# 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

  1 2 3 4 5 6 7 8 9 10 11 12 13 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

  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 # 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.