How to estimate the parameters of a process that is a Poisson times a constant?

Suppose that we have a sequence of observations $y_1, y_2, ... y_n$ that we know is generated by $ax_1, ax_2, ...ax_n$ where $X \sim \mathrm{Poisson}$. Can we use MLE to estimate $a$ and the $\lambda$ of the Poisson distribution?

EDIT: I deleted my attempt at the MLE, it was not helpful.

If $Y_i = aX_i$ where each $X_i \sim \operatorname{Poisson}(\lambda)$, then $\Pr[Y_i = y] = \Pr[X_i = y/a]$, with the obvious ramification that if $y/a$ is not a nonnegative integer, the probability is zero. Thus the joint likelihood is $$\mathcal L(a, \lambda \mid \boldsymbol y) = \prod_{i=1}^n e^{-\lambda} \frac{\lambda^{y_i/a}}{(y_i/a)!} \mathbb 1 (y_i/a \in \mathbb Z^+ \cup \{0\}).$$
From a practical perspective, if $a$ could be any real number, given a sufficiently large $n$ we can exploit the integer support of $X$ to estimate $a$ with great precision. If $a$ is itself an integer, we simply look for the least common divisor of the observations.
• Aha, I see it, thank you. Supposing that $a$ can indeed be any real number, do you mind going through how to estimate $a$? – Ben S. Feb 22 '18 at 1:59
• Think about it. If I gave you the sample $$\{66.0599, 28.3114, 0., 9.43712, 28.3114, 28.3114, 9.43712, 18.8742, 28.3114, 18.8742\}$$ and told you it came from such a distribution, what would your estimate of $a$ be? Wouldn't your first step be to figure out the pairwise ratios? – heropup Feb 22 '18 at 2:03
• Yes. I guess I should fess up that I have a more complicated estimation of parameters in mind than how I constructed the question (each observation is, instead of $aX$, made from $aX + bY + cZ$ where all of $X$, $Y$, and $Z$ are Poisson. I was hoping understanding how to get $a$ by MLE in the simpler case would let me see how to solve the more complicated question. – Ben S. Feb 22 '18 at 2:10