A Note on the Complementary Mixture Pareto II Distribution

Abstract

We introduce a new survival distribution, of Pareto type, that arises from a cure-mixture frailty model. We describe its properties and demonstrate connections with familiar distributions including the Pareto and exponential. We derive its characteristic function and moments.

1. Introduction

Recently, Leemis and McQueston (2008) presented a graphic describing the relationships among a large number of known univariate distributions. A distribution that we encountered in our applied work, which we denote the Complementary Mixture Pareto II (CMPII), was not included in the summary, nor does it appear in a popular compendium of univariate distributions (Johnson et al., 1995). Therefore, we believe that CMPII is new to the statistical literature and in this note we present our derivation of it, describe connections with known distributions, and derive its characteristic function and moments.

We came upon CMPII while developing a model to accommodate the unique characteristics of data from smoking cessation clinical trials (Li et al., 2010). As the durations of quit attempts in such trials are characterized by heterogeneity and the possibility of permanent success, an appropriate statistical model should incorporate both latent frailties and the probability of cure. Our construction of such a model led us directly to CMPII, as we demonstrate below.

Let $X$ denote an event time and $S\left(x\right)$ its survival function. We assume that with probability $\rho$ the event time is infinite, so that $S\left(x\right)$ takes the form

$$S\left(x\right)=\rho +\left(1-\rho \right)S^{*}\left(x\right),\rho \in \left(\mathrm{0,1}\right),$$

where $S^{*}\left(x\right)$ is a proper survival function. We assume moreover that, conditional on a latent frailty $g>0$, the cure probability and survival function given not cured take the forms

$$\rho \left(g\right)=\text{exp}\left(-g{e}^{-{\eta }_{\rho }}\right),S^{*}(x\mid g)=\text{exp}\left(-xg{e}^{{\eta }_{\sigma }}\right),$$ (1.1)

where ${\eta }_{\rho }$ is a linear predictor for the cure probability and ${\eta }_{\sigma }$ is a linear predictor for event rate given not cured. That is, we assume that: (i) conditionally on the frailty $g$ the probability of cure follows a generalized linear model with a complementary log-log (cloglog) link; (ii) survival given not cured is exponential with a proportional hazard regression; and (iii) the natural log of $g$ is an offset in both linear predictors. Moreover, we assume that $g$ is a random draw from a gamma distribution with shape $1/\theta$ and scale $\theta$, ensuring that $\text{E}\left(g\right)=1$ and $\text{Var}\left(g\right)=\theta (\theta >0)$. If the cure probability were fixed at 0, this would be the standard model of a gamma-distributed exponential hazard (Duchateau and Janssen, 2008).

This specification is unique among cure-mixture frailty models in using a cloglog rather than a logistic link for the cure probability and in allowing a common frailty to affect both the cure probability and the survival given not cured. Integrating out the frailty, we obtain the marginal survival function

$$\tilde{S}\left(x\right)=\int S\left(x\right)dF\left(g\right)=\tilde{\rho }+\left(1-\tilde{\rho }\right)\widetilde{S^{*}}\left(x\right),$$

where

$$\tilde{\rho }={\left(\frac{1}{1+\theta e^{-{\eta }_{\rho }}}\right)}^{1/\theta },$$

and

$$\widetilde{S^{*}}\left(x\right)=\frac{{\left(1+\theta x{e}^{{\eta }_{\sigma }}\right)}^{-1/\theta }-{\left(1+\theta e^{-{\eta }_{\rho }}+\theta x{e}^{{\eta }_{\sigma }}\right)}^{-1/\theta }}{1-{\left(1+\theta e^{-{\eta }_{\rho }}\right)}^{-1/\theta }}.$$

$\widetilde{S^{*}}\left(x\right)$ is a proper survival function in that it is monotone decreasing with $\widetilde{S^{*}}\left(0\right)=1$ and $\widetilde{S^{*}}(+\infty )=0$. Therefore, the marginal cure probability is $\tilde{\rho }$, and the marginal survival function given not cured is $\widetilde{S^{*}}\left(x\right)$. Note that the marginal cure probability $\tilde{\rho }$ differs from the conditional cure probability $\rho$, and the marginal non cured survival $\widetilde{S^{*}}\left(x\right)$ differs from $\int S^{*}\left(x\right)dF\left(g\right)$—the integral of the conditional non cured survival. The distribution $\widetilde{S^{*}}\left(x\right)$ is the subject of this article.

As we indicated, our research suggests that $\widetilde{S^{*}}\left(x\right)$ has not previously been identified as a distribution. We therefore devote the remainder of this note to an investigation of its properties and its relationship to other distributions. Note that our purpose here is simply to elucidate basic theoretical properties; we refer readers to Li et al. (2010) for an application.

Our argument invokes the following densities.

Exponential $\left(\lambda \right)$:

$$f\left(x\right)=\lambda e^{-\lambda x},x\ge 0,\lambda \in (0,\infty );$$

Pareto I $(\alpha ,\sigma )$:

$$f\left(x\right)=\frac{\alpha }{\sigma }{\left(\frac{x}{\sigma }\right)}^{-(\alpha +1)},x\ge \sigma ,\alpha ,\sigma \in (0,\infty );$$

Pareto II $(\mu ,\sigma ,\xi )$:

$$f\left(x\right)=\frac{1}{\sigma }{\left(1+\xi \frac{x-\mu }{\sigma }\right)}^{-(1/\xi +1)},x\ge \mu \in (-\infty ,\infty ),\sigma ,\xi \in (0,\infty ).$$

2. General Characteristics

For simplicity, we denote $a=e^{{\eta }_{\sigma }},b=e^{{\eta }_{\rho }},c=1-(1+\theta /b{)}^{-1/\theta }$; then,

$$\widetilde{S^{*}}\left(x\right)=\frac{1}{c}\left[(1+\theta ax{)}^{-1/\theta }-(1+\theta /b+\theta ax{)}^{-1/\theta }\right],x\ge 0,a,b,\theta \in (0,\infty ).$$ (2.1)

Here, $a$ is a parameter associated with the conditional hazard (conditional on the frailty) given not cured; a larger $a$ implies a higher conditional hazard of experiencing the event. Parameter $b$ is associated with the conditional cure probability, where a larger value implies a larger conditional cure probability. The parameter $\theta$ is the variance of the frailty, and $c$ is a normalizing constant. The corresponding marginal density is

$$\widetilde{f^{*}}\left(x\right)=\frac{a}{c}\left[(1+\theta ax{)}^{-1/\theta -1}-(1+\theta /b+\theta ax{)}^{-1/\theta -1}\right],x\in (0,\infty ),$$ (2.2)

and the hazard function is

$$\widetilde{h^{*}}\left(x\right)=a\frac{(1+\theta ax{)}^{-1/\theta -1}-(1+\theta /b+\theta ax{)}^{-1/\theta -1}}{(1+\theta ax{)}^{-1/\theta }-(1+\theta /b+\theta ax{)}^{-1/\theta }}x\in (0,\infty ).$$ (2.3)

We rewrite (2.2) as

$$\widetilde{f^{*}}\left(x\right)=\alpha f_1\left(x\right)-\beta f_2\left(x\right),$$ (2.4)

where

$$\begin{gathered} \alpha =\frac{1}{c}=\frac{1}{1-(1+\theta /b{)}^{-1/\theta }},\beta =1-\frac{1}{c}=\frac{(1+\theta /b{)}^{-\frac{1}{\theta }}}{1-(1+\theta /b{)}^{-1/\theta }}; \\ {f}_1~\text{Pareto}\text{II}(\mu =0,\sigma =1/a,\xi =\theta ), \\ {f}_2~\text{Pareto}\text{II}(\mu =0,\sigma =(1+\theta /b)/a,\xi =\theta ). \end{gathered}$$

and

$$\alpha >1,\beta >0,\alpha -\beta =1.$$

Because the density is a weighted difference of Pareto II densities, we denote our model as the Complementary Mixture Pareto II (CMPII) distribution.

Note that the first derivative of the density is

$$\begin{gathered} {\tilde{f}}^{{*}^{\prime }}(x)=-\frac{a^2(1+\theta )}{c}\left[(1+\theta ax{)}^{-\left(\frac{1}{\theta }+2\right)}-(1+\theta ax+\theta /b{)}^{-\left(\frac{1}{\theta }+2\right)}\right]<0, \\ a,b,\theta ,x\in (0,\infty ) \end{gathered}$$

so that the density function is always decreasing with time $x$.

Figure 1 plots the log density and log hazard for different values of $a$ and $\theta$ when $b=1$. Figure 2 depicts how the density and hazard vary with $b$ when $a$ and $\theta$ are held fixed. Note that the hazard increases with $b$ and appears to converge to a distribution as $b\uparrow \infty$. In fact, by L’Hôpital’s rule,

$$\underset{b\uparrow \infty }{\text{lim}}{\tilde{S}}^{*}\left(x\right)=(1+\theta ax{)}^{-\frac{1}{\theta }-1}=\text{Pareto}\text{II}(\mu =0,\sigma =1/\left[a\right(1+\theta \left)\right],\xi =\theta /(1+\theta )).$$

Figure 1.

Log density and hazard function when $b=1$.

Figure 2.

Log density and hazard function when $a=1$ and $\theta =1$.

As the hazard function (2.3) is analytically complicated, we explored permissible shapes by computing its derivative numerically over a wide range of values of $a,b$, and $\theta$. In all cases the derivative was negative for all $x$, suggesting that the hazard is non increasing with time under a wide range of parameter values. This is not surprising, as the special case of the Pareto II can be shown to have a non-increasing hazard (Johnson et al., 1995).

3. The Special Case of $\mathit{b}\downarrow 0$

Letting $b\downarrow 0$ in (2.1), the marginal non cured survival function becomes

$$\underset{b\downarrow 0}{\text{lim}}\widetilde{S^{*}}\left(x\right)=(1+\theta ax{)}^{-\frac{1}{\theta }},$$

the Pareto II $(\mu =0,\sigma =1/a,\xi =\theta )$ that arises from the standard frailty model with exponential baseline hazard (Duchateau and Janssen, 2008). This is because as $b\downarrow 0$, the conditional cure probability in (1.1) goes to 0, so that our model reduces to a standard frailty model with constant hazard conditional on the frailty. The density is then

$$\widetilde{f^{*}}\left(x\right)=a(1+\theta ax{)}^{-\left(\frac{1}{\theta }+1\right)},$$

with first derivative

$${\tilde{f}}^{{*}^{\prime }}\left(x\right)=-a^2(1+\theta )(1+\theta ax{)}^{-\left(\frac{1}{\theta }+2\right)}<0,a,\theta ,x\in (0,\infty )$$

so that the density function is always decreasing with time $x$.

Figures 3 and 4 depict the log density and hazard at various values of $a$ and $\theta$ for $b\downarrow 0$. We see that when $\theta$ is fixed and $a$ decreases, the density declines at a slower rate, and the hazard is smaller at any given time. This is because a smaller value of $a$ implies a smaller individual-level conditional hazard, which manifests itself as a smaller population-level marginal hazard. In particular, when $a\downarrow 0$, the marginal hazard also goes to 0. When $a\uparrow \infty$, the marginal non-cured survival becomes the Pareto I $(\alpha =1/\theta ,\sigma =1/(\theta a\left)\right)$:

$$\underset{b\downarrow 0,a\uparrow \infty }{\text{lim}}\widetilde{S^{*}}\left(x\right)=(\theta ax{)}^{-\frac{1}{\theta }}.$$

Figure 3.

Log density function when $b\downarrow 0$.

Figure 4.

Hazard function when $b\downarrow 0$.

Similarly, the density and hazard function vary with $\theta$ for fixed $a$. When $\theta$ is small, the log density approaches a linear function of time and the hazard approaches a constant. This is because $\theta \downarrow 0$ means that the variance of the frailty approaches 0, which implies that there is no between-subject heterogeneity, and therefore the population survival function equals the individual survival function, which is exponential:

$$\underset{b\downarrow 0,\theta \downarrow 0}{\text{lim}}\widetilde{S^{*}}\left(x\right)=e^{-ax}.$$

We summarize the relationships of CMPII with other known distributions in Fig. 5.

Figure 5.

Relationships of CMPII to other distributions.

4. Characteristic Function and Moments

We first derive the expectation and variance of CMPII.

Theorem 4.1. The expectation of a variate X following the CMPII distribution is

$$\text{E}\left(X\right)=\frac{1}{a\left(1-\theta \right)}\cdot \frac{1-(1+\theta /b{)}^{-\frac{1}{\theta }+1}}{1-(1+\theta /b{)}^{-\frac{1}{\theta }}},\theta <1;$$

the second moment is

$$\text{E}\left(X^2\right)=\frac{2}{a^2(1-\theta )(1-2\theta )}\cdot \frac{1-(1+\theta /b{)}^{-\frac{1}{\theta }+2}}{1-(1+\theta /b{)}^{-\frac{1}{\theta }}},\theta <1/2;$$

and therefore the variance is

$$\begin{gathered} \text{Var}(X)=\frac{2}{a^2(1-\theta )(1-2\theta )}\cdot \frac{1-(1+\theta /b{)}^{-\frac{1}{\theta }+2}}{1-(1+\theta /b{)}^{-\frac{1}{\theta }}} \\ -\frac{1}{a^2(1-\theta {)}^2}\cdot {\left[\frac{1-(1+\theta /b{)}^{-\frac{1}{\theta }+1}}{1-(1+\theta /b{)}^{-\frac{1}{\theta }}}\right]}^2,\theta <1/2. \end{gathered}$$

See Appendix A for a proof. We note that the second factor in the first moment formula is always smaller than 1, giving

$$\text{E}\left(X\right)<\frac{1}{a\left(1-\theta \right)}=\text{E}\left(X_1~f_1\right).$$

We next derive the characteristic function $\varphi \left(t\right)=\text{E}\left(e^{itX}\right)$.

Theorem 4.2. The characteristic function of CMPII is

$$\begin{gathered} \varphi \left(t\right)=\frac{1}{c\theta }{\left(-\frac{it}{\theta a}\right)}^{\frac{1}{\theta }}\left[\text{exp}\left(-\frac{it}{\theta a}\right)\Gamma \left(-\frac{1}{\theta },-\frac{it}{\theta a}\right)\right. \\ \left.-\text{exp}\left(-\frac{it(1+\theta /b)}{\theta a}\right)\Gamma \left(-\frac{1}{\theta },-\frac{it(1+\theta /b)}{\theta a}\right)\right]. \end{gathered}$$

Here, $\Gamma (\cdot ,\cdot )$ is the upper incomplete Gamma function with complex arguments:

$$\Gamma (s,z)={\int }_z^{\infty }{t}^{s-1}{e}^{-t}dt$$

(Abramowitz and Stegun, 1965; DiDonato and Morris, 1986; Temme, 1996). See Appendix B for a proof. Repeated differentiation gives

Theorem 4.3. The nth moment of the CMPII distribution is

$$\text{E}{X}^n=\frac{n!}{c{a}^n(1-\theta )(1-2\theta )\cdots (1-n\theta )}-\frac{n!(1+\theta /b{)}^{n-\frac{1}{\theta }}}{c{a}^n(1-\theta )(1-2\theta )\cdots (1-n\theta )},\theta <\frac{1}{n}.$$

See Appendix C for a proof. One can readily verify that Theorem 4.3 implies Theorem 4.1.

5. Discussion

We have described CMPII, a three-parameter family that arises as the marginal distribution of survival given not cured in a cure-mixture frailty model. Several known distributions are limiting cases.

The CMPII has the curious property that its density equals a linear combination of two Pareto II densities, which would appear to offer a clear path to its characteristic function and moments. Ironically, we were unable to find formulas for the Pareto II characteristic function, so we derived terms for CMPII directly (see Appendices B and C). The derivations are tedious but may serve as templates for similar derivations of other distributions.

Pareto distributions have been known for several decades (Arnold and Laguna, 1977; Arnold, 1983) and have found application in various areas of science, especially in economics to describe income and in medical statistics to describe survival. Naturally, one can apply CMPII in these areas as well. Our applied experience (Li et al., 2010) suggests its usefulness specifically in smoking cessation research.

Appendix A: Proof of Theorem 4.1

Johnson et al. (1995) showed that if $U$ is Pareto II $(\mu ,\sigma ,\xi )$, then

$$\text{E}\left(U\right)=\frac{\sigma }{1-\xi },\xi <1$$

and

$$\text{E}\left(U^2\right)=\frac{2{\sigma }^2}{(1-\xi )(1-2\xi )},\xi <1/2.$$

Recall that the density of a CMPII variate $X$ is

$$\widetilde{f^{*}}\left(x\right)=\alpha f_1\left(x\right)-\beta f_2\left(x\right),$$

where

$$\alpha =\frac{1}{1-(1+\theta /b{)}^{-1/\theta }}=1-\beta ,$$

$f_1$ is the density of a Pareto II $(0,1/a,\theta )$, and $f_2$ is the density of a Pareto II $(0,(1+\theta /b)/a,\theta )$. One can therefore derive $\text{E}\left(X\right)$ and $\text{E}\left(X^2\right)$ by taking linear combinations of the moments of Pareto II, and $\text{Var}\left(X\right)$ by subtraction.

Appendix B: Proof of Theorem 4.2

Here, we use the fact that

$${\int }_0^{\infty }(x+w{)}^v{e}^{-ux}dx=u^{-v-1}{e}^{wu}\Gamma \left(v+1,wu\right),$$

where $u,v,w$ are complex numbers with $\left|\text{arg}w\right|<\pi ,\text{Re}u>0$; and $\Gamma (\cdot ,\cdot )$ is the upper incomplete Gamma function (Gradshteyn and Ryzhik, 1980).

Equation (2.4) implies that

$$\varphi \left(t\right)=\text{E}\left(e^{itX}\right)=\alpha \text{E}\left(e^{itU},U~f_1\right)-\beta \text{E}\left(e^{itV},V~f_2\right).$$

Now

$$\begin{gathered} \text{E}\left(e^{itU},U~f_1\right)=a{\int }_0^{\infty }{e}^{itu}(1+\theta au{)}^{-\frac{1}{\theta }-1}du \\ =a(\theta a{)}^{-\frac{1}{\theta }-1}{\int }_0^{\infty }{\left(u+\frac{1}{\theta a}\right)}^{-\frac{1}{\theta }-1}{e}^{itu}du \\ =a\left(\theta a{)}^{-\frac{1}{\theta }-1}\right(-it{)}^{\frac{1}{\theta }}\text{exp}\left(-\frac{it}{\theta a}\right)\Gamma \left(-\frac{1}{\theta },-\frac{it}{\theta a}\right), \\ =\frac{1}{\theta }{\left(-\frac{it}{\theta a}\right)}^{\frac{1}{\theta }}\text{exp}\left(-\frac{it}{\theta a}\right)\Gamma \left(-\frac{1}{\theta },-\frac{it}{\theta a}\right), \end{gathered}$$

and similarly,

$$\begin{gathered} \text{E}\left(e^{itV},V~f_2\right)=\frac{a}{1+\theta /b}{\int }_0^{\infty }{e}^{itv}{\left(1+\frac{\theta a}{1+\theta /b}v\right)}^{-\frac{1}{\theta }-1}dv \\ =\frac{1}{\theta }{\left(-\frac{it}{\theta a}\right)}^{\frac{1}{\theta }}(1+\theta /b{)}^{\frac{1}{\theta }}\text{exp}\left(-\frac{it(1+\theta /b)}{\theta a}\right)\Gamma \left(-\frac{1}{\theta },-\frac{it(1+\theta /b)}{\theta a}\right); \end{gathered}$$

Therefore,

$$\begin{gathered} \varphi \left(t\right)=\frac{1}{c\theta }{\left(-\frac{it}{\theta a}\right)}^{\frac{1}{\theta }}[\text{exp}\left(-\frac{it}{\theta a}\right)\Gamma \left(-\frac{1}{\theta },-\frac{it}{\theta a}\right) \\ -\text{exp}\left(-\frac{it\left(1+\theta /b\right)}{\theta a}\right)\Gamma \left(-\frac{1}{\theta },-\frac{it\left(1+\theta /b\right)}{\theta a}\right)] \end{gathered}$$

Appendix C: Proof of Theorem 4.3

We first note a series of Lemmas.

Lemma A.1.

$$\frac{{\partial }^n\left(z^{-s}\Gamma (s,z)\right)}{{\partial }^{n}z}=(-1{)}^n{z}^{-s-n}\Gamma (s+n,z)$$

Proof. See Gradshteyn and Ryzhik (1980). □

Lemma A.2.

$$-{\left.z^{-s}\Gamma \left(s,z\right)\right|}_{z=0}=\frac{1}{s},\left(s<0\right).$$

Proof.

$$\begin{gathered} z^{-s}\Gamma (s,z)=z^{-s}{\int }_z^{\infty }{t}^{s-1}{e}^{-t}dt={\int }_1^{\infty }{y}^{s-1}{e}^{zy}dy, \\ -{\left.z^{-s}\Gamma (s,z)\right|}_{z=0}={\int }_1^{\infty }{y}^{s-1}dy=-\frac{1}{s},\left(s<0\right). \end{gathered}$$

□

Lemma A.3. Define $g(z,s)=e^z{z}^{-s}\Gamma (s,z)$. Then,

$$\begin{gathered} \frac{\partial g(z,s)}{\partial z}=-g(z,s+1)+g(z,s); \\ {\left.\frac{\partial g(z,s)}{\partial z}\right|}_{z=0}=\frac{-1}{s(s+1)},(s<-1). \end{gathered}$$

Proof.

$$\begin{gathered} \frac{\partial g(z,s)}{\partial z}=\frac{\partial \left(e^z{z}^{-s}\Gamma (s,z)\right)}{\partial z}=e^z\frac{\partial \left(z^{-s}\Gamma (s,z)\right)}{\partial z}+e^z\left(z^{-s}\Gamma (s,z)\right) \\ =e^z\left(-z^{-s-1}\Gamma (s+1,z)\right)+e^z\left(z^{-s}\Gamma (s,z)\right)\left(\text{by}\text{Lemma}\text{A.1}\right) \\ =-g(z,s+1)+g(z,s); \\ {\left.\frac{\partial g(z,s)}{\partial z}\right|}_{z=0}=-{\left.z^{-s-1}\Gamma (s+1,z)\right|}_{z=0}+{\left.z^{-s}\Gamma (s,z)\right|}_{z=0} \\ =\frac{1}{s+1}-\frac{1}{s}\left(\text{by}\text{Lemma}\text{A.2}\right) \\ =\frac{-1}{s(s+1)},\left(s<-1\right). \end{gathered}$$

□

Lemma A.4.

$$\frac{\partial g^n(z,s)}{\partial z^n}=\sum _{k=0}^n(-1{)}^k\left(\genfrac{}{}{0pt}{}{n}{k}\right)g(z,s+k).$$

Proof. Denoting $A_n=\partial g^n(z,s)/\partial z^n$, we prove the lemma by induction. Supposing it is true for $n$, then for $n+1$,

$$\begin{gathered} A_{n+1}=\frac{\partial A_n}{\partial z}=\sum _{k=0}^n{(-1)}^k\left(\begin{array}{c}n\\ k\end{array}\right)\frac{\partial g\left(z,s+k\right)}{\partial z} \\ =\sum _{k=0}^n{(-1)}^k\left(\begin{array}{c}n\\ k\end{array}\right)\left[g\left(z,s+k\right)-g\left(z,s+k+1\right)\right]\left(\text{by}\text{Lemma}\text{A.3}\right) \\ =\sum _{k=0}^n{(-1)}^k\left(\begin{array}{c}n\\ k\end{array}\right)g\left(z,s+k\right)+\sum _{l=1}^{n+1}{(-1)}^l\left(\begin{array}{c}n\\ l-1\end{array}\right)g\left(z,s+l\right) \\ =g\left(z,s\right)+\sum _{k=1}^n{(-1)}^k\left(\begin{array}{c}n\\ k\end{array}\right)g\left(z,s+k\right) \\ +\sum _{l=1}^n{(-1)}^l\left(\begin{array}{c}n\\ l-1\end{array}\right)g\left(z,s+l\right)+{(-1)}^{n+1}g\left(z,s+n+1\right) \\ =g\left(z,s\right)+\sum _{k=1}^n{(-1)}^k\left(\begin{array}{c}n\\ k\end{array}\right)g\left(z,s+k\right) \\ +\sum _{l=1}^n{(-1)}^l\left(\begin{array}{c}n\\ l\end{array}\right)\frac{l}{n-l+1}g\left(z,s+l\right)+{(-1)}^{n+1}g\left(z,s+n+1\right) \\ =g\left(z,s\right)+\sum _{k=1}^n{(-1)}^k\left(\begin{array}{c}n+1\\ k\end{array}\right)g\left(z,s+k\right)+{(-1)}^{n+1}g(z,s+n+1) \\ =\sum _{k=0}^{n+1}{(-1)}^k\left(\begin{array}{c}n+1\\ k\end{array}\right)g\left(z,s+k\right), \end{gathered}$$

so it is true for $n+1$ as well. □

Lemma A.5.

$${\left.\frac{\partial g^n(z,s)}{\partial z^n}\right|}_{z=0}=\sum _{k=0}^{n-1}(-1{)}^k\left(\genfrac{}{}{0pt}{}{n-1}{k}\right)\frac{1}{\left(s+k\right)\left(s+k+1\right)},\left(s+n<0\right).$$

Proof.

$$\begin{gathered} {\left.\frac{\partial g^n(z,s)}{\partial z^n}\right|}_{z=0}={\left.\frac{\partial }{\partial z}\left(\frac{\partial g^{n-1}(z,s)}{\partial z^{n-1}}\right)\right|}_{z=0}={\left.\frac{\partial }{\partial z}\sum _{k=0}^{n-1}(-1{)}^k\left(\genfrac{}{}{0pt}{}{n-1}{k}\right)g(z,s+k)\right|}_{z=0} \\ (\text{by}\text{Lemma}\text{A}.4) \\ ={\left.\sum _{k=0}^{n-1}(-1{)}^k\left(\genfrac{}{}{0pt}{}{n-1}{k}\right)\frac{\partial g(z,s+k)}{\partial z}\right|}_{z=0} \\ =\sum _{k=0}^{n-1}(-1{)}^k\left(\genfrac{}{}{0pt}{}{n-1}{k}\right)\frac{-1}{\left(s+k\right)\left(s+k+1\right)}\left(\text{by}\text{Lemma}\text{A.3}\right). \end{gathered}$$

□

Lemma A.6.

$${\left.\frac{\partial g^n(z,s)}{\partial z^n}\right|}_{z=0}=\frac{-n!}{s(s+1)\cdots (s+n)},\left(s+n<0\right).$$

Proof. Setting $B_n\left(s\right)=\partial g^n(z,s)/{\left.\partial z^n\right|}_{z=0}$, we again prove the lemma using induction. If it is true for $n$, then for $n+1$,

$$\begin{gathered} B_{n+1}\left(s\right)=\sum _{k=0}^n{(-1)}^k\left(\begin{array}{c}n\\ k\end{array}\right)\frac{1}{\left(s+k\right)\left(s+k+1\right)}\left(\text{by}\text{Lemma}\text{A.5}\right) \\ \text{=}\frac{1}{s\left(s+1\right)}+\sum _{k=1}^{n-1}{(-1)}^k\left(\begin{array}{c}n\\ k\end{array}\right)\frac{1}{\left(s+k\right)\left(s+k+1\right)}+{(-1)}^n\frac{1}{\left(s+n\right)\left(s+n+1\right)} \\ =\frac{1}{s\left(s+1\right)}+\sum _{k=1}^{n-1}{(-1)}^k\left[\left(\begin{array}{c}n-1\\ k\end{array}\right)+\left(\begin{array}{c}n-1\\ k\end{array}\right)\frac{k}{n-k}\right]\frac{1}{\left(s+k\right)\left(s+k+1\right)} \\ +\frac{{(-1)}^n}{\left(s+n\right)\left(s+n+1\right)} \\ =\frac{1}{s\left(s+1\right)}+\sum _{k=1}^{n-1}{(-1)}^k\left(\begin{array}{c}n-1\\ k\end{array}\right)\frac{1}{\left(s+k\right)\left(s+k+1\right)} \\ +\sum _{k=1}^{n-1}{(-1)}^k\left(\begin{array}{c}n-1\\ k\end{array}\right)\frac{k}{n-k}\frac{1}{\left(s+k\right)\left(s+k+1\right)}+{(-1)}^n\frac{1}{\left(s+n\right)\left(s+n+1\right)} \\ =B_n\left(s\right)+\sum _{k=1}^{n-1}{(-1)}^k\left(\begin{array}{c}n-1\\ k-1\end{array}\right)\frac{1}{\left(s+k\right)\left(s+k+1\right)}+{(-1)}^n\frac{1}{\left(s+n\right)\left(s+n+1\right)} \\ =B_n\left(s\right)-\sum _{l=0}^{n-2}{(-1)}^l\left(\begin{array}{c}n-1\\ l\end{array}\right)\frac{1}{\left(s+l+1\right)\left(s+l+2\right)}+{(-1)}^n\frac{1}{\left(s+n\right)\left(s+n+1\right)} \\ =B_n\left(s\right)-\sum _{l=0}^{n-1}{(-1)}^l\left(\begin{array}{c}n-1\\ l\end{array}\right)\frac{1}{\left(s+l+1\right)\left(s+l+2\right)} \\ =B_n\left(s\right)-B_n\left(s+1\right) \\ =\frac{-n!}{s\left(s+1\right)\cdots \left(s+n\right)}-\frac{-n!}{\left(s+1\right)\left(s+2\right)\cdots \left(s+n+1\right)}\left(\text{because}\text{it}\text{is}\text{true}\text{for}n\right) \\ =\frac{-\left(n+1\right)!}{s\left(s+1\right)\cdots \left(s+n+1\right)}, \end{gathered}$$

so it is true for $n+1$ as well. □

Based on these Lemmas we can prove Theorem 4.3 easily. Denote $z_1=-\left(it\right)/\left(\theta a\right)$, $z_2=-\left[it\right(1+\theta /b\left)\right]/\left(\theta a\right)$ and $s=-1/\theta$, then

$$\varphi \left(t\right)=\frac{\alpha }{\theta }{z}_1^{-s}{e}^{z_1}\Gamma \left(s,z_1\right)-\frac{\beta }{\theta }{z}_2^{-s}{e}^{z_2}\Gamma \left(s,z_2\right)=\frac{\alpha }{\theta }g\left(z_1,s\right)-\frac{\beta }{\theta }g\left(z_2,s\right),$$

$$\begin{gathered} {\left.\frac{\partial {\varphi }^n}{\partial t^n}\right|}_{t=0}={\left.\frac{\alpha }{\theta }\frac{\partial g^n\left(z_1,s\right)}{\partial z_1^n}\right|}_{z_1=0}{\left(\frac{\partial z_1}{\partial t}\right)}^n-{\left.\frac{\beta }{\theta }\frac{\partial g^n\left(z_2,s\right)}{\partial z_2^n}\right|}_{z_2=0}{\left(\frac{\partial z_2}{\partial t}\right)}^n \\ =\frac{\alpha }{\theta }\frac{-n!}{s(s+1)\cdots (s+n)}{\left(\frac{-i}{\theta a}\right)}^n-\frac{\beta }{\theta }\frac{-n!}{s(s+1)\cdots (s+n)}{\left(\frac{-i(1+\theta /b)}{\theta a}\right)}^n \\ (\text{by}\text{Lemma}\text{A}.6) \\ =\alpha \frac{i^n}{a^n}\frac{n!}{(1-\theta )(1-2\theta )\cdots (1-n\theta )}-\beta \frac{i^n}{a^n}(1+\theta /b{)}^n\frac{n!}{(1-\theta )(1-2\theta )\cdots (1-n\theta )}, \end{gathered}$$

$$\begin{gathered} \text{E}{X}^n={\left.i^{-n}\frac{\partial {\varphi }^n}{\partial t^n}\right|}_{t=0} \\ =\alpha \frac{1}{a^n}\frac{n!}{(1-\theta )(1-2\theta )\cdots (1-n\theta )}-\beta \frac{1}{a^n}(1+\theta /b{)}^n\frac{n!}{(1-\theta )(1-2\theta )\cdots (1-n\theta )} \\ =\frac{n!}{c{a}^n(1-\theta )(1-2\theta )\cdots (1-n\theta )}-\frac{n!(1+\theta /b{)}^{n-\frac{1}{\theta }}}{c{a}^n(1-\theta )(1-2\theta )\cdots (1-n\theta )},\left(\theta <\frac{1}{n}\right). \end{gathered}$$