Bootstrap Tests for the Location Parameter under the Skew-Normal Population with Unknown Scale Parameter and Skewness Parameter

Rendao Ye,Kun Luo,Yiting Lu,Bingni Fang,Weixiao Du

doi:10.3390/math10060921

Abstract

In this paper, the inference on location parameter for the skew-normal population is considered when the scale parameter and skewness parameter are unknown. Firstly, the Bootstrap test statistics and Bootstrap confidence intervals for location parameter of single population are constructed based on the methods of moment estimation and maximum likelihood estimation, respectively. Secondly, the Behrens-Fisher type and interval estimation problems of two skew-normal populations are discussed. Thirdly, by the Monte Carlo simulation, the proposed Bootstrap approaches provide the satisfactory performances under the senses of Type I error probability and power in most cases regardless of the moment estimator or ML estimator. Further, the Bootstrap test based on the moment estimator is better than that based on the ML estimator in most situations. Finally, the above approaches are applied to the real data examples of leaf area index, carbon fibers’ strength and red blood cell count in athletes to verify the reasonableness and effectiveness of the proposed approaches.

Highlights

We only provide the steps of the Bootstrap approach based on the moment estimators for hypothesis testing problem (19)
The Bootstrap test based on the moment estimator is better than that based on the maximum likelihood (ML) estimator in most situations, which can provide a useful approach for the inference on location parameter in the real data examples
By using the centered parameterization and Bootstrap approaches, we study the hypothesis testing and interval estimation problems of location parameters for single and two skew-normal populations with unknown scale parameters and skewness parameters

Summary

Introduction

The real-data distribution tends to be skew with unimodal and asymmetrical characteristics such as dental plaque index data [1], freeway speed data [2] and polarizer manufacturing process data [3]. Some recent studies include: characterizations of distribution [6,7], characteristic functions [8], sampling distributions [9], distribution of quadratic forms [10–12], measures of skewness and divergence [13,14], asymptotic expansions for moments of the extremes [15], rates of convergence of the extremes [16], exact density of the sum of independent random variables [17], identifiability of finite mixtures of the skew-normal distributions [18], etc On this basis, we can use the skew-normal distribution as the fitted distribution of real data and establish a statistical model to solve the practical problem.

Inference on the Location Parameter of Single Skew-Normal Population

Inference on the Location Parameters of Two Skew-Normal Populations

Simulation Results and Discussion

Illustrative Examples

Conclusions