Description Usage Arguments Details Value References
Density (dgh
), distribution function (pgh
), quantile function (qgh
),
random generation (rgh
), and bounds of the support (infgh
and supgh
)
of the Tukey's g-and-h distribution \insertCitetukey1977tukeyGH. All
functions with the exception of rgh
are vectorized with respect to all
arguments on the Tukey's distribution (x
, q
, p
, a
, b
, g
, h
).
1 2 3 4 5 6 7 8 9 10 11 | dgh(x, a = 0, b = 1, g = 0, h = 0.2, log = FALSE, ...)
pgh(q, a = 0, b = 1, g = 0, h = 0.2, lower.tail = TRUE, log.p = FALSE, ...)
qgh(p, a = 0, b = 1, g = 0, h = 0.2, lower.tail = TRUE, log.p = FALSE)
rgh(n, a = 0, b = 1, g = 0, h = 0.2)
infgh(a = 0, b = 1, g = 0, h = 0.2)
supgh(a = 0, b = 1, g = 0, h = 0.2)
|
x, q |
vector of quantiles. |
a |
location parameter(s). |
b |
scale parameter(s). |
g |
skewness parameter(s). |
h |
heavy-taildness parameter(s). Only non-negative values will be accepted (see Details). |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
... |
arguments passed to |
lower.tail |
logical; if TRUE (default), probabilities are P[X ≤ x] otherwise, P[X > x]. |
p |
vector of probabilities. |
n |
number of observations. If |
Given a Gaussian random variable Z\sim\mathcal{N}(0, 1), the following transformation:
X=a+b\,\frac{e^{gZ}-1}{g}\,e^{\frac{hZ^2}{2}}
defines the Tukey's g-and-h distribution. Hence X\sim gh(a, b, g, h) denotes a random variable distributed according to the Tukey's g-and-h distribution function, where a\in\mathbf{R} is the location parameter, b\in\mathbf{R}^+ is the scale parameter, g\in\mathbf{R} is the skewness parameter, and h\in\mathbf{R}^+ is the shape parameter.
In principle, the shape parameter h may also take negative values, however, in such a case, the above transformation is not monotone. All functions on this page require that h≥q0.
Note that, when g=0, the limit for g\to 0 of the previous transformation is considered:
X=\lim_{g\to0}≤ft(a+b\,\frac{e^{gZ}-1}{g}\,e^{\frac{hZ^2}{2}}\right)= a+b\,Z\,e^{\frac{hZ^2}{2}}
so that X\sim gh(a, b, 0, h).
dgh
gives the density, pgh
gives the distribution function, qgh
gives
the quantile function, and rgh
generates random numbers.
The length of the result is determined by n
for rgh
, 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. Only the first elements of the logical arguments are used.
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