Argue that the proposed estimators 𝜆ˆ and 𝜆˜ below are both consistent and asymptotically normal. Then, give their asymptotic variances 𝑉(𝜆ˆ) and 𝑉(𝜆˜) , and decide if one of them is always bigger than the other.

Question

Let 𝑋1,…,𝑋𝑛∼𝑖.𝑖.𝑑.𝖯𝗈𝗂𝗌𝗌(𝜆) , for some 𝜆>0 . Let 𝜆̂ =𝑋⎯⎯⎯⎯⎯𝑛 and 𝜆̃ =−ln(𝑌⎯⎯⎯⎯𝑛) , where 𝑌𝑖=1{𝑋𝑖=0},𝑖=1,…,𝑛 .

𝑉(𝜆ˆ) =? and 𝑉(𝜆˜) =?