I am trying to do a likelihood analysis on a variable, $Z$, which is defined as
$(1)$ $Z = X - cY$
where $X$ and $Y$ are both independent Poisson distributions with rate parameters $\lambda_{x}, \lambda_{y}$ and $c$ is a constant scaling factor such that $0 < c < 1$.
The underlying physical problem, is trying to isolate the rate of a specific source of photons observed by a detector. $X$ is the counts recorded in the region-of-interest, $Y$ is the counts recorded in a nearby region to determine the background rate, and $c$ is a scaling factor based on the difference in exposure, which gives us our final source rate above $(1)$. Typical values are something like $\lambda_{x}\sim 10−100,\lambda_{y}\sim 100−1000,c\sim0.01−0.001$.
Initially, when working on this with my advisor (neither of us are statisticians, so bear with us), we determined that $cY$ still obeyed a Poisson distribution, and thus $Z$ followed a Skellam distribution. I have since looked deeper into this issue, and found that this is incorrect as the scaled Poisson distribution is fundamentally different from a Poisson distribution.
Is there an appropriate distribution I can use to describe or approximate the resulting variable $Z$?
The best I can understand so far is that $Z$ is some convolution of the Poisson and scaled Poisson processes so I could write that
$(2)$ $p(z;c,\lambda_{x},\lambda_{y}) = \Sigma_{n=0}^{\infty} p_{X}(z+n;\lambda_{x})\cdot p_{Y}(n;c,\lambda_{y}) = \Sigma_{n=0}^{\infty} \frac{e^{-\lambda_{x}}\cdot \lambda_{x}^{z+n}}{(z+n)!}\cdot\frac{e^{-\lambda_{y}}\cdot \lambda_{y}^{n/c}}{(n/c)!} $
where I have simply substituted the pdf for the Poisson and scaled Poission distribution into the summation.
My main questions are have I formulated the convolution correctly and are there any ways to simplify or approximate the resulting pdf?
As an aside, the ultimate goal here is to use this pdf in R to evaluate the log-probability of a small set of observed values and use this to determine a maximum log-likelihood rate for the parameter $Z$. I am also wondering if Wilks' theorem would be appropriate for determining the $\chi^{2}$ value in this case.
Apologies if any of this is naive or miscommunicated, statistics is not my forte.
