Package 'gentransmuted'

Choose a compound distribution.

Description

choose.compound select a combination of baseline and compounding distributions in the class of compound distribution. See details for supported distributions.

Usage

choose.compound(x, type = "positive", criteria = "AIC")

Arguments

x	the vector of values to be fitted.
type	Support of the x's. Avaliable options: positive (default), unit, real.
criteria	model selection criteria to be applied for the selection. Avaliable options: AIC (default, Akaike's information criteria) and BIC (Bayesian's information criteria).

Details

The compound distribution has cumulative distribution function

$F(x;\gamma,\beta,\theta_1,\theta_2)=G_2(G_1(F(x;\gamma,\beta),\theta_1),\theta_2),$

where $F$ is related to the baseline distribution and $G_1, G_2$ are related to compounding models. For positive values, the options assessed for $F$ are exponential, gamma, log-normal, paretoII and Birnbaum-Saunders. For unit values, the options for $F$ are beta and Kumaraswamy. For real values, the options for $F$ are normal, logistic, Cauchy and Gumbel. For $G_1$ and $G_2$ are assessed all the combinations among the exponentiated, exponentiated of second kind, Marshall-Olkin, Marshall-Olkin of the second kind and

Value

A list containing the following components:

coefficients	A matrix with the estimates and standard errors.
logLik	The log-likelihood function evaluated in the estimated parameters
AIC	Akaike's Information Criterion
BIC	Bayesian's Information Criterion

Author(s)

Yolanda M. Gomez, Diego I. Gallardo, Hector W. Gomez and Barry Arnold

Examples

set.seed(2100)
y=rcompound(100, 1.2, 1.4, 1, 0.8, dist="exp", comp1="EXP", comp2="MO")

choose.compound(y, type="positive")

---

The Exponentiated and Exponentiated2 distributions

Description

Density, distribution function, quantile function and random generation for the compound distributions.

Usage

dcompound(x, dist = "exp", comp1 = as.null(), comp2 = as.null(), gamma = 1, beta = 1, 
	theta1 = 1, theta2 = 1, log = FALSE)
pcompound(q, dist = "exp", comp1 = as.null(), comp2 = as.null(), gamma = 1, beta = 1, 
	theta1 = 1, theta2 = 1, lower.tail = TRUE, log.p = FALSE)
qcompound(p, dist = "exp", comp1 = as.null(), comp2 = as.null(), gamma = 1, beta = 1, 
	theta1 = 1, theta2 = 1, lower.tail = TRUE, log.p = FALSE)
rcompound(n, dist = "exp", comp1 = as.null(), comp2 = as.null(), gamma = 1, beta = 1, 
	theta1 = 1, theta2 = 1)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required.
dist	baseline distribution. Avaliable options: exp (exponential), gamma, lnorm (log-normal), paretoII, bisa (Birnbaum-Saunders), lomax, beta, kumar (Kumaraswamy), norm (normal), logis (logistic), cauchy, gumbel. See details for parameterizations of these distributions.
comp1, comp2	compounding distributions. Avaliable options: EXP (Exponentiated), EXP2 (Exponentiated of the second kind), MO (Marshall-Olkin), MO2 (Marshall-Olkin of the second kind), SB (Shaw and Buckley).
gamma, beta	parameters for the baseline distribution.
theta1, theta2	shape parameter for the comp1 and comp2 distributions, respectively.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X\leq x]$, otherwise $P[X>x]$.

Details

The compound distribution has cumulative distribution function

$F(x;\gamma,\beta,\theta_1,\theta_2)=G_2(G_1(F(x;\gamma,\beta),\theta_1),\theta_2),$

where $F$ is related to dist, $G_1$ is related to comp1 and $G_2$ is related to comp2. The support for $x$ depends on the baseline distribution. For exp, gamma, lnorm, paretoII, bisa and lomax, the support is $(0,\infty)$; for beta and kumar is $(0,1)$; for norm, logis, cauchy and gumbel is $(-\infty,\infty)$. The parameter space for $\gamma$ and $\beta$ also depend on the baseline distribution. For exp, $\gamma>0$; for gamma, paretoII, bisa, lomax, beta and kumar $\gamma,\beta >0$; for lnorm, norm, logis, cauchy and gumbel $\gamma \in \mathbb{R}, \beta>0$. The parameter space for $\theta_1$ and $\theta_2$ depend on comp1 and comp2. For EXP, EXP2, MO and MO2 options the corresponding parameter space is $(0,\infty)$, whereas for SB option is $(-1,1)$. The probability density function for each of the baseline distribution is given below. exp

$f(x)=\gamma e^{-\gamma x}$

gamma

$f(x)=\frac{\beta^\gamma}{\Gamma(\gamma)}x^{\gamma-1} e^{-\beta x}$

lnorm

$f(x)=\frac{1}{x\sqrt{2\pi \beta^2}}\exp\left(-\frac{(\log(x)-\gamma)^2}{\beta^2}\right)$

paretoII

$f(x)=\frac{\gamma}{\beta}\left(1+\frac{x}{\beta}\right)^{-(\gamma+1)}$

bisa

$f(x)=\frac{1}{2\sqrt{2\pi}\gamma\beta}\left[\left(\frac{\beta}{x}\right)^{1/2}+\left(\frac{\beta}{x}\right)^{3/2}\right]\exp\left[-\frac{1}{2\gamma^2}\left(\frac{x}{\beta}+\frac{\beta}{x}-2\right)\right]$

beta

$f(x)=\frac{1}{B(\gamma,\beta)}x^{\gamma-1}(1-x)^{\beta-1}$

kumar

$f(x)=\gamma \beta x^{\gamma-1}(1-x^\gamma)^{\beta-1}$

For norm, logis, cauchy and gumbel, the probability density function is given by

$f(x)=\frac{1}{\beta} g\left(\frac{x-\gamma}{\beta}\right)$

where $g$ is given by norm

$g(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$

logis

$g(x)=\frac{e^{x}}{(1+e^x)^2}$

cauchy

$g(x)=\frac{1}{\pi (1+x^2)}$

gumbel

$g(x)=\exp\left(-(x+e^{-x})\right)$

Value

dcompound gives the density, pcompound gives the distribution function, qcompound gives the quantile function, and rcompound generates random deviates. The length of the result is determined by n for rcompound, and is the maximum of the lengths of the numerical arguments for the other functions. The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

Author(s)

Yolanda M. Gomez, Diego I. Gallardo, Hector W. Gomez and Barry Arnold

Examples

set.seed(2100)
y=rcompound(100, 1.2, 1.4, 1, 0.8, dist="exp", comp1="EXP", comp2="MO")

---

Fitting a compound distribution.

Description

estimate.compound computes the maximum likelihood estimates for a compound distribution. See arguments for supported distributions.

Usage

estimate.compound(x, dist = "exp", comp1 = as.null(), comp2 = as.null(), est.var = TRUE)

Arguments

x	the vector of values to be fitted.
dist	baseline distribution. Avaliable options: exp (exponential), gamma, lnorm (log-normal), paretoII, bisa (Birnbaum-Saunders), lomax, beta, kumar (Kumaraswamy), norm (normal), logis (logistic), cauchy, gumbel. See details for parameterizations of these distributions.
comp1, comp2	compounding distributions. Avaliable options: EXP (Exponentiated), EXP2 (Exponentiated of the second kind), MO (Marshall-Olkin), MO2 (Marshall-Olkin of the second kind), SB (Shaw and Buckley).
est.var	Logical. If TRUE the standard errors are estimated.

Details

The parameterization for the different distributions is given in .

Value

A list containing the following components:

coefficients	A matrix with the estimates and standard errors.
logLik	The log-likelihood function evaluated in the estimated parameters
AIC	Akaike's Information Criterion
BIC	Bayesian's Information Criterion

Author(s)

Yolanda M. Gomez, Diego I. Gallardo, Hector W. Gomez and Barry Arnold

Examples

set.seed(2100)
y=rcompound(100, 1.2, 1.4, 1, 0.8, dist="exp", comp1="EXP", comp2="MO")
estimate.compound(y, dist="exp", comp1="EXP", comp2="MO")

---

The Exponentiated and Exponentiated2 distributions

Description

Density, distribution function, quantile function and random generation for the Exponentiated (EXP) and Exponentiated of the second kind (EXP2) distributions.

Usage

dEXP(x, alpha = 1, log = FALSE)
pEXP(q, alpha = 1, lower.tail = TRUE, log.p = FALSE)
qEXP(p, alpha = 1, lower.tail = TRUE, log.p = FALSE)
rEXP(n, alpha = 1)
dEXP2(x, alpha = 1, log = FALSE)
pEXP2(q, alpha = 1, lower.tail = TRUE, log.p = FALSE)
qEXP2(p, alpha = 1, lower.tail = TRUE, log.p = FALSE)
rEXP2(n, alpha = 1)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required.
alpha	shape parameter (by default is 1).
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X\leq x]$, otherwise $P[X>x]$.

Details

The EXP model has cumulative distribution function

$F(x;\alpha)=x^\alpha, \quad x \in (0,1), \alpha>0,$

whereas the EXP2 model has cumulative distribution function

$F(x;\alpha)=1-(1-x)^\alpha, \quad x \in (0,1), \alpha>0,$

Value

dEXP and dEXP2 give the density, pEXP and pEXP2 give the distribution function, qEXP and qEXP2 give the quantile function, and rEXP and rEXP2 generate random deviates. The length of the result is determined by n for rcompound, and is the maximum of the lengths of the numerical arguments for the other functions. The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

Author(s)

Yolanda M. Gomez, Diego I. Gallardo, Hector W. Gomez and Barry Arnold

Examples

set.seed(2100)
y=rEXP(100, alpha = 1.2)

---

The Marshall-Olkin and Marshall-Olkin of the second kind distributions

Description

Density, distribution function, quantile function and random generation for the Marshall-Olkin (MO) and Marshall-Olkin of the second kind (MO2) distributions.

Usage

dMO(x, theta = 1, log = FALSE)
pMO(q, theta = 1, lower.tail = TRUE, log.p = FALSE)
qMO(p, theta = 1, lower.tail = TRUE, log.p = FALSE)
rMO(n, theta = 1)
dMO2(x, theta = 1, log = FALSE)
pMO2(q, theta = 1, lower.tail = TRUE, log.p = FALSE)
qMO2(p, theta = 1, lower.tail = TRUE, log.p = FALSE)
rMO2(n, theta = 1)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required.
theta	shape parameter (by default is 1).
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X\leq x]$, otherwise $P[X>x]$.

Details

The MO model has cumulative distribution function

$F(x;\alpha)=\frac{\theta x}{1-(1-\theta)x}, \quad x \in (0,1), \theta>0,$

whereas the MO2 model has cumulative distribution function

$F(x;\alpha)=\frac{x}{1-(1-\theta)(1-x)}, \quad x \in (0,1), \alpha>0,$

Value

dEXP and dEXP2 give the density, pEXP and pEXP2 give the distribution function, qEXP and qEXP2 give the quantile function, and rEXP and rEXP2 generate random deviates. The length of the result is determined by n for rcompound, and is the maximum of the lengths of the numerical arguments for the other functions. The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

Author(s)

Yolanda M. Gomez, Diego I. Gallardo, Hector W. Gomez and Barry Arnold

Examples

set.seed(2100)
y=rMO(100, theta = 1.2)

---

The Shaw and Buckley (transmuted) distribution

Description

Density, distribution function, quantile function and random generation for the Shaw and Buckley (SB) distribution.

Usage

dSB(x, lambda = 0, log = FALSE)
pSB(q, lambda = 0, lower.tail = TRUE, log.p = FALSE)
qSB(p, lambda = 0, lower.tail = TRUE, log.p = FALSE)
rSB(n, lambda = 0)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required.
lambda	shape parameter (by default is 0).
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X\leq x]$, otherwise $P[X>x]$.

Details

The SB model has cumulative distribution function

$F(x;\delta)=\delta x^2+(1-\delta)[1-(1-x)^2], \quad x \in (0,1),$

where $\delta=(1-\lambda)/2$ and $\lambda \in (-1,1)$.

Value

dSB gives the density, pSB gives the distribution function, qSB gives the quantile function, and rSB generates random deviates. The length of the result is determined by n for rcompound, and is the maximum of the lengths of the numerical arguments for the other functions. The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

Author(s)

Yolanda M. Gomez, Diego I. Gallardo, Hector W. Gomez and Barry Arnold

Examples

set.seed(2100)
y=rSB(100, lambda = 0.7)

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