(-\frac{x_i}{{\beta }^2}\right)-\frac{1}{{\beta }^2}\left(1+\frac{x_i+1}{\beta }\right)\exp \left(-\frac{1}{\beta }\right)+\left(\frac{x_i+1}{{\beta }^2}\right)\exp \left(-\frac{1}{\beta }\right)}{\left(1+\frac{x_i}{\beta }\right)-\left(1+\frac{x_i+1}{\beta }\right)\text{exp}\left(-\frac{1}{\beta }\right)}=0.$$

The MLE of $\beta$ cannot be calculated explicitly. Hence, numerical methods can be adopted to obtain it.

3.2 Methods of Anderson-Darling and right-tail Anderson-Darling

Let $X_{i:n}$ be the $i$ th order statistic in a sample of size $n$ . The Anderson-Darling (AD) test is proposed by (Anderson and Darling, 1952). Let $\widehat{\beta }$ be the AD estimator (ADE) which is obtained by minimizing the following equation $$A\left(\beta \right)=-\sum _{i=1}^n\frac{2_i-1}{n}\left[\log \left(1-\left(1+\frac{1+x_{i:n}}{\beta }\right){\text{e}}^{-\left(\frac{1+x_{i:n}}{\beta }\right)}\right)+\log \left(1-\left(1+\frac{1+x_{n+1-i:n}}{\beta }\right){\text{e}}^{-\left(\frac{1+x_{n+1-i:n}}{\beta }\right)}\right)\right]-n.$$

This estimator of $\beta$ can also be derived by solving $$\sum _{i=1}^{n}2\left(i-1\right)\left[\frac{\phi \left(x_{i:n}|\beta \right)}{F\left(x_{i:n}|\beta \right)}-\frac{\phi \left(x_{n+1-i:n}|\beta \right)}{\overline{F}\left(x_{n+1-i:n}|\beta \right)}\right]=0,$$ where $\phi \left(x_{i:n}|\beta \right)=\frac{d}{d\beta }F\left(x_{i:n}|\beta \right)$ and it reduces to

The right-tail Anderson-Darling (RADE) of the parameter $\beta$ follows by minimizing $$RT\left(\beta \right)=\frac{n}{2}-2\sum _{i=1}^n\left[1-\left(1+\frac{1+x_{i:n}}{\beta }\right)e^{-\left(\frac{1+x_{i:n}}{\beta }\right)}\right]-\frac{1}{n}\sum _{i=1}^{n}2\left(i-1\right)\log \left[\left(1+\frac{1+x_{n+1-i:n}}{\beta }\right)e^{-\left(\frac{1+x_{n+1-i:n}}{\beta }\right)}\right]$$

The RADE is also calculated by solving $$-2\sum _{i=1}^n\varphi \left(x_{i:n}|\beta \right)+\frac{1}{n}\sum _{i=1}^n\left(2i-1\right)\frac{\phi \left(x_{n+1-1:n}|\beta \right)}{1-F\left(x_{n+1-i:n}|\beta \right)}=0$$

3.3 Method of Cramèr-von-Misses

Macdonald (Macdonald, 1971) proved that the Cramèr-von-Mises estimator (CVME) has a smaller bias as compared to other minimum distance type estimators. The CVME of $\beta$ follows by minimizing $$C\left(\beta \right)=\sum _{i=1}^n{\left[1-\frac{2_i-1}{2n}-\left(1+\frac{1+x_{i:n}}{\beta }\right){\text{e}}^{-\left(\frac{1+x_{i:n}}{\beta }\right)}\right]}^2+\frac{1}{12n}.$$

The CVME of $\beta$ is also obtained by solving $$\sum _{i=1}^n\left[1-\frac{2i-1}{2n}-\left(1+\frac{1+x_{i:n}}{\beta }\right)e^{-\left(\frac{1+x_{i:n}}{\beta }\right)}\right]\phi \left(x_{i:n}|\beta \right)=0$$

3.4 Methods of least-squares and weighted least-squares

The least-squares estimator (LSE) of $\beta$ follows by minimizing $$LS\left(\beta \right)=\sum _{i=1}^n{\left[1-\frac{i}{1+n}-\left(1+\frac{1+x_{i:n}}{\beta }\right){\text{e}}^{-\left(\frac{1+x_{i:n}}{\beta }\right)}\right]}^2,$$ with respect to $\beta$ . Moreover, the LSE of $\beta$ is also obtained by solving $$\sum _{i=1}^m\left[1-\frac{i}{1+n}-\left(1+\frac{1+x_{i:n}}{\beta }\right)e^{-\left(\frac{1+x_{i:n}}{\beta }\right)}\right]\phi \left(x_{i:n}|\beta \right)=0,$$

The weighted least-squares estimator (WLSE) of $\beta$ follows by minimizing $$W\left(\beta \right)=\sum _{i=1}^n{1\frac{{\left(1+n\right)}^2\left(2+n\right)}{i\left(n-i+1\right)}\left[1-\frac{i}{1+n}-\left(1+\frac{1+x_{i:n}}{\beta }\right)e^{-\left(\frac{1+x_{i:n}}{\beta }\right)}\right]}^2$$

The WLSE of $\beta$ can also be obtained by solving $$\sum _{i=1}^{n}1\frac{{\left(1+n\right)}^2\left(2+n\right)}{i\left(n-i+1\right)}\left[-\frac{i}{1+n}-\left(1+\frac{1+x_{i:n}}{\beta }\right)e^{-\left(\frac{1+x_{i:n}}{\beta }\right)}\right]\phi \left(x_{i:n}|\beta \right)=0,$$

In which $\phi \left(x_{i:n}|\beta \right)$ is presented in (6).

3.5 Methods of percentiles

Suppose that $u_i=i/\left(m+1\right)$ is an unbiased estimator of $F\left(x_{i:n}|\beta \right)$ . Hence, the percentiles estimator (PCE) (Kao, 1959) of the parameter $\beta$ can be derived by minimizing the following equation $$P\left(\theta \right)=\sum _{i=1}^m{\left\{x_{i:n}+1+\beta +\beta W\left(\frac{\left(u_i-1\right)}{\text{exp}\left(1\right)}\right)\right\}}^2,$$ where $W\left(.\right)$ is the negative branch of the Lambert $W$ function.