d_rsize_Weibull {RSizeBiased}

Weibull size biased distribution of order r.

Description

Calculates the density of the r-size biased Weibull distribution.

Usage

d_rsize_Weibull(x,TRpar,r) 

Arguments

x

Grid points where the functional is being calculated.

TRpar

A vector of length 2, containing the shape and scale parameters of the distribution.

r

The size (order) of the distribution. The special cases r=1,2,3 correspond to length, area, volume biased samples respectively and are the most frequently encountered in practice. The case r=0 corresponds to random samples from the Weibull distribution.

Details

The r-size density of the observed biased sample X_1, \dots, X_n is defined by

f_r(x; \theta)=\frac{x^r f(x; \theta)}{E(X^r)}

where f(x; \theta) is the density of the Weibull distribution and \theta the vector of the shape and scale parameters of the distribution.

Value

A vector of length equal to the length of x.

Author(s)

Polychronis Economou

R implementation and documentation: Polychronis Economou <[e-mail]>

References

Economou et. al. (2021). Hypothesis testing for the population mean and variance based on r-size biased samples, under review.

See Also

p_rsize_Weibull, r_rsize_Weibull

Examples

# example of r-size Weibull distribution, r=0,1,2
x<- seq(0, 10, length=50)
dens.0.size<-d_rsize_Weibull(x,c(2,3),0)
dens.1.size<-d_rsize_Weibull(x,c(2,3),1)
dens.2.size<-d_rsize_Weibull(x,c(2,3),2)
plot(x, dens.0.size, type="l", ylab="r-denisty")
lines(x, dens.1.size, col=2)
lines(x, dens.2.size, col=3)
legend("topright",legend=c("r= 0","r= 1","r= 2"),
       col=c("black","red","green"),lty=c(1,1,1))