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

Isqeel Ogunsola, Abosede Akintunde, Kehinde Yusuff, Basirat Adetona, Faheez Abdulrasaq

Abstract

Lately, a New Transmuted Logistic-exponential (NTLE) distribution was introduced and studied as an extension of the Logistic-Exponential Distribution (LED) with wider applicability in lifetime modelling. However, the maximum likelihood estimates (MLE) of NTLE are not in closed form, and the consistency of the estimates was not examined. Furthermore, some other important properties of NTLE, namely the Shannon entropy, Rényi entropy, stochastic ordering, mode, stress-strength reliability measure, residual life functions (mean and reverse), incomplete moments, Bonferroni and Lorenz curves are yet to be derived. Motivated by this, we derived and studied these important properties and evaluated the performance of ten estimation methods (Maximum Likelihood, Moments, Least Squares, Weighted Least Squares, Maximum product of Spacings, Anderson-Darling, Cramer-von Mises, percentile estimation, and Maximum Goodness-of-Fit methods) for NTLE parameters via Monte Carlo simulation using bias, mean square error, and root mean square error as evaluation criteria. Real-life infectious mortality data fitted to the distributions showed that NTLE has a better fit compared to its base distributions (Exponential and Logistic-Exponential). This finding contributes valuable insights for researchers and practitioners when selecting the appropriate estimation methods, especially for NTLE and some similar distributions in non-closed form.

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

Abstract

Lately, a New Transmuted Logistic-exponential (NTLE) distribution was introduced and studied as an extension of the Logistic-Exponential Distribution (LED) with wider applicability in lifetime modelling. However, the maximum likelihood estimates (MLE) of NTLE are not in closed form, and the consistency of the estimates was not examined. Furthermore, some other important properties of NTLE, namely the Shannon entropy, Rényi entropy, stochastic ordering, mode, stress-strength reliability measure, residual life functions (mean and reverse), incomplete moments, Bonferroni and Lorenz curves are yet to be derived. Motivated by this, we derived and studied these important properties and evaluated the performance of ten estimation methods (Maximum Likelihood, Moments, Least Squares, Weighted Least Squares, Maximum product of Spacings, Anderson-Darling, Cramer-von Mises, percentile estimation, and Maximum Goodness-of-Fit methods) for NTLE parameters via Monte Carlo simulation using bias, mean square error, and root mean square error as evaluation criteria. Real-life infectious mortality data fitted to the distributions showed that NTLE has a better fit compared to its base distributions (Exponential and Logistic-Exponential). This finding contributes valuable insights for researchers and practitioners when selecting the appropriate estimation methods, especially for NTLE and some similar distributions in non-closed form.
Paper Structure (29 sections, 9 theorems, 213 equations, 2 figures, 12 tables)

This paper contains 29 sections, 9 theorems, 213 equations, 2 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Some New properties of NTLE
  3. Shannon entropy
  4. Rényi entropy
  5. Stochastic ordering
  6. Mode of the NTLE distribution
  7. Stress-strength reliability
  8. Residual life functions
  9. Incomplete moments
  10. Bonferroni and Lorenz curves
  11. Methods of parameter estimation for NTLE
  12. Maximum Likelihood estimation
  13. Observed Fisher Information Matrix
  14. Method of Moments estimation
  15. Least Squares Estimation (LSE)
  16. ...and 14 more sections

Key Result

Theorem 1

Let $Y\sim NTLE(\lambda, \beta, \delta)$ with probability density function where $\lambda>0$, $\beta>0$ and $-1<\delta<1$. Then, the Shannon entropy is in the form where and For $\delta\neq 0$, the term $K_{\delta}$ has the closed form while, by implication, as a result, for $\delta\neq 0$,

Figures (2)

  • Figure 1: Histogram of COVID - 19 deaths in Egypt with fitted density functions of the competing models.
  • Figure 2: Empirical cumulative distribution function and fitted cumulative distribution functions of the competing models.

Theorems & Definitions (21)

  • Theorem 1
  • proof
  • Remark 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Remark 2
  • Theorem 4
  • proof
  • ...and 11 more