This paper proposes confidence intervals for a single mean and difference of two means of normal distributions with unknown coefficients of variation (CVs). The generalized confidence interval (GCI) approach and large sample (LS) approach were proposed to construct confidence intervals for the single normal mean with unknown CV. These confidence intervals were compared with existing confidence interval for the single normal mean based on the Student’s t-distribution (small sample size case) and the z-distribution (large sample size case). Furthermore, the confidence intervals for the difference between two normal means with unknown CVs were constructed based on the GCI approach, the method of variance estimates recovery (MOVER) approach and the LS approach and then compared with the Welch–Satterthwaite (WS) approach. The coverage probability and average length of the proposed confidence intervals were evaluated via Monte Carlo simulation. The results indicated that the GCIs for the single normal mean and the difference of two normal means with unknown CVs are better than the other confidence intervals. Finally, three datasets are given to illustrate the proposed confidence intervals.

In this article, we propose approaches for constructing confidence intervals for the single signal-to-noise ratio (SNR) of a log-normal distribution and the difference in the SNRs of two log-normal distributions. The performances of all of the approaches were compared, in terms of the coverage probability and average length, using Monte Carlo simulations for varying values of the SNRs and sample sizes. The simulation studies demonstrate that the generalized confidence interval (GCI) approach performed well, in terms of coverage probability and average length. As a result, the GCI approach is recommended for the confidence interval estimation for the SNR and the difference in SNRs of two log-normal distributions.

The log-normal distribution (skewed distribution or asymmetry distribution) is used to describe random variables comprising positive real values. It is well known that the logarithm values of these are normally distributed (symmetry distribution). Positively right-skewed data applicable to the log-normal distribution are frequently observed in the fields of environmental studies, biology, and medicine. The number of zero observations follows a binomial distribution. However, problems can arise in the analysis of data containing zero observations along with log-normally distributed data, for which the delta-lognormal distribution is often referred to for using the analysis of the data. In statistics, the percentile provides the relative standing of a numerical data point when compared to all of the others in a distribution with reference to the observations at or below it. In this study, estimates for the confidence interval for the ratio of the percentiles of two delta-lognormal distributions are constructed using fiducial generalized confidence interval approaches based on the fiducial quantity and the optimal generalized fiducial quantity, the Bayesian approach, and the parametric bootstrap method. As assessed by Monte Carlo simulations using the RStudio programming in terms of the coverage probability and the average length, the Bayesian approach performed quite well by providing adequate coverage probabilities along with the shortest average lengths in all of the scenarios tested. Daily rainfall data contain both zero and positive values. The daily rainfall data can usually be fitted to the delta-lognormal distribution. Their application to rainfall data is also provided to illustrate their efficacies with real data. The efficacy of the approach is used to compare two rainfall dispersion populations.

The Birnbaum–Saunders distribution is of particular interest for statistical inference. This distribution represents the failure time distribution in engineering. In addition, the Birnbaum–Saunders distribution is commonly used in different areas of science and engineering. Percentiles are a frequently employed statistical concept. Percentiles help ascertain the position of an observation concerning the percentage of data points below it. These percentiles serve as indicators of both the central tendency and the dispersion of data. While comparing two data distributions, the mean is typically the most dependable parameter for describing the population. However, in situations where the distribution exhibits significant skewness, percentiles may sometimes offer a more reliable representation. Herein, the confidence intervals for the difference between percentiles of Birnbaum–Saunders distributions were constructed by the generalized confidence interval (GCI) approach, the bootstrap approach, the Bayesian approach, and the highest posterior density (HPD) approach. A Monte Carlo simulation was conducted to evaluate the performance of the confidence intervals. The performance was considered via coverage probability and average width. The findings suggest that utilizing the GCI approach is advisable for estimating confidence intervals for the disparity between two percentiles. Ultimately, the outcomes of the simulation investigation, coupled with an application in the field of environmental sciences, were outlined.