# Asymptotic distribution of maximum likelihood

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

## 1 Answer

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