I got a question: We let $X$ and $Y$ be independent random variables with $X$ Poisson distributed with mean $\lambda$ and $Y$ exponentially distributed with rate $\lambda>0$ and we let $(X_1,Y_1),\ldots,(X_n,Y_n)$ be a sample from this distribution.

  1. I have to find the asymptotic distribution for the maximum likelihood estimator.

I think we can use that $\tilde{\lambda} \sim(\lambda, ni(\lambda)^{-1} )$. But how can I find the Fisher Information $I(\lambda)$? Can anyone help me. Maybe this can help: Finding moment estimator and its asymptotic distribution


You can try doing it directly. Let $f(\lambda)$ be the log-likelihood after ignoring the additive terms that do not involve $\lambda$, then (please check my calculation)

$$ f(\lambda) = -n\lambda+\ln{\lambda}\sum {x_i} +n\ln{\lambda} -\lambda\sum{y_i}. $$ Then differentiating with respect to $\lambda$ and equating it to zero get

$$ \hat{\lambda}= \frac{1+\bar{X}}{1+\bar{Y}}. $$ By taking a second derivative you can see that it is indeed the maximum. Then you can obtain the asymptotic distribution of $\hat{\lambda}$ in two steps

  1. Use the central limit theorem to get the joint asymptotic distribution of $$ \sqrt{n}\left((\bar{X},\bar{Y})^{\top}-\left(\lambda,\frac{1}{\lambda}\right)^{\top}\right) $$

  2. Use delta method to get the asymptotic distribution of

$$ \sqrt{n}\left(g(\bar{X},\bar{Y})^{\top}-g\left(\lambda,\frac{1}{\lambda}\right)^{\top}\right), $$

where $g(a,b):= \frac{1+a}{1+b}$.


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