Constructing a confidence interval for the ratio of bivariate normal means is a classical problem in statistics. Several methods have been proposed in the literature. The Fieller method is known as an exact method, but can produce an unbounded confidence interval if the denominator of the ratio is not significantly deviated from 0; while the delta and some numeric methods are all bounded, they are only first-order correct. Motivated by a real-world problem, we propose the penalized Fieller method, which employs the same principle as the Fieller method, but adopts a penalized likelihood approach to estimate the denominator. The proposed method has a simple closed form, and can always produce a bounded confidence interval by selecting a suitable penalty parameter. Moreover, the new method is shown to be second-order correct under the bivariate normality assumption, that is, its coverage probability will converge to the nominal level faster than other bounded methods. Simulation results show that our proposed method generally outperforms the existing methods in terms of controlling the coverage probability and the confidence width and is particularly useful when the denominator does not have adequate power to reject being 0. Finally, we apply the proposed approach to the interval estimation of the median response dose in pharmacology studies to show its practical usefulness.

中文翻译：

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双变量正态均值比率的惩罚 Fieller 置信区间

构建二元正态均值比的置信区间是统计学中的经典问题。文献中已经提出了几种方法。Fieller 方法被称为精确方法，但如果比率的分母没有显着偏离 0，则可以产生无界置信区间；虽然 delta 和一些数值方法都是有界的，但它们只是一阶正确的。受现实世界问题的启发，我们提出了惩罚 Fieller 方法，该方法采用与 Fieller 方法相同的原理，但采用惩罚似然法来估计分母。所提出的方法具有简单的封闭形式，并且通过选择合适的惩罚参数总是可以产生有界置信区间。而且，在二元正态假设下，新方法被证明是二阶正确的，即其覆盖概率将比其他有界方法更快地收敛到名义水平。仿真结果表明，我们提出的方法在控制覆盖概率和置信宽度方面通常优于现有方法，并且在分母没有足够的能力拒绝为 0 时特别有用。最后，我们将提出的方法应用于区间在药理学研究中估计中值反应剂量以显示其实际用途。