# GAD: Generalized quantile-based asymmetric family In QBAsyDist: Asymmetric Distributions and Quantile Estimation

## 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

 ```1 2 3 4 5 6 7 8``` ```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 η. `phi` This is the scale parameter φ. `alpha` This is the index parameter α. `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_{α}^g(y;η,φ), default -Inf. `upper` This is the upper limit of the domain (support of the random variable) f_{α}^g(y;η,φ), 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

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54``` ```# 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 ```

QBAsyDist documentation built on Sept. 4, 2019, 1:05 a.m.