Parametric Estimation For The Birnbaum-Saunders Distribution Under The Optimal Sampling Design

Birnbaum-Saunders distribution(BS(α,β))is proposed by Birnbaum and Saunders(1969).BS(α,β)is one of the reliability analysis distribution,and it is commonly used in reliability analysis.In the reliability life test,the failure of product has various reasons.BS(α,β)is used,in particular,to model the lifetime of materials subjected to a cycle stress pattern where ultimate failure is due to growth of a dominant crack.Ranked set sampling(RSS)is an effective sampling method suitable for situations where it is relatively easy to use subjective experience to judge the relative size of individuals to be tested.RSS was originally estimated by McIntyre(1952,2005).Takehasi et al.(1968)began to establish a mathematical theory,because of its high efficiency,it has been widely used in the fields of economy,environment and medicine.The current paper devotes to seek optimal sampling designs and optimal estimators for BS(α,β)distribution:(1)Maximum likelihood estimators(MLEs)of parameter from BS(α,β)distribution in RSS are studied.Furthermore,median ranked set sampling and minimum ranked set sampling based on maximizing the Fisher information are proposed.MLEs of parameter in median ranked set sampling and minimum ranked set sampling are studied.The results of research show that the estimators under RSS are more efficient than the ones under SRS,and RSS carry more information about parameters than SRS.(2)Based on an effective ranked set sample,optimal estimators of parameter from BS(α,β)in RSS are studied.Furthermore,optimal estimators of parameter in median ranked set sampling and minimum ranked set sampling are studied.The results of research show that the estimators under RSS are more efficient than the ones under SRS,and median ranked set sampling and minimum ranked set sampling carry more information about parameters than RSS.(3)In order to reduce the error,Al-Odat et al.(2001)introduced a modification of RSS called moving extremes RSS(MERSS),MLEs of parameter from BS(α,β)in MERSS are studied.The results of research show that the estimators under MERSS are more efficient than the ones under SRS,and MERSS carry more information about parameters than SRS.(4)Based on an effective moving extremes ranked set sample,optimal estimators of parameter from BS(α,β)in MERSS are studied.The results of research show that the estimators under MERSS are more efficient than the ones under SRS.(5)Fisher information in the corresponding samples about the skewness parameter from skew-normal distribution under SRS,RSS,extreme RSS(ERSS)and median RSS(MRSS)will be respectively studied.The results of research show that the ranked set sample,the extreme ranked set sample and the median ranked set sample carry more information than a simple random sample of equivalent size.Because the density function of BS(α,β)distribution is complicated,it is difficult to estimate its parameters.It is still of great theoretical significance and research value to find an effective parameter estimation method.