asymmetry.measures (version 0.2)

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

Description

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

Usage

eta.w.breve(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 \(\breve{\eta}\).

Details

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

where

$$U_i = \hat{f}(X_i), \; W_i =F_n(X_i), \; \bar{U}= n^{-1}\sum_{i=1}^n U_i, \; \bar{W}=n^{-1} \sum_{i=1}^{n} W_i. $$

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

Examples

# NOT RUN {
eta.w.breve(GDP.Per.head.dist.1995,Epanechnikov)
0.329707 #estimate of etabreve

  
# }