Package 'BvCh'

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

x1, x2	numeric vectors of first and second component.
alpha1, alpha2, alpha3, beta, lambda	positive parameters.
log	logical; if TRUE the log density is returned.
n	number of observations.

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

z1, z2	numeric vectors of first and second component.
alpha1, alpha2, alpha3, beta	positive parameters.
log	logical; if TRUE the log density is returned.
n	number of observations.

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

z1, z2	numeric vectors of first and second component.
alpha1, alpha2, alpha3, beta	positive parameters.

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

x, q	vector of quantiles.
beta	shape parameter.
lambda	scale parameter.
log, log.p	logical; if TRUE probabilities are returned as log(p).
lower.tail	logical; if TRUE (default) P[X <= x], otherwise P[X > x].
p	vector of probabilities.
n	number of observations.
t	age component for the conditional reliability function.

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

...	fitted model objects from fitbvch, fitbec or fitifgmchen.

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

x, q	vector of quantiles.
alpha, lambda, beta	positive parameters.
log, log.p	logical; if TRUE probabilities are returned as log(p).
lower.tail	logical; if TRUE (default) P[X <= x], otherwise P[X > x].
p	vector of probabilities.
n	number of observations.

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

x1, x2	numeric vectors of observations, or a two column matrix.
start	optional starting values on the natural scale in the order alpha1, alpha2, alpha3, beta, lambda.
singular	logical; whether tied observations contribute the singular density.
method	optimisation method passed to optim.
control	optional control list for optim.
se	logical; whether standard errors should be computed.

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

x1, x2	numeric vectors of observations, or a two column matrix.
start	optional numeric vector of starting values on the natural scale, in the order alpha1, alpha2, alpha3, beta.
singular	logical; whether tied observations contribute the singular density.
method	optimisation method passed to optim.
control	optional control list for optim.
se	logical; whether standard errors should be computed.

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

x1, x2	numeric vectors of observations, or a two column matrix.
start	optional starting values on the natural scale in the order theta1, beta1, theta2, beta2, gamma, omega.
method	optimisation method passed to optim.
control	optional control list for optim.
se	logical; whether standard errors should be computed.

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

x1, x2	numeric vectors of first and second component.
alpha1, alpha2, alpha3, beta, lambda	positive parameters.

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

u, v	numeric vectors of copula arguments in $[0,1]$.
gamma, omega	copula parameters, each in $[0,1]$.
n	number of observations.

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

x, y	numeric vectors of first and second component.
theta1, beta1, theta2, beta2	positive marginal parameters.
gamma, omega	copula parameters in $[0,1]$.
log	logical; if TRUE the log density is returned.
n	number of observations.

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

fit	a fitted model from fitbvch.
x1, x2	numeric vectors of observations, or a two column matrix.

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)

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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

par	numeric vector of parameters on the log scale.
z1, z2	numeric vectors of observations.
singular	logical; if TRUE ties (z1 == z2) contribute the singular density. Use FALSE to drop tied observations.

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])

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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

x, object	a fitted model object.
digits	number of significant digits to print.
...	further arguments, currently unused.
signif.stars	logical; whether to print significance stars.

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)

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