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Compounded Linear Failure Rate Distribution: Properties, Simulation and Analysis

Suchismita Das, Akul Ameya, Cahyani Karunia Putri

TL;DR

The paper introduces the Compounded Linear Failure Rate Distribution (CLFRD), a three-parameter extension of the classical LFRD obtained by Poisson-compounding the minimum of an i.i.d. LFRD sample. It derives the CLFRD's survival and density, establishes a full set of statistical properties (MRL, MIT, moments, quantiles, order statistics, and stochastic orders), and develops maximum likelihood estimation with an observed-information-based asymptotic theory. Through extensive simulations and three real-data studies, CLFRD demonstrates superior or competitive fit relative to LFRD, RD, ED, and GED, with hazard shapes that include increasing, bathtub, and inverse bathtub. The results suggest CLFRD as a versatile, tractable tool for reliability and survival analysis, capable of capturing complex aging patterns and supporting maintenance and risk assessment decisions.

Abstract

This paper proposes a new extension of the linear failure rate (LFR) model to better capture real-world lifetime data. The model incorporates an additional shape parameter to increase flexibility. It helps model the minimum survival time from a set of LFR distributed variables. We define the model, derive certain statistical properties such as the mean residual life, the mean inactivity time, moments, quantile, order statistics and also discuss the results on stochastic orders of the proposed distribution. The proposed model has increasing, bathtub shaped and inverse bathtub shaped hazard rate function. We use the method of maximum likelihood estimation to estimate the unknown parameters. We conduct simulation studies to examine the behavior of the estimators. We also use three real datasets to evaluate the model, which turns out superior compared to classical alternatives.

Compounded Linear Failure Rate Distribution: Properties, Simulation and Analysis

TL;DR

The paper introduces the Compounded Linear Failure Rate Distribution (CLFRD), a three-parameter extension of the classical LFRD obtained by Poisson-compounding the minimum of an i.i.d. LFRD sample. It derives the CLFRD's survival and density, establishes a full set of statistical properties (MRL, MIT, moments, quantiles, order statistics, and stochastic orders), and develops maximum likelihood estimation with an observed-information-based asymptotic theory. Through extensive simulations and three real-data studies, CLFRD demonstrates superior or competitive fit relative to LFRD, RD, ED, and GED, with hazard shapes that include increasing, bathtub, and inverse bathtub. The results suggest CLFRD as a versatile, tractable tool for reliability and survival analysis, capable of capturing complex aging patterns and supporting maintenance and risk assessment decisions.

Abstract

This paper proposes a new extension of the linear failure rate (LFR) model to better capture real-world lifetime data. The model incorporates an additional shape parameter to increase flexibility. It helps model the minimum survival time from a set of LFR distributed variables. We define the model, derive certain statistical properties such as the mean residual life, the mean inactivity time, moments, quantile, order statistics and also discuss the results on stochastic orders of the proposed distribution. The proposed model has increasing, bathtub shaped and inverse bathtub shaped hazard rate function. We use the method of maximum likelihood estimation to estimate the unknown parameters. We conduct simulation studies to examine the behavior of the estimators. We also use three real datasets to evaluate the model, which turns out superior compared to classical alternatives.
Paper Structure (16 sections, 4 theorems, 56 equations, 3 figures, 19 tables)

This paper contains 16 sections, 4 theorems, 56 equations, 3 figures, 19 tables.

Table of Contents

  1. Introduction
  2. The Compounded Linear Failure Rate Distribution
  3. Statistical Properties
  4. Mean Residual Life Function
  5. Mean Inactivity Time Function
  6. Moments
  7. Quantile and Median
  8. Stochastic Orders
  9. Order Statistics
  10. Estimation
  11. Simulation Study
  12. Data Analysis
  13. Sample Data 1
  14. Sample Data 2
  15. Sample Data 3
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $U$ be a random variable having CLFRD$(\alpha, \beta, \lambda)$. The PDF of $U$ is:

Figures (3)

  • Figure 1: Plot corresponding to CLFRD Density Functions for different values of parameters
  • Figure 2: Plot of the Hazard Rate functions for different values of parameters
  • Figure 3: Survival Functions Fitted vs. Empirical

Theorems & Definitions (9)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3