Robustness of confidence intervals for a normal mean and some interval estimators, powerful unbiased tests under skew-normal model

• Main Authors: Chien-Chih Liao, 廖芊帙
• Other Authors: Yu-Ling Tseng
• Format: Others
• Language: en_US
• Published: 2009
• Online Access: http://ndltd.ncl.edu.tw/handle/60341660789345941535

碩士 === 國立東華大學 === 應用數學系 === 97 === In this thesis, we first study the robustness of confidence intervals for normal mean against skew-normality, then we study some inferential problems for the skewness parameter under skew-normal model. The confidence intervals of normal means are widely used in statistical analyses. Various robustness against the departure of normality has been considered in the literature. Here we consider the robustness of confidence intervals of normal mean under skew-normality. Based on theoretical derivations and simulation studies, we find that the coverage probabilities of the confidence intervals vary with respect to the claimed confidence coefficients. The discrepancies are larger as the skew parameters become larger or as the sample sizes become smaller. In particular, the coverage probabilities sometimes can be much smaller than the nominal confidence coefficient, hence we find confidence intervals for normal mean, one or two-sided, are not robust against skew-normality. Any type of statistical inferences for the skewness parameter under skew-normal model is difficult due to the unusual form of the joint density function of the related random samples, hence, to our knowledge, no related work has been found in the literature. We first consider the problem of constructing 1-α confidence intervals for the skewness parameter. With numerical results and some theoretical derivations, successfully, our proposed one-sided and two-sided intervals do retain the nominal confidence coefficient 1-α. Next, we address the problems of finding tests for choosing normal versus skew-normal model. We first successfully derive the uniformly most powerful level α one-sided tests by using the Neyman-Pearson fundamental lemma for the case when the sample size n=1. It is interesting to notice that these one-sided uniformly most powerful tests can be induced, in certain way, from our proposed one-sided confidence intervals for the skewness parameter. Such approach is, then, employed to induce our proposed tests, one or two-sided, for any given n. Empirical evidence and analytical derivations show our tests have level α, are unbiased and have very good power performance. Especially, powers of our tests are proved to tend to 1 when n → ∞; as a result, it may be difficult to find better tests than ours.