Bivariate Laplace Transforms, Stochastic Orders, and Entropy Measures in Reliability.
BivLaplaceRL 
BivLaplaceRL is an R package for bivariate and univariate Laplace transforms of residual lives, stochastic ordering concepts, and entropy measures in reliability analysis.
Residual Life Analysis
Bivariate Laplace transforms of residual lives — closed-form Gumbel results, general numerical integration, nonparametric estimation, and NBUHR/NWUHR aging class characterisation.
Reversed Residual Lives
BLt-Rrl framework: reversed hazard gradient, reversed mean residual life, and closed-form transforms for FGM and bivariate power distributions.
Univariate Methods
Univariate LT of residual life, hazard rate, mean residual life, and three stochastic orders (Lt-rl, hazard rate, MRL) with nonparametric estimation.
Stochastic Orders
Seven bivariate and three univariate stochastic order checks, each returning a logical flag and supporting diagnostic values.
Research Basis
| Paper | Journal | Authors |
|---|---|---|
| Bivariate Laplace transform of residual lives and their properties | Communications in Statistics — Theory and Methods (2022) | Jayalekshmi S., Rajesh G., Nair N.U. |
| Bivariate Laplace transform order and ordering of reversed residual lives | Int. J. Reliability, Quality and Safety Engineering | Jayalekshmi S., Rajesh G. |
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 functionblt_residual_gumbel()— closed-form for Gumbel distributionbiv_hazard_gradient()— bivariate hazard gradientbiv_mean_residual()— bivariate mean residual lifenbuhr_test()— NBUHR/NWUHR aging class testnp_blt_residual()— nonparametric estimatorsim_blt_residual()— Monte-Carlo simulation study
Bivariate Laplace Transform of Reversed Residual Lives
blt_reversed()— for any joint CDFblt_reversed_fgm()— closed form for FGMblt_reversed_power()— for bivariate power distributionbiv_rhazard_gradient()— reversed hazard gradientbiv_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 orderblt_order_reversed()— BLt-Rrl orderbiv_whr_order()— weak bivariate hazard rate orderbiv_wmrl_order()— weak bivariate MRL orderbiv_brlmr_order()— bivariate relative MRL orderbiv_wrhr_order()— weak bivariate reversed hazard rate orderbiv_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, thr_order()— hazard rate order: h_X(t) ≤ h_Y(t) for all tmrl_order()— MRL order: m_X(t) ≤ m_Y(t) for all t
Entropy Measures
shannon_entropy()— Shannon differential entropyinfo_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.