Bivariate Laplace Transforms, Stochastic Orders, and Entropy Measures in Reliability

Implements methods for bivariate and univariate Laplace transforms of residual lives and reversed residual lives, associated stochastic ordering concepts, and entropy measures for reliability analysis. The package covers: (1) Bivariate Laplace transform of residual lives and stochastic comparisons based on the bivariate Laplace transform order of residual lives (BLt-rl), including weak bivariate hazard rate, mean residual life, and relative mean residual life orders, nonparametric estimation, and NBUHR/NWUHR aging class characterisation; Jayalekshmi, Rajesh, and Nair (2022) "Bivariate Laplace Transform of Residual Lives and Their Properties" ; (2) Bivariate Laplace transform order of reversed residual lives (BLt-Rrl), reversed hazard gradient, reversed mean residual life, and the associated stochastic orders (weak bivariate reversed hazard rate, weak bivariate reversed mean residual life); Jayalekshmi, Rajesh, and Nair (2022) "Bivariate Laplace Transform Order and Ordering of Reversed Residual Lives" ; (3) Univariate Laplace transform of residual life, hazard rate, mean residual life, and the corresponding stochastic orders (Lt-rl order, hazard rate order, MRL order), together with a nonparametric estimator. Shannon entropy and Golomb's (1966) information generating function are also provided. Parametric families supported include the Gumbel bivariate exponential, Farlie-Gumbel-Morgenstern (FGM), bivariate power, and Schur-constant distributions. Plotting utilities and a simulation framework for evaluating estimator performance are also provided.

BivLaplaceRL

BivLaplaceRL is an R package for bivariate and univariate Laplace transforms of residual lives, stochastic ordering concepts, and entropy measures in reliability analysis.

Research Basis

Features

Parametric Distributions

• Gumbel bivariate exponential (dgumbel_biv, sgumbel_biv, rgumbel_biv, pgumbel_biv)
• Farlie-Gumbel-Morgenstern — FGM (dfgm_biv, pfgm_biv, sfgm_biv, rfgm_biv)
• Bivariate power distribution (dbivpower, pbivpower, sbivpower, rbivpower)
• Schur-constant distribution (sschur_biv, rschur_biv)

Bivariate Laplace Transform of Residual Lives

• blt_residual() — numerical computation for any survival function
• blt_residual_gumbel() — closed-form for Gumbel distribution
• biv_hazard_gradient() — bivariate hazard gradient
• biv_mean_residual() — bivariate mean residual life
• nbuhr_test() — NBUHR/NWUHR aging class test
• np_blt_residual() — nonparametric estimator
• sim_blt_residual() — Monte-Carlo simulation study

Bivariate Laplace Transform of Reversed Residual Lives

• blt_reversed() — for any joint CDF
• blt_reversed_fgm() — closed form for FGM
• blt_reversed_power() — for bivariate power distribution
• biv_rhazard_gradient() — reversed hazard gradient
• biv_rmrl() — reversed mean residual life

Univariate Residual Life Analysis

• lt_residual() — LT of residual life: E[e^{-sX} | X > t]
• hazard_rate() — hazard rate h(t) = f(t)/S(t)
• mean_residual() — mean residual life m(t) = E[X-t | X>t]
• np_lt_residual() — nonparametric estimator

Stochastic Orders (Bivariate)

• blt_order_residual() — BLt-rl order
• blt_order_reversed() — BLt-Rrl order
• biv_whr_order() — weak bivariate hazard rate order
• biv_wmrl_order() — weak bivariate MRL order
• biv_brlmr_order() — bivariate relative MRL order
• biv_wrhr_order() — weak bivariate reversed hazard rate order
• biv_wrmrl_order() — weak bivariate reversed MRL order

Stochastic Orders (Univariate)

• lt_rl_order() — Lt-rl order: L_X(s,t) ≤ L_Y(s,t) for all s, t
• hr_order() — hazard rate order: h_X(t) ≤ h_Y(t) for all t
• mrl_order() — MRL order: m_X(t) ≤ m_Y(t) for all t

Entropy Measures

• shannon_entropy() — Shannon differential entropy
• info_gen_function() — Golomb information generating function

Plotting

• plot_blt_residual(), plot_blt_reversed()

Installation

# Install from CRAN
install.packages("BivLaplaceRL")

# Development version from GitHub
# install.packages("devtools")
devtools::install_github("itsmdivakaran/BivLaplaceRL")

Quick Start

library(BivLaplaceRL)

# 1. Simulate from Gumbel bivariate exponential
set.seed(42)
dat <- rgumbel_biv(500, k1 = 1, k2 = 1, theta = 0.5)

# 2. Nonparametric estimate of BLT of residual lives
np_blt_residual(dat, s1 = 1, s2 = 1, t1 = 0.3, t2 = 0.3)

# 3. Compare with closed-form
blt_residual_gumbel(s1 = 1, s2 = 1, t1 = 0.3, t2 = 0.3, k1 = 1, k2 = 1, theta = 0.5)

# 4. Univariate LT of residual life for Exp(1)
f  <- function(x) dexp(x, 1)
Fb <- function(x) pexp(x, 1, lower.tail = FALSE)
lt_residual(f, Fb, s = 1, t = 0.5)

# 5. Hazard rate and MRL
hazard_rate(f, Fb, t = c(0.5, 1, 2))
mean_residual(Fb, t = c(0, 0.5, 1, 2))

# 6. Check univariate stochastic orders: Exp(2) <=_hr Exp(1)?
f2  <- function(x) dexp(x, 2)
Fb2 <- function(x) pexp(x, 2, lower.tail = FALSE)
hr_order(f2, Fb2, f, Fb, t_grid = c(0.5, 1, 2))$order_holds

Authors

Mahesh Divakaran (maintainer) Research Scholar, Amity School of Applied Sciences, Amity University Lucknow [email protected]

S. Jayalekshmi, G. Rajesh, N. Unnikrishnan Nair Department of Statistics, Cochin University of Science and Technology

References

Jayalekshmi S., Rajesh G., Nair N.U. (2022). Bivariate Laplace transform of residual lives and their properties. Communications in Statistics — Theory and Methods. https://doi.org/10.1080/03610926.2022.2085874

Jayalekshmi S., Rajesh G. Bivariate Laplace transform order and ordering of reversed residual lives. International Journal of Reliability, Quality and Safety Engineering. https://doi.org/10.1142/S0218539322500061

Belzunce F., Ortega E., Ruiz J.M. (1999). The Laplace order and ordering of residual lives. Statistics & Probability Letters, 42(2), 145--156. https://doi.org/10.1016/S0167-7152(98)00202-8

License

GPL-3 © 2024 Mahesh Divakaran