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

In QBAsyDist: Asymmetric Distributions and Quantile Estimation

Description

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

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,…) for calculating μ_k=\int_{0}^{∞} 2s^k f(s) ds and γ_k=\int_{0}^{∞}s^{k-1}\frac{[f'(s)]^2}{f(s)}ds.
mu	This is the location parameter μ.
phi	This is the scale parameter φ.
alpha	This is the index parameter α.
mu_1	This is the quantity \int_{0}^{∞} 2 s f(s) ds.
mu_2	This is the quantity \int_{0}^{∞} 2 s^2 f(s) ds.
mu_3	This is the quantity \int_{0}^{∞} 2 s^3 f(s) ds.
mu_4	This is the quantity \int_{0}^{∞} 2 s^4 f(s) ds.
r	This is a value which is used to calculate the rth moment about μ.

Value

mu_k provides the quantity \int_{0}^{∞} 2s^k f(s) ds, gamma_k provides the quantity \int_{0}^{∞}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 rth moment about the location parameter μ 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

# 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)

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