Package {BvCh}

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

Description

Implements probability density, survival and hazard functions, random generation, graphical displays and maximum likelihood estimation for a family of bivariate Chen type lifetime models.

Models

Bivariate Chen (BvCh)

Marshall-Olkin type model of Gupta, Kumar and co-authors. With U_i \sim Chen(\alpha_i, \beta) independent and Z_1 = \min(U_1, U_3), Z_2 = \min(U_2, U_3), the pair (Z_1,Z_2) is BvCh(\alpha_1,\alpha_2,\alpha_3,\beta). The distribution is a mixture of an absolutely continuous part of total mass (\alpha_1+\alpha_2)/(\alpha_1+\alpha_2+\alpha_3) and a singular part of mass \alpha_3/(\alpha_1+\alpha_2+\alpha_3) carried on the diagonal.

Bivariate Extended Chen (BEC)

Marshall-Olkin type model using extended Chen latent variables, adding a shape parameter \lambda > 0.

Bivariate Inverse FGM Chen (IFGM-Chen)

Copula construction pairing the Inverse FGM copula with Chen marginals. This model is absolutely continuous and has no singular component.

Univariate Chen

All univariate Chen building blocks are taken from the reliaR package rather than being re-implemented. In particular reliaR::schen(x, beta, lambda) satisfies schen(x, beta, lambda) == exp(lambda * (1 - exp(x^beta)), which is the S_{Ch}(z;\alpha,\beta) used throughout the bivariate formulae.

Author(s)

Maintainer: Puneet Kumar Gupta puneetstat999@gmail.com (ORCID iD: https://orcid.org/0000-0002-3458-5655)

Authors:

• Puneet Kumar Gupta puneetstat999@gmail.com (ORCID iD: https://orcid.org/0000-0002-3458-5655)

• Mukul Bijalwan (ORCID iD: https://orcid.org/0009-0001-3040-6912)

References

Chen, Z. (2000). A new two-parameter lifetime distribution with bathtub shape or increasing failure-rate function. Statistics & Probability Letters, 49(2), 155–161. doi:10.1016/S0167-7152(00)00044-4.

Marshall, A. W., & Olkin, I. (1967). A multivariate Cauchy distribution and a generalization of the multivariate exponential distribution. Biometrika, 54(3), 573–588. doi:10.1093/biomet/54.3-4.573.

Nelsen, R. B. (2006). An Introduction to Copulas (2nd ed.). Springer, New York.

Kumar, V. & Ligges, U. (2025). reliaR: Comprehensive Tools for Some Probability Distributions. R package version 0.2. doi:10.32614/CRAN.package.reliaR. https://CRAN.R-project.org/package=reliaR.

See Also

Useful links:

• https://github.com/puneetstat999/BvCh

• Report bugs at https://github.com/puneetstat999/BvCh/issues

Bivariate Extended Chen distribution: density, survival, hazard and random generation

Description

Probability density, survival function, hazard function and random generation for the Marshall-Olkin type Bivariate Extended Chen distribution BEC(\alpha_1,\alpha_2,\alpha_3,\beta,\lambda).

Usage

dbec(x1, x2, alpha1, alpha2, alpha3, beta, lambda, log = FALSE)

sbec(x1, x2, alpha1, alpha2, alpha3, beta, lambda)

hbec(x1, x2, alpha1, alpha2, alpha3, beta, lambda)

rbec(n, alpha1, alpha2, alpha3, beta, lambda)

Arguments

Details

With U_i \sim EC(\alpha_i,\beta,\lambda) independent and X_1=\min(U_1,U_3), X_2=\min(U_2,U_3), the joint survival function is

S(x_1,x_2)=\begin{cases} S_{EC}(x_1;\alpha_1,\beta,\lambda)\, S_{EC}(x_2;\alpha_2+\alpha_3,\beta,\lambda) & x_2>x_1\\ S_{EC}(x_1;\alpha_1+\alpha_3,\beta,\lambda)\, S_{EC}(x_2;\alpha_2,\beta,\lambda) & x_1>x_2\\ S_{EC}(x;\alpha_1+\alpha_2+\alpha_3,\beta,\lambda) & x_1=x_2=x,\end{cases}

where S_{EC}(t;\alpha,\beta,\lambda)=[1+\lambda(e^{t^\beta}-1)]^{-\alpha} (see sech). The joint density is

f(x_1,x_2)=\begin{cases} f_1 & x_2>x_1\\ f_2 & x_1>x_2\\ f_3 & x_1=x_2=x,\end{cases}

with

f_1=\alpha_1(\alpha_2+\alpha_3)\lambda^2\beta^2 x_1^{\beta-1}x_2^{\beta-1}e^{x_1^{\beta}+x_2^{\beta}} [1+\lambda(e^{x_1^{\beta}}-1)]^{-(\alpha_1+1)} [1+\lambda(e^{x_2^{\beta}}-1)]^{-(\alpha_2+\alpha_3+1)},

f_2=\alpha_2(\alpha_1+\alpha_3)\lambda^2\beta^2 x_1^{\beta-1}x_2^{\beta-1}e^{x_1^{\beta}+x_2^{\beta}} [1+\lambda(e^{x_1^{\beta}}-1)]^{-(\alpha_1+\alpha_3+1)} [1+\lambda(e^{x_2^{\beta}}-1)]^{-(\alpha_2+1)},

f_3(x)=\alpha_3\beta\lambda x^{\beta-1}e^{x^{\beta}} [1+\lambda(e^{x^{\beta}}-1)]^{-(\alpha_1+\alpha_2+\alpha_3+1)}.

The three regions carry mass \alpha_i/(\alpha_1+\alpha_2+\alpha_3), so the density integrates to one. The hazard function is f/S,

h(x_1,x_2)=\begin{cases} \dfrac{\alpha_1(\alpha_2+\alpha_3)\lambda^2\beta^2x_1^{\beta-1} x_2^{\beta-1}e^{x_1^{\beta}+x_2^{\beta}}} {[1+\lambda(e^{x_1^{\beta}}-1)][1+\lambda(e^{x_2^{\beta}}-1)]} & x_2>x_1\\[2ex] \dfrac{\alpha_2(\alpha_1+\alpha_3)\lambda^2\beta^2x_1^{\beta-1} x_2^{\beta-1}e^{x_1^{\beta}+x_2^{\beta}}} {[1+\lambda(e^{x_1^{\beta}}-1)][1+\lambda(e^{x_2^{\beta}}-1)]} & x_1>x_2\\[2ex] \dfrac{\alpha_3\beta\lambda x^{\beta-1}e^{x^{\beta}}} {1+\lambda(e^{x^{\beta}}-1)} & x_1=x_2=x.\end{cases}

The componentwise hazard gradients are available separately through hbec1 and hbec2.

Value

A numeric vector, or for rbec a two column matrix with columns x1 and x2.

See Also

sbec, hbec, rbec, fitbec, sech

Examples

dbec(0.5, 0.9, alpha1 = 1.5, alpha2 = 2.5, alpha3 = 0.5, beta = 1.8, lambda = 2)
sbec(0.5, 0.9, alpha1 = 1.5, alpha2 = 2.5, alpha3 = 0.5, beta = 1.8, lambda = 2)
hbec(0.5, 0.9, alpha1 = 1.5, alpha2 = 2.5, alpha3 = 0.5, beta = 1.8, lambda = 2)
set.seed(1)
rbec(5, alpha1 = 1.5, alpha2 = 2.5, alpha3 = 0.5, beta = 1.8, lambda = 2)

Bivariate Chen distribution: density, survival, hazard and random generation

Description

Probability density, survival function, hazard function and random generation for the Marshall-Olkin type Bivariate Chen distribution BvCh(\alpha_1,\alpha_2,\alpha_3,\beta).

Usage

dbvch(z1, z2, alpha1, alpha2, alpha3, beta, log = FALSE)

sbvch(z1, z2, alpha1, alpha2, alpha3, beta)

hbvch(z1, z2, alpha1, alpha2, alpha3, beta)

rbvch(n, alpha1, alpha2, alpha3, beta)

Arguments

Details

Let U_i \sim Chen(\alpha_i,\beta), i=1,2,3, be independent and set Z_1=\min(U_1,U_3), Z_2=\min(U_2,U_3). Then the joint survival function is

S(z_1,z_2) = \exp\left\{\alpha_1\left(1-e^{z_1^{\beta}}\right) \right\}\exp\left\{\alpha_2\left(1-e^{z_2^{\beta}}\right)\right\} \exp\left\{\alpha_3\left(1-e^{w^{\beta}}\right)\right\},

where w=\max(z_1,z_2). Each factor is obtained from schen. The joint density is piecewise,

f(z_1,z_2) = \begin{cases} f_1(z_1,z_2) & z_1<z_2\\ f_2(z_1,z_2) & z_1>z_2\\ f_0(z) & z_1=z_2=z,\end{cases}

with

f_1=\alpha_1(\alpha_2+\alpha_3)\beta^2 z_1^{\beta-1}z_2^{\beta-1} \exp\left\{z_1^{\beta}+z_2^{\beta}+\alpha_1(1-e^{z_1^{\beta}}) +(\alpha_2+\alpha_3)(1-e^{z_2^{\beta}})\right\},

f_2=\alpha_2(\alpha_1+\alpha_3)\beta^2 z_1^{\beta-1}z_2^{\beta-1} \exp\left\{z_1^{\beta}+z_2^{\beta}+\alpha_2(1-e^{z_2^{\beta}}) +(\alpha_1+\alpha_3)(1-e^{z_1^{\beta}})\right\},

f_0(z) = \alpha_3\beta z^{\beta-1} \exp\left\{z^{\beta}+(\alpha_1+\alpha_2+\alpha_3)(1-e^{z^{\beta}})\right\}.

The mass carried by the three regions is respectively \alpha_1/(\alpha_1+\alpha_2+\alpha_3), \alpha_2/(\alpha_1+\alpha_2+\alpha_3) and \alpha_3/(\alpha_1+\alpha_2+\alpha_3), so the density integrates to one. The hazard function is the ratio f/S, that is

h(z_1,z_2)=\begin{cases} \alpha_1(\alpha_2+\alpha_3)\beta^2 z_1^{\beta-1}z_2^{\beta-1} e^{z_1^{\beta}+z_2^{\beta}} & z_1<z_2\\ \alpha_2(\alpha_1+\alpha_3)\beta^2 z_1^{\beta-1}z_2^{\beta-1} e^{z_1^{\beta}+z_2^{\beta}} & z_1>z_2\\ \alpha_3\beta z^{\beta-1}e^{z^{\beta}} & z_1=z_2=z.\end{cases}

The model is a mixture (Proposition 1) of an absolutely continuous part of weight (\alpha_1+\alpha_2)/(\alpha_1+\alpha_2+\alpha_3) and a singular part of weight \alpha_3/(\alpha_1+\alpha_2+\alpha_3) supported on the diagonal; see dbvch_ac and dbvch_sing.

Value

A numeric vector, or for rbvch a two column matrix with columns z1 and z2.

See Also

sbvch, hbvch, rbvch, fitbvch, plot_surface_bvch

Examples

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)
set.seed(1)
rbvch(5, alpha1 = 1.5, alpha2 = 2.5, alpha3 = 0.5, beta = 1.8)

Absolutely continuous and singular components of the BvCh density

Description

Decomposition of the Bivariate Chen density into the two components of the mixture representation f = \frac{\alpha_1+\alpha_2}{\alpha_1+\alpha_2+\alpha_3}f_a + \frac{\alpha_3}{\alpha_1+\alpha_2+\alpha_3}f_s.

Usage

dbvch_ac(z1, z2, alpha1, alpha2, alpha3, beta)

dbvch_sing(z1, z2, alpha1, alpha2, alpha3, beta)

Arguments

Details

dbvch_ac is the properly normalised absolutely continuous component of Proposition 1; it integrates to one over the plane, and dbvch_sing integrates to one along the diagonal. Consequently dbvch is recovered exactly as (alpha1 + alpha2)/(alpha1 + alpha2 + alpha3) * dbvch_ac(...) plus alpha3/(alpha1 + alpha2 + alpha3) * dbvch_sing(...).

Value

A numeric vector.

See Also

dbvch

Examples

a <- 1.5; b2 <- 2.5; c3 <- 0.5; bt <- 1.8
zz1 <- c(0.4, 0.9, 0.7); zz2 <- c(0.6, 0.4, 0.7)
ac <- dbvch_ac(zz1, zz2, a, b2, c3, bt)
sg <- dbvch_sing(zz1, zz2, a, b2, c3, bt)
k <- (a + b2 + c3)
mix <- (a + b2) / k * ac + c3 / k * sg
all.equal(mix, dbvch(zz1, zz2, a, b2, c3, bt))

Univariate Chen distribution

Description

Density, distribution function, quantile function, random generation, survival and hazard function for the univariate Chen distribution with shape parameter beta and scale parameter lambda.

Usage

dchen(x, beta, lambda, log = FALSE)

pchen(q, beta, lambda, lower.tail = TRUE, log.p = FALSE)

qchen(p, beta, lambda, lower.tail = TRUE, log.p = FALSE)

rchen(n, beta, lambda)

schen(x, beta, lambda)

hchen(x, beta, lambda)

hra.chen(x, beta, lambda)

crf.chen(x, t = 0, beta, lambda)

Arguments

Details

These are the reliaR implementations, re-exported by BvCh so that the Chen building blocks used in the bivariate formulae are available without attaching a second package. The density is

f(x;\lambda,\beta) = \lambda\beta x^{\beta-1} \exp\left(x^{\beta}\right) \exp\left[\lambda\left\{1-\exp\left(x^{\beta}\right)\right\}\right],

and the survival function is

S(x;\lambda,\beta) = \exp\left[\lambda\left\{1-\exp(x^{\beta})\right\}\right].

Taking \lambda = \alpha gives the S_{Ch}(z;\alpha,\beta) that appears in the bivariate Chen formulae.

Value

dchen gives the density, pchen the distribution function, qchen the quantile function, rchen random deviates, schen the survival function and hchen the hazard function.

Random generation

rchen is not a direct call to rchen. In reliaR 0.2 that function does not agree with the package's own distribution, quantile and survival functions: for \beta=1.5, \lambda=2 its output has Kolmogorov-Smirnov distance about 0.38 from pchen (p<10^{-16}), and the discrepancy grows with \beta (distance about 0.85 at \beta=2.5); it happens to be correct only when \beta=1. Since qchen is the verified analytic inverse of pchen (agreement to machine precision), rchen here draws qchen(runif(n), beta, lambda), which keeps all six univariate Chen routines mutually consistent. Everything is still computed by reliaR.

See Also

Chen, Chensurvival

Examples

dchen(0.5, beta = 1.5, lambda = 2)
pchen(0.5, beta = 1.5, lambda = 2)
qchen(0.5, beta = 1.5, lambda = 2)
schen(0.5, beta = 1.5, lambda = 2)
hchen(0.5, beta = 1.5, lambda = 2)

Model comparison table

Description

Collects fitted models into a single table of -\log L, AIC and BIC.

Usage

compare_models(...)

Arguments

Value

A data frame with columns Model, nobs, neglogL, AIC and BIC, sorted by increasing AIC so that the best model is on the first row.

Examples

set.seed(1)
a <- fitbvch(rbvch(200, 1.5, 2.5, 0.5, 1.8))
b <- fitbec(rbec(200, 1.5, 2.5, 0.5, 1.8, 2))
compare_models(a, b)

Univariate Extended Chen distribution

Description

Density, distribution function, quantile function, random generation, survival and hazard function for the univariate Extended Chen distribution used as the latent building block of the Bivariate Extended Chen (BEC) model.

Usage

sech(x, alpha, beta, lambda)

hech(x, alpha, beta, lambda)

dech(x, alpha, beta, lambda, log = FALSE)

pech(q, alpha, beta, lambda, lower.tail = TRUE, log.p = FALSE)

qech(p, alpha, beta, lambda, lower.tail = TRUE, log.p = FALSE)

rech(n, alpha, beta, lambda)

Arguments

Details

The Extended Chen survival function is

S_{EC}(x;\alpha,\beta,\lambda) = \left[1+\lambda\left(\exp(x^{\beta})-1\right)\right]^{-\alpha},

giving hazard

h_{EC}(x;\alpha,\beta,\lambda) = \frac{\alpha\lambda\beta x^{\beta-1}\exp(x^{\beta})} {1+\lambda\left(\exp(x^{\beta})-1\right)}

and density f_{EC}(x) = S_{EC}(x)\,h_{EC}(x). The quantile function is available in closed form,

Q(p) = \left[\log\left\{1+\frac{p^{-1/\alpha}-1}{\lambda}\right\}\right]^{1/\beta}.

Unlike the univariate Chen distribution, which reliaR supplies, no Extended Chen implementation exists there, so it is provided here.

Value

dech gives the density, pech the distribution function, qech the quantile function, rech random deviates, sech the survival function and hech the hazard function.

Examples

sech(0.5, alpha = 1.5, beta = 1.5, lambda = 2)
hech(0.5, alpha = 1.5, beta = 1.5, lambda = 2)
qech(c(0.25, 0.5, 0.75), alpha = 1.5, beta = 1.5, lambda = 2)

Maximum likelihood estimation for the BEC model

Description

Fits the Bivariate Extended Chen model by maximum likelihood.

Usage

fitbec(
  x1,
  x2 = NULL,
  start = NULL,
  singular = TRUE,
  method = "BFGS",
  control = list(),
  se = TRUE
)

Arguments

Value

An object of class fitbec with the same components as fitbvch.

See Also

fitbvch, logLik_bec

Examples

set.seed(1)
d <- rbec(400, alpha1 = 1.5, alpha2 = 2.5, alpha3 = 0.5, beta = 1.8,
          lambda = 2)
fit <- fitbec(d)
fit

Maximum likelihood estimation for the BvCh model

Description

Fits the Marshall-Olkin type Bivariate Chen model by maximum likelihood.

Usage

fitbvch(
  x1,
  x2 = NULL,
  start = NULL,
  singular = TRUE,
  method = "BFGS",
  control = list(),
  se = TRUE
)

Arguments

Details

Observations with z1 == z2 fall on the singular component of the model. Whether they contribute the singular density f_0 or are discarded is controlled by singular; the counts are always reported so the choice stays visible. Standard errors come from the numerical Hessian of the log-likelihood and are on the original (untransformed) scale.

Value

An object of class fitbvch: a list with components coefficients, vcov, se, loglik, aic, bic, nobs, counts, converged, message and call.

See Also

fitbec, fitifgmchen, logLik_bvch

Examples

set.seed(1)
d <- rbvch(400, alpha1 = 1.5, alpha2 = 2.5, alpha3 = 0.5, beta = 1.8)
fit <- fitbvch(d)
fit
coef(fit)

Maximum likelihood estimation for the IFGM-Chen model

Description

Fits the Bivariate Inverse FGM Chen model by maximum likelihood.

Usage

fitifgmchen(
  x1,
  x2 = NULL,
  start = NULL,
  method = "BFGS",
  control = list(),
  se = TRUE
)

Arguments

Details

The copula parameters gamma and omega are constrained to [0,1] and are estimated on the logit scale; the remaining parameters are estimated on the log scale. All reported values are on the natural scale.

Value

An object of class fitifgmchen with the same components as fitbvch.

Identifiability

The four marginal parameters are estimated very precisely, but gamma and omega are only weakly identified: with n=5000 simulated from (\gamma,\omega)=(0.6,0.4) the marginal parameters have standard errors around 0.014 while gamma and omega have standard errors around 0.54 and 1.29. The log-likelihood surface is nearly flat in that direction, so individual values of gamma and omega should not be over-interpreted; logLik_ifgmchen is exported so the profile likelihood can be inspected and start used to explore it.

See Also

fitbvch, logLik_ifgmchen

Examples

set.seed(1)
d <- rifgmchen(400, theta1 = 1.2, beta1 = 1.3, theta2 = 1.5,
               beta2 = 1.1, gamma = 0.6, omega = 0.4)
fit <- fitifgmchen(d)
fit

Componentwise hazard gradients of the BEC model

Description

The hazard gradients of the Bivariate Extended Chen model, that is the hazard of the i-th marginal conditional on the other component.

Usage

hbec1(x1, x2, alpha1, alpha2, alpha3, beta, lambda)

hbec2(x1, x2, alpha1, alpha2, alpha3, beta, lambda)

Arguments

Details

h_{X_1}(x_1,x_2)=\begin{cases} \dfrac{\alpha_1\beta\lambda x_1^{\beta-1}e^{x_1^{\beta}}} {1+\lambda(e^{x_1^{\beta}}-1)} & x_2>x_1\\ \dfrac{(\alpha_1+\alpha_3)\beta\lambda x_1^{\beta-1}e^{x_1^{\beta}}} {1+\lambda(e^{x_1^{\beta}}-1)} & x_1>x_2\\ \dfrac{(\alpha_1+\alpha_2+\alpha_3)\beta\lambda x_1^{\beta-1} e^{x_1^{\beta}}}{1+\lambda(e^{x_1^{\beta}}-1)} & x_1=x_2,\end{cases}

with h_{X_2} obtained by interchanging the roles of x_1, x_2 and \alpha_1, \alpha_2.

Value

A numeric vector.

See Also

hbec

Examples

hbec1(0.5, 0.9, alpha1 = 1.5, alpha2 = 2.5, alpha3 = 0.5, beta = 1.8, lambda = 2)
hbec2(0.5, 0.9, alpha1 = 1.5, alpha2 = 2.5, alpha3 = 0.5, beta = 1.8, lambda = 2)

Inverse FGM copula

Description

The Inverse FGM (IFGM) copula, its density, dependence measures and a random generator.

Usage

cifgm(u, v, gamma, omega)

cifgmdens(u, v, gamma, omega)

rifgm(n, gamma, omega)

tau_ifgm(gamma, omega)

rho_ifgm(gamma, omega)

Arguments

Details

For 0\le\gamma,\omega\le 1 the Inverse FGM copula is

C(u,v)=\frac{uv}{\sqrt{1+\gamma(1-u)(1-v)+\omega uv(1-u)(1-v)}}.

Writing D=1+\gamma(1-u)(1-v)+\omega uv(1-u)(1-v) and P=uv, its density has the closed form

c(u,v)=D^{-1/2} -\frac{v D_v + u D_u}{2D^{3/2}} +\frac{3P D_u D_v}{4D^{5/2}} -\frac{P D_{uv}}{2D^{3/2}},

with D_u=-\gamma(1-v)+\omega(1-2u)v(1-v), D_v=-\gamma(1-u)+\omega u(1-2v)(1-u) and D_{uv}=\gamma+\omega(1-2u)(1-2v).

Kendall's tau and Spearman's rho are computed as

\tau=4\int_0^1\!\int_0^1 C(u,v)c(u,v)\,du\,dv-1,\qquad \rho=12\int_0^1\!\int_0^1 uv\,c(u,v)\,du\,dv-3.

Both are obtained by numerical quadrature and reproduce Monte Carlo estimates from rifgm.

Value

cifgm gives the copula, cifgmdens its density, rifgm a two column matrix of copula draws, and tau_ifgm and rho_ifgm scalar dependence measures.

Sign of the dependence

With the +\,\gamma(1-u)(1-v)+\omega uv(1-u)(1-v) denominator given above, the copula lies below the independence copula uv (C(0.5,0.5)=0.2306<0.25 at \gamma=0.6,\omega=0.4), so \tau and \rho are non-positive for \gamma,\omega\ge0: numerically \tau=-0.0699 and \rho=-0.1049 at \gamma=0.6,\omega=0.4. This is the usual behaviour of an inverse FGM type copula. Both measures are exactly zero at \gamma=\omega=0. Note that the summary document this package was written from quotes \tau\approx 0.156>0 for the same parameters, which is inconsistent with the copula formula it also quotes; the formula has been implemented as given, and tau_ifgm/rho_ifgm are provided so the dependence can be checked directly.

See Also

difgmchen, hifgmchen

Examples

cifgm(0.3, 0.7, gamma = 0.6, omega = 0.4)
cifgmdens(0.3, 0.7, gamma = 0.6, omega = 0.4)
tau_ifgm(0.6, 0.4)
rho_ifgm(0.6, 0.4)
set.seed(1)
head(rifgm(5, gamma = 0.6, omega = 0.4))

Bivariate IFGM-Chen distribution

Description

Reliability function, density, hazard function and random generation for the Bivariate Inverse FGM Chen model, obtained by combining the Inverse FGM copula with Chen marginals.

Usage

relifgmchen(x, y, theta1, beta1, theta2, beta2, gamma, omega)

difgmchen(x, y, theta1, beta1, theta2, beta2, gamma, omega, log = FALSE)

hifgmchen(x, y, theta1, beta1, theta2, beta2, gamma, omega)

rifgmchen(n, theta1, beta1, theta2, beta2, gamma, omega)

Arguments

Details

The marginals are Chen distributions F_X(x)=1-e^{-\theta_1(e^{x^{\beta_1}}-1)} and F_Y(y)=1-e^{-\theta_2(e^{y^{\beta_2}}-1)}, which are exactly reliaR::pchen(x, beta1, theta1) and reliaR::pchen(y, beta2, theta2). With C the Inverse FGM copula of cifgm and c its density, the joint reliability function and density are

R(x,y)=C(F_X(x),F_Y(y))-F_X(x)-F_Y(y)+1,

f(x,y)=c(F_X(x),F_Y(y))\,f_X(x)\,f_Y(y),

where f_X = dchen(x,\beta_1,\theta_1) and f_Y = dchen(y,\beta_2,\theta_2), where dchen is dchen. The hazard function is the ratio f/R.

Value

A numeric vector, or for rifgmchen a two column matrix with columns x and y.

Remarks

Unlike the Marshall-Olkin type models, this construction is absolutely continuous: it has no mass on the diagonal and therefore no singular component. The survival function is 1 - pchen(x, beta1, theta1) - pchen(y, beta2, theta2) + R(x, y) and can be negative by cancellation in the far tail.

See Also

cifgm, fitifgmchen, plot_reliability_ifgmchen

Examples

difgmchen(0.5, 0.9, theta1 = 1.2, beta1 = 1.3, theta2 = 1.5,
          beta2 = 1.1, gamma = 0.6, omega = 0.4)
relifgmchen(0.5, 0.9, theta1 = 1.2, beta1 = 1.3, theta2 = 1.5,
            beta2 = 1.1, gamma = 0.6, omega = 0.4)
hifgmchen(0.5, 0.9, theta1 = 1.2, beta1 = 1.3, theta2 = 1.5,
          beta2 = 1.1, gamma = 0.6, omega = 0.4)
set.seed(1)
head(rifgmchen(5, theta1 = 1.2, beta1 = 1.3, theta2 = 1.5,
               beta2 = 1.1, gamma = 0.6, omega = 0.4))

Kolmogorov-Smirnov goodness of fit for the BvCh marginals

Description

Tests the fitted marginals of a Bivariate Chen fit against the data using ks.chen.

Usage

ksfit_bvch(fit, x1, x2 = NULL)

Arguments

Details

The two components are compared with Chen(\alpha_1+\alpha_3,\beta) and Chen(\alpha_2+\alpha_3,\beta), since Z_1=\min(U_1,U_3) and Z_2=\min(U_2,U_3), and their minimum with Chen(\alpha_1+\alpha_2+\alpha_3,\beta), which is the marginal of \min(Z_1,Z_2)=\min(U_1,U_2,U_3). Each of these is a Chen distribution with the same shape \beta, because the minimum of independent Chen variables is Chen in the summed scale parameter.

Value

A data frame with one row per series (Z1, Z2 and min(Z1,Z2)), giving the fitted alpha and beta, the Kolmogorov-Smirnov distance D and its p-value p.value, together with the sample mean and standard deviation.

Examples

set.seed(1)
d <- rbvch(200, alpha1 = 1.5, alpha2 = 2.5, alpha3 = 0.5, beta = 1.8)
fit <- fitbvch(d)
ksfit_bvch(fit, d)

Log-likelihood functions of the bivariate Chen models

Description

Log-likelihoods used by fitbvch, fitbec and fitifgmchen.

Usage

logLik_bvch(par, z1, z2, singular = TRUE)

logLik_bec(par, z1, z2, singular = TRUE)

logLik_ifgmchen(par, z1, z2)

Arguments

Details

For the Marshall-Olkin type models the log-likelihood is the exact observed-data log-likelihood: the contribution of an observation is entirely determined by whether z_1<z_2, z_1>z_2 or z_1=z_2, so the complete-data log-likelihood coincides with the observed-data one. Direct numerical maximisation is therefore used; it reaches the same fixed point as the EM algorithm.

Value

The sum of the log densities, or -Inf when any contribution is not finite.

Examples

set.seed(1)
d <- rbvch(50, alpha1 = 1.5, alpha2 = 2.5, alpha3 = 0.5, beta = 1.8)
logLik_bvch(log(c(1.5, 2.5, 0.5, 1.8)), d[, 1], d[, 2])

Surface and contour plots for the bivariate Chen models

Description

Graphical displays of the density, survival or reliability surfaces of the BvCh, BEC and IFGM-Chen models, using base R graphics only.

Usage

plot_surface_bvch(
  alpha1 = 1.5,
  alpha2 = 2.5,
  alpha3 = 0.5,
  beta.vec = c(0.8, 1.8, 5, 10),
  what = c("density", "survival", "hazard"),
  part = c("ac", "singular", "all"),
  n.grid = 61L,
  range = 3,
  theta = 30,
  phi = 25,
  expand.fac = 0.6,
  col.colours = NULL,
  n.col = 100,
  main = NULL,
  ...
)

plot_surface_bec(
  alpha1 = 1.5,
  alpha2 = 2.5,
  alpha3 = 0.5,
  beta = 1.8,
  lambda = 2,
  what = c("density", "survival", "hazard"),
  n.grid = 61L,
  range = 3,
  theta = 30,
  phi = 25,
  expand.fac = 0.6,
  col.colours = NULL,
  n.col = 100,
  main = NULL,
  ...
)

plot_reliability_ifgmchen(
  theta1 = 1.2,
  beta1 = 1.3,
  theta2 = 1.5,
  beta2 = 1.1,
  gamma = 0.6,
  omega = 0.4,
  n.grid = 61L,
  range = 3,
  theta = 30,
  phi = 25,
  expand.fac = 0.6,
  col.colours = NULL,
  n.col = 100,
  main = NULL,
  ...
)

plot_contour_bvch(
  alpha1 = 1.5,
  alpha2 = 2.5,
  alpha3 = 0.5,
  beta = 1.8,
  n.grid = 81L,
  range = 3,
  n.levels = 12,
  lab = NULL,
  add = FALSE,
  draw.singular = TRUE,
  main = NULL,
  ...
)

Arguments

Details

plot_surface_bvch draws the absolutely continuous part of the Bivariate Chen density, which is the surface shown for varying \beta in the source paper: as \beta increases the surface changes from flat or monotone to peaked near the origin. part selects between the absolutely continuous part, the singular part and the full piecewise density.

Value

Invisibly, the matrix of values plotted. plot_surface_bvch returns one matrix per element of beta.vec.

See Also

dbvch, dbec, difgmchen

Examples

plot_surface_bvch(alpha1 = 1.5, alpha2 = 2.5, alpha3 = 0.5,
                  beta.vec = c(0.8, 1.8, 5, 10))
plot_surface_bec(alpha1 = 1.5, alpha2 = 2.5, alpha3 = 0.5, beta = 1.8,
                lambda = 2)
plot_reliability_ifgmchen(theta1 = 1.2, beta1 = 1.3, theta2 = 1.5,
                          beta2 = 1.1, gamma = 0.6, omega = 0.4)
plot_contour_bvch(alpha1 = 1.5, alpha2 = 2.5, alpha3 = 0.5, beta = 1.8)

Print, summary and extractor methods for fitted bivariate Chen models

Description

Methods for the objects returned by fitbvch, fitbec and fitifgmchen.

Usage

## S3 method for class 'fitbvch'
print(x, digits = 4, ...)

## S3 method for class 'fitbec'
print(x, digits = 4, ...)

## S3 method for class 'fitifgmchen'
print(x, digits = 4, ...)

## S3 method for class 'fitbvch'
summary(object, digits = 4, signif.stars = FALSE, ...)

## S3 method for class 'fitbec'
summary(object, digits = 4, signif.stars = FALSE, ...)

## S3 method for class 'fitifgmchen'
summary(object, digits = 4, signif.stars = FALSE, ...)

## S3 method for class 'fitbvch'
coef(object, ...)

## S3 method for class 'fitbec'
coef(object, ...)

## S3 method for class 'fitifgmchen'
coef(object, ...)

## S3 method for class 'fitbvch'
vcov(object, ...)

## S3 method for class 'fitbec'
vcov(object, ...)

## S3 method for class 'fitifgmchen'
vcov(object, ...)

## S3 method for class 'fitbvch'
nobs(object, ...)

## S3 method for class 'fitbec'
nobs(object, ...)

## S3 method for class 'fitifgmchen'
nobs(object, ...)

## S3 method for class 'fitbvch'
logLik(object, ...)

## S3 method for class 'fitbec'
logLik(object, ...)

## S3 method for class 'fitifgmchen'
logLik(object, ...)

## S3 method for class 'fitbvch'
print(x, digits = 4, ...)
## S3 method for class 'fitbec'
print(x, digits = 4, ...)
## S3 method for class 'fitifgmchen'
print(x, digits = 4, ...)
## S3 method for class 'fitbvch'
summary(object, digits = 4, signif.stars = FALSE, ...)
## S3 method for class 'fitbvch'
coef(object, ...)
## S3 method for class 'fitbec'
coef(object, ...)
## S3 method for class 'fitifgmchen'
coef(object, ...)
## S3 method for class 'fitbvch'
vcov(object, ...)
## S3 method for class 'fitbec'
vcov(object, ...)
## S3 method for class 'fitifgmchen'
vcov(object, ...)
## S3 method for class 'fitbvch'
nobs(object, ...)
## S3 method for class 'fitbec'
nobs(object, ...)
## S3 method for class 'fitifgmchen'
nobs(object, ...)
## S3 method for class 'fitbvch'
logLik(object, ...)
## S3 method for class 'fitbec'
logLik(object, ...)
## S3 method for class 'fitifgmchen'
logLik(object, ...)

Arguments

Details

The fitted objects also carry aic and bic components, and compare_models tabulates them across models. Because aic and bic are not S3 generics that a package can register methods for, no aic or bic methods are provided.

Value

The printing methods return their argument invisibly; coef, vcov, nobs and logLik return the corresponding component of the fit.

Examples

set.seed(1)
d <- rbvch(300, alpha1 = 1.5, alpha2 = 2.5, alpha3 = 0.5, beta = 1.8)
fit <- fitbvch(d)
print(fit)
summary(fit)
coef(fit)
vcov(fit)
nobs(fit)