Consider a discrete probability distribution with the following probability mass function $f(y;\lambda ,\nu) =\frac{\lambda^y}{A(\lambda,\nu)(y!)^\nu}$, $y= 0,1,2,...,$ with parameters $\lambda \gt 0,\nu \ge 0$ and where $A(\lambda,\nu) =\sum_{i=0}^\infty\frac {\lambda^y}{(y!)^\nu}$ is a normalizing constant.(Note: When $\nu= 0$ the distribution is only defined for $0\lt \lambda \lt 1 $.)
(1) Specify the canonical link function of a corresponding generalized linear model.(Hint: The canonical link function cannot be expressed in closed form for every value of $\nu$.)
(2) Which known discrete probability distributions correspond to $\nu= 0$, $\nu= 1$?
ANSWER(1): Here, in GLM, the link function connects the expected value of the response variable $(y)$ to the linear predictor $\eta$ . The canonical link function is the one that directly relates the expected value to the natural parameter of the exponential family distribution.
In my case, the discrete probability distribution belongs to the exponential family with the mean parameter $\lambda$ and dispersion parameter $\nu$ . The natural parameter is defined as: $$\theta =\log \lambda$$ The expected value of the response variable (y) is: $$E(y) =\lambda$$
Therefore, to obtain the canonical link function, we need to find a function $g(·)$ such that:$$g(E(y)) =\lambda$$
and due to the presence of the dispersion parameter $\nu$, this relationship cannot be expressed in closed form for every value of $\nu$.
ANSWER (2): When $\nu = 0$, the distribution is only defined for $0\lt \lambda \lt 1$. This suggests a truncated geometric distribution, where the probability mass function is given by $f(y;\lambda,0) = (1-\lambda)^y \cdot \lambda$.
When $\nu = 1$, the known discrete probability distribution that corresponds to it is the Poisson distribution. The probability mass function of the Poisson distribution is given by $f(y;\lambda,1) = \frac{e^{-\lambda} \cdot \lambda^y}{y!}$.
I am trying to solve this question. But I am not totally sure on it, anyone please clear me if i am wrong here .