# Calculates the exact value $\rho_p$, used in the implementation of the strong asymmetry measure $\eta(X)$. Returns $\rho_p$, used in the calculation of the strong asymetry measure $\eta(X)$. ```r Rho.p.exact(xin, p.param, dist, p1=0, p2=1) ``` ## 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: $$ \frac{ 2\sqrt{3}}{p} \frac{-\int_{-\infty}^{\xi_p} f^2(x)F(x)\,dx - \frac{p}{2}\int_{-\infty}^{\xi_p} f^2(x)\,dx}{ \left \{ p\int_{-\infty}^{\xi_p} f^3(x)\,dx-(\int_{-\infty}^{\xi_p} f^2(x)\,dx)^2 \right \}^{1/2} } $$ defined on page 6 [Patil, Bagkavos and Wood](https://link.springer.com/article/10.1007/s13171-013-0034-z), see also (4) in [Bagkavos, Patil and Wood](https://link.springer.com/chapter/10.1007/978-3-319-41582-6_1) . This implementation uses exact calculation of the functionals in the definition of $\rho_p$. ## Returns Returns a scalar, the exact value of $\rho_p$. ## 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.](https://link.springer.com/article/10.1007/s13171-013-0034-z) * c("[list(\"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.\")](https://link.springer.com/chapter/10.1007/978-3-319-41582-6_1)", "[\n](https://link.springer.com/chapter/10.1007/978-3-319-41582-6_1)", "[ ](https://link.springer.com/chapter/10.1007/978-3-319-41582-6_1)") ## Author(s) Dimitrios Bagkavos and Lucia Gamez Gallardo R implementation and documentation: Dimitrios Bagkavos , Lucia Gamez Gallardo ## See Also `Rho.p,Rhostar.p, Rhostar.p.exact` ## Examples ```r 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 Rho.p.exact(xout,param,selected.r,shape,scale) # calculate Rho.p.exact #-0.06665222 # 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 Rho.p.exact(xout,param,selected.r2,mean,sd) # calculate Rho.p.exact #-0.2384271 # 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 Rho.p.exact(xout,param,selected.r3,location,scale) # calculate Rho.p.exact #-0.02340374 # returns the result ```

Rho.p.exact function