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; $$h\left( \beta ,\alpha \right) =\frac{\alpha }{\beta ^{\alpha }} \left(e^{\left( x/\beta \right) ^{\alpha }}-1\right) x^{\alpha -1}$$