# GTEF: Generalized tick-exponential family In QBAsyDist: Asymmetric Distributions and Quantile Estimation

## Description

Density, cumulative distribution function, quantile function and random sample generation from the generalized tick-exponential family (GTEF) of densities discusse in Gijbels et al. (2019b).

## Usage

 ```1 2 3 4 5 6 7``` ```dGTEF(y, eta, phi, alpha, p, g) pGTEF(q, eta, phi, alpha, p, g) qGTEF(beta, eta, phi, alpha, p, g, lower = -Inf, upper = Inf) rGTEF(n, eta, phi, alpha, p, g, lower = -Inf, upper = Inf) ```

## 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. `p` This is the shape parameter, which must be positive. `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. `beta` This is a vector of probabilities. `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.

## Value

`dGTEF` provides the density, `pGTEF` provides the cumulative distribution function, `qGTEF` provides the quantile function, and `rGTEF` generates a random sample from the generalized tick-exponential family of densities. The length of the result is determined by n for `rGTEF`, and is the maximum of the lengths of the numerical arguments for the other functions.

## 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``` ```# For identiy link function y=rnorm(100) g_id<-function(y){y} dGTEF(y,eta=0,phi=1,alpha=0.5,p=2,g=g_id) # cumulative distribution function pGTEF(q=y,eta=10,phi=1,alpha=0.5,p=2,g=g_id) # Quantile function beta=c(0.25,0.5,0.75) qGTEF(beta=beta,eta=10,phi=1,alpha=0.5,p=2,g=g_id) # random sample generation rGTEF(n=100,eta=10,phi=1,alpha=.5,p=2,g=g_id,lower = -Inf, upper = Inf) # For log link function y=rexp(100) g_log<-function(y){log(y)} dGTEF(y,eta=10,phi=1,alpha=0.5,p=2,g=g_log) # cumulative distribution function pGTEF(q=y,eta=10,phi=1,alpha=0.5,p=2,g=g_log) # Quantile function g_log<-function(y){log(y)} #' beta=c(0.25,0.5,0.75) qGTEF(beta=beta,eta=10,phi=1,alpha=0.5,p=2,g=g_log,lower = 0, upper = Inf) # random sample generation rGTEF(n=100,eta=10,phi=1,alpha=.5,p=2,g=g_log,lower = 0, upper = Inf) ```

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