Bivariate Chen Distribution: Density, Survival, Hazard, Plots and Estimates

Probability density, survival and hazard functions, random generation, graphical displays and maximum likelihood estimation for bivariate Chen type models. Three constructions are covered: the Bivariate Chen distribution with a singular component on the diagonal, the Bivariate Extended Chen distribution, and the Bivariate Inverse copula Chen distribution obtained through an inverse copula with Chen marginals. Univariate Chen density, distribution, quantile, survival, hazard and random generation routines are obtained from the 'reliaR' package.

BvCh

Probability density, survival function, hazard function, random generation, graphical displays and maximum likelihood estimation for bivariate Chen type lifetime models.

Installation

# install.packages("remotes")
remotes::install_github("puneetstat999/BvCh")

The univariate Chen building blocks come from reliaR, which is installed automatically.

The three models

Model	Construction	Singular part
BvCh	Marshall-Olkin type: $Z_1=\min(U_1,U_3)$, $Z_2=\min(U_2,U_3)$ with $U_i\sim\mathrm{Chen}(\alpha_i,\beta)$	yes, on the diagonal
BEC	same construction with extended Chen variables, adding $\lambda$	yes, on the diagonal
IFGM-Chen	Inverse FGM copula with Chen marginals	no

Quick start

library(BvCh)

## density, survival and hazard
dbvch(0.5, 0.9, alpha1 = 1.5, alpha2 = 2.5, alpha3 = 0.5, beta = 1.8)
sbvch(0.5, 0.9, alpha1 = 1.5, alpha2 = 2.5, alpha3 = 0.5, beta = 1.8)
hbvch(0.5, 0.9, alpha1 = 1.5, alpha2 = 2.5, alpha3 = 0.5, beta = 1.8)

## simulation
set.seed(1)
d <- rbvch(500, alpha1 = 1.5, alpha2 = 2.5, alpha3 = 0.5, beta = 1.8)

## maximum likelihood estimation
fit <- fitbvch(d)
fit
ksfit_bvch(fit, d)

## model comparison
compare_models(fit,
               fitbec(rbec(500, 1.5, 2.5, 0.5, 1.8, 2)),
               fitifgmchen(rifgmchen(500, 1.2, 1.3, 1.5, 1.1, 0.6, 0.4)))

## surfaces
plot_surface_bvch(alpha1 = 1.5, alpha2 = 2.5, alpha3 = 0.5,
                  beta.vec = c(0.8, 1.8, 5, 10))
plot_reliability_ifgmchen(gamma = 0.6, omega = 0.4)

Univariate Chen

All univariate Chen routines are the reliaR implementations, re-exported for convenience:

dchen(x, beta, lambda)   # density
pchen(q, beta, lambda)   # distribution function
qchen(p, beta, lambda)   # quantile function
rchen(n, beta, lambda)   # random generation
schen(x, beta, lambda)   # survival function
hchen(x, beta, lambda)   # hazard function

reliaR::schen(x, beta, lambda) equals exp(lambda * (1 - exp(x^beta))), which is the $S_{Ch}(z;\alpha,\beta)$ used throughout the bivariate formulae, and pchen + schen == 1 exactly.

The extended Chen distribution, which reliaR does not provide and which the BEC model needs, is implemented here as sech, hech, dech, pech, qech, rech.

Two things worth knowing

rchen uses reliaR::qchen. In reliaR 0.2 reliaR::rchen does not agree with the package's own distribution function — its Kolmogorov-Smirnov distance from pchen is about 0.38 at beta = 1.5, lambda = 2 and about 0.85 at beta = 2.5, and it is correct only when beta = 1. Since qchen is the verified analytic inverse of pchen, BvCh::rchen draws qchen(runif(n), beta, lambda) so that all six univariate Chen routines are mutually consistent. See ?chen.
The IFGM copula gives negative dependence for positive parameters. With the formula as written in the source document, C(u,v) = uv / sqrt(1 + gamma*(1-u)*(1-v) + omega*u*v*(1-u)*(1-v)), the copula lies below the independence copula uv, so tau_ifgm and rho_ifgm are non-positive for gamma, omega >= 0 (tau = -0.0699, rho = -0.1049 at gamma = 0.6, omega = 0.4). The source quotes tau = 0.156 > 0 for the same parameters, which contradicts the copula formula it also quotes. The formula has been implemented as given; see ?cifgm.