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A New SMP Transformed Standard Weibull Distribution for Health Data Modelling

Isqeel Ogunsola, Nurudeen Ajadi, Gboyega Adepoju

TL;DR

This work addresses the need for flexible time-to-event modelling in health data by extending the Weibull distribution through the SMP transformation. The authors introduce the SMPtW distribution with parameters λ and φ, derive its full set of statistical properties (reliability, quantiles, moments, MGF, CF, mode, mean waiting time, mean residual life, stress-strength, order statistics, Renyi entropy), and implement inverse-transform sampling for random number generation. They assess parameter estimation via simulation, showing consistent MLE behavior as sample size increases, and demonstrate superior fit of the three-parameter SMPtW to a fatigue-fracture health dataset compared with several competing distributions using information criteria. Overall, the SMPtW provides a versatile and tractable framework for modelling health data and invites further exploration of SMP-based extensions to other base distributions.

Abstract

New methods of extending base distributions are always invoke to increase their adaptability in modeling real life data. Recently, SMP method was introduced but Weibull distribution is yet to be explored through this method. First, we provide updated review on SMP transformed distributions. We then proposed and developed another extended Weibull distribution through this technique named SMPtW. Importantly, twelve of its statistical properties - reliability measures, quantile function, moment, stress-strength, mean waiting time, moment generating function, characteristics function, renyi entropy, order statistics, mean residual life and mode, were derived and studied extensively. The hazard function has a decreasing, increasing and constant shapes. We found a relation between the quantile of SMPtW and that of SMP Pareto distribution despite their difference in density functions. We adopt the inverse transform approach in random number generation and through simulation we evaluate maximum likelihood estimates (MLE) performance of its parameters. The result showed that MLE is consistent all through. The performance of the distribution was then examined using health dataset compared with five similar distributions. The results showed that three parameters SMPtW performed best among the competing models.

A New SMP Transformed Standard Weibull Distribution for Health Data Modelling

TL;DR

This work addresses the need for flexible time-to-event modelling in health data by extending the Weibull distribution through the SMP transformation. The authors introduce the SMPtW distribution with parameters λ and φ, derive its full set of statistical properties (reliability, quantiles, moments, MGF, CF, mode, mean waiting time, mean residual life, stress-strength, order statistics, Renyi entropy), and implement inverse-transform sampling for random number generation. They assess parameter estimation via simulation, showing consistent MLE behavior as sample size increases, and demonstrate superior fit of the three-parameter SMPtW to a fatigue-fracture health dataset compared with several competing distributions using information criteria. Overall, the SMPtW provides a versatile and tractable framework for modelling health data and invites further exploration of SMP-based extensions to other base distributions.

Abstract

New methods of extending base distributions are always invoke to increase their adaptability in modeling real life data. Recently, SMP method was introduced but Weibull distribution is yet to be explored through this method. First, we provide updated review on SMP transformed distributions. We then proposed and developed another extended Weibull distribution through this technique named SMPtW. Importantly, twelve of its statistical properties - reliability measures, quantile function, moment, stress-strength, mean waiting time, moment generating function, characteristics function, renyi entropy, order statistics, mean residual life and mode, were derived and studied extensively. The hazard function has a decreasing, increasing and constant shapes. We found a relation between the quantile of SMPtW and that of SMP Pareto distribution despite their difference in density functions. We adopt the inverse transform approach in random number generation and through simulation we evaluate maximum likelihood estimates (MLE) performance of its parameters. The result showed that MLE is consistent all through. The performance of the distribution was then examined using health dataset compared with five similar distributions. The results showed that three parameters SMPtW performed best among the competing models.
Paper Structure (21 sections, 7 theorems, 80 equations, 4 figures, 5 tables, 1 algorithm)

This paper contains 21 sections, 7 theorems, 80 equations, 4 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Review of SMP transformed distributions
  3. SMP transformed standard Weibull distribution
  4. Properties of SMPtW
  5. Reliability Measures: Survival and Hazard functions of SMPtW
  6. Quantile Function
  7. Moment
  8. Moment Generating Function of SMPtW
  9. Characteristics Function of SMPtW
  10. Mode of SMPtW
  11. Mean Waiting Time
  12. Mean Residual Life
  13. Stress-Strength Reliability
  14. Order Statistics of SMPtW
  15. Renyi Entropy
  16. ...and 6 more sections

Key Result

Theorem 1

If $Y\sim SMPtW(\lambda,\phi)$ distribution, then the quantile function of $Y$ is given as

Figures (4)

  • Figure 1: PDF of SMPtW
  • Figure 2: CDF of SMPtW
  • Figure 3: Hazard function of SMPtW
  • Figure 4: Survival function of SMPtW

Theorems & Definitions (14)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 4 more