0
$\begingroup$

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

$\endgroup$
1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.