论文标题
关于伽马和beta分布的点估计器
On point estimators for Gamma and Beta distributions
论文作者
论文摘要
令$ x_1，\ ldots，x_n $为伽马分布中的随机示例，密度$ f（x）=λ^αx^{α-1} e^{ - λx}/γ（α）$，$ x> 0 $，在$α> 0 $（形状参数）和$λ> 0 $的情况下，均为参数（cam）。主要结果表明，当且仅当$ n \ geq 4 $时，存在形状参数的最小值无偏估计器（UMVUE），即$α$；此外，仅当$ n \ geq 6 $时，它具有有限的差异。更确切地说，为所有参数函数$α$，$λ$，$ 1/α$和$ 1/λ$提供了UMVUE的形式。此外，还给出了两参数beta分布的高效估计程序。这是基于Beta分布的Stein型协方差身份，然后应用了$ U $统计学和Delta-Method的理论。 MSC：主要62F10; 62F12;中学62E15。 关键词和短语：公正的估计；伽马分布； beta分布； ye-chen型封闭式估计器；渐近效率； $ u $ - 统计； Stein型协方差身份； Delta-Method。
Let $X_1,\ldots,X_n$ be a random sample from the Gamma distribution with density $f(x)=λ^αx^{α-1}e^{-λx}/Γ(α)$, $x>0$, where both $α>0$ (the shape parameter) and $λ>0$ (the reciprocal scale parameter) are unknown. The main result shows that the uniformly minimum variance unbiased estimator (UMVUE) of the shape parameter, $α$, exists if and only if $n\geq 4$; moreover, it has finite variance if and only if $n\geq 6$. More precisely, the form of the UMVUE is given for all parametric functions $α$, $λ$, $1/α$ and $1/λ$. Furthermore, a highly efficient estimating procedure for the two-parameter Beta distribution is also given. This is based on a Stein-type covariance identity for the Beta distribution, followed by an application of the theory of $U$-statistics and the delta-method. MSC: Primary 62F10; 62F12; Secondary 62E15. Key words and phrases: unbiased estimation; Gamma distribution; Beta distribution; Ye-Chen-type closed-form estimators; asymptotic efficiency; $U$-statistics; Stein-type covariance identity; delta-method.
