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Odd Log-Logistic Rayleigh Distribution with Applications

This paper introduces the two-parameter Odd Log-Logistic Rayleigh (OLLR) distribution as a flexible extension of the Rayleigh model, deriving its statistical properties and demonstrating its superior performance in modeling lifetime data through applications on two real-world datasets.

Original authors: Sadiq Abubakar Ismaila, Bello Ibrahim Monday

Published 2026-08-05

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CC BY 4.0

Original authors: Sadiq Abubakar Ismaila, Bello Ibrahim Monday

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to predict how long things will last. Maybe it's a lightbulb, a car engine, or even a human life. In the world of statistics, there's a special tool called a "distribution." Think of a distribution as a master blueprint or a mold that statisticians use to shape data. It helps them answer questions like, "What are the odds this machine breaks down tomorrow?" or "How likely is it that a patient survives another year?"

For a long time, detectives have relied on a classic, simple blueprint called the Rayleigh distribution. It's like a basic, one-size-fits-all cookie cutter. It works great for simple shapes, but real life is messy. Sometimes things break down quickly, sometimes they last forever, and sometimes the risk of breaking changes over time. The old cookie cutter just can't handle those weird, wiggly shapes. To fix this, statisticians have started building "generator families"—think of these as fancy, adjustable cookie cutters with extra knobs and dials. These tools can twist and stretch the basic blueprint to fit almost any shape of data, from the smooth curves of a rolling hill to the jagged spikes of a mountain range. This paper dives into one of these new, super-flexible tools to see if it can solve a mystery that the old, simple tools couldn't crack.

The authors of this paper, Sadiq Abubakar Ismaila and Bello Ibrahim Monday, decided to take a specific, powerful generator called the "Odd Log-Logistic" family and mix it with the classic Rayleigh distribution. The result is a brand-new blueprint they call the Odd Log-Logistic Rayleigh (OLLR) distribution. You can think of the OLLR as a "smart cookie cutter" that has two main knobs to turn: one controls the scale (how wide the data spreads out) and the other controls the shape (how weird or curved the data looks). By adding that second knob, the authors hoped to create a tool that could hug the data much tighter than the old, rigid Rayleigh model.

To test if their new invention actually works, the authors didn't just sit in a lab; they put it to the test with real-world data. They simulated thousands of scenarios using a computer method called Monte Carlo simulation. Imagine rolling a digital dice 1,000 times to see if the new tool guesses the right answer. The results were promising: as they fed the tool more data (increasing the sample size), its guesses got closer and closer to the truth, with very little error. This suggests the tool is reliable and consistent.

But the real magic happened when they applied the OLLR to two very different, real-life stories. The first story was about air conditioning systems in airplanes. They looked at the time intervals between failures, which ranged from a tiny 9 hours to a massive 447 hours. The second story was about bladder cancer patients, tracking how long it took for their cancer to go into remission, with times ranging from a few weeks to over 79 months.

The authors compared their new OLLR model against two other popular models: the standard Rayleigh and the Logistic Rayleigh. They used a scoring system called "information criteria" (like AIC and BIC) to judge which model fit the data best. Think of it like a judge scoring a gymnastics routine: the lower the score, the better the performance. In both the airplane and the cancer patient datasets, the OLLR model scored the lowest, meaning it fit the messy, real-world data better than its competitors. It also passed a "goodness-of-fit" test, which checks if the model's predictions are statistically close enough to reality to be trusted.

The paper concludes that the Odd Log-Logistic Rayleigh distribution is a superior, more flexible alternative for modeling lifetime data. It successfully captured the complex patterns in both the mechanical failures and the medical remission times that the older models missed. While the authors don't claim it solves every problem in the universe, their simulations and real-world tests strongly suggest that this new "smart cookie cutter" is a powerful addition to the statistician's toolkit, ready to help us understand the unpredictable rhythms of how long things last.

Technical Summary: Odd Log-Logistic Rayleigh Distribution with Applications

Problem Statement
Classical lifetime distributions, such as the Rayleigh distribution, often lack the flexibility required to model real-world phenomena characterized by diverse skewness, kurtosis, and varying hazard rate shapes. While the Rayleigh distribution is valued for its simplicity in reliability engineering and survival analysis, its single-scale parameter limits its ability to capture complex hazard behaviors. Existing generalized families, such as the Odd Log-Logistic Generalized (OLL-G) family introduced by Gleaton and Lynch (2006), offer a mechanism to introduce additional shape parameters to baseline distributions, thereby improving tail behavior and adaptability. However, the specific application of the OLL-G family to the Rayleigh distribution to create a more flexible model for lifetime data had not been fully explored in this context.

Methodology
This study proposes the Odd Log-Logistic Rayleigh (OLLR) distribution, a two-parameter extension derived by incorporating the Rayleigh distribution as the baseline within the OLL-G framework. The methodology involves the following steps:

Mathematical Formulation: The Cumulative Distribution Function (CDF) and Probability Density Function (PDF) of the OLLR are derived by substituting the Rayleigh CDF and PDF into the general OLL-G equations. The resulting distribution is defined by a scale parameter ( $\theta$ ) and a shape parameter ( $\beta$ ).
Statistical Properties: The paper derives several structural properties, including:
- The Survival Function and Hazard Rate Function.
- The Quantile Function, used for simulation and random variate generation.
- Infinite series representations of the CDF and PDF using the generalized binomial identity.
- Raw moments and the Moment Generating Function (MGF).
- The distribution of order statistics.
Parameter Estimation: The parameters $\beta$ and $\theta$ are estimated using the Maximum Likelihood Estimation (MLE) method. The log-likelihood function is constructed, and the resulting non-linear score equations are solved numerically using iterative methods.
Simulation Study: A Monte Carlo simulation study with 1,000 replications was conducted to assess the consistency of the MLEs. The study evaluated performance across sample sizes of 50, 100, 200, and 500 using metrics such as Average Estimate, Bias, and Root Mean Squared Error (RMSE).
Real-Life Applications: The model's practical utility was tested by fitting the OLLR distribution to two real-world datasets:
- Dataset 1: Failure times of air-conditioning systems in aircraft (Proschan, 1963).
- Dataset 2: Remission times of bladder cancer patients (Lee & Wang, 2003).
  The OLLR model was compared against the Logistic Rayleigh (LR) and standard Rayleigh distributions using information criteria (AIC, BIC, AICc), Log-Likelihood values, and the Kolmogorov-Smirnov (K-S) test.

Key Contributions and Results

Model Development: The paper successfully establishes the OLLR distribution as a valid probability model, proving that its PDF integrates to unity.
Simulation Consistency: The simulation results demonstrate that the parameter estimates are consistent. As the sample size increases, the bias and RMSE for both $\beta$ and $\theta$ decay toward zero, indicating that the average estimates converge to the true parameter values.
Superiority in Fit: In the application to the two real-life datasets, the OLLR distribution outperformed its competitors:
- Dataset 1: The OLLR achieved the lowest AIC (-170.669), BIC (345.339), and AICc (348.142) values, along with the lowest K-S statistic (0.2993), compared to the LR and Rayleigh models.
- Dataset 2: Similarly, the OLLR provided the best fit with the lowest information criteria and a K-S statistic of 0.2397, slightly outperforming the LR model (0.2399) and significantly outperforming the standard Rayleigh model (0.3521).
Visual Validation: Histograms with density plots for both datasets confirm that the OLLR distribution closely follows the empirical data patterns.

Significance and Claims
The paper claims that the OLLR distribution is a competitive and flexible alternative for modeling lifetime data. By introducing an additional shape parameter to the Rayleigh distribution, the OLLR overcomes the rigidity of the classical model, allowing it to better capture complex hazard rate behaviors observed in reliability engineering and biomedical research. The results indicate that the OLLR provides a superior fit relative to the Logistic Rayleigh and standard Rayleigh distributions across the tested datasets, suggesting its potential utility in fields requiring precise survival and reliability modeling. The study concludes that the null hypothesis—that the datasets follow the OLLR distribution—cannot be rejected at standard significance levels (1%, 5%, or 10%).

Technical Summary: Odd Log-Logistic Rayleigh Distribution with Applications

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