pdfsqcdfstar: Calculate $f^2(x)(1-F(x))$.

Description

Return the product $f^2(x)(1-F(x))$.

Usage

pdfsqcdfstar(s,dist, p1,p2)

Arguments

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).

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

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

Value

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

Details

Based on user-specified argument dist, the function returns the value of $f^2(x)(1-F(x))dx$, 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.

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.

Examples

Run this code

# NOT RUN {
  selected.dens <- "weib" #select Weibull
  shape <- 2  # specify shape parameter
  scale <- 1 # specify scale parameter
  xout <- seq(0.1,5,length=50)  #design point
  pdfsqcdfstar(xout,selected.dens,shape,scale)  #return f^2(xout)F(xout)
# }

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pdfsqcdfstar: Calculate \(f^2(x)(1-F(x))\).