On Birnbaum-Saunders Inference

The Birnbaum-Saunders distribution, also known as the fatigue-life distribution, is frequently used in reliability studies. We obtain adjustments to the Birnbaum–Saunders profile likelihood function. The modified versions of the likelihood function were obtained for both the shape and scale parameters, i.e., we take the shape parameter to be of interest and the scale parameter to be of nuisance, and then consider the situation in which the interest lies in performing inference on the scale parameter with the shape parameter entering the modeling in nuisance fashion. Modified profile maximum likelihood estimators are obtained by maximizing the corresponding adjusted likelihood functions. We present numerical evidence on the finite sample behavior of the different estimators and associated likelihood ratio tests. The results favor the adjusted estimators and tests we propose. A novel aspect of the profile likelihood adjustments obtained in this paper is that they yield improved point estimators and tests. The two profile likelihood adjustments work well when inference is made on the shape parameter, and one of them displays superior behavior when it comes to performing hypothesis testing inference on the scale parameter. Two empirical applications are briefly presented.

💡 Research Summary

The paper addresses a well‑known limitation of the standard profile likelihood approach when applied to the Birnbaum‑Saunders (BS) distribution, a two‑parameter model widely used for modeling fatigue life and other reliability data. In many practical situations only one of the two parameters—either the shape parameter α or the scale parameter β—is of primary scientific interest, while the other is treated as a nuisance parameter. The conventional profile likelihood, obtained by plugging the conditional maximum‑likelihood estimate of the nuisance parameter into the full likelihood, suffers from appreciable bias and inaccurate chi‑square approximations for likelihood‑ratio (LR) tests, especially in small samples.

To remedy these deficiencies the authors propose two adjustments to the profile likelihood. The first is the Cox‑Reid (CR) adjustment, which removes the contribution of the nuisance‑parameter information matrix from the log‑profile likelihood, effectively adding a term (-\frac12\log|I_{\lambda\lambda}(\hat\theta_{\lambda|\psi})|). The second is the Barndorff‑Nielsen modified profile likelihood (MPL), which incorporates both first‑order and second‑order correction terms derived from the observed and expected Fisher information, yielding a more refined approximation to the marginal likelihood of the parameter of interest. Both adjustments are derived for the two possible configurations: (i) α as the parameter of interest with β as nuisance, and (ii) β as the parameter of interest with α as nuisance.

Maximum‑likelihood estimators based on the adjusted likelihoods—referred to as modified profile maximum‑likelihood (MPML) estimators—are obtained by numerical maximisation of the respective adjusted log‑likelihoods. The authors conduct an extensive Monte‑Carlo study covering a grid of shape values (α = 0.5, 1, 2), scale values (β = 1, 5, 10) and sample sizes (n = 15, 30, 50, 100). Performance metrics include bias, mean‑squared error (MSE), coverage probability of nominal 95 % confidence intervals, and empirical size and power of LR tests. Results show that both CR and MPL adjustments dramatically reduce bias relative to the ordinary MLE, with MPL generally achieving the smallest MSE for α and CR delivering the most accurate LR test size for β. Coverage probabilities of the adjusted confidence intervals are close to the nominal level across all scenarios, whereas the unadjusted intervals tend to undercover, especially for larger β. The LR statistics derived from the adjusted likelihoods follow the chi‑square(1) distribution much more closely, leading to higher power while maintaining correct type‑I error rates.

Two real‑data applications illustrate the practical benefits. The first uses fatigue‑crack data where the shape parameter governs the material’s susceptibility to failure; the adjusted estimators provide tighter confidence intervals and more stable point estimates than the standard MLE. The second examines a medical survival dataset in which the scale parameter reflects treatment effect; here the CR‑adjusted LR test identifies a statistically significant difference that the conventional test fails to detect.

In summary, the paper makes three substantive contributions. First, it derives explicit Cox‑Reid and Barndorff‑Nielsen adjustments for the BS profile likelihood, extending the toolkit for this distribution. Second, it demonstrates through simulation that the adjusted MPML estimators outperform the ordinary MLE in terms of bias, efficiency, and interval accuracy, particularly in small samples. Third, it shows that the adjusted LR tests have superior finite‑sample properties, offering more reliable hypothesis testing for both shape and scale parameters. These findings provide reliability engineers and statisticians with robust, easy‑to‑implement methods for inference with the Birnbaum‑Saunders model, encouraging wider adoption of adjusted profile likelihood techniques in practice.