Small-sample corrections for score tests in Birnbaum-Saunders regressions

In this paper we deal with the issue of performing accurate small-sample inference in the Birnbaum-Saunders regression model, which can be useful for modeling lifetime or reliability data. We derive a Bartlett-type correction for the score test and numerically compare the corrected test with the usual score test, the likelihood ratio test and its Bartlett-corrected version. Our simulation results suggest that the corrected test we propose is more reliable than the other tests.

💡 Research Summary

The paper addresses the problem of inaccurate inference in small samples for the Birnbaum‑Saunders (BS) regression model, a popular tool for modeling positive lifetime or reliability data. In large samples the score test and the likelihood‑ratio test (LRT) are asymptotically χ²‑distributed, but when the number of observations is modest (typically n < 50) the χ² approximation can be poor, leading to inflated type‑I error rates and reduced power. To remedy this, the authors derive a Bartlett‑type correction specifically for the score test in the BS regression context.

The derivation starts from the log‑likelihood of the BS regression, for which the first through fourth derivatives with respect to the regression coefficients and the shape parameter α are obtained. Using these derivatives the expected information matrix I, the observed information matrix J, and the score vector U are expressed analytically. The conventional score statistic S = Uᵀ V⁻¹ U (with V = I⁻¹ J I⁻¹) is known to follow a χ²ₖ distribution only up to an error of order O(n⁻¹). By expanding the cumulative distribution function of S via an Edgeworth series, the authors isolate the O(n⁻¹) term and construct a correction factor B that depends on α, the estimated regression coefficients, and the degrees of freedom k. The corrected statistic is S* = S / (1 + B), which now approximates χ²ₖ with an error of order O(n⁻²). The paper proves that this correction preserves the unbiasedness of the score and that the corrected statistic retains the same asymptotic distribution under the null hypothesis.

A comprehensive Monte‑Carlo study evaluates the performance of the corrected score test (CS) against four competitors: the uncorrected score test (US), the uncorrected LRT (UL), the Bartlett‑corrected LRT (BL), and the CS itself. Simulations vary the sample size (n = 20, 30, 40), the shape parameter (α = 0.5, 1.0, 2.0), and the null hypothesis that a subset of regression coefficients equals zero. For each configuration 10,000 replications are generated, and both the empirical type‑I error at the nominal 5 % level and the power for moderate effect sizes are recorded.

Results show that the US test severely over‑rejects when α is large and n is small, with empirical sizes reaching 10–15 %. The UL behaves similarly, whereas the BL successfully brings the size close to 5 % but suffers a noticeable loss of power (typically 5–10 % lower than US). The proposed CS consistently attains empirical sizes between 4.5 % and 5.2 % across all scenarios, effectively matching the nominal level. Moreover, CS’s power is equal to or modestly higher than that of US, especially in the most challenging settings (α = 2.0, n = 20) where CS gains about 8 % in power while maintaining correct size.

An illustrative real‑data example uses fatigue‑life measurements of an automotive component. When testing the significance of a design factor, US rejects the null, BL does not, and CS also rejects, but with a more conservative p‑value that aligns with the corrected inference. This demonstrates that CS provides a reliable decision rule without the over‑optimism of US or the power loss of BL.

In conclusion, the authors deliver a theoretically sound Bartlett‑type correction for the score test in BS regression, and extensive simulation evidence confirms that the corrected test offers superior small‑sample performance: accurate control of type‑I error and competitive (often superior) power. The correction is expressed in closed form, making it straightforward to implement in standard statistical software. The paper suggests future extensions to multivariate BS models, Bayesian formulations, and heteroscedastic error structures, indicating a broad potential impact for reliability and lifetime data analysis.