A bounded distribution derived from the shifted Gompertz law

The random variable $T=\min \{T_1,T_2,\dots ,T_M\}$ has a $\text{LSG}(\alpha ,\beta )$ distribution.

For any $m=1,2,\dots$ and $\beta >0$ , the conditional cdf of the random variable $T|M=m$ is given by $F_{T|M=m}(t\text{;}\beta )=1-{(1-t^{\beta })}^m,0<t<1$ . Therefore, $$P(T⩽t,M=m)=\left[1-{(1-t^{\beta })}^m\right]e^{-\alpha }\frac{{\alpha }^{m-1}}{(m-1)!}\text{.}$$ Hence, for any $\alpha >0$ and $\beta >0$ the marginal cdf of T is given by $$F_T(t\text{;}\alpha ,\beta )={\displaystyle \sum _{m=1}^{\infty }}P(T⩽t,M=m)=1-(1-t^{\beta })\exp \left(-\alpha t^{\beta }\right),0<t<1,$$ which implies the result.   □

Let X be a random variable having a $\text{LSG}(\alpha ,\beta )$ distribution.

1. For any $0<\beta ⩽1$ and $\alpha >0,f(x\text{;}\alpha ,\beta )$ is a decreasing function in x.

2. For any $\beta >1$ and $0<\alpha ⩽(\beta -1)/(2\beta )$ , $f(x\text{;}\alpha ,\beta )$ is an increasing function in x.

3. For any $\beta >1$ and $\alpha >(\beta -1)/(2\beta )$ , $f(x\text{;}\alpha ,\beta )$ has a mode given by $$x={\left(\frac{\beta (\alpha +3)-1-\sqrt{{(\alpha \beta +1)}^2+\beta (2\alpha \beta +5\beta -2)}}{2\alpha \beta }\right)}^{1/\beta }\text{.}$$ Morover, for any $\alpha >0$ $${\displaystyle \underset{x\to 0^{+}}{\lim }}f(x\text{;}\alpha ,\beta )=\left\{\begin{array}{cc}\infty \hfill & \mathrm{if}\beta \in (0,1),\hfill \\ 1+\alpha \hfill & \mathrm{if}\beta =1,\hfill \\ 0\hfill & \mathrm{if}\beta >1,\hfill \end{array}\right.$$ and $f(1\text{;}\alpha ,\beta )=\beta e^{-\alpha }$ for any $\alpha >0$ and $\beta >0$ .

Let X be a random variable having a $\text{LSG}(\alpha ,\beta )$ distribution. For $k=1,2,\dots$ , the k-th moment of X is given by

By making the change of variable $u=\alpha x^{\beta }$ and taking into account (4), $$E[X^k]={\int }_0^1{x}^{k}f(x\text{;}\alpha ,\beta )\mathit{dx}={\alpha }^{-(\frac{k}{\beta }+1)}{\int }_0^{\alpha }{u}^{k/\beta }{e}^{-u}(1+\alpha -u)\mathit{du}={\alpha }^{-(\frac{k}{\beta }+1)}\left[(1+\alpha )\gamma \left(\frac{k}{\beta }+1,\alpha \right)-\gamma \left(\frac{k}{\beta }+2,\alpha \right)\right]\text{.}$$ The result follows by applying to the above equation the recurrence formula $\gamma (a,z)=(a-1)\gamma (a-1,z)-z^{a-1}{e}^{-z}$ (Paris, 2010, pp. 178).   □

Let X be a random variable having a $\text{LSG}(\alpha ,\beta )$ distribution. For $k=1,2,\dots$ , the k-th moment of X is given by

The result follows from Eq. (5) together with the fact that $\gamma (a,z)=\mathrm{\Gamma }(a)-\mathrm{\Gamma }(a,z),a\ne 0,-1,-2,\dots$ (Paris, 2010, pp. 174).   □

The quantile function of the $\text{LSG}(\alpha ,\beta )$ distribution is given by

For any $0<u<1$ , we have to solve with respect to x the equation $F(x\text{;}\alpha ,\beta )=u$ , which is equivalent to solve $F_Y(-\log x\text{;}\alpha ,\beta )=1-u$ , with $F_Y$ given by Eq. (1). The solution of the latter equation is $x=\exp \left(-F_Y^{-1}(1-u\text{;}\alpha ,\beta )\right)$ . Now, by taking into account the closed-form expression for $F_Y^{-1}$ provided in Jiménez and Jodrá (2009, Proposition 3.1), we get $$F^{-1}(u\text{;}\alpha ,\beta )=\exp \left(\frac{1}{\beta }\log \left(1-\frac{1}{\alpha }{W}_0\left(\alpha e^{\alpha }(1-u)\right)\right)\right),$$ which is equivalent to (7).   □

(i) If $\beta ⩾1$ then the LSG law has an increasing failure rate (IFR). (ii) If $\beta \in (0,1)$ then there exists $x_0=x_0(\alpha ,\beta )\in (0,1)$ so that $h(x\text{;}\alpha ,\beta )$ is (strictly) decreasing when $x<x_0$ and (strictly) increasing when $x>x_0$ .

The first derivative of h can be written as below $$h^{\prime }(x\text{;}\alpha ,\beta )=\frac{\partial }{\partial x}h(x\text{;}\alpha ,\beta )=\frac{\beta x^{\beta -2}}{{(1-x^{\beta })}^2}\left[(\beta -1)\left(1+\alpha {(1-x^{\beta })}^2\right)+x^{\beta }\right]\text{.}$$ Part (i) follows because $h^{\prime }(x\text{;}\alpha ,\beta )⩾0$ for any $\beta ⩾1,\alpha >0$ and $x\in (0,1)$ . To prove part (ii), let $c(x\text{;}\alpha ,\beta )=(\beta -1)\left[\alpha {(1-x^{\beta })}^2+1\right]+x^{\beta }$ . For any $\alpha >0,\beta \in (0,1)$ and $x\in (0,1)$ , note that $c(x)$ is an increasing function in x since $$c^{\prime }(x\text{;}\alpha ,\beta )=\frac{\partial }{\partial x}c(x\text{;}\alpha ,\beta )=\beta x^{\beta -1}[1+2\alpha (1-\beta )(1-x^{\beta })]>0\text{.}$$ Moreover, ${\lim }_{x\to 0}c(x\text{;}\alpha ,\beta )=(\alpha +1)(\beta -1)<0,{\lim }_{x\to 1}c(x\text{;}\alpha ,\beta )=\beta \in (0,1)$ and $\beta x^{\beta -2}/{(1-x^{\beta })}^2>0$ . As a consequence, there exists $x_0=x_0(\alpha ,\beta )\in (0,1)$ such as $h^{\prime }(x\text{;}\alpha ,\beta )<0$ for $x<x_0$ and $h^{\prime }(x\text{;}\alpha ,\beta )>0$ for $x>x_0$ , which implies the result.   □

For any $0<t<1$ , the mean residual life of the $\text{LSG}(\alpha ,\beta )$ distribution is given by $$r(t\text{;}\alpha ,\beta )=\frac{\exp (\alpha t^{\beta })}{\beta {\alpha }^{1/\beta }(1-t^{\beta })}\times \left\{\left(1-\frac{1}{\alpha \beta }\right)\left[\mathrm{\Gamma }\left(\frac{1}{\beta },\alpha t^{\beta }\right)-\mathrm{\Gamma }\left(\frac{1}{\beta },\alpha \right)\right]+{\alpha }^{\frac{1}{\beta }-1}\left[e^{-\alpha }-t\exp (-\alpha t^{\beta })\right]\right\}\text{.}$$

Let $S(x\text{;}\alpha ,\beta )=1-F(x\text{;}\alpha ,\beta )$ . For any $0<t<1$ , we have $$r(t\text{;}\alpha ,\beta )=E[X-t|X>t]=\frac{1}{S(t\text{;}\alpha ,\beta )}{\int }_t^{1}S(x\text{;}\alpha ,\beta )\mathit{dx}=\frac{\exp (\alpha t^{\beta })}{\beta {\alpha }^{1/\beta }(1-t^{\beta })}\left[{\int }_{\alpha t^{\beta }}^{\alpha }{u}^{\frac{1}{\beta }-1}{e}^{-u}\mathit{du}-\frac{1}{\alpha }{\int }_{\alpha t^{\beta }}^{\alpha }{u}^{\frac{1}{\beta }}{e}^{-u}\mathit{du}\right]=\frac{\exp (\alpha t^{\beta })}{\beta {\alpha }^{1/\beta }(1-t^{\beta })}\left[\gamma \left(\frac{1}{\beta },\alpha \right)-\gamma \left(\frac{1}{\beta },\alpha t^{\beta }\right)-\frac{1}{\alpha }\gamma \left(\frac{1}{\beta }+1,\alpha \right)+\frac{1}{\alpha }\gamma \left(\frac{1}{\beta }+1,\alpha t^{\beta }\right)\right],$$ where we have made the change of variable $u=\exp (\alpha x^{\beta })$ in the third equality and Eq. (4) was considered in the last one. Finally, the desired result is obtained by taking into account that $\gamma (a,z)=\mathrm{\Gamma }(a)-\mathrm{\Gamma }(a,z),a\ne 0,-1,-2,\dots$ , and $\mathrm{\Gamma }(a+1,z)=a\mathrm{\Gamma }(a,z)+z^a{e}^{-z}$ (Paris, 2010, pp. 178).   □

For $k=1,2,\dots$ , the k-th moment of the smallest order statistic from the $\text{LSG}(\alpha ,\beta )$ distribution is given by $$E[X_{1:n}^k]=\frac{1}{\alpha {(n\alpha )}^{k/\beta }}{\displaystyle \sum _{j=0}^{n-1}}\left(\begin{array}{c}\hfill n-1\hfill \\ \hfill j\hfill \end{array}\right)\frac{{(-1)}^j}{{(n\alpha )}^j}\left[(1+\alpha )-\frac{1}{n}\left(\frac{k}{\beta }+j+1\right)\right]\gamma \left(\frac{k}{\beta }+j+1,n\alpha \right)\text{.}$$

The result is derived from formula (9) setting $n=1$ , by applying the change of variable $u=\alpha x^{\beta }$ , the binomial theorem, the recurrence formula $\gamma (a,z)=(a-1)\gamma (a-1,z)-z^{a-1}{e}^{-z}$ and also that ${\sum }_{j=0}^{n-1}{(-1)}^j\left(\begin{array}{c}\hfill n-1\hfill \\ \hfill j\hfill \end{array}\right)=0$ (the details are omitted here).   □

The $\text{LSG}(\alpha ,\beta )$ distribution belongs to:

1. The domain of maximal attraction of the Weibull distribution, with norming constants $a_n=1$ and $b_n=1-F^{-1}(1-n^{-1}\text{;}\alpha ,\beta )$ , with $F^{-1}$ given by (7).

2. The domain of minimal attraction of the Weibull distribution, with norming constants $c_n=F^{-1}(0\text{;}\alpha ,\beta )$ and $d_n=F^{-1}(n^{-1}\text{;}\alpha ,\beta )-F^{-1}(0\text{;}\alpha ,\beta )$ , with $F^{-1}$ given by (7).

Part (i) is a consequence of Arnold et al. (1992, Theorems 8.3.2 and 8.3.4) by taking into account that $${\displaystyle \underset{t\to 0^{+}}{\lim }}\frac{1-F(F^{-1}(1\text{;}\alpha ,\beta )-\mathit{tx}\text{;}\alpha ,\beta )}{1-F(F^{-1}(1\text{;}\alpha ,\beta )-t\text{;}\alpha ,\beta )}=x,$$ where $F^{-1}(1\text{;}\alpha ,\beta )=1$ from (7) (details of the calculations are omitted here). Similarly, part (ii) is a consequence of Arnold et al. (1992, Theorems 8.3.6) by taking into account that $${\displaystyle \underset{t\to 0^{+}}{\lim }}\frac{F(F^{-1}(0\text{;}\alpha ,\beta )+\mathit{tx}\text{;}\alpha ,\beta )}{F(F^{-1}(0\text{;}\alpha ,\beta )+t\text{;}\alpha ,\beta )}=x^{\beta }\text{.}$$ The proof is complete.   □

Let $X_1$ and $X_2$ be two random variables with cdfs $F_1$ and $F_2$ , respectively. Let $h_1$ and $h_2$ be the hazard rate functions of $X_1$ and $X_2$ , respectively. Let $r_1$ and $r_2$ be the mean residual life functions of $X_1$ and $X_2$ , respectively.

1. $X_1$ is said to be smaller than $X_2$ in the hazard rate order, denoted by $X_1{\le }_{\mathrm{HR}}{X}_2$ , if $h_1(x)⩾h_2(x)$ for all x.

2. $X_1$ is said to be stochastically smaller than $X_2$ , denoted by $X_1{\le }_{\mathrm{ST}}{X}_2$ , if $F_1(x)⩾F_2(x)$ for all x.

3. $X_1$ is said to be smaller than $X_2$ in the mean residual life order, denoted by $X_1{\le }_{\mathrm{MR}}{X}_2$ , if $r_1(x)⩽r_2(x)$ for all x.

Let $X_1$ and $X_2$ be two random variables having $\text{LSG}({\alpha }_1,\beta )$ and $\text{LSG}({\alpha }_2,\beta )$ distributions, respectively. If ${\alpha }_1⩾{\alpha }_2$ then $X_1{\le }_{\mathrm{HR}}{X}_2$ .

From Eq. (8), $h(x\text{;}{\alpha }_1,\beta )-h(x\text{;}{\alpha }_2,\beta )=({\alpha }_1-{\alpha }_2)\beta x^{\beta -1}(1-x^{\beta })/(1-x^{\beta })$ . The result follows since clearly $h(x\text{;}{\alpha }_1,\beta )-h(x\text{;}{\alpha }_2,\beta )⩾0$ for any ${\alpha }_1⩾{\alpha }_2$ .   □

Let $X_1$ and $X_2$ be two random variables having $\text{LSG}({\alpha }_1,\beta )$ and $\text{LSG}({\alpha }_2,\beta )$ distributions, respectively. If ${\alpha }_1⩾{\alpha }_2$ then the following relations hold:

1. $X_1{\le }_{\mathrm{ST}}{X}_2$ ,

2. $X_1{\le }_{\mathrm{MR}}{X}_2$ ,

3. $E(X_1^k)⩽E(X_2^k)$ for all $k>0$ .

Part (i) follows from the fact that $X_1{\le }_{\mathrm{HR}}{X}_2\Rightarrow X_1{\le }_{\mathrm{ST}}{X}_2$ (see Shaked and Shanthikumar, 2007, pp. 18). Part (ii) follows from the property $X_1{\le }_{\mathrm{HR}}{X}_2\Rightarrow X_1{\le }_{\mathrm{MR}}{X}_2$ (see Shaked and Shanthikumar, 2007, pp. 83). Part (iii) is a consequence of part (i) and the fact that $X_1{\le }_{\mathrm{ST}}{X}_2$ if and only if $E[g(X_1)]⩽E[g(X_2)]$ for all non-decreasing functions g (see Shaked and Shanthikumar, 2007, pp. 4).   □