QBAsyDist (version 0.1.2)

GAD: Generalized quantile-based asymmetric family

Description

Density, cumulative distribution function, quantile function and random sample generation from the generalized quantile-based asymmetric family of densities defined in Gijbels et al. (2019b).

Usage

dGAD(y, eta, phi, alpha, f, g)
pGAD(q, eta, phi, alpha, F, g)
qGAD(beta, eta, phi, alpha, F, g, QF = NULL, lower = -Inf,
  upper = Inf)
rGAD(n, eta, phi, alpha, F, g, lower = -Inf, upper = Inf, QF = NULL)

Arguments

y, q

These are each a vector of quantiles.

eta

This is the location parameter \(\eta\).

phi

This is the scale parameter \(\phi\).

alpha

This is the index parameter \(\alpha\).

f

This is the reference density function \(f\) which is a standard version of a unimodal and symmetric around 0 density.

g

This is the "link" function. The function \(g\) is to be differentiated. Therefore, \(g\) must be written as a function. For example, g<-function(y){log(y)} for log link function.

F

This is the cumulative distribution function \(F\) of the unimodal and symmetric around 0 reference density function \(f\).

beta

This is a vector of probabilities.

QF

This is the quantile function of the reference density \(f\).

lower

This is the lower limit of the domain (support of the random variable) \(f_{\alpha}^g(y;\eta,\phi)\), default -Inf.

upper

This is the upper limit of the domain (support of the random variable) \(f_{\alpha}^g(y;\eta,\phi)\), default Inf.

n

This is the number of observations, which must be a positive integer that has length 1.

References

Gijbels, I., Karim, R. and Verhasselt, A. (2019b). Quantile estimation in a generalized asymmetric distributional setting. To appear in Springer Proceedings in Mathematics & Statistics, Proceedings of `SMSA 2019', the 14th Workshop on Stochastic Models, Statistics and their Application, Dresden, Germany, in March 6--8, 2019. Editors: Ansgar Steland, Ewaryst Rafajlowicz, Ostap Okhrin.

Examples

# NOT RUN {
# Example 1: Let F be a standard normal cumulative distribution function then
f_N<-function(s){dnorm(s, mean = 0,sd = 1)} # density function of N(0,1)
F_N<-function(s){pnorm(s, mean = 0,sd = 1)} # distribution function of N(0,1)
QF_N<-function(beta){qnorm(beta, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)}

# For identiy link function
g_id<-function(y){y}
# For log-link function
g_log<-function(y){log(y)}

rnum<-rnorm(100)
beta=c(0.25,0.50,0.75)

# Density
dGAD(y=rnorm(100),eta=10,phi=1,alpha=0.5,f=f_N,g=g_id) # For identity link
dGAD(y=rexp(100,0.1),eta=10,phi=1,alpha=0.5,f=f_N,g=g_log) # For log-link

# Distribution function
pGAD(q=rnorm(100),eta=0,phi=1,alpha=.5,F=F_N,g=g_id) # For identity link
pGAD(q=rexp(100,0.1),eta=10,phi=1,alpha=.5,F=F_N,g=g_log) # For log-link

# Quantile function
qGAD(beta=beta,eta=0,phi=1,alpha=0.5,F=F_N,g=g_id) # For identity link
qGAD(beta=beta,eta=10,phi=1,alpha=0.5,F=F_N,g=g_log,lower = 0, upper = Inf)  # For log-link

# random sample generation
rGAD(n=100,eta=0,phi=1,alpha=.5,F=F_N,g=g_id ,lower = -Inf, upper = Inf,QF=NULL) # For identity link
rGAD(n=100,eta=10,phi=1,alpha=.5,F=F_N,g=g_log ,lower =0, upper = Inf,QF=NULL)   # For log-link


# Example 2: Let F be a standard Laplace cumulative distribution function then
f_La<-function(s){0.5*exp(-abs(s))} # density function of Laplace(0,1)
F_La<-function(s){0.5+0.5*sign(s)*(1-exp(-abs(s)))} # distribution function of Laplace(0,1)
QF_La<-function(beta){-sign(beta-0.5)*log(1-2*abs(beta-0.5))}

# For identiy link function
g_log<-function(y){log(y)}
beta=c(0.25,0.50,0.75)

# Density
dGAD(y=rnorm(100),eta=10,phi=1,alpha=0.5,f=f_La,g=g_id) # For identity-link
dGAD(y=rexp(100,0.1),eta=10,phi=1,alpha=0.5,f=f_La,g=g_log) # For log-link

# Distribution function
pGAD(q=rnum,eta=0,phi=1,alpha=.5,F=F_La,g=g_id) # For identity-link
pGAD(q=rexp(100,0.1),eta=10,phi=1,alpha=.5,F=F_La,g=g_log) # For log-link

# Quantile function
qGAD(beta=beta,eta=0,phi=1,alpha=0.5,F=F_La,g=g_id,lower = -Inf, upper = Inf) # For identity link
qGAD(beta=beta,eta=10,phi=1,alpha=0.5,F=F_La,g=g_log,lower = 0, upper = Inf) # For log-link

# random sample generation
rGAD(n=100,eta=0,phi=1,alpha=.5,F=F_La,g=g_id) # For identity link
rGAD(n=100,eta=10,phi=1,alpha=.5,F=F_La,g=g_log ,lower =0, upper = Inf,QF=NULL)   # For log-link




# }