Empirical powers for testing versus when .

The three-parameter Poisson Inverse Weibull (PIW) distribution offers enhanced flexibility for modeling system failure times. This study introduces the signed log-likelihood ratio test (SLRT) for hypothesis testing of the scale parameter () in the PIW distribution and compares its performance with the test based on the asymptotic normality of maximum likelihood estimators (ANMLE). Simulation studies show that the SLRT consistently maintains type I error rates within the acceptable range of 0.04 to 0.06 at a significance level of 0.05, satisfying Cochran’s criterion across various sample sizes and parameter configurations. In contrast, the ANMLE method tends to be conservative, often underestimating the nominal significance level. In terms of empirical power, the SLRT outperforms the ANMLE, particularly in small-sample scenarios (n = 10, 15), and maintains superior power across all tested configurations. For example, when testing against with , and n = 10, the SLRT achieves a power of 0.6621, compared to 0.4181 for the ANMLE, demonstrating the SLRT’s robustness and reliability in limited-data. Moreover, the ANMLE generally exhibits low power in most cases, indicating reduced sensitivity to detecting true effects in small samples. However, with medium and large sample sizes (n = 30, 50, 80 and 100), the power of the ANMLE begins to approach that of the SLRT. Despite this, the ANMLE never outperforms the SLRT, highlighting a fundamental limitation of this method. Additionally, varying the shape parameter while fixing showed a negligible impact on power, further confirming the robustness of the SLRT. Sensitivity analyses also validate the reliability of the SLRT under extreme values of and across different sample sizes. To support practical application, the PIW4LIFETIME web application (accessible at https://jularatchumnaul.shinyapps.io/PIW4LIFETIME/) was developed to enable users to assess whether data fit the PIW distribution, estimate model parameters using maximum likelihood, and perform two-sided test for the scale parameter using SLRT. The performance of the proposed method and the PIW4LIFETIME web application was demonstrated through a real-world example.三参数泊松逆威布尔（Poisson Inverse Weibull, PIW）分布在系统失效时间建模中具备更优异的灵活性。本研究提出了针对PIW分布尺度参数的符号对数似然比检验（Signed Log-likelihood Ratio Test, SLRT），并将其性能与基于最大似然估计渐近正态性（Asymptotic Normality of Maximum Likelihood Estimators, ANMLE）的检验方法进行了对比。 模拟研究结果表明，在显著性水平为0.05的情况下，SLRT的一类错误率始终维持在0.04至0.06的可接受范围内，在各类样本量与参数配置下均符合科克伦准则。与之相对，ANMLE方法往往过于保守，经常低估名义显著性水平。 在经验功效方面，SLRT的表现优于ANMLE，尤其在小样本场景（样本量n=10、15）中优势显著，且在所有测试配置下均保持更优的功效。例如，当针对尺度参数进行检验，在给定参数配置与n=10的场景下，SLRT的功效可达0.6621，而ANMLE仅为0.4181，这体现了SLRT在有限数据场景下的稳健性与可靠性。此外，ANMLE在多数场景下功效普遍偏低，表明其在小样本中检测真实效应的灵敏度不足。不过当样本量达到中大型（n=30、50、80与100）时，ANMLE的功效开始逐渐趋近于SLRT。即便如此，ANMLE始终未能超越SLRT，这凸显了该方法的固有局限性。 另外，在固定某一形状参数的前提下调整另一形状参数，对检验功效的影响微乎其微，这进一步验证了SLRT的稳健性。敏感性分析同样证实了SLRT在极端参数值与不同样本量下的可靠性。 为支撑实际应用，本研究开发了PIW4LIFETIME网页应用（访问地址：https://jularatchumnaul.shinyapps.io/PIW4LIFETIME/），用户可通过该工具评估数据是否符合PIW分布、采用最大似然法估计模型参数，并借助SLRT完成尺度参数的双侧检验。本研究通过一则实际案例验证了所提方法与PIW4LIFETIME网页应用的性能。