Consider a discrete probability distribution with the following probability mass function $$f(y;\lambda ,ν) =\frac{\lambda^y}{A(\lambda,ν)(y!)^ν}$$, $y= 0,1,2,...,$ with parameters $\lambda \gt 0,ν \ge 0$ and where $A(\lambda,ν) =\sum_{i=0}^\infty\frac {\lambda^y}{(y!)^ν}$ is a normalizing constant.(Note: When $ν= 0$ the distribution is only defined for $0\lt \lambda \lt 1$.) Assume the value of $ν$ is known. Show that the given probability distribution is a natural exponential family and identify the canonical parameter $\theta$ and the cumulant function $κ(\theta)$
Answer: Here, we have to express the PMF in the form of $$f(y;\theta) = a(y)e^{\theta y−κ(\theta)}$$ to show that the given probability distribution is a natural exponential family.
Given probability mass function: $$f(y;\lambda ,ν) =\frac{\lambda^y}{A(\lambda,ν)(y!)^ν}$$ This can be written as :$$f(y;\lambda ,ν) =\frac{e^{y\log\lambda}}{A(\lambda,ν)(y!)^ν}$$ To express it in the natural exponential family,
let’s define: $$\theta = \log(\lambda)$$ Now, the PMF can be written as:$$f(y;\theta,\nu) = e^{y\theta - \log(A(e^{\theta},\nu)) - \nu\log(y!)}$$ This fits the format of the natural exponential family: $f(y;\theta) = a(y)e^{\theta T(y) - \kappa(\theta)}$
Where: $\theta = \log(\lambda)$ is the canonical parameter
$T(y) = y$ is the sufficient statistic
$\kappa(\theta) = \log(A(e^{\theta},\nu))$ is the cumulant function.
Thus, in this case, the canonical parameter is $\theta = \log(\lambda)$ and the cumulant function is $\kappa(\theta) = \log(A(e^{\theta},\nu))$.
I am trying to solve this problem by using definition from book;A natural exponential family (NEF) is a family ${f(.;\theta)|\theta \in Θ}$ of PDFs or PMFs of the form $f(y;\theta) = a(y)e^{\theta y−κ(\theta)}$ for $y\in\Bbb R$ or $y\in \Bbb N_0$, where Θ open is the parameter space.$\theta \in Θ$ is called canonical parameter and $κ(\theta)$ is called cumulant function with $exp(κ(\theta)) =\int_{-\infty}^\infty a(y)e^{\theta y} dy$ for y continuous and $exp(κ(\theta)) = \sum_{i=0}^\infty a(y)e^{\theta y}$ for y discrete.
Can anyone help me to know , am i in the right track?