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A New Sine Type II Topp-Leone Frechet Distribution for Modelling Life-Time Data: Theoretical Development and Applications

This paper introduces the three-parameter Sine Type II Topp-Leone Frechet (STIITLF) distribution, derives its key statistical properties and parameter estimation methods, and demonstrates its superior flexibility and goodness-of-fit compared to existing models through Monte Carlo simulations and applications to real-life lifetime data.

Original authors: Waheed Babatunde Yahya, Alhaji Mustapha Mahmud, Emmanuel Shammah Chaku

Published 2026-07-07✓ Author reviewed ⓘ

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CC BY 4.0

Original authors: Waheed Babatunde Yahya, Alhaji Mustapha Mahmud, Emmanuel Shammah Chaku

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 by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict how long a lightbulb will last, or how strong a specific type of carbon fiber is before it snaps. In the world of statistics, we use mathematical "shapes" called distributions to map out these possibilities. Think of a distribution as a custom-made mold that fits the shape of your data.

For decades, statisticians have used a mold called the Fréchet distribution to handle data about extreme events, like the strongest winds in a storm or the longest survival times of patients. It's a reliable, classic mold, but sometimes real-world data is messy, wiggly, and complex. The old mold doesn't fit perfectly; it leaves gaps or bulges where it shouldn't.

The New "Super-Mold"

The authors of this paper, Waheed Babatunde Yahya and his colleagues, decided to build a better, more flexible mold. They created a new shape called the Sine Type II Topp-Leone Fréchet (STIITLF) distribution.

Here is how they did it, using a simple analogy:

The Base: They started with the classic Fréchet mold.
The Trigonometric Twist: They added a "sine" function (the wavy line you see in trigonometry) to the mix. Imagine taking a straight ruler and bending it into a gentle wave to better hug the curves of your data.
The Topp-Leone Layer: They then wrapped this in a "Topp-Leone" layer, which acts like a special coating that allows the mold to stretch and shrink in specific ways to fit tight corners.

The result is a three-parameter super-mold. While the old Fréchet mold had two knobs to turn to adjust its shape, this new one has three. This extra knob gives the statisticians much more control to twist and turn the mold until it fits the data perfectly.

What Did They Do with It?

The paper is essentially a "user manual" and a "proof of performance" for this new mold.

The Blueprint (Theory): The authors wrote down the exact mathematical formulas for this new shape. They calculated how to find the average, the median, and how likely something is to fail at a certain time (reliability). They even figured out how the mold behaves when you look at the "best" or "worst" cases in a group (order statistics).
The Stress Test (Simulation): Before using it on real data, they ran a computer simulation. They generated thousands of fake datasets and tried to fit the new mold to them. They found that as they fed the computer more data, the mold's settings became incredibly accurate, proving it was a reliable tool.
The Real-World Trial (Applications): Finally, they tested the mold on two real-life scenarios:
- Carbon Fibers: They looked at data on how much tension carbon fibers could take before breaking.
- Breast Cancer Patients: They analyzed data on how long 121 patients survived after a diagnosis.

The Verdict

In both tests, the new STIITLF mold outperformed four other popular molds (including the original Fréchet and some other modern variations).

The authors measured success using two main tools:

Log-Likelihood: A score that says, "How well does this shape explain the data?" The new mold got the highest score.
AIC (Akaike Information Criteria): A score that penalizes models for being too complicated. The new mold got the lowest score, meaning it was the most efficient fit.

The Bottom Line

The paper concludes that if you have complex, "wiggly" data about how long things last (like machine parts or human lifespans), this new Sine Type II Topp-Leone Fréchet distribution is a superior tool. It fits the data tighter and more accurately than the older, simpler models they compared it against.

The authors recommend that statisticians and researchers use this new tool whenever they need a highly efficient and flexible way to model complex life-time data.

Technical Summary: A New Sine Type II Topp-Leone Frechet Distribution for Modelling Life-Time Data

Problem Statement
The Frechet distribution, introduced by Fréchet in 1924, is a standard tool for modeling extreme values and life-time data across engineering, actuarial science, and medical fields. However, as noted in the literature, traditional distributions often lack the flexibility required to capture the complexity of modern, diverse datasets. While numerous extensions of the Frechet distribution exist (e.g., exponentiated, beta, and transmuted variants), there remains a need for more adaptable models that can better fit complex life-time phenomena. This study addresses the need for a more flexible distribution capable of modeling complex life-time data with higher efficiency than existing two-parameter Frechet models.

Methodology
The authors introduce a new continuous probability distribution termed the Sine Type II Topp-Leone Frechet (STIITLF) distribution. This model is constructed by applying the Sine Type II Topp-Leone-G (STIITL-G) family generator, proposed by Isa et al. (2023), to the baseline Frechet distribution.

Theoretical Formulation: The STIITLF distribution is defined by three positive parameters: two shape parameters ( $\alpha, \lambda$ ) and one scale parameter ( $\theta$ ). The cumulative distribution function (CDF) and probability density function (PDF) are derived by substituting the Frechet CDF and PDF into the STIITL-G generator equations.
Mathematical Derivations: The paper utilizes Maclaurin series expansions for sine and cosine functions, combined with binomial expansions, to derive linear representations of the PDF and CDF. These expansions facilitate the derivation of key statistical properties, including:
- Moments and Moment-Generating Function (MGF): Explicit formulas for the $r$ -th moments and the MGF are derived using the linearized PDF.
- Reliability and Hazard Functions: Formulas for the survival function and the hazard rate are established.
- Quantile Function: The inverse CDF is derived to allow for the calculation of quantiles and the median.
- Order Statistics: The PDFs for the $r$ -th order statistic, as well as the minimum and maximum order statistics, are formulated.
Parameter Estimation: The parameters of the STIITLF distribution are estimated using the Maximum Likelihood Estimation (MLE) method. The log-likelihood function is derived, and the resulting non-linear simultaneous equations are solved numerically using the Newton-Raphson algorithm via the optim function in R software.
Validation: A Monte Carlo simulation study is conducted to evaluate the performance of the MLEs. The study assesses bias and Root Mean Square Error (RMSE) across varying sample sizes ( $n = 20$ to $250$) and parameter settings.

Key Contributions

New Distribution: The primary contribution is the proposal of the STIITLF distribution, which extends the classical Frechet distribution by incorporating two additional parameters derived from the Sine Type II Topp-Leone family.
Statistical Characterization: The paper provides a comprehensive theoretical framework for the new distribution, including closed-form expressions (via infinite series) for its moments, MGF, reliability, hazard, quantile, and order statistics.
Estimation Framework: The study establishes the MLE procedure for the new model and validates the consistency and efficiency of the estimators through simulation.

Results

Simulation Study: The simulation results (Table 1) demonstrate that as the sample size increases, the bias and RMSE of the estimated parameters decrease, approaching zero. This indicates that the MLEs for the STIITLF distribution are consistent and efficient.
Real-Data Applications: The model's performance was tested on two real-life datasets:
1. Carbon Fiber Strength: Tensile strength data of carbon fibers (Bi and Gui, 2017).
2. Breast Cancer Survival Times: Survival data of 121 patients (Ramos et al., 2013).
Comparative Analysis: The STIITLF distribution was compared against four competing models: the Gompertz Frechet (GoFr), Exponentiated Frechet (ExFr), Type II Topp-Leone Frechet (TIITLF), and the classical Frechet (Fr) distribution.
- Dataset 1: The STIITLF model yielded the highest Log-Likelihood (LL = -56.4231) and the lowest Akaike Information Criterion (AIC = 118.8462), outperforming all competitors.
- Dataset 2: Similarly, for the breast cancer data, the STIITLF model achieved the best fit with an LL of -596.3071 and an AIC of 1198.614, significantly lower than the AIC values of the TIITLF, GoFr, ExFr, and Fr models.
- Visual inspections of the fitted PDFs (Figures 3 and 4) further confirmed the superior fit of the STIITLF distribution.

Significance and Claims
The paper claims that the STIITLF distribution offers a significant improvement in modeling efficiency for complex life-time data compared to existing Frechet-based models. By extending the two-parameter Frechet distribution to a three-parameter model, the authors argue that the new distribution provides greater flexibility in capturing diverse data characteristics, particularly skewed data.

The authors conclude that the STIITLF distribution is a robust and versatile tool for statistical modeling. Based on the goodness-of-fit statistics (Log-Likelihood and AIC) and the simulation results, they recommend the adoption of the STIITLF distribution for modeling complex life-time datasets where high model efficiency and reliability are desired. The study positions the STIITLF as a superior alternative to the classical Frechet and its recent variants for applications in engineering, medical sciences, and reliability analysis.

The New "Super-Mold"

What Did They Do with It?

The Verdict

The Bottom Line

Technical Summary: A New Sine Type II Topp-Leone Frechet Distribution for Modelling Life-Time Data

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