Confidence intervals for the signal-to-noise ratio of a log-normal distribution based on the normal approximation

The objective of this research is to construct the confidence intervals (CIs) for the signal-to-noise ratio (SNR) of a log-normal distribution. The proposed CIs are based on the normal approximation and the delta methods. Three point estimators for the SNR used in this study are the naive estimator, logarithmic estimator and the special estimator. Naive estimator can be applied to any distribution (not necessary to the log-normal distribution). In addition, the special estimator works only for the log-normal distribution. There are five approaches for constructing CIs which are (i) CI for the SNR based on the naive estimator constructed from any distribution, (ii) logarithmic CI for the SNR based on the naive estimator constructed from any distribution, (iii) CI for the SNR based on the naive estimator constructed from log-normal distribution, (iv) logarithmic CI for the SNR based on the naive estimator constructed from log-normal distribution, and (v) CI for the SNR based on the special estimator constructed from log-normal distribution. The efficacy of all approaches is determined in terms of the coverage probability and average width. Monte Carlo simulation method is applied to investigate the performance of all five CIs. These CIs were applied to real data in terms of the SNR of the survival times of small-cell lung cancer (SCLC) patients in Canada and real data of the daily particulate matters 2.5 (PM2.5) level in Bangkok, Thailand.

本研究的目标是构建对数正态分布（log-normal distribution）的信噪比（signal-to-noise ratio, SNR）的置信区间（confidence interval, CI）。所提出的置信区间基于正态近似法（normal approximation）和Delta方法（delta method）。本研究中使用的三种信噪比点估计量为朴素估计量（naive estimator）、对数估计量（logarithmic estimator）和特殊估计量（special estimator）。朴素估计量可应用于任何分布（不一定局限于对数正态分布）。此外，特殊估计量仅适用于对数正态分布。构建置信区间的方法共有五种，分别为：（i）基于适用于任何分布的朴素估计量构建的信噪比置信区间；（ii）基于适用于任何分布的朴素估计量构建的对数信噪比置信区间；（iii）基于适用于对数正态分布的朴素估计量构建的信噪比置信区间；（iv）基于适用于对数正态分布的朴素估计量构建的对数信噪比置信区间；（v）基于适用于对数正态分布的特殊估计量构建的信噪比置信区间。所有方法的有效性通过覆盖概率（coverage probability）和平均宽度（average width）来评估。本研究采用蒙特卡洛模拟法（Monte Carlo simulation method）来考察这五种置信区间的性能。这些置信区间被应用于实际数据，包括加拿大小细胞肺癌（small-cell lung cancer, SCLC）患者生存时间的信噪比数据，以及泰国曼谷的每日细颗粒物2.5（particulate matter 2.5, PM2.5）水平数据。