Density, distribution function, quantile function and random generation for
the Power Muth distribution with parameters shape and scale.
Usage
dtpmd(x, beta = 1, alpha, log = FALSE)
ptpmd(q, beta = 1, alpha, lower.tail = TRUE, log.p = FALSE)
qtpmd(p, beta = 1, alpha, lower.tail = TRUE)
rtpmd(n, beta = 1, alpha)Arguments
- x, q
vector of quantiles.
- beta
a scale parameter.
- alpha
a shape parameter.
- log, log.p
logical; if TRUE, probabilities p are given as log(p).
- lower.tail
logical; if TRUE (default), probabilities are \(P\left[ X\leq x\right]\), otherwise, \(P\left[ X>x\right] \).
- p
vector of probabilities.
- n
number of observations. If
length(n) > 1, the length is taken to be the number required.
Value
dtpmd gives the density, ptpmd gives the distribution
function, qtpmd gives the quantile function and rtpmd generates
random deviates.
Details
The Power Muth distribution with shape parameter \(\alpha\) and
scale parameter \(\beta\) has density
$$f\left( x\right) =\frac{\alpha }{\beta ^\alpha }x^{\alpha -1}
\left( e^{\left(x/\beta \right) ^{\alpha }}-1\right)
\left( e^{\left( x/\beta \right) ^{\alpha }-
\left( e^{\left( x/\beta \right) ^{\alpha }}-1\right) }\right), $$
where
$$x>0,~\alpha ,\beta>0.$$
Note
Hazard function; $$h\left( \beta ,\alpha \right) =\frac{\alpha }{\beta ^{\alpha }} \left(e^{\left( x/\beta \right) ^{\alpha }}-1\right) x^{\alpha -1}$$