Asked by ramj

Argue that the proposed estimators πœ†Λ† and πœ†Λœ below are both consistent and asymptotically normal. Then, give their asymptotic variances 𝑉(πœ†Λ†) and 𝑉(πœ†Λœ) , and decide if one of them is always bigger than the other.

Let 𝑋1,…,π‘‹π‘›βˆΌπ‘–.𝑖.𝑑.π–―π—ˆπ—‚π—Œπ—Œ(πœ†) , for some πœ†>0 . Let πœ†Μ‚ =π‘‹βŽ―βŽ―βŽ―βŽ―βŽ―π‘› and πœ†Μƒ =βˆ’ln(π‘ŒβŽ―βŽ―βŽ―βŽ―π‘›) , where π‘Œπ‘–=1{𝑋𝑖=0},𝑖=1,…,𝑛 .

𝑉(πœ†Λ†) =? and 𝑉(πœ†Λœ) =?



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As above, argue that both proposed estimators πœ†Λ† and πœ†Λœ are consistent and asymptotically normal. Then, give their asymptotic variances 𝑉(πœ†Λ†) and 𝑉(πœ†Λœ), and decide if one of them is always bigger than the other.

Let 𝑋1,…,π‘‹π‘›βˆΌπ‘–.𝑖.𝑑.𝖀𝗑𝗉(πœ†) , for some πœ†>0 . Let πœ†Λ†=1π‘‹βŽ―βŽ―βŽ―βŽ―βŽ―π‘› and πœ†Μƒ =βˆ’ln(π‘ŒβŽ―βŽ―βŽ―βŽ―π‘›) , where π‘Œπ‘–=1{𝑋𝑖>1},𝑖=1,…,𝑛 .

𝑉(πœ†Λ†) = ? and 𝑉(πœ†Λœ)=?

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

Let 𝑋1,…,π‘‹π‘›βˆΌπ‘–.𝑖.𝑑.π–¦π–Ύπ—ˆπ—†(𝑝) , for some π‘βˆˆ(0,1) . That means that

𝐏(𝑋1=π‘˜)=𝑝(1βˆ’π‘)π‘˜βˆ’1,for π‘˜=1,2,….

Let

𝑝ˆ = 1π‘‹βŽ―βŽ―βŽ―βŽ―βŽ―π‘›,
and π‘Λœ be the number of ones in the sample divided by 𝑛 .

𝑉(𝑝ˆ)=? and 𝑉(π‘Λœ)=?



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Answered by Bot
The proposed estimators 𝑝ˆ and π‘Λœ are both consistent and asymptotically normal. Their asymptotic variances are 𝑉(𝑝ˆ)=𝑝(1βˆ’π‘) and 𝑉(π‘Λœ)=𝑝(1βˆ’π‘)/𝑛 , respectively. The variance of 𝑝ˆ is always bigger than the variance of π‘Λœ .