Rhostar.p.exact {asymmetry.measures}

Calculates the exact value of \rho_p^*, used in the implementation of the strong asymmetry measure \eta(X).

Description

Returns \rho_p^*, used in the calculation of the strong asymetry measure \eta(X).

Usage

  Rhostar.p.exact(xin, p.param, dist, p1, p2)

Arguments

xin

A vector of data points - the available sample.

p.param

A parameter with the value greater than or equal to 1/2 and less than 1.

dist

Character string, specifies selected distribution function.

p1

A scalar. Parameter 1 (vector or object) of the selected distribution.

p2

A scalar. Parameter 2 (vector or object) of the selected distribution.

Details

Implements the quantity

\frac{ 2\sqrt{3}}{p} \frac{-\int_{\xi_{1-p}}^{\infty}{f^2(x)(1-F(x))\,dx}+\frac{p}{2}\int_{\xi_{1-p}}^{\infty} f^2(x)\,dx}{ \left \{ p\int_{\xi_{1-p}}^{\infty}f^3(x)\,dx-(\int_{\xi_{1-p}}^{\infty}f^2(x)\,dx)^2 \right \}^{1/2} }

defined on page 6 Patil, Bagkavos and Wood (2014), see also (5) in Bagkavos, Patil and Wood (2016). This implementation uses exact calculation of the functionals in the definition of \rho_p^*.

Value

Returns a scalar, the exact value of \rho_p^*.

Author(s)

Dimitrios Bagkavos and Lucia Gamez Gallardo

R implementation and documentation: Dimitrios Bagkavos <[e-mail]> , Lucia Gamez Gallardo <[e-mail]>

References

• Patil P.N., Bagkavos D. and Wood A.T.A., (2014). A measure of asymmetry based on a new necessary and sufficient condition for symmetry, Sankhya A, 76, 123–145.

• Bagkavos D., Patil P.N., Wood A.T.A. (2016), A Numerical Study of the Power Function of a New Symmetry Test. In: Cao R., Gonzalez Manteiga W., Romo J. (eds), Nonparametric Statistics. Springer Proceedings in Mathematics and Statistics, vol 175, Springer.

See Also

Rho.p, Rhostar.p, Rho.p.exact

Examples

set.seed(1234)

selected.r <- "weib" #select Weibull as the distribution
shape <- 1 # specify shape parameter
scale <- 1  # specify scale parameter
n <- 100    # specify sample size
param <- 0.9 # specify parameter
xout<-r.sample(n,selected.r,shape,scale) # specify sample
Rhostar.p.exact(xout,param,selected.r,shape,scale)  # calculate Rhostar.p.exact
#-0.05206678  # returns the result

selected.r2 <- "norm" #select Normal as the distribution
n <- 100    # specify sample size
mean <- 0 # specify the mean
sd <- 1 # specify the variance
param <- 0.9 # specify parameter
xout <-r.sample(n,selected.r2,mean,sd) # specify sample
Rhostar.p.exact(xout,param,selected.r2,mean,sd) # calculate Rhostar.p.exact
#-0.008687447 # returns the result


selected.r3 <- "cauchy" #select Cauchy as the distribution
n <- 100    # specify sample size
location <- 0 # specify the location parameter
scale <- 1 # specify the scale parameter
param <- 0.9 # specify parameter
xout<-r.sample(n,selected.r3,location,scale) # specify sample
Rhostar.p.exact(xout,param,selected.r3,location,scale) # calculate Rhostar.p.exact
#0.0280602  # returns the result