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Transmuted logistic-exponential distribution - some new properties, estimation methods and application with infectious disease mortality data

This paper investigates the New Transmuted Logistic-exponential (NTLE) distribution by deriving previously unexplored properties, evaluating ten estimation methods through Monte Carlo simulations, and demonstrating its superior fit for infectious disease mortality data compared to its base distributions.

Original authors: Isqeel Ogunsola, Abosede Akintunde, Kehinde Yusuff, Basirat Adetona, Faheez Abdulrasaq

Published 2026-03-19

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CC BY 4.0

Original authors: Isqeel Ogunsola, Abosede Akintunde, Kehinde Yusuff, Basirat Adetona, Faheez Abdulrasaq

Original paper licensed under CC BY 4.0 (http://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 trying to predict when a lightbulb will burn out, or how long a patient might survive after a diagnosis. In the world of statistics, we use mathematical "shapes" (called distributions) to model these lifetimes.

For a long time, statisticians have used simple shapes like the Exponential (a straight slide) or the Logistic-Exponential (a slightly curved slide) to make these predictions. But real life is messy. Sometimes data has a weird "hump," sometimes it drags out a long tail, and sometimes it spikes unexpectedly. Simple shapes just can't capture that complexity.

This paper introduces a new, super-flexible shape called the New Transmuted Logistic-Exponential (NTLE) distribution. Think of it as a "Swiss Army Knife" of statistical shapes. It can twist, stretch, and bend to fit almost any kind of messy real-world data.

Here is a breakdown of what the researchers did, using simple analogies:

1. The Problem: The "One-Size-Fits-All" Suit Doesn't Work

The old models (like the standard Exponential) are like a basic t-shirt. It fits okay for some people, but if you have broad shoulders or a long torso, it looks terrible. The researchers wanted to create a "custom-tailored suit" (the NTLE) that could fit any body type (any data pattern).

They had already invented the suit, but they hadn't fully measured it yet. They didn't know:

How "uncertain" it is (Entropy).
Where the "average" failure time actually sits (Mode).
How it behaves when stressed (Reliability).

The Paper's First Job: They took a ruler and a magnifying glass to this new suit, measuring every single property to understand exactly how it behaves. They calculated things like "Shannon Entropy" (which is just a fancy way of saying "how surprised are we by the outcome?") and "Stress-Strength" (the chance that a system survives a shock).

2. The Challenge: The "Hidden Treasure" Hunt

Now that they had the suit, they needed to figure out how to find the right settings for it when looking at real data. This is called Parameter Estimation.

Imagine you have a puzzle with three missing pieces (the parameters $\lambda$ , $\beta$ , and $\delta$ ). You have a pile of data (the puzzle box picture), and you need to find the three pieces that make the picture complete.

The problem? The math to find these pieces is like a maze with no exit sign. You can't just write down a simple formula to solve it; you have to guess and check, over and over, until you get it right.

The Paper's Second Job: They tested 10 different detectives (estimation methods) to see which one was best at finding those missing puzzle pieces.

The Veteran: Maximum Likelihood (MLE) – The old reliable detective.
The Newcomers: Methods like "Maximum Goodness-of-Fit" (MGFE) and "Bayesian" – Newer, sometimes smarter detectives.

They ran a massive simulation (a computer game) where they generated thousands of fake datasets and asked each detective to solve the puzzle. They measured who was the fastest, who made the fewest mistakes, and who stayed calm when the data was small and messy.

The Result: For small groups of data, the MGFE detective was the superstar, finding the pieces with the least error. As the data got bigger, the MLE detective caught up and became very competitive.

3. The Real-World Test: The "COVID-19" Case Study

To prove the suit actually works in the real world, the researchers took a real dataset: COVID-19 death records from Egypt in 2020.

They tried to fit three different suits to this data:

The Basic T-Shirt: The Exponential distribution.
The Standard Suit: The Logistic-Exponential distribution.
The Custom Suit: The new NTLE distribution.

The Outcome:
The "Basic T-Shirt" and "Standard Suit" looked a bit baggy and didn't match the shape of the data well. They missed the peaks and valleys of the death rates.
The NTLE (Custom Suit), however, hugged the data perfectly. It captured the sharp rise and the slow decline of the mortality rates much better than the others.

The Big Takeaway

This paper tells us two main things:

We have a better tool: The NTLE distribution is a powerful new tool for statisticians to model complex, messy lifetimes (like how long a machine lasts or how long a disease lasts).
We know how to use it: They figured out that while the "Maximum Likelihood" method is great for big data, the "Maximum Goodness-of-Fit" method is often the best choice when you have limited data.

In a nutshell: The researchers built a better, more flexible mathematical model for predicting lifetimes, figured out the best way to tune it, and proved it works better than the old models when looking at real-life tragedy like the pandemic. It's a win for anyone trying to understand the unpredictable nature of time and survival.

1. Problem Statement

The study addresses two primary gaps in the statistical literature regarding the New Transmuted Logistic-Exponential (NTLE) distribution, an extension of the Logistic-Exponential (LE) distribution proposed by Adesegun et al. (2023):

Theoretical Gaps: While the NTLE distribution was introduced for modeling lifetime data, several critical mathematical properties remained unexplored. Specifically, the Shannon entropy, Rényi entropy, stochastic ordering, mode, stress-strength reliability, residual life functions, incomplete moments, and concentration curves (Bonferroni and Lorenz) had not been derived.
Estimation and Consistency Gaps: The Maximum Likelihood Estimators (MLE) for NTLE parameters do not have closed-form solutions, and the consistency of these estimates had not been rigorously examined. Furthermore, there was a lack of comparative studies evaluating the performance of NTLE against a wide range of estimation techniques (including Bayesian, goodness-of-fit, and moment-based methods) to determine the most robust approach for parameter estimation, particularly for non-closed-form models.

2. Methodology

The research employs a three-pronged methodological approach: theoretical derivation, simulation-based evaluation, and real-world application.

A. Theoretical Derivation

The authors derived and analyzed several new analytical properties of the NTLE distribution defined by parameters $\lambda$ (scale), $\beta$ (shape), and $\delta$ (transmutation). Key derivations include:

Entropy Measures: Closed-form and integral representations for Shannon and Rényi entropies to quantify uncertainty.
Stochastic Ordering: Proof that the distribution is stochastically smaller as $\lambda$ or $\delta$ increases (for fixed $\beta$ ).
Mode Analysis: Characterization of the mode based on the value of $\beta$ and $\delta$ , identifying conditions for unbounded density at the origin versus interior modes.
Reliability Metrics: Derivation of the stress-strength reliability parameter ( $R = P(Y < X)$ ) and residual life functions (Mean and Reversed).
Inequality Measures: Formulation of incomplete moments, Bonferroni, and Lorenz curves for economic and reliability inequality analysis.

B. Parameter Estimation and Simulation

The study evaluated ten distinct estimation methods for the parameters $(\lambda, \beta, \delta)$ :

Maximum Likelihood Estimation (MLE)
Method of Moments (MME)
Least Squares (LSE)
Weighted Least Squares (WLSE)
Maximum Product of Spacings (MPS)
Bayesian Estimation (using Gamma priors and Metropolis-Hastings MCMC)
Anderson-Darling (ADE)
Cramér-von Mises (CVME)
Percentile Estimation (PCE)
Maximum Goodness-of-Fit Estimation (MGFE)

Simulation Design:

Data Generation: Random samples were generated via the inverse transform method for sample sizes $n = 20, 50, 100, 200, 500, 1000$ .
Parameter Settings: Two scenarios were tested: $(\lambda=1, \beta=1.5, \delta=0.5)$ and $(\lambda=0.5, \beta=1.5, \delta=0.2)$ .
Evaluation Metrics: Performance was assessed using Bias, Mean Squared Error (MSE), and Root Mean Square Error (RMSE).

C. Real-World Application

The NTLE model was applied to a real dataset: COVID-19 mortality data in Egypt (2020).

Comparators: The NTLE was compared against its base distributions: the Exponential and the standard Logistic-Exponential (LE).
Fitting: NTLE parameters were fitted using both MLE and MGFE; base models used MLE.
Criteria: Model fit was evaluated using Log-likelihood, AIC, BIC, CAIC, HQIC, and the Kolmogorov-Smirnov (K-S) test with p-values.

3. Key Contributions

Mathematical Expansion: The paper provides the first comprehensive derivation of entropy, stochastic ordering, and reliability measures for the NTLE distribution, filling a significant theoretical void.
Comparative Estimation Framework: It offers a rare, comprehensive comparison of ten estimation methods for a transmuted distribution. This is particularly valuable for distributions where MLE lacks closed-form solutions.
Identification of Optimal Estimators: The study identifies that while MLE is standard, Maximum Goodness-of-Fit Estimation (MGFE) and Bayesian methods often outperform others in small-to-moderate sample sizes, while MPS and PCE show strong performance in large samples.
Practical Validation: It demonstrates the superiority of the NTLE model over standard exponential and logistic-exponential models in modeling skewed infectious disease mortality data.

4. Results

Simulation Findings

Sample Size Effect: As expected, Bias, MSE, and RMSE decreased for all methods as sample size increased, confirming consistency.
Small Samples ( $n=20$ ): The MGFE method demonstrated superior performance, yielding the lowest MSE and RMSE for $\lambda$ and $\delta$ . The Bayesian method performed best for estimating $\beta$ .
Large Samples ( $n \ge 500$ ): MLE, MPS, and PCE became highly competitive, often outperforming other methods.
Parameter Difficulty: Estimating the transmutation parameter $\delta$ was consistently more challenging (higher variance/bias) than estimating $\lambda$ and $\beta$ , especially in smaller samples.

Application Findings (Egypt COVID-19 Data)

Model Fit: The NTLE distribution provided a significantly better fit than the Exponential and LE distributions.
Statistical Evidence:
- The NTLE (MGFE) model achieved the lowest Kolmogorov-Smirnov statistic ( $D_m = 0.0791$ ) and the highest p-value ($0.7216$), indicating the closest agreement with the empirical data.
- While NTLE (MLE) had the lowest AIC, the MGFE variant was superior in terms of distributional fit (K-S test).
Visual Confirmation: Histograms and Empirical CDF plots confirmed that the NTLE density curve closely followed the data's shape and tail behavior, outperforming the base models.

5. Significance

This paper makes a substantial contribution to the field of statistical reliability and survival analysis:

For Theorists: It establishes a complete mathematical foundation for the NTLE distribution, enabling its use in complex uncertainty and reliability modeling.
For Practitioners: It provides a "user guide" for estimation. The finding that MGFE is robust for small samples is crucial for researchers working with limited data (common in epidemiology or niche engineering failures).
For Epidemiology: The successful application to COVID-19 mortality data validates the NTLE as a flexible tool for modeling infectious disease dynamics, particularly where data exhibits heavy tails or skewness that standard exponential models fail to capture.

In conclusion, the study confirms that the NTLE distribution is a versatile extension of the Logistic-Exponential model, offering enhanced flexibility for real-world data, provided that appropriate estimation methods (specifically MGFE for small samples) are selected.