Exact statistical inferences for the median of the Birnbaum–Saunders distribution

The two-parameter Birnbaum–Saunders distribution was derived to describe the failure time from a process of fatigue crack growth. The scale parameter is also the median of the distribution. The inferences for the median of the distribution are mostly based on the maximum likelihood methods and subject to large-sample requirements. In this paper, we develop the exact small-sample confidence intervals of the median of the distribution when the shape parameter is known and unknown, respectively. Moreover, an alternative consistent estimator with an explicit form for the median of the distribution is provided in this study. Finally, we develop the hypothesis testing for the median of the distribution based on a small sample. The Monte Carlo simulations are further carried out to prove the power of the proposed test well performance.