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A Novel Unified Computational Approach to Constructing Shortest-Length or Equal Tails Confidence Intervals in Terms of Pivotal Quantities and Quantile Functions
Automatic Control and Computer Sciences Pub Date : 2021-03-22 , DOI: 10.3103/s0146411621010065
N. A. Nechval , G. Berzins , K. N. Nechval , Zh. Tsaurkubule

Abstract

A confidence interval is a range of values that gives the user a sense of how precisely a statistic estimates a parameter. In the present paper, a new simple computational method is proposed for simultaneous constructing and comparing confidence intervals of shortest length and equal tails in order to make efficient decisions under parametric uncertainty. This method represents a simple and computationally attractive numerical technique for finding the shortest-length and equal tails confidence intervals using pivotal quantities, which are developed from either maximum likelihood estimates or sufficient statistics. In statistics, a pivotal quantity (or pivot) is a function of observations and unobservable parameters such that the function’s probability distribution does not depend on the unknown parameters (including nuisance parameters). Finding a pivotal quantity is not discussed, but the choice a “good” pivotal quantity is essential for the resulting confidence interval to be useful. The unified computational technique yields intervals in several situations which have previously required separate analyses using more advanced techniques and tables for numerical solutions. Unlike the Bayesian approach, the proposed approach is independent of the choice of priors and represents a novelty in the theory of statistical decisions. It allows one to eliminate nuisance parameters from the problem via the technique of invariant statistical embedding and averaging in terms of pivotal quantities (ISE&APQ). It should be noted that the well-known classical approach to constructing confidence intervals of the shortest length considers at least three versions of possible solutions and is in need of information about the form of the probability distribution of pivotal quantity in order to determine an adequate version of the correct solution. The proposed method does not need such information. It receives this information through the quantiles of the probability distribution of pivotal quantity. Therefore, the proposed method automatically recognizes an adequate version of the correct solution. To illustrate this method, numerical examples are given. In detail, the Pareto distribution is discussed.



中文翻译:

基于枢轴数量和分位数函数构造最短或相等尾巴置信区间的新颖的统一计算方法

摘要

置信区间是一个范围的值,它使用户可以了解统计量估计参数的精确程度。本文提出了一种新的简单计算方法,用于同时构造和比较最短长度和相等尾部的置信区间,以便在参数不确定性下做出有效决策。该方法代表了一种简单且在计算上具有吸引力的数值技术,该技术可使用枢轴量来找到最短长度和等尾的置信区间,该枢轴量是根据最大似然估计或足够的统计量得出的。在统计中,枢轴数量(或枢轴)是观察值和不可观察参数的函数,因此该函数的概率分布不依赖于未知参数(包括讨厌的参数)。没有讨论找到关键量,但是选择“好”关键量对于所得到的置信区间有用是必不可少的。统一的计算技术会在几种情况下产生间隔,这些间隔以前需要使用更高级的技术和数值解算表来进行单独分析。与贝叶斯方法不同,所提出的方法与先验的选择无关,并且代表了统计决策理论的新颖性。它允许通过不变的统计嵌入和按枢轴数量取平均值(ISE&APQ)的技术从问题中消除令人讨厌的参数。应该注意的是,构建最短长度的置信区间的众所周知的经典方法考虑了至少三种可能的解决方案,并且需要有关枢轴数量的概率分布形式的信息才能确定适当的版本正确的解决方案。所提出的方法不需要这样的信息。它通过枢轴数量的概率分布的分位数来接收此信息。因此,提出的方法会自动识别正确解决方案的适当版本。为了说明该方法,给出了数值示例。详细地,讨论了帕累托分布。所提出的方法不需要这样的信息。它通过枢轴数量的概率分布的分位数来接收此信息。因此,提出的方法会自动识别正确解决方案的适当版本。为了说明该方法,给出了数值示例。详细地,讨论了帕累托分布。所提出的方法不需要这样的信息。它通过枢轴数量的概率分布的分位数来接收此信息。因此,提出的方法会自动识别正确解决方案的适当版本。为了说明该方法,给出了数值示例。详细地,讨论了帕累托分布。

更新日期:2021-03-22
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