Rhostar.p {asymmetry.measures}

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

Description

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

Usage

  Rhostar.p(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). Estimation of the p.d.f. and c.d.f. functions is currently performed by maximum likelihood as e.g. kernel estimates inherit large amount of variance to \rho_p^*.

Value

Returns a scalar, the 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.exact, 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(xout,param,selected.r,shape,scale)  # calculate Rhostar.p
#-0.08936363  # 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(xout,param,selected.r2,mean,sd) # calculate Rhostar.p
#-0.02302223 # 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(xout,param,selected.r3,location,scale) # calculate Rhostar.p
#0.02043852  # returns the result