pdfthird {asymmetry.measures}

Calculate f^3(x)

Description

Return the value of f^3(x).

Usage

pdfthird(s,dist, p1,p2)

Arguments

s

A scalar or vector: the x-axis grid points where the probability density function will be evaluated.

dist

Character string, used as a switch to the user selected distribution function (see details below).

p1

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

p2

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

Details

Based on user-specified argument dist, the function returns the value of f^3(x)dx, used in the definitions of \rho_p^*, \rho_p and their exact versions.

Supported distributions (along with the corresponding dist values) are:

• weib: The weibull distribution is implemented as

  f(s;p_1,p_2)= \frac{p_1}{p_2} \left (\frac{s}{p_2}\right )^{p_1-1} \exp \left \{- \left (\frac{s}{p_2}\right )^{p_1} \right \}

  with s \ge 0 where p_1 is the shape parameter and p_2 the scale parameter.

• lognorm: The lognormal distribution is implemented as

  f(s) = \frac{1}{p_2s\sqrt{2\pi}}e^{-\frac{(log s -p_1)^2}{2p_2^2}}

  where p_1 is the mean and p_2 is the standard deviation of the distirbution.

• norm: The normal distribution is implemented as

  f(s) = \frac{1}{p_2\sqrt{2 \pi}}e^{-\frac{ (s - p_1)^2 }{ 2p_2^2 }}

  where p_1 is the mean and the p_2 is the standard deviation of the distirbution.

• uni: The uniform distribution is implemented as

  f(s) = \frac{1}{p_2-p_1}

  for p_1 \le s \le p_2.

• cauchy: The cauchy distribution is implemented as

  f(s)=\frac{1}{\pi p_2 \left \{1+( \frac{s-p_1}{p_2})^2\right \} }

  where p_1 is the location parameter and p_2 the scale parameter.

• fnorm: The half normal distribution is implemented as

  2 f(s)-1

  where

  f(s) = \frac{1}{sd\sqrt{2 \pi} }e^{-\frac{s^2}{2 sd^2 }},

  and sd=\sqrt{\pi/2}/p_1.

• normmixt:The normal mixture distribution is implemented as

  f(s)=p_1\frac{1}{p_2[2] \sqrt{2\pi} } e^{- \frac{ (s - p_2[1])^2}{2p_2[2]^2}} +(1-p_1)\frac{1}{p_2[4]\sqrt{2\pi}} e^{-\frac{(s - p_2[3])^2}{2p_2[4]^2 }}

  where p1 is a mixture component(scalar) and p_2 a vector of parameters for the mean and variance of the two mixture components p_2= c(mean1, sd1, mean2, sd2).

• skewnorm: The skew normal distribution with parameter p_1 is implemented as

  f(s)=2\phi(s)\Phi(p_1s)

  .

• fas: The Fernandez and Steel distribution is implemented as

  f(s; p_1, p_2) = \frac{2}{p_1+\frac{1}{p_1}} \left \{ f_t(s/p_1; p_2) I_{\{s \ge 0\}} + f_t(p_1s; p_2)I_{\{s<0 \}}\right \}

  where f_t(x;\nu) is the p.d.f. of the t distribution with \nu = 5 degrees of freedom. p_1 controls the skewness of the distribution with values between (0, +\infty) and p_2 denotes the degrees of freedom.

• shash: The Sinh-Arcsinh distribution is implemented as

  f(s;\mu, p_1, p_2, \tau) = \frac{ce^{-r^2/2}}{\sqrt{2\pi }} \frac{1}{p_2} \frac{1}{2} \sqrt{1+z^2}

  where r=\sinh(\sinh(z)-(-p_1)), c=\cosh(\sinh(z)-(-p_1)) and z=((s-\mu)/p2). p_1 is the vector of skewness, p_2 is the scale parameter, \mu=0 is the location parameter and \tau=1 the kurtosis parameter.

Value

A vector containing the user selected density values at the user specified points s.

Author(s)

Dimitrios Bagkavos and Lucia Gamez Gallardo

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

References

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

pdfsq, pdfsqcdf, pdfsqcdfstar

Examples

  selected.dens <- "weib" #select Weibull
  shape <- 2  # specify shape parameter
  scale <- 1 # specify scale parameter
  xout <- seq(0.1,5,length=50)  #design point
  pdfthird(xout,selected.dens,shape,scale)  # calculate density to the cube at xout