QBAsyDist (version 0.1.2)

momentQBAD: Moment estimation for the quantile-based asymmetric family of distributions.

Description

Mean, variance, skewness, kurtosis and moments about the location parameter (i.e., \(\alpha\)th quantile) of the quantile-based asymmetric family of densities defined in Gijbels et al. (2019a) useful for quantile regression with location parameter equal to \(\mu\), scale parameter \(\phi\) and index parameter \(\alpha\).

Usage

mu_k(f, k)
gamma_k(f, k)
meanQBAD(mu, phi, alpha, mu_1)
varQBAD(mu, phi, alpha, mu_1, mu_2)
skewQBAD(alpha, mu_1, mu_2, mu_3)
kurtQBAD(alpha, mu_1, mu_2, mu_3, mu_4)
momentQBAD(phi, alpha, f, r)

Arguments

f

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

k

This is an integer value (\(k=1,2,3,\ldots\)) for calculating \(\mu_k=\int_{0}^{\infty} 2s^k f(s) ds\) and \(\gamma_k=\int_{0}^{\infty}s^{k-1}\frac{[f'(s)]^2}{f(s)}ds\).

mu

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

phi

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

alpha

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

mu_1

This is the quantity \(\int_{0}^{\infty} 2 s f(s) ds\).

mu_2

This is the quantity \(\int_{0}^{\infty} 2 s^2 f(s) ds\).

mu_3

This is the quantity \(\int_{0}^{\infty} 2 s^3 f(s) ds\).

mu_4

This is the quantity \(\int_{0}^{\infty} 2 s^4 f(s) ds\).

r

This is a value which is used to calculate the \(r\)th moment about \(\mu\).

Value

mu_k provides the quantity \(\int_{0}^{\infty} 2s^k f(s) ds\), gamma_k provides the quantity \(\int_{0}^{\infty}s^{k-1}\frac{[f'(s)]^2}{f(s)}ds\), meanQBAD provides the mean, varQBAD provides the variance, skewQBAD provides the skewness, kurtQBAD provides the kurtosis, and momentQBAD provides the \(r\)th moment about the location parameter \(\mu\) of the asymmetric family of distributions.

References

Gijbels, I., Karim, R. and Verhasselt, A. (2019a). On quantile-based asymmetric family of distributions: properties and inference. International Statistical Review, https://doi.org/10.1111/insr.12324.

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)
mu_k(f=f_N,k=1)
gamma_k(f=f_N,k=1)
mu.1_N=sqrt(2/pi)
mu.2_N=1
mu.3_N=2*sqrt(2/pi)
mu.4_N=4
meanQBAD(mu=0,phi=1,alpha=0.5,mu_1=mu.1_N)
varQBAD(mu=0,phi=1,alpha=0.5,mu_1=mu.1_N,mu_2=mu.2_N)
skewQBAD(alpha=0.5,mu_1=mu.1_N,mu_2=mu.2_N,mu_3=mu.3_N)
kurtQBAD(alpha=0.5,mu_1=mu.1_N,mu_2=mu.2_N,mu_3=mu.3_N,mu_4=mu.4_N)
momentQBAD(phi=1,alpha=0.5,f=f_N,r=1)


# 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)
mu_k(f=f_La,k=1)
gamma_k(f=f_La,k=1)
mu.1_La=1
mu.2_La=2
mu.3_La=6
mu.4_La=24
meanQBAD(mu=0,phi=1,alpha=0.5,mu_1=mu.1_La)
varQBAD(mu=0,phi=1,alpha=0.5,mu_1=mu.1_La,mu_2=mu.2_La)
skewQBAD(alpha=0.5,mu_1=mu.1_La,mu_2=mu.2_La,mu_3=mu.3_La)
kurtQBAD(alpha=0.5,mu_1=mu.1_La,mu_2=mu.2_La,mu_3=mu.3_La,mu_4=mu.4_La)
momentQBAD(phi=1,alpha=0.5,f=f_La,r=1)
# }