View source: R/momIntegrated.R
momIntegrated | 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 |
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 |
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)
.
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
### 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))
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