Package 'BivLaplaceRL'

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

Description

BivLaplaceRL provides a unified framework for reliability analysis covering bivariate and univariate Laplace transforms of residual lives, associated stochastic orders, and entropy measures:

1. Bivariate Laplace Transform of Residual Lives

Implements the vector Laplace transform of bivariate residual lives $(L_{X_{t_1|t_2}}(s_1),\,L_{X_{t_2|t_1}}(s_2))$, the associated stochastic ordering BLt-rl, and its relationships with the weak bivariate hazard rate order, weak bivariate mean residual life order, and bivariate relative mean residual life order. Nonparametric estimation and NBUHR/NWUHR aging class tests are included. Reference: Jayalekshmi, Rajesh, and Nair (2022) doi:10.1080/03610926.2022.2085874.

2. Bivariate Laplace Transform of Reversed Residual Lives

Implements the bivariate Laplace transform of reversed (past) residual lives, the BLt-Rrl stochastic order, and connections with weak bivariate reversed hazard rate and reversed mean residual life orders. Reference: Jayalekshmi, Rajesh, and Nair (2022) doi:10.1142/S0218539322500061.

3. Univariate Residual Life Analysis

Implements the univariate Laplace transform of residual life $L_X(s,t) = E[e^{-sX} \mid X > t]$, hazard rate, mean residual life, and the corresponding stochastic orders (Lt-rl order, hazard rate order, MRL order), together with a nonparametric estimator.

Parametric families

Gumbel bivariate exponential, Farlie-Gumbel-Morgenstern (FGM), bivariate power, and Schur-constant distributions are built in.

Plotting

plot_blt_residual and plot_blt_reversed provide ready-made visualisations.

Author(s)

Maintainer: Mahesh Divakaran [email protected] (ORCID) (Amity School of Applied Sciences, Amity University Lucknow)

Authors:

• S. Jayalekshmi (Department of Statistics, Cochin University of Science and Technology) [contributor]

• G. Rajesh [email protected] (Department of Statistics, Cochin University of Science and Technology)

• 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*. doi:10.1080/03610926.2022.2085874

Jayalekshmi S., Rajesh G., Nair N.U. (2022). Bivariate Laplace transform order and ordering of reversed residual lives. *International Journal of Reliability, Quality and Safety Engineering*. doi: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. doi:10.1016/S0167-7152(98)00202-8

Golomb S.W. (1966). The information generating function of a probability distribution. *IEEE Transactions on Information Theory*, 12(1), 75–77.

See Also

Useful links:

• https://itsmdivakaran.github.io/BivLaplaceRL/

• https://github.com/itsmdivakaran/BivLaplaceRL

• Report bugs at https://github.com/itsmdivakaran/BivLaplaceRL/issues

---

Bivariate Relative Mean Residual Life Order

Description

Checks whether $X \le_{\rm brlmr} Y$: the ratio $m_{i,Y}(t_1,t_2) / m_{i,X}(t_1,t_2)$ is increasing in $t_i$.

Usage

biv_brlmr_order(
  surv_X,
  surv_Y,
  t2_fixed = 0.5,
  t1_grid = seq(0.2, 3, by = 0.3),
  tol = 1e-06
)

Arguments

surv_X, surv_Y	Joint survival functions.
t2_fixed	Fixed value of $t_2$ for the univariate slices.
t1_grid	Grid of $t_1$ values.
tol	Tolerance.

Value

List with order_holds and ratio_values.

References

Kayid M., Izadkhah S., Alshami S. (2016). Residual probability function, associated orderings, and related aging classes. *Statistics and Probability Letters*, 116, 37–47.

Examples

sX <- function(x1, x2) sgumbel_biv(x1, x2, k1 = 2, k2 = 1)
sY <- function(x1, x2) sgumbel_biv(x1, x2, k1 = 1, k2 = 1)
biv_brlmr_order(sX, sY, t2_fixed = 0.5)

---

Bivariate Hazard Gradient

Description

Computes the bivariate hazard gradient

$h_i(t_1,t_2) = -\frac{\partial}{\partial t_i}\log\bar{F}(t_1,t_2), \quad i = 1, 2.$

Usage

biv_hazard_gradient(
  t1,
  t2,
  surv_fn = NULL,
  k1 = 1,
  k2 = 1,
  theta = 0,
  eps = 1e-07
)

Arguments

t1, t2	Evaluation points (non-negative).
surv_fn	A function function(x1, x2) returning the joint survival probability. Defaults to Gumbel bivariate exponential.
k1, k2, theta	Parameters for the default Gumbel distribution.
eps	Step size for numerical differentiation (default 1e-7).

Value

A named numeric vector (h1, h2).

References

Jayalekshmi S., Rajesh G., Nair N.U. (2022). doi:10.1080/03610926.2022.2085874

Examples

biv_hazard_gradient(t1 = 1, t2 = 1)
biv_hazard_gradient(t1 = 0.5, t2 = 0.5, k1 = 2, k2 = 1.5, theta = 0.3)

---

Bivariate Mean Residual Life Function

Description

Computes the bivariate mean residual life (MRL) function

$m_1(t_1,t_2) = E(X_{t_1|t_2}) = \frac{\int_{t_1}^{\infty} \bar{F}(x_1,t_2)\,dx_1}{\bar{F}(t_1,t_2)}$

and analogously $m_2(t_1,t_2)$.

Usage

biv_mean_residual(
  t1,
  t2,
  surv_fn = NULL,
  k1 = 1,
  k2 = 1,
  theta = 0,
  upper = 100
)

Arguments

t1, t2	Non-negative thresholds.
surv_fn	Joint survival function; defaults to Gumbel bivariate exponential.
k1, k2, theta	Gumbel parameters (used when surv_fn = NULL).
upper	Upper integration bound (default 100).

Value

A named numeric vector (m1, m2).

References

Jayalekshmi S., Rajesh G., Nair N.U. (2022). doi:10.1080/03610926.2022.2085874

Examples

biv_mean_residual(t1 = 0.5, t2 = 0.5)
biv_mean_residual(t1 = 1, t2 = 0.5, k1 = 1, k2 = 2, theta = 0.2)

---

Bivariate Reversed Hazard Gradient

Description

Computes the bivariate reversed hazard gradient

$h_i(x_1,x_2) = \frac{\partial}{\partial x_i}\log F(x_1,x_2), \quad i = 1, 2.$

Usage

biv_rhazard_gradient(x1, x2, cdf_fn = NULL, theta = 0, eps = 1e-07)

Arguments

x1, x2	Positive evaluation points.
cdf_fn	Joint CDF function; defaults to FGM with parameter theta.
theta	FGM parameter (used when cdf_fn = NULL).
eps	Numerical differentiation step.

Value

Named numeric vector (rh1, rh2).

References

Jayalekshmi S., Rajesh G. — IJRQSE, Section 2.

Examples

biv_rhazard_gradient(x1 = 0.5, x2 = 0.5, theta = 0.3)

---

Bivariate Reversed Mean Residual Life Function

Description

Computes the bivariate reversed mean residual life (RMRL):

$m_1(x_1,x_2) = \frac{\int_0^{x_1} F(t_1,x_2)\,dt_1}{F(x_1,x_2)}, \quad m_2(x_1,x_2) = \frac{\int_0^{x_2} F(x_1,t_2)\,dt_2}{F(x_1,x_2)}.$

Usage

biv_rmrl(x1, x2, cdf_fn = NULL, theta = 0)

Arguments

x1, x2	Positive evaluation points.
cdf_fn	Joint CDF; defaults to FGM.
theta	FGM parameter.

Value

Named numeric vector (m1, m2).

References

Jayalekshmi S., Rajesh G. — IJRQSE, Section 2.

Examples

biv_rmrl(x1 = 0.5, x2 = 0.5, theta = 0.3)
biv_rmrl(x1 = 0.7, x2 = 0.4, theta = -0.2)

---

Weak Bivariate Hazard Rate Order

Description

Checks whether $X \le_{\rm whr} Y$: the ratio $\bar{F}_Y(x_1,x_2) / \bar{F}_X(x_1,x_2)$ is increasing in $(x_1,x_2)$, i.e.\ $h_{i,X}(t_1,t_2) \ge h_{i,Y}(t_1,t_2),\; i=1,2$.

Usage

biv_whr_order(
  surv_X,
  surv_Y,
  t1_grid = seq(0.1, 3, by = 0.5),
  t2_grid = seq(0.1, 3, by = 0.5),
  tol = 1e-06
)

Arguments

surv_X, surv_Y	Joint survival functions.
t1_grid, t2_grid	Evaluation grids.
tol	Tolerance.

Value

A list: order_holds (logical), violations (data frame).

References

Shaked M., Shanthikumar J.G. (2007). *Stochastic Orders*. Springer. Jayalekshmi S., Rajesh G., Nair N.U. (2022). doi:10.1080/03610926.2022.2085874

Examples

sX <- function(x1, x2) sgumbel_biv(x1, x2, k1 = 1, k2 = 1, theta = 0)
sY <- function(x1, x2) sgumbel_biv(x1, x2, k1 = 2, k2 = 2, theta = 0)
biv_whr_order(sX, sY)

---

Weak Bivariate Mean Residual Life Order

Description

Checks whether $X \le_{\rm wmrl} Y$: $m_{i,X}(t_1,t_2) \le m_{i,Y}(t_1,t_2),\; i=1,2$.

Usage

biv_wmrl_order(
  surv_X,
  surv_Y,
  t1_grid = seq(0.1, 2, by = 0.5),
  t2_grid = seq(0.1, 2, by = 0.5),
  tol = 1e-06
)

Arguments

surv_X, surv_Y	Joint survival functions.
t1_grid, t2_grid	Evaluation grids.
tol	Tolerance.

Value

List with order_holds and violations.

References

Jayalekshmi S., Rajesh G., Nair N.U. (2022). doi:10.1080/03610926.2022.2085874

Examples

sX <- function(x1, x2) sgumbel_biv(x1, x2, k1 = 1.5, k2 = 1.5)
sY <- function(x1, x2) sgumbel_biv(x1, x2, k1 = 1, k2 = 1)
biv_wmrl_order(sX, sY)

---

Weak Bivariate Reversed Hazard Rate Order

Description

Checks $X \le_{\rm wrhr} Y$: the ratio $F_X(x_1,x_2)/F_Y(x_1,x_2)$ is decreasing in $(x_1,x_2)$, i.e.\ $h_{i,X}(x_1,x_2) \le h_{i,Y}(x_1,x_2),\; i=1,2$.

Usage

biv_wrhr_order(
  cdf_X,
  cdf_Y,
  x1_grid = seq(0.1, 0.9, by = 0.2),
  x2_grid = seq(0.1, 0.9, by = 0.2),
  tol = 1e-06
)

Arguments

cdf_X, cdf_Y	Joint CDFs.
x1_grid, x2_grid	Evaluation grids (positive values).
tol	Tolerance.

Value

List with order_holds and violations.

References

Jayalekshmi S., Rajesh G. — IJRQSE, Section 2.

Examples

cX <- function(x1, x2) pfgm_biv(x1, x2, theta = 0.2)
cY <- function(x1, x2) pfgm_biv(x1, x2, theta = 0.5)
biv_wrhr_order(cX, cY)

---

Weak Bivariate Reversed Mean Residual Life Order

Description

Checks $X \le_{\rm wrmrl} Y$: $m_{i,X}(x_1,x_2) \ge m_{i,Y}(x_1,x_2),\; i=1,2$.

Usage

biv_wrmrl_order(
  cdf_X,
  cdf_Y,
  x1_grid = seq(0.2, 0.8, by = 0.2),
  x2_grid = seq(0.2, 0.8, by = 0.2),
  tol = 1e-06
)

Arguments

cdf_X, cdf_Y	Joint CDFs.
x1_grid, x2_grid	Evaluation grids.
tol	Tolerance.

Value

List with order_holds and violations.

References

Jayalekshmi S., Rajesh G. — IJRQSE, Section 2.

Examples

cX <- function(x1, x2) pfgm_biv(x1, x2, theta = 0.1)
cY <- function(x1, x2) pfgm_biv(x1, x2, theta = 0.8)
biv_wrmrl_order(cX, cY)

---

Bivariate Power Distribution

Description

Distribution function, survival function, density, and random generation for the bivariate power distribution:

$F(x_1,x_2) = x_1^{p_1 + \theta \log x_2}\, x_2^{p_2},\quad 0 \le x_1,x_2 \le p_2,\; p_1,p_2 > 0,\; 0 \le \theta \le 1.$

Usage

pbivpower(x1, x2, p1 = 1, p2 = 1, theta = 0)

sbivpower(x1, x2, p1 = 1, p2 = 1, theta = 0)

dbivpower(x1, x2, p1 = 1, p2 = 1, theta = 0)

rbivpower(n, p1 = 1, p2 = 1, theta = 0)

Arguments

x1, x2	Values in the support.
p1, p2	Positive shape parameters.
theta	Association parameter, $0 \le \theta \le 1$.
n	Number of random observations.

Value

Numeric vector or two-column matrix (rbivpower).

References

Jayalekshmi S., Rajesh G. Bivariate Laplace transform order and ordering of reversed residual lives. *International Journal of Reliability, Quality and Safety Engineering*.

Examples

pbivpower(0.4, 0.5, p1 = 2, p2 = 2, theta = 0.3)
set.seed(7); head(rbivpower(30, p1 = 2, p2 = 2, theta = 0.2))

---

Bivariate Laplace Transform Order of Residual Lives (BLt-rl)

Description

Checks whether random vector $X$ is smaller than $Y$ in the bivariate Laplace transform order of residual lives (BLt-rl).

$X \le_{\rm BLt\text{-}rl} Y$ if and only if for all $(t_1,t_2)$ in the support:

$\frac{\int_{t_1}^\infty e^{-s_1 x_1}\bar{F}_X(x_1,t_2)\,dx_1}{ e^{-s_1 t_1}\bar{F}_X(t_1,t_2)} \ge \frac{\int_{t_1}^\infty e^{-s_1 x_1}\bar{F}_Y(x_1,t_2)\,dx_1}{ e^{-s_1 t_1}\bar{F}_Y(t_1,t_2)}$

i.e.\ $L^*_{X_{t_1|t_2}}(s_1) \ge L^*_{Y_{t_1|t_2}}(s_1)$ for all $t_1,t_2,s_1$. The function evaluates this inequality at a grid.

Usage

blt_order_residual(
  surv_X,
  surv_Y,
  s1 = 1,
  s2 = 1,
  t1_grid = seq(0.1, 3, by = 0.5),
  t2_grid = seq(0.1, 3, by = 0.5),
  tol = 1e-06
)

Arguments

surv_X, surv_Y	Joint survival functions for $X$ and $Y$ respectively, each of the form function(x1, x2).
s1, s2	Laplace parameters to evaluate at.
t1_grid, t2_grid	Grids of truncation times (default 0.1 to 3).
tol	Tolerance for declaring inequality (default 1e-6).

Value

A list with elements:

order_holds

Logical; TRUE if $X \le Y$ at all grid points.

violations

Data frame of grid points where the ordering fails.

ratio_matrix

Matrix of $L^*_X / L^*_Y$ values.

References

Jayalekshmi S., Rajesh G., Nair N.U. (2022), Definition 4.1 & Prop. 4.1. doi:10.1080/03610926.2022.2085874

See Also

blt_residual, biv_whr_order

Examples

# Compare two Gumbel distributions
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)
blt_order_residual(sX, sY, s1 = 1, s2 = 1,
                   t1_grid = c(0.1, 0.5, 1), t2_grid = c(0.1, 0.5))

---

Bivariate Laplace Transform Order of Reversed Residual Lives (BLt-Rrl)

Description

Checks whether $X \le_{\rm BLt\text{-}Rrl} Y$: for all $(t_1,t_2)$, $L^{X}_{t_1|t_2}(s_1) \ge L^{Y}_{t_1|t_2}(s_2)$.

Usage

blt_order_reversed(
  cdf_X,
  cdf_Y,
  s1 = 1,
  s2 = 1,
  t1_grid = seq(0.2, 0.8, by = 0.2),
  t2_grid = seq(0.2, 0.8, by = 0.2),
  tol = 1e-06
)

Arguments

cdf_X, cdf_Y	Joint CDF functions for $X$ and $Y$.
s1, s2	Laplace parameters.
t1_grid, t2_grid	Grids of truncation times.
tol	Tolerance.

Value

Same structure as blt_order_residual.

References

Jayalekshmi S., Rajesh G. — IJRQSE, Definition 2.

See Also

blt_reversed, biv_wrhr_order

Examples

cX <- function(x1, x2) pfgm_biv(x1, x2, theta = 0.3)
cY <- function(x1, x2) pfgm_biv(x1, x2, theta = 0.5)
blt_order_reversed(cX, cY, s1 = 1, s2 = 1,
                   t1_grid = c(0.2, 0.5), t2_grid = c(0.2, 0.5))

---

Bivariate Laplace Transform of Residual Lives

Description

Computes the bivariate Laplace transform of residual lives $L_{X_{t_1|t_2}}(s_1)$ and $L_{X_{t_2|t_1}}(s_2)$ defined as

$L_{X_{t_1|t_2}}(s_1) = \frac{\int_{t_1}^{\infty} e^{-s_1 x_1} f(x_1 | X_2 > t_2)\,dx_1}{e^{-s_1 t_1} \bar{F}(t_1 | X_2 > t_2)}$

and analogously for the second component.

The *star* version (used for ordering) is

$L^*_{X_{t_1|t_2}}(s_1) = \frac{1 - L_{X_{t_1|t_2}}(s_1)}{s_1} = \frac{\int_{t_1}^{\infty} e^{-s_1 x_1} \bar{F}(x_1, t_2)}{e^{-s_1 t_1}\bar{F}(t_1,t_2)}\,dx_1.$

Usage

blt_residual(
  s1,
  s2,
  t1 = 0,
  t2 = 0,
  surv_fn = NULL,
  k1 = 1,
  k2 = 1,
  theta = 0,
  upper = 50,
  star = FALSE
)

Arguments

s1, s2	Positive Laplace parameters.
t1, t2	Non-negative time thresholds (truncation ages).
surv_fn	A function function(x1, x2) returning the joint survival probability $\bar{F}(x_1, x_2)$. Defaults to the Gumbel bivariate exponential with k1, k2, theta.
k1, k2	Rate parameters (used only when surv_fn = NULL).
theta	Association parameter (used only when surv_fn = NULL).
upper	Integration upper bound (default 50).
star	Logical; if TRUE returns the star version $L^*$.

Value

A named numeric vector with elements L1 and L2 (or L1_star and L2_star when star = TRUE).

References

Jayalekshmi S., Rajesh G., Nair N.U. (2022). doi:10.1080/03610926.2022.2085874

See Also

blt_residual_gumbel, blt_order_residual, np_blt_residual

Examples

# Gumbel bivariate exponential, default parameters
blt_residual(s1 = 1, s2 = 1, t1 = 0.5, t2 = 0.5)

# Star version used in ordering
blt_residual(s1 = 0.5, s2 = 0.5, t1 = 1, t2 = 1, star = TRUE)

# User-supplied survival function (Schur-constant exponential)
surv <- function(x1, x2) exp(-(x1 + x2))
blt_residual(s1 = 1, s2 = 1, t1 = 0.3, t2 = 0.3, surv_fn = surv)

---

Closed-Form Bivariate Laplace Transform of Residual Lives (Gumbel)

Description

Returns the *star* bivariate Laplace transform of residual lives for the Gumbel bivariate exponential distribution in closed form:

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

Usage

blt_residual_gumbel(s1, s2, t1 = 0, t2 = 0, k1 = 1, k2 = 1, theta = 0)

Arguments

s1, s2	Positive Laplace parameters.
t1, t2	Non-negative truncation ages.
k1, k2	Positive rate parameters.
theta	Non-negative association parameter.

Value

A named numeric vector (L1_star, L2_star).

References

Jayalekshmi S., Rajesh G., Nair N.U. (2022), Example 3.1. doi:10.1080/03610926.2022.2085874

See Also

blt_residual

Examples

blt_residual_gumbel(s1 = 1, s2 = 1, t1 = 0.5, t2 = 0.5)
blt_residual_gumbel(s1 = 2, s2 = 0.5, t1 = 0, t2 = 1, k1 = 2, k2 = 1, theta = 0.3)

---

Bivariate Laplace Transform of Reversed Residual Lives

Description

Computes the bivariate Laplace transform of reversed (past) residual lives. For a random pair $(X_1,X_2)$ with joint distribution function $F(x_1,x_2)$, the transform is

$L_{t_1|t_2}(s_1) = e^{-s_1 t_1} + \frac{s_1 \int_0^{t_1} e^{-s_1 x_1} F(x_1, t_2)\,dx_1}{F(t_1, t_2)}$

and the associated $G$ function (useful for ordering) is

$G_1(t_1,t_2) = \frac{\int_0^{t_1} e^{-s_1 x_1} F(x_1, t_2)\,dx_1}{ e^{-s_1 t_1} F(t_1,t_2)}.$

Usage

blt_reversed(s1, s2, t1, t2, cdf_fn = NULL, theta = 0, g_form = FALSE)

Arguments

s1, s2	Positive Laplace parameters.
t1, t2	Positive truncation times ($t_i > 0$).
cdf_fn	A function function(x1, x2) returning the joint CDF $F(x_1,x_2)$. Defaults to the FGM distribution.
theta	FGM association parameter (used when cdf_fn = NULL).
g_form	Logical; if TRUE returns the $G$ form instead of $L$.

Value

A named numeric vector (L1, L2) or (G1, G2).

References

Jayalekshmi S., Rajesh G. Bivariate Laplace transform order and ordering of reversed residual lives. *International Journal of Reliability, Quality and Safety Engineering*.

See Also

blt_reversed_fgm, blt_order_reversed

Examples

# FGM distribution (default)
blt_reversed(s1 = 1, s2 = 1, t1 = 0.6, t2 = 0.6)

# G form
blt_reversed(s1 = 1, s2 = 1, t1 = 0.6, t2 = 0.6, g_form = TRUE)

# User-supplied CDF
cdf <- function(x1, x2) pfgm_biv(x1, x2, theta = 0.5)
blt_reversed(s1 = 1, s2 = 1, t1 = 0.5, t2 = 0.5, cdf_fn = cdf)

---

Closed-Form BLT of Reversed Residual Lives for the FGM Distribution

Description

Computes the bivariate Laplace transform of the FGM distribution in closed form (Jayalekshmi and Rajesh, Example 1):

$L_X(s_1,s_2) = \frac{(1-e^{-s_1})(1-e^{-s_2})}{s_1 s_2} + \theta \Phi(s_1,s_2)$

where $\Phi$ captures the dependence correction.

Usage

blt_reversed_fgm(s1, s2, theta = 0)

Arguments

s1, s2	Positive Laplace parameters.
theta	FGM association parameter, $|\theta| \le 1$.

Value

Scalar numeric; the joint bivariate Laplace transform.

References

Jayalekshmi S., Rajesh G. Bivariate Laplace transform order and ordering of reversed residual lives. *International Journal of Reliability, Quality and Safety Engineering*, Example 1.

Examples

blt_reversed_fgm(s1 = 1, s2 = 1, theta = 0.5)
blt_reversed_fgm(s1 = 2, s2 = 0.5, theta = -0.3)

---

Bivariate Laplace Transform of Reversed Residual Lives — Power Distribution

Description

Computes the $G_1$ function for the bivariate power distribution:

$G_1(t_1,t_2) = \frac{\int_0^{t_1} e^{-s_1 x_1} x_1^{p_1+\theta\log t_2}\,dx_1}{e^{-s_1 t_1}\, t_1^{p_1+\theta\log t_2}}$

evaluated numerically.

Usage

blt_reversed_power(s1, t1, t2, p1 = 1, p2 = 1, theta = 0)

Arguments

s1	Positive Laplace parameter.
t1, t2	Positive truncation times.
p1, p2	Positive shape parameters.
theta	Association parameter, $0 \le \theta \le 1$.

Value

Named numeric vector (G1, reversed_hazard_h1).

References

Jayalekshmi S., Rajesh G. — IJRQSE, Example 2.

Examples

blt_reversed_power(s1 = 1, t1 = 0.5, t2 = 0.5, p1 = 2, p2 = 2, theta = 0.3)

---

Farlie-Gumbel-Morgenstern (FGM) Bivariate Distribution

Description

Density, distribution function, survival function, and random generation for the FGM bivariate distribution on $[0,1]^2$:

$F(x_1, x_2) = x_1 x_2 \bigl[1 + \theta(1-x_1)(1-x_2)\bigr], \quad 0 \le x_1, x_2 \le 1,\; |\theta| \le 1.$

Usage

pfgm_biv(x1, x2, theta = 0)

dfgm_biv(x1, x2, theta = 0)

sfgm_biv(x1, x2, theta = 0)

rfgm_biv(n, theta = 0)

Arguments

x1, x2	Values in $[0,1]$.
theta	Association parameter, $|\theta| \le 1$.
n	Number of random observations.

Value

Numeric vector (scalar functions) or two-column matrix (rfgm_biv).

References

Jayalekshmi S., Rajesh G. Bivariate Laplace transform order and ordering of reversed residual lives. *International Journal of Reliability, Quality and Safety Engineering*.

Examples

pfgm_biv(0.4, 0.6, theta = 0.5)
dfgm_biv(0.4, 0.6, theta = 0.5)
set.seed(1); head(rfgm_biv(50, theta = 0.5))

---

Gumbel Bivariate Exponential Distribution

Description

Density, distribution (survival) function, and random generation for the Gumbel bivariate exponential distribution with parameters $k_1$, $k_2$, and association parameter $\theta$.

The joint survival function is

$\bar{F}(x_1,x_2) = \exp(-k_1 x_1 - k_2 x_2 - \theta x_1 x_2), \quad x_1,x_2 > 0,\; k_1,k_2 > 0,\; 0 \le \theta \le k_1 k_2.$

Usage

dgumbel_biv(x1, x2, k1 = 1, k2 = 1, theta = 0, log.p = FALSE)

sgumbel_biv(x1, x2, k1 = 1, k2 = 1, theta = 0, log.p = FALSE)

pgumbel_biv(x1, x2, k1 = 1, k2 = 1, theta = 0)

rgumbel_biv(n, k1 = 1, k2 = 1, theta = 0)

Arguments

x1, x2	Non-negative numeric values or vectors.
k1, k2	Positive rate parameters.
theta	Non-negative association parameter; must satisfy $0 \le \theta \le k_1 k_2$.
log.p	Logical; if TRUE probabilities are given as $\log(p)$.
n	Number of random observations.

Value

dgumbel_biv

Numeric vector of density values.

pgumbel_biv

Numeric vector of joint CDF values.

sgumbel_biv

Numeric vector of joint survival function values.

rgumbel_biv

A two-column matrix with columns X1 and X2 containing the simulated observations.

References

Gumbel E.J. (1960). Bivariate exponential distributions. *Journal of the American Statistical Association*, 55(292), 698–707.

Jayalekshmi S., Rajesh G., Nair N.U. (2022). doi:10.1080/03610926.2022.2085874

Examples

# Survival function
sgumbel_biv(1, 2, k1 = 1, k2 = 1, theta = 0.5)

# Density
dgumbel_biv(0.5, 0.5, k1 = 1, k2 = 1.5, theta = 0.3)

# Random sample
set.seed(42)
dat <- rgumbel_biv(100, k1 = 1, k2 = 1, theta = 0.5)
head(dat)

---

Univariate Hazard Rate Function

Description

Computes the hazard rate (failure rate) of a non-negative continuous random variable:

$h(t) = \frac{f(t)}{\bar{F}(t)}, \quad t \geq 0.$

Usage

hazard_rate(dens_fn, surv_fn = NULL, t, upper = 100)

Arguments

dens_fn	Density function $f(t)$.
surv_fn	Survival function $\bar{F}(t)$; computed by numerical integration if NULL.
t	Scalar or numeric vector of time points.
upper	Upper integration limit (used only when surv_fn = NULL).

Value

Numeric vector of hazard rates at t.

See Also

mean_residual, hr_order

Examples

# Exp(1): constant hazard rate = 1
f  <- function(x) dexp(x, 1)
Fb <- function(x) pexp(x, 1, lower.tail = FALSE)
hazard_rate(f, Fb, t = c(0.5, 1, 2))

# Gamma(2,1): increasing hazard rate
fG  <- function(x) dgamma(x, shape = 2, rate = 1)
FbG <- function(x) pgamma(x, shape = 2, rate = 1, lower.tail = FALSE)
hazard_rate(fG, FbG, t = c(0.5, 1, 2))

---

Hazard Rate Order

Description

Checks whether $X \leq_{\rm hr} Y$ (hazard rate order): the hazard rate of $X$ is pointwise no greater than that of $Y$:

$h_X(t) \leq h_Y(t) \quad \forall\, t \geq 0.$

Under this order $X$ is stochastically longer-lived than $Y$. For exponential distributions, $\mathrm{Exp}(\lambda_1) \leq_{\rm hr} \mathrm{Exp}(\lambda_2)$ iff $\lambda_1 \leq \lambda_2$.

Usage

hr_order(
  dens_fn_X,
  surv_fn_X = NULL,
  dens_fn_Y,
  surv_fn_Y = NULL,
  t_grid = seq(0.1, 3, by = 0.5),
  upper = 100
)

Arguments

dens_fn_X, dens_fn_Y	Density functions of $X$ and $Y$.
surv_fn_X, surv_fn_Y	Survival functions; computed if NULL.
t_grid	Grid of time points.
upper	Integration upper bound.

Value

A list with:

order_holds

Logical.

max_violation

Maximum of $h_X(t) - h_Y(t)$ over the grid.

hazard_X

Hazard rate of $X$ at t_grid.

hazard_Y

Hazard rate of $Y$ at t_grid.

See Also

hazard_rate, mrl_order, lt_rl_order

Examples

# Exp(1) <=_hr Exp(2): h_X(t)=1 <= 2=h_Y(t)
fX  <- function(x) dexp(x, 1)
FbX <- function(x) pexp(x, 1, lower.tail = FALSE)
fY  <- function(x) dexp(x, 2)
FbY <- function(x) pexp(x, 2, lower.tail = FALSE)
hr_order(fX, FbX, fY, FbY, t_grid = c(0.5, 1, 2))$order_holds

---

Golomb Information Generating Function

Description

Computes the information generating function (IGF) introduced by Golomb (1966):

$\mathcal{I}_\alpha(f) = \int_0^\infty f^\alpha(x)\,dx, \quad \alpha > 0.$

When $\alpha \to 1$, $-d\mathcal{I}_\alpha / d\alpha|_{\alpha=1} = H(f)$.

Usage

info_gen_function(dens_fn, alpha = 1, upper = 100)

Arguments

dens_fn	A function of one argument returning the density.
alpha	Positive parameter (default 1).
upper	Upper integration limit.

Value

Scalar numeric.

References

Golomb S.W. (1966). The information generating function of a probability distribution. *IEEE Transactions on Information Theory*, 12(1), 75–77.

See Also

shannon_entropy

Examples

# Exponential(1) with alpha=1 gives 1
info_gen_function(function(x) dexp(x, rate = 1), alpha = 1)

# alpha = 2
info_gen_function(function(x) dexp(x, rate = 1), alpha = 2)

---

Univariate Laplace Transform of Residual Life

Description

Computes the Laplace transform of the residual life of a non-negative continuous random variable conditioned on survival past time $t$:

$L_X(s,t) = E[e^{-sX} \mid X > t] = \frac{1}{\bar{F}(t)}\int_t^\infty e^{-sx} f(x)\,dx, \quad s \geq 0,\; t \geq 0.$

At $t = 0$ this reduces to the standard Laplace transform $L_X(s) = E[e^{-sX}]$.

Usage

lt_residual(dens_fn, surv_fn = NULL, s, t = 0, upper = 100)

Arguments

dens_fn	Density function $f(x)$.
surv_fn	Survival function $\bar{F}(x)$; computed by numerical integration of dens_fn if NULL.
s	Non-negative Laplace parameter.
t	Truncation time (default 0).
upper	Upper integration limit.

Value

Scalar numeric.

References

Belzunce F., Ortega E., Ruiz J.M. (1999). The Laplace order and ordering of residual lives. *Statistics & Probability Letters*, 42(2), 145–156.

See Also

hazard_rate, mean_residual, np_lt_residual, lt_rl_order, blt_residual

Examples

# Exp(1): L_X(s, 0) = 1/(1+s) = 0.5 at s=1
f  <- function(x) dexp(x, 1)
Fb <- function(x) pexp(x, 1, lower.tail = FALSE)
lt_residual(f, Fb, s = 1, t = 0)

# Memoryless property: L_X(s,t) should equal L_X(s,0) for Exp
lt_residual(f, Fb, s = 1, t = 0.5)

---

Univariate Laplace Transform Order of Residual Lives

Description

Checks whether $X \leq_{\rm Lt\text{-}rl} Y$: the Laplace transform of the residual life of $X$ is dominated by that of $Y$ pointwise over a grid of $(s, t)$ values:

$L_X(s,t) \leq L_Y(s,t) \quad \forall\, s \geq 0,\; t \geq 0.$

The order is verified numerically on s_grid x t_grid.

Usage

lt_rl_order(
  dens_fn_X,
  surv_fn_X = NULL,
  dens_fn_Y,
  surv_fn_Y = NULL,
  s_grid = seq(0.5, 3, by = 0.5),
  t_grid = seq(0, 2, by = 0.5),
  upper = 100
)

Arguments

dens_fn_X, dens_fn_Y	Density functions of $X$ and $Y$.
surv_fn_X, surv_fn_Y	Survival functions; computed if NULL.
s_grid	Numeric vector of Laplace parameter values to check.
t_grid	Numeric vector of truncation times to check.
upper	Integration upper bound.

Value

A list with:

order_holds

Logical; TRUE if the order holds at all grid points.

max_violation

Maximum violation $\max(L_X - L_Y, 0)$.

ratio_matrix

Matrix of $L_X(s,t)/L_Y(s,t)$ values (rows = s_grid, columns = t_grid).

See Also

lt_residual, hr_order, mrl_order, blt_order_residual

Examples

# Exp(1) <=_Lt-rl Exp(2): L_{Exp(lambda)}(s,t) = lambda*exp(-s*t)/(s+lambda)
# For s>0: 1/(s+1) < 2/(s+2), so Exp(1) has smaller LT of residual life
fX  <- function(x) dexp(x, 1)
FbX <- function(x) pexp(x, 1, lower.tail = FALSE)
fY  <- function(x) dexp(x, 2)
FbY <- function(x) pexp(x, 2, lower.tail = FALSE)
lt_rl_order(fX, FbX, fY, FbY,
            s_grid = c(0.5, 1, 2), t_grid = c(0, 0.5, 1))$order_holds

---

Univariate Mean Residual Life

Description

Computes the mean residual life (mean excess function):

$m(t) = E[X - t \mid X > t] = \frac{1}{\bar{F}(t)}\int_t^\infty \bar{F}(x)\,dx, \quad t \geq 0.$

Usage

mean_residual(surv_fn, t = 0, upper = 100)

Arguments

surv_fn	Survival function $\bar{F}(x)$.
t	Scalar or numeric vector of time points.
upper	Upper integration limit.

Value

Numeric vector of MRL values at t.

See Also

hazard_rate, mrl_order

Examples

# Exp(1): constant MRL = 1 (memoryless)
Fb <- function(x) pexp(x, 1, lower.tail = FALSE)
mean_residual(Fb, t = c(0, 0.5, 1, 2))

# Gamma(2,1): decreasing MRL
FbG <- function(x) pgamma(x, shape = 2, rate = 1, lower.tail = FALSE)
mean_residual(FbG, t = c(0, 0.5, 1, 2))

---

Mean Residual Life Order

Description

Checks whether $X \leq_{\rm mrl} Y$ (mean residual life order): the MRL of $X$ is pointwise no greater than that of $Y$:

$m_X(t) \leq m_Y(t) \quad \forall\, t \geq 0.$

Usage

mrl_order(surv_fn_X, surv_fn_Y, t_grid = seq(0, 3, by = 0.5), upper = 100)

Arguments

surv_fn_X, surv_fn_Y	Survival functions of $X$ and $Y$.
t_grid	Grid of time points.
upper	Integration upper bound.

Value

A list with:

order_holds

Logical.

max_violation

Maximum of $m_X(t) - m_Y(t)$ over the grid.

mrl_X

MRL of $X$ at t_grid.

mrl_Y

MRL of $Y$ at t_grid.

See Also

mean_residual, hr_order, lt_rl_order

Examples

# Exp(2) <=_mrl Exp(1): m_X(t)=0.5 <= 1=m_Y(t)
FbX <- function(x) pexp(x, 2, lower.tail = FALSE)
FbY <- function(x) pexp(x, 1, lower.tail = FALSE)
mrl_order(FbX, FbY, t_grid = c(0, 0.5, 1, 2))$order_holds

---

Test NBUHR / NWUHR Aging Class

Description

Checks whether a bivariate lifetime distribution belongs to the NBUHR (New Better than Used in Hazard Rate) or NWUHR (New Worse than Used) aging class. A distribution is NBUHR if

$h_1(0,t_2) \ge h_1(t_1,t_2)\;\text{ for all }t_1 > 0$

and similarly for the second component. The function evaluates this at a grid of $t_1$ values.

Usage

nbuhr_test(
  t2 = 1,
  t1_grid = seq(0.1, 5, by = 0.1),
  surv_fn = NULL,
  k1 = 1,
  k2 = 1,
  theta = 0
)

Arguments

t2	Fixed value of the second age coordinate.
t1_grid	Numeric vector of $t_1$ values to check (default 0.1 to 5 in steps of 0.1).
surv_fn	Joint survival function; defaults to Gumbel bivariate exponential.
k1, k2, theta	Gumbel parameters.

Value

A list with components:

class1

Character: "NBUHR", "NWUHR", or "neither" for the first component.

class2

Same for the second component.

h1_grid

Numeric vector of $h_1(t_1,t_2)$ values.

h2_grid

Numeric vector of $h_2(t_1,t_2)$ values.

References

Jayalekshmi S., Rajesh G., Nair N.U. (2022), Definition 3.2. doi:10.1080/03610926.2022.2085874

Examples

nbuhr_test(t2 = 1, k1 = 1, k2 = 1, theta = 0.3)
nbuhr_test(t2 = 0.5,
           surv_fn = function(x1, x2) exp(-(x1 + x2)))

---

Nonparametric Estimator for the Bivariate Laplace Transform of Residual Lives

Description

Given a bivariate sample $(X_{1i}, X_{2i}),\; i=1,\ldots,n$, estimates

$\hat{L}^*_{1}(s_1; t_1, t_2) = \frac{\sum_{i:X_{1i}>t_1,\,X_{2i}>t_2} \int_{t_1}^{X_{1i}} e^{-s_1 u}\,du}{e^{-s_1 t_1} \cdot \#\{X_{1i}>t_1,X_{2i}>t_2\}}$

and analogously for the second component, using the empirical survival function as described in Jayalekshmi et al. (2022), Section 6.

Usage

np_blt_residual(data, s1, s2, t1 = 0, t2 = 0)

Arguments

data	A two-column numeric matrix or data frame with columns for $X_1$ and $X_2$.
s1, s2	Positive Laplace parameters.
t1, t2	Non-negative truncation ages.

Value

A named numeric vector (L1_hat, L2_hat).

References

Jayalekshmi S., Rajesh G., Nair N.U. (2022), Section 6. doi:10.1080/03610926.2022.2085874

See Also

blt_residual_gumbel, sim_blt_residual

Examples

set.seed(123)
dat <- rgumbel_biv(200, k1 = 1, k2 = 1, theta = 0.5)
np_blt_residual(dat, s1 = 1, s2 = 1, t1 = 0.3, t2 = 0.3)

# Compare with closed form
blt_residual_gumbel(s1 = 1, s2 = 1, t1 = 0.3, t2 = 0.3, theta = 0.5)

---

Nonparametric Estimator for the Univariate Laplace Transform of Residual Life

Description

Given a sample $X_1, \ldots, X_n$, estimates the Laplace transform of the residual life using the empirical survival function:

$\hat{L}_X(s,t) = \frac{\displaystyle\sum_{i:\,X_i > t} e^{-s X_i}} {\#\{i : X_i > t\}}.$

Usage

np_lt_residual(x, s, t = 0)

Arguments

x	Numeric vector; observed sample.
s	Non-negative Laplace parameter.
t	Truncation time (default 0).

Value

Scalar numeric estimate of $L_X(s,t)$.

See Also

lt_residual

Examples

set.seed(1)
x <- rexp(300, rate = 1)

# Estimate at s=1, t=0: true value 1/(1+1) = 0.5
np_lt_residual(x, s = 1, t = 0)

# Estimate at s=1, t=0.5: true value still approx 0.5 (memoryless)
np_lt_residual(x, s = 1, t = 0.5)

---

Plot Bivariate Laplace Transform of Residual Lives

Description

Plots the star Laplace transform of residual lives $L^*_{X_{t_1|t_2}}(s_1)$ as a function of $t_1$ for fixed $s_1$, $t_2$. Optionally overlays two distributions for visual comparison of the BLt-rl order.

Usage

plot_blt_residual(
  surv_fn,
  surv_fn2 = NULL,
  s1 = 1,
  t2 = 0.5,
  t1_grid = seq(0.1, 3, by = 0.1),
  k1 = 1,
  k2 = 1,
  theta = 0,
  xlab = expression(t[1]),
  ylab = expression(L^"*"[X[t[1] * "|" * t[2]]](s[1])),
  main = "Bivariate LT of Residual Lives",
  col1 = "steelblue",
  col2 = "firebrick",
  lwd = 2,
  legend_labels = c("Distribution 1", "Distribution 2")
)

Arguments

surv_fn	Joint survival function. If a second distribution is to be overlaid, pass it as surv_fn2.
surv_fn2	Optional second survival function for comparison.
s1	Laplace parameter (default 1).
t2	Fixed second truncation age (default 0.5).
t1_grid	Grid of first truncation ages.
k1, k2, theta	Parameters for the default Gumbel distribution; used only when surv_fn = NULL.
xlab, ylab, main	Plot labels.
col1, col2	Line colours.
lwd	Line width.
legend_labels	Length-2 character vector for legend (ignored if surv_fn2 = NULL).

Value

Invisibly returns the data frame used for plotting.

Examples

sX <- function(x1, x2) sgumbel_biv(x1, x2, k1 = 1, k2 = 1, theta = 0.3)
sY <- function(x1, x2) sgumbel_biv(x1, x2, k1 = 2, k2 = 1, theta = 0.3)
plot_blt_residual(sX, sY, s1 = 1, t2 = 0.5,
                  legend_labels = c("k1=1", "k1=2"))

---

Plot Bivariate Laplace Transform of Reversed Residual Lives

Description

Plots the reversed-life Laplace transform $L_{t_1|t_2}(s_1)$ as a function of $t_1$ for fixed $s_1$ and $t_2$.

Usage

plot_blt_reversed(
  cdf_fn,
  cdf_fn2 = NULL,
  s1 = 1,
  t2 = 0.5,
  t1_grid = seq(0.1, 0.9, by = 0.05),
  theta = 0,
  xlab = expression(t[1]),
  ylab = expression(L[t[1] * "|" * t[2]](s[1])),
  main = "Bivariate LT of Reversed Residual Lives",
  col1 = "darkgreen",
  col2 = "darkorange",
  lwd = 2,
  legend_labels = c("Distribution 1", "Distribution 2")
)

Arguments

cdf_fn	Joint CDF function.
cdf_fn2	Optional second CDF for comparison.
s1	Laplace parameter.
t2	Fixed second truncation time.
t1_grid	Grid of first truncation times.
theta	FGM parameter (used if cdf_fn = NULL).
xlab, ylab, main	Plot labels.
col1, col2	Line colours.
lwd	Line width.
legend_labels	Legend labels.

Value

Invisibly returns the data frame used for plotting.

Examples

cX <- function(x1, x2) pfgm_biv(x1, x2, theta = 0.2)
cY <- function(x1, x2) pfgm_biv(x1, x2, theta = 0.7)
plot_blt_reversed(cX, cY, s1 = 1, t2 = 0.5,
                  legend_labels = c("theta=0.2", "theta=0.7"))

---

Schur-Constant Bivariate Distribution

Description

Random generation and survival function for a Schur-constant bivariate distribution with survival function

$\bar{F}(x_1,x_2) = S(x_1 + x_2),\quad x_1,x_2 > 0,$

where $S$ is a given univariate survival function. The default marginal is exponential with rate lambda.

Usage

sschur_biv(x1, x2, lambda = 1)

rschur_biv(n, lambda = 1)

Arguments

x1, x2	Non-negative values.
lambda	Exponential rate parameter for the generating survival function.
n	Number of random observations.

Value

Numeric vector (sschur_biv) or two-column matrix (rschur_biv).

References

Barlow R.E., Mendel M.B. (1992). De Finetti-type representations for life distributions. *Journal of the American Statistical Association*, 87(420), 1116–1122.

Examples

sschur_biv(0.5, 1, lambda = 1)
set.seed(2); head(rschur_biv(40, lambda = 1))

---

Shannon Differential Entropy

Description

Computes the Shannon differential entropy

$H(f) = -\int_0^\infty f(x)\log f(x)\,dx$

for a non-negative continuous random variable with density dens_fn.

Usage

shannon_entropy(dens_fn, upper = 100)

Arguments

dens_fn	A function of one argument returning the density $f(x)$.
upper	Upper integration limit (default 100).

Value

Scalar numeric.

References

Shannon C.E. (1948). A mathematical theory of communication. *Bell System Technical Journal*, 27(3), 379–423.

See Also

info_gen_function

Examples

# Exponential(1): H = 1
shannon_entropy(function(x) dexp(x, rate = 1))

# Exponential(2): H = 1 - log(2)
shannon_entropy(function(x) dexp(x, rate = 2))

---

Monte-Carlo Simulation Study for the BLT Residual Estimator

Description

Evaluates the performance of np_blt_residual via repeated simulation from the Gumbel bivariate exponential distribution and compares estimates to the closed-form blt_residual_gumbel values. Returns bias, variance, and mean squared error (MSE).

Usage

sim_blt_residual(
  n_obs = 200,
  n_sim = 100,
  s1 = 1,
  s2 = 1,
  t1 = 0.3,
  t2 = 0.3,
  k1 = 1,
  k2 = 1,
  theta = 0.5,
  seed = 42L
)

Arguments

n_obs	Sample size per replicate.
n_sim	Number of simulation replicates.
s1, s2	Laplace parameters.
t1, t2	Truncation ages.
k1, k2, theta	Gumbel parameters.
seed	Random seed for reproducibility.

Value

A data frame with columns component, true_value, mean_est, bias, variance, mse.

References

Jayalekshmi S., Rajesh G., Nair N.U. (2022), Section 6. doi:10.1080/03610926.2022.2085874

See Also

np_blt_residual

Examples

sim_blt_residual(n_obs = 100, n_sim = 50, s1 = 1, s2 = 1,
                 t1 = 0.3, t2 = 0.3, k1 = 1, k2 = 1, theta = 0.5)

---