Density, distribution function, quantile function and
random generation for the four parameter Beta distribution
with minimum value min and scale scale.
dbeta4(x, min, max, shape1, shape2, gap = 0)pbeta4(q, min, max, shape1, shape2, gap = 0)
qbeta4(p, min, max, shape1, shape2)
rbeta4(n, min, max, shape1, shape2, seed = 42)
Vector of quantiles
The minumum value on which the distribution is defined
The maximum value on which the distribution is defined
Shape parameter
Shape parameter
Spacing from min and max
Vector of quantiles
Vector of probabilities
Number of observations
A numeric value for the seed of the random number generator
dbeta4 gives the density,
pbeta4 gives the distribution function,
qbeta4 gives the quantile function, and
rbeta4 generates random observations.
The length of the result is determined by n
for rbeta4, and is the maximum of the lengths
of the numerical arguments for the other functions.
The numerical arguments other than n are
recycled to the length of the result.
If shape is not specified, a default
value of 1 is used.
The Birmbaum-Saunders distribution with shape \(\beta\) and scale \(\theta\) has density
$$f(x;\theta,\beta) = \frac{\sqrt{\frac{x}{\theta}}+\sqrt{\frac{\theta}{x}}}{2\beta x}\phi_{_{NOR}(z)},\quad x \ge 0 $$
where \(\phi_{_{NOR}}(z)\) is the density of the standard normal distribution and
$$z = \frac{1}{\beta}\left(\sqrt{\frac{x}{\theta}}-\sqrt{\frac{\theta}{x} } \right)$$.