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Introduction to BivLaplaceRL

Mahesh Divakaran

2026-04-29

Overview

BivLaplaceRL is an R package that implements statistical methods from three research papers on bivariate Laplace transforms and entropy measures in reliability theory:

  1. Bivariate Laplace transform of residual lives (Jayalekshmi, Rajesh & Nair, 2022)
  2. Bivariate Laplace transform order of reversed residual lives (Jayalekshmi & Rajesh)
  3. Residual entropy generating function (Smitha, Rajesh & Jayalekshmi, 2024)
library(BivLaplaceRL)

Parametric Distributions

The package provides four bivariate distributions commonly used in reliability modelling.

Gumbel Bivariate Exponential

# Simulate 200 observations
set.seed(42)
dat <- rgumbel_biv(200, k1 = 1, k2 = 1.5, theta = 0.3)
head(dat)
#>             X1          X2
#> [1,] 0.1983368 0.520919155
#> [2,] 0.6608953 0.076339189
#> [3,] 0.2834910 0.156729738
#> [4,] 0.0381919 0.294297662
#> [5,] 0.4731766 0.305411898
#> [6,] 1.4636271 0.000850079

# Survival function at a point
sgumbel_biv(1, 1, k1 = 1, k2 = 1.5, theta = 0.3)
#> [1] 0.06081006

Farlie-Gumbel-Morgenstern (FGM)

set.seed(1)
dat_fgm <- rfgm_biv(100, theta = 0.5)
head(dat_fgm)
#>             X1        X2
#> [1,] 0.2655087 0.4050251
#> [2,] 0.3721239 0.4433111
#> [3,] 0.5728534 0.5392890
#> [4,] 0.9082078 0.8448911
#> [5,] 0.2016819 0.3851136
#> [6,] 0.8983897 0.2511579
pfgm_biv(0.5, 0.6, theta = 0.5)
#> [1] 0.33

Schur-Constant

set.seed(2)
dat_sc <- rschur_biv(100, lambda = 1)
# Marginals are standard exponential
summary(rowSums(dat_sc))  # T = X1 + X2 ~ Exp(lambda)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#> 0.00394 0.34161 0.82448 1.00673 1.29280 4.86222

Bivariate Laplace Transform of Residual Lives

Closed-Form (Gumbel Distribution)

The key result from Jayalekshmi et al. (2022) is:

\[L^*_{X_{t_1|t_2}}(s_1) = \frac{1}{k_1 + s_1 + \theta t_2}, \quad L^*_{X_{t_2|t_1}}(s_2) = \frac{1}{k_2 + s_2 + \theta t_1}.\]

blt_residual_gumbel(s1 = 1, s2 = 1, t1 = 0.5, t2 = 0.5,
                    k1 = 1, k2 = 1, theta = 0.5)
#>   L1_star   L2_star 
#> 0.4444444 0.4444444

Numerical Computation

For a user-supplied survival function:

my_surv <- function(x1, x2) exp(-x1 - x2 - 0.3 * x1 * x2)
blt_residual(s1 = 1, s2 = 1, t1 = 0.5, t2 = 0.5,
             surv_fn = my_surv, star = TRUE)
#>   L1_star   L2_star 
#> 0.4651163 0.4651163

Nonparametric Estimation

set.seed(123)
dat <- rgumbel_biv(500, k1 = 1, k2 = 1, theta = 0.5)
np_blt_residual(dat, s1 = 1, s2 = 1, t1 = 0.3, t2 = 0.3)
#>    L1_hat    L2_hat 
#> 0.4743203 0.3849056

# True value
blt_residual_gumbel(s1 = 1, s2 = 1, t1 = 0.3, t2 = 0.3, theta = 0.5)
#>   L1_star   L2_star 
#> 0.4651163 0.4651163

Simulation Study

sim_blt_residual(n_obs = 200, n_sim = 50, s1 = 1, s2 = 1,
                 t1 = 0.3, t2 = 0.3, k1 = 1, k2 = 1, theta = 0.5)
#>         component true_value  mean_est         bias     variance          mse
#> L1_star   L1_star  0.4651163 0.4682817  0.003165417 0.0009658125 0.0009758324
#> L2_star   L2_star  0.4651163 0.3951277 -0.069988580 0.0004963515 0.0053947528

Bivariate Hazard Gradient and MRL

# Hazard gradient
biv_hazard_gradient(t1 = 1, t2 = 1, k1 = 1, k2 = 1, theta = 0.3)
#>  h1  h2 
#> 1.3 1.3

# Mean residual life
biv_mean_residual(t1 = 0.5, t2 = 0.5, k1 = 1, k2 = 1, theta = 0)
#> m1 m2 
#>  1  1

NBUHR / NWUHR Aging Classification

res <- nbuhr_test(t2 = 0.5, k1 = 1, k2 = 1, theta = 0.3,
                  t1_grid = seq(0.1, 3, 0.3))
cat("Component 1:", res$class1, "\n")
#> Component 1: neither
cat("Component 2:", res$class2, "\n")
#> Component 2: NWUHR

Bivariate Laplace Transform of Reversed Residual Lives

# FGM distribution
blt_reversed(s1 = 1, s2 = 1, t1 = 0.5, t2 = 0.5, theta = 0.3)
#>       L1       L2 
#> 0.791495 0.791495

# G form (used for characterisation)
blt_reversed(s1 = 1, s2 = 1, t1 = 0.5, t2 = 0.5, theta = 0.3, g_form = TRUE)
#>        G1        G2 
#> 0.3049546 0.3049546

# Reversed hazard gradient
biv_rhazard_gradient(x1 = 0.5, x2 = 0.5, theta = 0.3)
#>      rh1      rh2 
#> 1.860465 1.860465

# Reversed MRL
biv_rmrl(x1 = 0.5, x2 = 0.5, theta = 0.3)
#>       m1       m2 
#> 0.255814 0.255814

Stochastic Orders

BLt-rl Order

sX <- function(x1, x2) sgumbel_biv(x1, x2, k1 = 1, k2 = 1, theta = 0.2)
sY <- function(x1, x2) sgumbel_biv(x1, x2, k1 = 2, k2 = 2, theta = 0.2)
res <- blt_order_residual(sX, sY, s1 = 1, s2 = 1,
                          t1_grid = c(0.5, 1), t2_grid = c(0.5, 1))
cat("X <=_BLt-rl Y:", res$order_holds, "\n")
#> X <=_BLt-rl Y: TRUE

Weak Bivariate Hazard Rate Order

res <- biv_whr_order(sX, sY, t1_grid = c(0.5, 1), t2_grid = c(0.5, 1))
cat("X <=_whr Y:", res$order_holds, "\n")
#> X <=_whr Y: FALSE

Residual Entropy Generating Function

Shannon Entropy and Golomb IGF

# Shannon entropy of Exp(1) = 1
shannon_entropy(function(x) dexp(x, 1))
#> [1] 1

# Golomb IGF of Exp(1) at alpha=2: lambda/2 = 0.5
info_gen_function(function(x) dexp(x, 1), alpha = 2)
#> [1] 0.5

Dynamic REGF

f  <- function(x) dexp(x, 1)
Fb <- function(x) pexp(x, 1, lower.tail = FALSE)

# REGF at various t values
t_vals <- seq(0, 2, 0.5)
sapply(t_vals, function(t) residual_info_gen(f, Fb, alpha = 2, t = t))
#> [1] 0.5 0.5 0.5 0.5 0.5

Nonparametric Estimation

set.seed(7)
x <- rexp(300, rate = 1)
np_residual_info_gen(x, alpha = 2, t = 0.5)
#> [1] 0.5736774

# True value
residual_info_gen(f, Fb, alpha = 2, t = 0.5)
#> [1] 0.5

REGF Profile and Characterisation

prof <- regf_profile(f, Fb, t = 0.5, plot = TRUE)
REGF profile across alpha
REGF profile across alpha
head(prof)
#>   alpha      REGF
#> 1   0.1 9.9995227
#> 2   0.3 3.3333333
#> 3   0.5 2.0000000
#> 4   0.7 1.4285714
#> 5   0.9 1.1111111
#> 6   1.1 0.9090909
# Log-linearity in t characterises exponential distribution
ch <- regf_characterise(f, Fb, alpha = 2, t_grid = seq(0, 2, 0.4))
ch
#>     t REGF      log_REGF        slope
#> 1 0.0  0.5  0.000000e+00 4.123686e-16
#> 2 0.4  0.5 -1.221245e-15 4.123686e-16
#> 3 0.8  0.5  6.661338e-16 4.123686e-16
#> 4 1.2  0.5  1.110223e-15 4.123686e-16
#> 5 1.6  0.5  1.110223e-15 4.123686e-16
#> 6 2.0  0.5 -3.330669e-16 4.123686e-16

Simulation Study for REGF Estimator

sim_regf(lambda = 1, alpha = 2, t = 0, n_obs = 200, n_sim = 50)
#>   true_value  mean_est        bias    variance         mse
#> 1        0.5 0.4158333 -0.08416671 0.001263665 0.008347701

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.

Smitha S., Rajesh G., Jayalekshmi S. (2024). On residual entropy generating function. Journal of the Indian Statistical Association, 62(1), 81–93.

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