Power Series Distribution
Let $T$ be any countable set of real numbers with no finite limit point and let $a(\cdot)$ be any real valued function defined on $T$. Then there exists a function $f(\theta)$ for all $\theta \in \Omega$ and $0<\theta<\infty$ such that it satisfies the power series expansion. Then the random variable $X$ with probability function
$$ \begin{equation}\label{psd1} P(X=x)= \frac{a_x \theta^x}{f(\theta)};\; x\in T. \end{equation} $$
for $T={ 0,1,2,\cdots }$ and $0 < \theta < \infty$. The function defined in \eqref{psd1} is called Power Series Distribution (P.S.D.).
Definition
A random variable $X$ is said to have Power series distribution if its probability mass function is given by
$$ \begin{eqnarray*} P(X=x) &=& \frac{a_x \theta^x}{f(\theta)}\\ & & \quad x\in T,\; T=\{ 0,1,2,\cdots \}\\ & & \quad 0 < \theta < \infty. \end{eqnarray*} $$
where $f(\theta)$ satisfies power series expansion $f(\theta)=\sum_{x\in T} a_x \theta^x$.
Moments of P.S.D.
Raw Moments
First Raw Moment
$$ \begin{eqnarray}\label{psd2}\nonumber \mu_1^\prime = E(X) &=& \sum_{x\in T} x\cdot P(X=x) \\\nonumber &=&\sum{x\in T} x\cdot\frac{a_x \theta^x}{f(\theta)}\\ &=&\frac{1}{f(\theta)}\sum_{x\in T} x\cdot a_x \theta^x \end{eqnarray} $$
We know that,
$$ \begin{equation*} f(\theta) = \sum_{x\in T} a_x \theta^x \end{equation*} $$
Differentiating w.r.t. $\theta$ we get
$$ \begin{eqnarray*} \frac{d}{d\theta}f(\theta)&=&f^\prime(\theta)\\ &=& \sum_{x\in T} x a_x \theta^{x-1} \end{eqnarray*} $$
Multiplying by $\theta$ on both the sides, we get
$$ \begin{equation}\label{psd3} \theta f^\prime(\theta) = \sum_{x\in T} x a_x \theta^{x} \end{equation} $$
Using \eqref{psd3} in (2), we get
$$ \begin{equation}\label{psd3a} \mu_1^\prime (\theta) = \frac{\theta f^\prime(\theta)}{f(\theta)}. \end{equation} $$
Second Raw Moment
The second raw moment is given by
$$ \begin{eqnarray}\label{psd4}\nonumber \mu_2^\prime = E(X^2) &=& \sum_{x\in T} x^2\\ &=&\frac{1}{f(\theta)}\sum_{x\in T} x^2\cdot a_x \theta^x \end{eqnarray} $$
Differentiating equation \eqref{psd3} w.r.t. $\theta$, we get
$$ \begin{equation*} f^\prime(\theta) + \theta f^{\prime\prime}(\theta) = \sum_{x\in T} x^2 a_x \theta^{x-1} \end{equation*} $$
Multiplying by $\theta/f(\theta)$ on both the sides, we have
$$ \begin{equation*} \frac{\theta f^\prime(\theta)}{f(\theta)} + \frac{\theta^2 f^{\prime\prime}(\theta)}{f(\theta)} = \sum_{x\in T} \frac{x^2 a_x \theta^{x}}{f(\theta)}=\mu_2^\prime. \end{equation*} $$
Hence, we have
$$ \begin{equation}\label{psd5} \mu_2^\prime= \frac{\theta f^\prime(\theta)}{f(\theta)} + \frac{\theta^2 f^{\prime\prime}(\theta)}{f(\theta)}. \end{equation} $$
Recurrence Relation for Raw moments of P.S.D.
The recurrence relation for raw moments of power series distribution is
$$ \begin{equation*} \mu_{r+1}^\prime(\theta) = \mu_1^\prime(\theta) \mu^\prime_r(\theta)+ \theta \frac{d}{d\theta} \mu_r^\prime (\theta). \end{equation*} $$
Proof
The $r^{th}$ raw moment is given by
$$ \begin{equation}\label{psd6} \mu_r^\prime (\theta) = E(X^r) = \sum_{x\in T} x^r\frac{a_x \theta^x}{f(\theta)} \end{equation} $$
i.e.
$$ \begin{equation}\label{psd7} f(\theta)\mu_r^\prime (\theta) =\sum_{x\in T} x^ra_x\theta^x \end{equation} $$
Differentiating \eqref{psd7} w.r.t. $\theta$, we get
$$ \begin{equation*} f^\prime(\theta) \mu^\prime_r(\theta) + f(\theta) \frac{d}{d\theta} \mu_r^\prime (\theta) = \sum_{x\in T} x^{r+1}a_x \theta^{x-1} \end{equation*} $$
Multiplying both the sides by $\theta/f(\theta)$, we get
$$ \begin{equation*} \frac{\theta f^\prime(\theta)\mu^\prime_r(\theta)}{f(\theta)} + \theta \frac{d}{d\theta} \mu_r^\prime (\theta) = \sum_{x\in T} x^{r+1}\frac{a_x \theta^{x}}{f(\theta)} \end{equation*} $$
Using equation \eqref{psd3a}, we have
$$ \begin{equation*} \mu_1^\prime(\theta) \mu^\prime_r(\theta) + \theta \frac{d}{d\theta} \mu_r^\prime (\theta) = \mu_{r+1}^\prime(\theta) \end{equation*} $$
Hence the recurrence relation for raw moments of P.S.D. is
$$ \begin{equation*} \mu_{r+1}^\prime(\theta) = \mu_1^\prime(\theta) \mu^\prime_r(\theta)+ \theta \frac{d}{d\theta} \mu_r^\prime (\theta). \end{equation*} $$
Recurrence Relation for Central Moments of P.S.D.
The recurrence relation for central moments of power series distribution is
$$ \begin{equation*} \mu_{r+1} = \theta \bigg(\frac{d\mu_r}{d\theta} + r \mu_{r-1}\frac{d\mu_1^\prime}{d\theta}\bigg) \end{equation*} $$
Proof
The $r^{th}$ central moment of a power series distribution is given by
$$ \begin{eqnarray*} \mu_r &=& E(X-\mu_1^\prime)^r \\ &=& \sum_{x\in T} (x-\mu_1^\prime)^r \frac{a_x\theta^x}{f(\theta)} \end{eqnarray*} $$
i.e.,
$$ \begin{equation*} f(\theta)\mu_r = E(X-\mu_1^\prime)^r = \sum_{x\in T} (x-\mu_1^\prime)^r a_x\theta^x \end{equation*} $$
Differentiating both the sides w.r.t. $\theta$, we get
$$ \begin{eqnarray*} \frac{d \mu_r}{d\theta} f(\theta) + \mu_r f^\prime(\theta) & = & \sum_{x\in T} (x-\mu_1^\prime)^r x a_x \theta^{x-1} - \sum_{x\in T} r(x-\mu_1^\prime)^{r-1} a_x \theta^x \frac{d \mu_1^\prime}{d\theta} \\ & = & \sum_{x\in T} (x-\mu_1^\prime+\mu_1^\prime)(x-\mu_1^\prime)^r a_x \theta^{x-1}\\ & & - r\frac{d \mu_1^\prime}{d\theta}\sum_{x\in T} (x-\mu_1^\prime)^{r-1} a_x \theta^x \\ & = & \sum_{x\in T} (x-\mu_1^\prime)^{r+1} a_x \theta^{x-1} +\mu_1^\prime \sum_{x\in T} (x-\mu_1^\prime)^{r} a_x \theta^{x-1}\\ & & -r\frac{d \mu_1^\prime}{d\theta}\sum_{x\in T} (x-\mu_1^\prime)^{r-1} a_x \theta^x \\ \end{eqnarray*} $$
Multiplying both the side by $\theta/f(\theta)$, we get
$$ \begin{eqnarray*} \theta \frac{d \mu_r}{d\theta} + \mu_r \frac{\theta f^\prime(\theta)}{f(\theta)} & = & \sum_{x\in T} (x-\mu_1^\prime)^{r+1} \frac{a_x \theta^{x}}{f(\theta)} +\mu_1^\prime \sum_{x\in T} (x-\mu_1^\prime)^{r} \frac{a_x \theta^{x}}{f(\theta)}\\ & & -r\theta\frac{d \mu_1^\prime}{d\theta}\sum_{x\in T} (x-\mu_1^\prime)^{r-1} \frac{a_x \theta^x}{f(\theta)} \\ \text{i.e., }\theta \frac{d \mu_r}{d\theta} + \mu_r \mu_1^\prime & = & \mu_{r+1}+\mu_1^\prime \mu_r-r\theta \mu_{r-1}\frac{d\mu_1^\prime}{d\theta}\\ \text{i.e., }\theta \frac{d \mu_r}{d\theta} & = & \mu_{r+1}-r\theta \mu_{r-1}\frac{d\mu_1^\prime}{d\theta}\\ \end{eqnarray*} $$
Hence the recurrence relation for the central moments of P.S.D. is
$$ \begin{equation*} \mu_{r+1} = \theta \bigg(\frac{d\mu_r}{d\theta} + r \mu_{r-1}\frac{d\mu_1^\prime}{d\theta}\bigg) \end{equation*} $$
M.G.F. of Power Series Distribution
The moment generating function of power series distribution is
$$M_X(t) = \frac{f(\theta e^t)}{f(\theta)}$$
Proof
The moment generating function of power series distribution is
$$ \begin{eqnarray*} M_X(t) &=& E(e^{tX}) \\ &=& \sum_{x\in T} e^{tx} \frac{a_x \theta^x}{f(\theta)}\\ &=& \frac{1}{f(\theta)}\sum_{x\in T} a_x (\theta e^{t})^x\\ &=& \frac{f(\theta e^t)}{f(\theta)}. \end{eqnarray*} $$
Moments of Power Series distribution can be obtained by differentiating the m.g.f with respect to $t$ and taking $t=0$.
The $r^{th}$ moment is given by
$$ \begin{equation*} \mu_r^\prime= \bigg[\frac{d^r}{dt^r} M_X(t)\bigg]_{t=0}. \end{equation*} $$
Hence,
$$ \begin{eqnarray*} \mu_1^\prime & = & \bigg[\frac{d}{dt} M_X(t)\bigg]_{t=0}\\ &=& \bigg[\frac{d}{dt} \frac{f(\theta e^t)}{f(\theta)}\bigg]_{t=0}.\\ & = & \bigg[\frac{f^\prime(\theta e^t)\theta e^t}{f(\theta)}\bigg]_{t=0}\\ &=&\theta\frac{f^\prime(\theta)}{f(\theta)}. \end{eqnarray*} $$
$$ \begin{eqnarray*} \mu_2^\prime &=& \bigg[\frac{d^2}{dt^2} M_X(t)\bigg]_{t=0}\\ &=& \bigg[\frac{d}{dt} \frac{f^\prime(\theta e^t)\theta e^t}{f(\theta)}\bigg]_{t=0}.\\ &=& \frac{\theta}{f(\theta)}\bigg[f^\prime(\theta e^t) e^t+ e^tf^{\prime\prime}(\theta e^t)\theta e^t\bigg]_{t=0}\\ &=&\frac{\theta}{f(\theta)}\bigg[f^\prime(\theta)+ f^{\prime\prime}(\theta)\theta\bigg]\\ &=&\frac{\theta f^\prime(\theta)}{f(\theta)}+\frac{\theta^2 f^{\prime\prime}(\theta)}{f(\theta)} \end{eqnarray*} $$
C.G.F. of Power Series Distribution
The cumulant generating function of power series distribution is
$$K_X(t)=\log_e f(\theta e^t) - \log_e f(\theta)$$
Proof
The cumulant generating function of power series distribution is
$$ \begin{eqnarray*} K_X(t) &=& \log_e M_X(t) \\ &=& \log_e \bigg(\frac{f(\theta e^t)}{f(\theta)}\bigg) \\ &=& \log_e f(\theta e^t) - \log_e f(\theta). \end{eqnarray*} $$
Cumulants of Power Series distribution can be obtained by differentiating the c.g.f with respect to $t$ and taking $t=0$.
The $r^{th}$ cumulant is given by
$$ \begin{equation*} \kappa_r= \bigg[\frac{d^r}{dt^r} K_X(t)\bigg]_{t=0}. \end{equation*} $$
Hence,
$$ \begin{eqnarray*} \kappa_1 & = & \bigg[\frac{d}{dt} K_X(t)\bigg]_{t=0} = \frac{d}{dt} \bigg[\log f(\theta e^t)-\log f(\theta)\bigg]_{t=0}.\\ & = & \bigg[\frac{f^\prime(\theta e^t)\theta e^t}{f(\theta e^t)}\bigg]_{t=0}=\theta\frac{f^\prime(\theta)}{f(\theta)}=\mu_1^\prime. \end{eqnarray*} $$
Recurrence Relation for cumulants
The recurrence relation for cumulants of power series distribution is
$$\kappa_{r+1} = \theta \frac{d\kappa_r}{d\theta}$$
Proof
The cumulant generating function of power series distribution is
$$ \begin{equation*} K_X(t) =\log_e f(\theta e^t) - \log_e f(\theta). \end{equation*} $$
The $r^{th}$ cumulants is obtained by
$$ \begin{eqnarray*} \kappa_r &=& \bigg[\frac{d^r K_X(t)}{dt^r} \bigg]_{t=0} \\ &=& \bigg[\frac{d^r}{dt^r}[\log_e f(\theta e^t) - \log_e f(\theta)] \bigg]_{t=0} \\ &=& \bigg[\frac{d^r}{dt^r}\log_e f(\theta e^t) \bigg]_{t=0} \\ \end{eqnarray*} $$
Differentiating above equation w.r.t. $\theta$, we get
$$ \begin{eqnarray}\label{psd8}\nonumber \frac{d\kappa_r}{d\theta} &=& \bigg[\frac{d}{d\theta}\bigg\{\frac{d^r}{dt^r}\log_e f(\theta e^t)\bigg\} \bigg]_{t=0} \\\nonumber &=& \bigg[\frac{d^r}{dt^r}\bigg\{\frac{d}{d\theta}\log_e f(\theta e^t)\bigg\} \bigg]_{t=0} \\ &=& \bigg[\frac{d^r}{dt^r}\bigg\{\frac{e^t f^\prime(\theta e^t)}{f(\theta e^t)} \bigg\} \bigg]_{t=0} \end{eqnarray} $$
Now, $(r+1)^{th}$ cumulants is given by
$$ \begin{eqnarray}\label{psd9}\nonumber \kappa_{r+1} &=& \bigg[\frac{d^{r+1} K_X(t)}{dt^{r+1}} \bigg]_{t=0} \\\nonumber &=& \bigg[\frac{d^r}{dt^r}\bigg\{\frac{d}{dt}\log_e f(\theta e^t) \bigg\}\bigg]_{t=0} \\\nonumber &=& \bigg[\frac{d^r}{dt^r}\bigg\{\frac{\theta e^t f^\prime(\theta e^t)}{f(\theta e^t)} \bigg\} \bigg]_{t=0}\\ &=& \theta \bigg[\frac{d^r}{dt^r}\bigg\{\frac{e^t f^\prime(\theta e^t)}{f(\theta e^t)} \bigg\}\bigg]_{t=0} \end{eqnarray} $$
From (9) and (10), we get
$$ \begin{equation*} \kappa_{r+1} = \theta \frac{d\kappa_r}{d\theta}. \end{equation*} $$
which is the recurrence relation for cumulants of P.S.D.
Hope you enjoyed this article. Do read my next article on Particular Cases of Power Series Distribution to understand some of the particular cases of Power Series Distributions.
If you have any doubt or queries on Power Series Distribution feel free to post them in the comment section.
