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˜) =?

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As above, argue that both proposed estimators 𝜆ˆ and 𝜆˜ are 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.

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

𝑉(𝜆ˆ) = ? and 𝑉(𝜆˜)=?

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As above, argue that both proposed estimators 𝑝ˆ,𝑝˜, are consistent and asymptotically normal. Then, give their asymptotic variances 𝑉(𝑝ˆ) and 𝑉(𝑝˜) and decide if one of them is always bigger than the other.

Let 𝑋1,…,𝑋𝑛∼𝑖.𝑖.𝑑.𝖦𝖾𝗈𝗆(𝑝) , for some 𝑝∈(0,1) . That means that

𝐏(𝑋1=𝑘)=𝑝(1−𝑝)𝑘−1,for 𝑘=1,2,….

Let

𝑝ˆ = 1𝑋⎯⎯⎯⎯⎯𝑛,
and 𝑝˜ be the number of ones in the sample divided by 𝑛 .

𝑉(𝑝ˆ)=? and 𝑉(𝑝˜)=?

Answer 1

The proposed estimators 𝑝ˆ and 𝑝˜ are both consistent and asymptotically normal. Their asymptotic variances are 𝑉(𝑝ˆ)=𝑝(1−𝑝) and 𝑉(𝑝˜)=𝑝(1−𝑝)/𝑛 , respectively. The variance of 𝑝ˆ is always bigger than the variance of 𝑝˜ .