# Signed log-likelihood ratio test for the scale parameter of Poisson Inverse Weibull distribution with the development of PIW4LIFETIME web application

## TL;DR

This paper introduces a new statistical test for analyzing system failure times using the Poisson Inverse Weibull distribution and provides a web app for practical use.

## Contribution

The study introduces the signed log-likelihood ratio test for the scale parameter of the Poisson Inverse Weibull distribution and develops a web application for its implementation.

## Key findings

- The SLRT maintains type I error rates within acceptable limits across various sample sizes and parameter configurations.
- The SLRT outperforms the ANMLE in empirical power, especially in small-sample scenarios.
- The PIW4LIFETIME web application enables users to fit data to the PIW distribution and perform hypothesis testing.

## Abstract

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 (ω) 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 H0:ω=0.25 against H1:ω=0.5 with β=0.5,λ=1, 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 β while fixing λ=1 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 ω 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.