Second Order Efficiency of the MLE with Respect to any Bounded Bowl-Shape Loss Function

Second Order Efficiency of the MLE with Respect to any Bounded Bowl-Shape Loss Function
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MLE 相对于任何有界碗形损失函数的二阶效率

DOI：
10.1214/aos/1176345005
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发表时间：
1980
期刊：
影响因子：
--
通讯作者：
H. Wieand
H. Wieand
中科院分区：
--
文献类型：
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作者：
J. Ghosh;B. K. Sinha;H. Wieand

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设X 1，X 2，.可以是i.i.d.序列。随机变量，每个变量具有 密度f（x，θ 0）其中{f（x，θ）}是关于a的密度族 主导措施。设n ½（θ θ- θ）和n ½（T - 其中θ是mle，T是任何其他有效估计，有Edgeworth 在θ 0的紧致邻域内均匀展开至o（n-1）。然后 （在某些正则性条件下）可以选择函数c（θ），使得θ = θ 0 + c（θ 1）/n满足Pθ 0 {-x 1 < n ½（θ 0 ′- θ 0）（I（θ 0））½ < x 2 } > Pθ 0 {-x 1 ½（T - θ 0）（I（θ 0））1/2 } + o（n-1）， 对于所有的x 1，x 2 > 0.这一结果意味着二阶效率的 mle关于任何有界损失函数Ln（θ，a）= h（n ½（a - θ））， 其为碗状即，其最小值在a - θ = 0处为零，并且随着|一种 θ|增大这就回答了C. R.拉奥（讨论埃夫隆教授的 纸）。
Let X 1 , X 2 , .. be a sequence of i.i.d. random variables, each having density f(x, θ 0 ) where {f(x, θ)} is a family of densities with respect to a dominating measure µ. Suppose n ½ (θˆ - θ) and n ½ (T - θ), where θˆ is the mle and T is any other efficient estimate, have Edgeworth expansions up to o(n -1 ) uniformly in a compact neighbourhood of θ 0 . Then (under certain regularity conditions) one can choose a function c(θ) such that θˆ = θˆ + c(θˆ)/n satisfies Pθ 0 {-x 1 < n ½ (θˆ' - θ 0 )(I(θ 0 )) ½ < x 2 } > Pθ 0 {-x 1 ½ (T - θ 0 )(I(θ 0 )) ½ 2 } + o(n -1 ), for all x 1 , x 2 > 0. This result implies the second order efficiency of the mle with respect to any bounded loss function Ln(θ, a) = h(n ½ (a - θ)), which is bowl-shaped i.e., whose minimum value is zero at a - θ = 0 and which increases as |a - θ| increases. This answers a question raised by C. R. Rao (Discussion on Professor Efron's paper).