QBAsyDist (version 0.1.2)

mleGAD: Maximum likelihood estimation (MLE) for the generalized quantile-based asymmetric family of distributions (GAD).

Description

The log-likelihood function \(\ell_n(\eta,\phi,\alpha)=\ln[L_n(\eta,\phi,\alpha)]\) and parameter estimation of \( \theta=(\eta,\phi,\alpha)\) in the three parameter generalized quantile-based asymmetric family of densities by using the maximum likelihood estimation are discussed in Gijbels et al. (2019b).

Usage

mleGAD(y, f, g, lower = -Inf, upper = Inf)

Arguments

y

This is a vector of quantiles.

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.

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.

Value

The maximum likelihood estimate of parameter \(\theta=(\eta,\phi,\alpha)\) of the generalized quantile-based asymmetric family of densities

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)
y<-rnorm(100)
g_id<-function(y){y}
g_log<-function(y){log(y)}
mleGAD(y,f=f_N,g=g_id) # For identity-link
mleGAD(rexp(100,0.1),f=f_N,g=g_log,lower = 0, upper = Inf) # 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)
mleGAD(y,f=f_La,g=g_id) # For identity-link
mleGAD(rexp(100,0.1),f=f_La,g=g_log,lower = 0, upper = Inf) # For log-link
# }