Mis-specification effects on mean and model choice: A comparison of log-normal and log-logistic distributions

The log-normal and log-logistic distributions are frequently used to model positively skewed data. Despite the availability of numerous model selection techniques, the risk of mis-specification between these two closely related models remains substantial. An incorrect model choice can distort the estimation of key statistical measures. This study investigates how model mis-specification affects mean estimation under mis-specification. We derive the maximum likelihood estimator (MLE) and quasi-MLE (QMLE) of the log-normal mean under a mis-specified log-logistic model and vice versa. We construct bootstrap confidence intervals for both MLE and QMLE to evaluate estimation uncertainty under model mis-specification. Analytical expressions for the ratio of biases and the ratio of mean squared errors are used to quantify the distortion introduced by mis-specification. Simulation and bootstrap experiments validate the theoretical findings and demonstrate the practical implications. A real-world dataset is also analysed to illustrate the applicability of the proposed model selection strategy.