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On improved estimation under Weibull model

机译：改进威布尔模型估计

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摘要

Consider a Weibull distribution W(δ, σ) where δ and σ are positive shape and scale parameters, respectively. The pth quantile and the Fisher information matrix of (δ, σ) per observation are defined. In this A_1 is the Euler's constant and ζ(z) is the Riemann zeta function. The article deals with the estimation problems of the shape parameter δ, the scale parameter σ and the pth quantile ζ. The Weibull distribution is originally used by a Swedish scientist to represent the probability distribution of the breaking strength of materials. Subsequently the use of this distribution has been extended to reliability studies, modeling wind-power and rainfall intensity. The maximum likelihood estimators (MLE) of the shape, scale parameters and the pth quantile of a Weibull distribution do not have closed expressions but can be solved using second-order risk functions under the squared loss function. This method provides the necessary and sufficient condition on a generalized for a single parameter exponential family of distributions. However, since the expressions do not have a closed form so the first-order expression of the MLE has to be obtained. Also the presence of nuisance parameter creates analytical complexities. The MLE of the shape parameter is second-order inadmissible (SOI). The new second-order superior estimator of the shape parameter is also of second-order and is admissible (SOA). The MLE of the scale parameter is second-order admissible and hence no new estimator is proposed. The MLE of the p-th quantile is SOI when p is extreme that is close to 0 or 1 and SOA otherwise. In these cases also improved estimators are obtained that are SOA.
机译：考虑分别δ和σ是正形和比例参数的威布尔分布W（δ，σ）。定义了每个观察的第patileile和Fisher信息矩阵（Δ，σ）。在此A_1中是欧拉的常量，ζ（z）是riemann zeta函数。该物品涉及形状参数δ，刻度参数σ和第patileiLeζ的估计问题。瑞典科学家最初使用威布尔分布来代表材料破碎强度的概率分布。随后，使用该分布的使用已经扩展到可靠性研究，建模风力和降雨强度。威布尔分布的形状，缩放参数和第Pathtile的最大似然估计（MLE）没有封闭表达式，但可以在平方损耗函数下使用二阶风险功能来解决。该方法为单个参数指数族的分布族提供必要和充分的条件。然而，由于表达不具有封闭形式，因此必须获得MLE的一阶表达式。此外，Nuisance参数的存在会产生分析复杂性。形状参数的MLE是二阶不予受理（SOI）。形状参数的新二阶卓越估计也是二阶，可允许（SOA）。比例参数的MLE是二阶可接受，因此提出了任何新的估算器。当P是极端的，第p分位数的MLE是靠近0或1和SOA的SOI。在这些情况下，还可以获得改进的估计器是SOA。

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