Likelihood‐based confidence intervals for a log‐normal mean

Abstract

To construct a confidence interval for the mean of a log-normal distribution in small samples, we propose likelihood-based approaches - the signed log-likelihood ratio and modified signed log-likelihood ratio methods. Extensive Monte Carlo simulation results show the advantages of the modified signed log-likelihood ratio method over the signed log-likelihood ratio method and other methods. In particular, the modified signed log-likelihood ratio method produces a confidence interval with a nearly exact coverage probability and highly accurate and symmetric error probabilities even for extremely small sample sizes. We then apply the methods to two sets of real-life data.

Noninformative priors for linear function of parameters in the lognormal distribution

“…We compare the confidence interval based on Cox' method (Land, 1972), the parametric bootstrap method (Angus, 1994), the modified signed log-likelihood ratio method (Wu et al, 2003) and Bayesian credible intervals for the lognormal mean. Two real examples are illustrated, and these examples analyzed by Wu et al (2003).…”

“…They showed that Cox's method and the parametric bootstrap method are better than the other methods by numerical studies. Wu et al (2003) revealed that Cox's method does not appropriate when the sample size is small, and the parametric bootstrap has good coverage in small variance cases, but has not good performance for large variance cases. However the modified signed log-likelihood ratio method of Wu et al (2003) gives a good coverage even if the variance increases.…”

“…Wu et al (2003) revealed that Cox's method does not appropriate when the sample size is small, and the parametric bootstrap has good coverage in small variance cases, but has not good performance for large variance cases. However the modified signed log-likelihood ratio method of Wu et al (2003) gives a good coverage even if the variance increases. Krishnamoorthy and Mathew (2003) derived the exact confidence interval and tests using the generalized p method, and showed that the coverage probabilities of confidence interval based on the generalized limits are close almost to the nominal coverage level.…”

Confidence Intervals for the Mean of Lognormal Data with Excess Zeros

“…We will demonstrate that the proposed approach consistently outperforms the three approaches proposed by Zhou and Tu (2000). The adjusted signed log-likelihood ratio statistics have been widely used in interval estimation and it has been shown that this approach produces confidence intervals with nearly exact coverage probabilities and highly accurate and symmetric error probabilities even at very small sample sizes by many authors; for example, see for its application in the confidence interval estimation of the ratio of two independent lognormal means; see Wong and Wu (2002) for its application in confidence interval estimation of coefficient of variation; see Wu, Wong and Jiang (2003) for its application in confidence interval estimation of log-normal means; see Tian and Wilding (2005) for its application in construction of a confidence interval of the ratio of two inverse Gaussian means.…”

Interval estimation for log‐normal mean with applications to water quality

“…In this case it is clear that the coverage probability of the Z interval will be below the nominal level since t a=2; ðnÀpÞ > z a=2 . Wu et al (2003) specialized the results of Barndorff-Nielsen (1986 and Barndorff-Nielsen and Cox (1994) to obtain the modified signed likelihood ratio…”