Estimations of Parameters under a Skew-Normal Model and Asymptotic Confidence Intervals for the Mean of a Population with Standardized Skew-Normal Error

• Main Authors: Kuei-Hsiung Chang, 張貴雄
• Other Authors: Yu-Ling Tseng
• Format: Others
• Language: en_US
• Published: 2011
• Online Access: http://ndltd.ncl.edu.tw/handle/64434829685386730742

碩士 === 國立東華大學 === 應用數學系 === 99 === The assumption of normality is widely applied in statistical models, however it is sometimes hard to be justified when analyzing real data. The skew normal distribution is therefore proposed as an extension of the normal distribution that includes a skewness parameter. Despite its many probabilistic properities being derived, there are still certain difficulties in statistical inferences under skew normal models. In particular, the complicated form of the likelihood function under a skew normal model causes numerical issues in finding maximum likelihood estimators (MLE) of parameters. In this study, we first study the choice of a crucial constant k to the two methods provided in Monti (2003) to ease above mentioned numerical difficulty in finding maximum likelihood estimators for skew normal parameters; k=n the sample size is used by Monti. The problem is scrutinized, first, under exponential, normal models where MLE's are known, then, of course, under the skew normal model. Our simulation results indicate that these two methods give better estimators in terms of mean squared error, with choices of k≠n for many of our studied cases, where some are under the skew normal model. Thus, taking into account our findings, the selection k=n taken in Monti (2003) is seriously questioned. Next, in this work, we take on the statistical inference issue, under a skew normal model, from a basic but important framework. More precisely, we provide one-sided or two-sided asymptotic confidence intervals for the mean of a population with standardized skew normal error distribution. Based on theoretical derivations and simulation results, we find that the performance of two-sided confidence interval is satisfactory for moderate to large sample sizes. Also, the coverage probabilities of confidence intervals, especially one-sided, vary drastically with the skewness parameters. Finally, we illustrate how to apply the asymptotic two-sided confidence interval for the population mean under a skew normal model to analyze the health care expenditure data from MEPS, which is analyzed, from confidence interval aspect, in Yu (2005) under normal, gamma, and lognormal models. The results we have here indicate that, successfully, we implement the skew normal model for real data analyses, and better off, our asymptotic two-sided confidence interval has higher coverage probabilities than those considered in Yu (2005) for reasonable sample sizes.