p.sample {asymmetry.measures}

Switch between a range of available cumulative distribution functions.

Description

Returns the value of the selected cumulative distribution function at user supplied grid points.

Usage

p.sample(s,dist, p1,p2)

Arguments

s

A scalar or vector: the x-axis grid points where the cumulative distribution function is 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 distribution.

p2

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

Details

Based on the user-specified argument dist, the function returns the value of the cumulative distribution function at s.

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

• weib: The Weibull distribution is implemented as

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

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

• lognorm: The lognormal distribution is implemented as

  F(s)=\Phi \left ( \frac{\ln s-p_1 }{p_2} \right )

  where p_1 is the mean,p_2 is the standard deviation and \Phi is the cumulative distribution function of the standard normal distribution.

• norm: The normal distribution is implemented as

  \Phi(s)={\frac {1}{\sqrt {2\pi}p_2 }}\int_{-\infty }^s e^{-\frac{(t-p_1)^2}{2p_2^2}}\,dt

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

• uni: The uniform distribution is implemented as

  F(s)=\frac{s-p_1}{p_2-p_1}

  for p_1 \le s \le p_2.

• cauchy: The cauchy distribution is implemented as

  F(s;p_1,p_2)=\frac{1}{\pi}\arctan \left ( \frac{s-p_1}{p_2} \right ) + \frac{1}{2}

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

• fnorm: The half normal distribution is implemented as

  F_S(s;\sigma)=\int_0^s \frac{\sqrt{2/\pi}}{\sigma} \exp \left \{ -\frac{x^2}{2\sigma^2} \right \} \,dx

  where mean=0 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}}\int_{-\infty }^{s}e^{-\frac{(t - p_2[1])^2}{2p_2[2]^2}}\,dt + (1-p_1) \frac{1}{p_2[4]\sqrt{2\pi}} \int_{-\infty }^s e^{-\frac{(t - p2[3])^2}{2p_2[4]^2}}\,dt

  where p_1 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 is implemented as

  F(y; p_1) = \Phi \left ( \frac{y-\xi}{\omega} \right )-2 T \left ( \frac{y-\xi}{\omega},p_1 \right )

  where location=\xi=0, scale=\omega=1, parameter=p_1 and T(h, a) is the Owens T function, defined by

  T(h,a) = \frac{1}{2\pi}\int_{0}^{a} \exp \left \{ \frac{- 0.5 h^2 (1+x^2) }{1+x^2} \right \} \,dx, -\infty \le h, a \le \infty

• fas: The Fernandez and Steel distribution is implemented as

  F(s;p_1,p_2) = \frac{2}{p_1+\frac{1}{p_1}} \left \{ \int_{-\infty}^s f_t(x/p_1; p_2)I_{\{x \ge 0\}} \,dx + \int_{-\infty}^s f_t(p_1 x; p_2)I_{\{x<0\}}\, dx \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 is the degrees of freedom.

• shash: The Sinh-Arcsinh distribution is implemented as

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

  where r=\sinh(\sinh(z)- p_1), c=\cosh(\sinh(z)- p_1) and z=(s-\mu)/p_2. 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 cumulative distribution function 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

r.sample, q.sample, d.sample

Examples

selected.d <- "weib" #select Weibull as the CDF
shape <- 2  # specify shape parameter
scale <- 1 # specify scale parameter
xout <- seq(0.1,5,length=50)  #design point where the CDF is evaluated
p.sample(xout,selected.d,shape,scale)  # calculate CDF at xout