momIntegrated {HyperbolicDist} | R Documentation |
Calculates moments and absolute moments about a given location for the generalized hyperbolic and related distributions.
momIntegrated(densFn, order, param = NULL, about = 0, absolute = FALSE)
densFn |
Character. The name of the density function whose moments are to be calculated. See Details. |
order |
Numeric. The order of the moment or absolute moment to be calculated. |
param |
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. |
about |
Numeric. The point about which the moment is to be calculated. |
absolute |
Logical. Whether absolute moments or ordinary moments
are to be calculated. Default is FALSE . |
Denote the density function by f. Then if
order
=k and about
=a,
momIntegrated
calculates
integral_{-infinity}^infinity (x - a)^k f(x) dx
when absolute = FALSE
and
integral_{-infinity}^infinity |x - a|^k f(x) dx
when absolute = TRUE
.
Only certain density functions are permitted.
When densFn="ghyp"
or "generalized hyperbolic"
the
density used is dghyp
. The default value for param
is
c(1,1,0,1,0)
.
When densFn="hyperb"
or "hyperbolic"
the density used is
dhyperb
. The default value for param
is
c(0,1,1,0)
.
When densFn="gig"
or "generalized inverse gaussian"
the
density used is dgig
. The default value for param
is
c(1,1,1)
.
When densFn="gamma"
the density used is dgamma
. The
default value for param
is c(1,1)
.
When densFn="invgamma"
or "inverse gamma"
the
density used is the density of the inverse gamma distribution given by
f(x) = u^alpha exp(-u)/(x Gamma(alpha)), u = theta/x
for x > 0, alpha > 0 and
theta > 0. The parameter vector
param = c(shape,rate)
where shape
=alpha and
rate
=1/theta. The default value for
param
is c(-1,1)
.
When densFn="vg"
or "variance gamma"
the density used is
dvg
from the package VarianceGamma. In this case, the package
VarianceGamma must be loaded or an error will result. The
default value for param
is c(0,1,0,1)
.
The value of the integral as specified in Details.
David Scott d.scott@auckland.ac.nz, Christine Yang Dong c.dong@auckland.ac.nz
dghyp
, dhyperb
,
dgamma
, dgig
,
VarianceGamma
### Calculate the mean of a generalized hyperbolic distribution ### Compare the use of integration and the formula for the mean m1 <- momIntegrated("ghyp", param = c(1/2,3,1,1,0), order = 1, about = 0) m1 ghypMean(c(1/2,3,1,1,0)) ### The first moment about the mean should be zero momIntegrated("ghyp", order = 1, param = c(1/2,3,1,1,0), about = m1) ### The variance can be calculated from the raw moments m2 <- momIntegrated("ghyp", order = 2, param = c(1/2,3,1,1,0), about = 0) m2 m2 - m1^2 ### Compare with direct calculation using integration momIntegrated("ghyp", order = 2, param = c(1/2,3,1,1,0), about = m1) momIntegrated("generalized hyperbolic", param = c(1/2,3,1,1,0), order = 2, about = m1) ### Compare with use of the formula for the variance ghypVar(c(1/2,3,1,1,0))