weibull4: Four-Parametric Weibull Function

Description

Four-parametric Weibull function and its definite integral.

Usage

fweibull4(x, p)
aweibull4(p, lower, upper)

Value

fweibull4 gives the Weibull function and aweibull4 its definite integral (cumulative sum or area under curve).

Arguments

x: vector of function arguments,
p: vector of function parameters with: p[1] vertical shift (zero offset), p[2] vertical scaling, p[3] shape parameter $a$ of dweibull, and p[4] scale parameter $b$ of dweibull,
lower: lower limit of the cumulative (integrated) function,
upper: upper limit of the cumulative (integrated) function.

Details

The four-parametric Weibull function is essentially based on the Weibull density function dweibull and its integral by the Weibull distribution function pweibull with two additional parameters for y scaling and zero offset. It can be given by: $$f(x) = p_1 + p_2 (p_3/p_4) {(x/p_4)}^{p_3-1} \exp (-{(x/b)}^{p_3})$$ for $x \ge 0$.

Examples

Run this code

x <- seq(0, 5, 0.02)
plot(x, fweibull4(x, c(0, 1, 2, 1)), type = "l", ylim = c(0, 2))
points(x, dweibull(x, 2, 1), pch = "+") ## identical to former

## shape
lines(x, fweibull4(x, c(0, 2, 1.5, 1)), type = "l", col = "orange")
## horizontal scaling
lines(x, fweibull4(x, c(0, 2, 2, 2)), type = "l", col = "green")
## shifting
lines(x, fweibull4(x, c(1, 1, 2, 1)), type = "l", col = "blue")
## vertical scaling
lines(x, fweibull4(x, c(0, 2, 2, 1)), type = "l", col = "red")


## definite integral
p <- c(0, 1, 2, 2)
plot(x, aweibull4(p, lower = 0, upper = x))

p <- c(0.1, 1, 2, 2)
plot(x, aweibull4(p, lower = 0, upper = x))