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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.

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  • $\begingroup$ It is impossible for $cY$ to have a Poisson distribution, because it has positive probabilities of attaining non-integral values (such as $c$ itself). However, you confusingly appear to refer to processes as if they were distributions or random variables. Could you clarify what you're trying to describe? $\endgroup$
    – whuber ♦
    Commented Jun 22, 2021 at 13:24
  • $\begingroup$ @whuber Thank you for the response. Okay, this makes sense. If c had an integer value, would $cY$ then be Poisson? I apologize for referencing processes; I think distributions is closest to what I was trying to convey. I will edit my original post to say this. Essentially, I am trying to find the pdf of $Z$ in as succinct a form as possible. $\endgroup$
    – Jack
    Commented Jun 22, 2021 at 16:13
  • $\begingroup$ The only way $cY$ can have a nondegenerate Poisson distribution when $Y$ has a nondegenerate Poisson distribution is when $c=1.$ The PDF of $Z=X-cY$ is complicated otherwise, because it has positive probabilities at all numbers of the form $m-cn$ for natural numbers $m,n$ and the probability at any value $x$ depends on the number of solutions to $m-cn=x.$ When $c$ is irrational, for instance, those counts are all $0$ or $1$ but $Z$ has a continuous yet not absolutely continuous distribution. Easy approximations are available for non-small $\lambda_x$ and $\lambda_y.$ $\endgroup$
    – whuber ♦
    Commented Jun 22, 2021 at 17:02
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    $\begingroup$ Sure. Basically, I am interested in the photon counts observed from a specific source ($Z$); however, there is a non-negligible background that I must account for. So, to find $Z$, I take the counts recorded in the "source" region ($X$) and the counts recorded in a separate "background" region ($Y$). The counts recorded in $Y$ are then weighted according to the difference in exposures, $c$ (hence: $Z = X - cY$). Typical values are something like $\lambda_{x} \sim 10-100, \lambda_{y} \sim 100-1000, c \sim 0.01 - 0.001$. I will include this in the original post as well. $\endgroup$
    – Jack
    Commented Jun 22, 2021 at 19:11
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    $\begingroup$ Unfortunately, that is not always the case. I will have to take the second option. Many thanks for the help. $\endgroup$
    – Jack
    Commented Jun 22, 2021 at 20:43

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