Given the density function of a unimodal distribution specified by the root of the density function name, returns the distribution function and quantile function of the specified distribution.
pDist(densFn = "norm", q, param = NULL, subdivisions = 100,
lower.tail = TRUE, intTol = .Machine$double.eps^0.25,
valueOnly = TRUE, ...)
qDist(densFn = "norm", p, param = NULL,
lower.tail = TRUE, method = "spline", nInterpol = 501,
uniTol = .Machine$double.eps^0.25,
subdivisions = 100, intTol = uniTol, ...)pDist gives the distribution function, qDist gives the
quantile function.
An estimate of the accuracy of the approximation to the distribution
function can be found by setting valueOnly = FALSE in the call to
pDist which returns a list with components value and
error.
Character. The name of the density function for which the distribution function or quantile function is required.
Vector of quantiles.
Vector of probabilities.
Numeric. A vector giving the parameter values for the
distribution specified by densFn. If no param values
are specified, then the default parameter values of each
distribution are used instead.
Character. If "spline" quantiles are found from a
spline approximation to the distribution function. If
"integrate", the distribution function used is always obtained
by integration.
Logical. If lower.tail = TRUE, the cumulative
density is taken from the lower tail.
The maximum number of subdivisions used to integrate the density and determine the accuracy of the distribution function calculation.
Value of rel.tol and hence abs.tol in calls
to integrate. See integrate.
Logical. If valueOnly = TRUE calls to
pDist only return the value obtained for the integral. If
valueOnly = FALSE an estimate of the accuracy of the numerical
integration is also returned.
Number of points used in qDist for cubic spline
interpolation of the distribution function.
Value of tol in calls to uniroot. See
uniroot.
Passes additional arguments to integrate,
distMode or distCalcRange. In particular,
the parameters of the distribution.
David Scott d.scott@auckland.ac.nz Joyce Li xli053@aucklanduni.ac.nz
The name of the unimodal density function must be supplied as the
characters of the root for that density (e.g. norm, ghyp).
pDist uses the function integrate to numerically
integrate the density function specified. The integration is from
-Inf to x if x is to the left of the mode, and from
x to Inf if x is to the right of the mode. The
probability calculated this way is subtracted from 1 if
required. Integration in this manner appears to make calculation of the
quantile function more stable in extreme cases.
qDist provides two methods to calculate quantiles both of which
use uniroot to find the value of \(x\) for which a given
\(q\) is equal to \(F(x)\) where \(F(.)\) denotes the distribution
function. The difference is in how the numerical approximation to
\(F\) is obtained. The more accurate method, which is specified as
"integrate", is to calculate the value of \(F(x)\) whenever it
is required using a call to pDist. It is clear that the time
required for this approach is roughly linear in the number of quantiles
being calculated. The alternative (and default) method is that for the
major part of the distribution a spline approximation to \(F(x)\) is
calculated and quantiles found using uniroot with this
approximation. For extreme values of some heavy-tailed distributions
(where the tail probability is less than \(10^(-7)\)), the
integration method is still used even when the method specified as
"spline".
If accurate probabilities or quantiles are required, tolerances
(intTol and uniTol) should be set to small values, i.e
\(10^{-10}\) or \(10^{-12}\) with
method = "integrate". Generally then accuracy might be expected
to be at least \(10^{-9}\). If the default values of the
functions are used, accuracy can only be expected to be around
\(10^{-4}\). Note that on 32-bit systems
.Machine$double.eps^0.25 = 0.0001220703 is a typical value.
pDist("norm", q = 2, mean = 1, sd = 1)
pDist("t", q = 0.5, df = 4)
require(GeneralizedHyperbolic)
pDist("ghyp", q = 0.1)
require(SkewHyperbolic)
qDist("skewhyp", p = 0.4, param = c(0, 1, 0, 10))
qDist("t", p = 0.2, df = 4)
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