asymmetry.measures (version 0.2)

eta.w.hat: Asymmetry coefficient \(\hat{\eta}\)

Description

Implements the asymmetry coefficient \(\hat{\eta}\) of Patil, Patil and Bagkavos (2012).

Usage

eta.w.hat(xin, kfun)

Arguments

xin

A vector of data points - the available sample.

kfun

The kernel to use in the density estimate.

Value

Returns a scalar, the estimate of \(\hat{\eta}\).

Details

Given a sample \(X_1, X_2,\dots, X_n\) from a continuous density function \(f(x)\) and distribution function \(F(x)\), \(\hat{\eta}\) is defined by $$\hat{\eta}=-\frac{\sum_{i=1}^n {U_iV_i}-n\bar{U}\bar{V}}{\sqrt{(\sum_{i=1}^n{U_i^2-n\bar{U^2}})(\sum_{i=1}^n{V_i^2-n\bar{V^2}})}}$$

where

$$U_i = \hat{f}(X_i), \; V_i =\hat{F}(X_i), \; \bar{U}=n^{-1}\sum_{i=1}^n U_i, \; \bar{V}=n^{-1}\sum_{i=1}^n V_i. $$

References

Patil, P.N., Patil, P.P. and Bagkavos, D., (2012), A measure of asymmetry. Stat. Papers, 53, 971-985.

See Also

eta.w.hat.bc, eta.w.breve, eta.w.breve.bc, eta.w.tilde,eta.w.tilde.bc

Examples

# NOT RUN {
eta.w.hat(GDP.Per.head.dist.1995,Epanechnikov)
0.3463025 #estimate of etahat

  
# }