1.4 Fourier Inversion
1.4.1 Error Types in Approximating the Quantile through Fourier Inversion
Let
denote a continuous, absolutely integrable function and
its Fourier transform. Then,
the inversion formula
 |
(1.16) |
holds.
The key to an error analysis of trapezoidal, equidistant approximations to the
integral (1.16)
 |
(1.17) |
is the Poisson summation formula
 |
(1.18) |
see (Abate and Whitt; 1992, p.22). If
is approximated by
,
the residual
 |
(1.19) |
is called the aliasing error, since different ``pieces'' of
are
aliased into the window
. Another suitable
choice is
:
 |
(1.20) |
If
is nonnegative,
. If
is decreasing
in
for
, then
holds for
. The aliasing error can be controlled by
letting
tend to 0. It decreases only slowly when
has ``heavy
tails'', or equivalently, when
has non-smooth features.
It is practical to first decide on
to control the aliasing error
and then decide on the cut-off in the sum (1.17):
 |
(1.21) |
Call
the
truncation error.
For practical purposes, the truncation error
essentially depends only on
and the decision on how to choose
and
can be decoupled.
converges to
 |
(1.22) |
for
. Using
and the convolution
theorem, one gets
 |
(1.23) |
which provides an explicit expression for the truncation error
in terms of
. It decreases only slowly with
(
) if
does not have infinitely many derivatives,
or equivalently,
has ``power tails''. The following lemma leads to the
asymptotics of the truncation error in this case.
LEMMA 1.1
If

,

, and

exists and is finite for some

, then
for

.
PROOF.
Under the given conditions, both the left and the right hand side
converge to 0, so l'Hospital's rule is applicable to the ratio of the left and
right hand sides.
THEOREM 1.1
If the asymptotic behavior of a Fourier transform

of a function

can be described as
 |
(1.25) |
with

, then the truncation error
(
1.22)
for

at all points

where

converges to

. (If in the first case

,
this shall mean that

.)
PROOF.
The previous lemma is applicable for all points

where the Fourier
inversion integral converges.
The theorem completely characterizes the truncation error for those cases,
where
has a ``critical point of non-smoothness'' and has a higher degree
of smoothness everywhere else. The truncation error decreases one power faster
away from the critical point than at the critical point. Its amplitude is
inversely proportional to the distance from the critical point.
Let
be a (continuous) approximation to a (differentiable) cdf
with
. Denote by
a known error-bound for the
cdf. Any solution
to
may be considered an
approximation to the true
-quantile
. Call
the
quantile error. Obviously, the quantile error can be bounded by
 |
(1.26) |
where
is a suitable neighborhood of
. Given a sequence of
approximations
with
,
 |
(1.27) |
holds.
FFT-based Fourier inversion yields approximations for the cdf
on
equidistant
-spaced grids. Depending on the smoothness of
,
linear or higher-order interpolations may be used. Any monotone interpolation
of
yields a quantile approximation whose
interpolation error can be bounded by
. This bound can be
improved if an upper bound on the density
in a suitable
neighborhood of the true quantile is known.
1.4.2 Tail Behavior
If
for some
, then
. In the following, we assume that
for all
. The norm of
has the form
The arg has the form
(for
). This motivates the following approximation for
:
is the location and
the ``weight'' of the singularity.
The multivariate delta-gamma-distribution is
except at
, where the highest continuous derivative of the cdf is of order
.
Note that
 |
(1.36) |
and
meets the assumptions of theorem 1.1.
1.4.3 Inversion of the cdf minus the Gaussian Approximation
Assume that
is a cdf with mean
and standard deviation
, then
 |
(1.37) |
holds, where
is the normal cdf with mean
and standard
deviation
and
its characteristic
function. (Integrating the inversion formula (1.16) w.r.t.
and applying Fubini's theorem leads to (1.38).) Applying the
Fourier inversion to
instead of
solves the
(numerical) problem that
has a pole at 0. Alternative
distributions with known Fourier transform may be chosen if they better
approximate the distribution
under consideration.
The moments of the delta-gamma-distribution can be derived from
(1.3) and (1.5):
and
Let
.
Since
, the truncated sum (1.21) can for
and
be written as
which can comfortably be computed by a FFT with modulus
:
 |
(1.38) |
with
and the last
components of
the input vector to the FFT are padded with zeros.
The aliasing error of the approximation (1.20)
applied to
is
^{j}.$](xfghtmlimg352.gif) |
(1.39) |
The cases
are the ones
with the fattest tails and are thus candidates for the worst case for
(1.40), asymptotically for
.
In these cases, (1.40) is eventually
an alternating sequence of decreasing absolute value
and thus
 |
(1.40) |
is an asymptotic bound for the aliasing error.
The truncation error (1.22) applied to
is
 |
(1.41) |
The Gaussian part plays no role asymptotically for
and
Theorem 1.1 applies with
.
The quantile error for a given parameter
is
 |
(1.42) |
asymptotically for
and
. (
denotes
the true 1%-quantile for the triplet
.)
The problem is now to find the right trade-off between ``aliasing error'' and
``truncation error'', i.e., to choose
optimally for a given
.
Empirical observation of the one- and two-factor cases shows that
has the smallest density
(
) at the 1%-quantile. Since
is the case with the
maximal ``aliasing error'' as well, it is the only candidate for the worst
case of the ratio of the ``aliasing error'' over the density (at the
1%-quantile).
The question which
is the worst case for the ratio of the
``truncation error'' over the density (at the 1%-quantile) is not as
clear-cut. Empirical observation shows that the case
is also the worst case for
this ratio over a range of parameters in one- and two-factor problems. This
leads to the following heuristic to choose
for a given
(
). Choose
such as to minimize the sum of
the aliasing and truncation errors for the case
, as approximated by the
bounds (1.41) and
 |
(1.43) |
with
,
, and the 1%-quantile
. (Note that this is suitable only for intermediate
, leading to
accuracies of 1 to 4 digits in the quantile. For higher
, other cases
become the worst case for the ratio of the truncation error over the density
at the quantile.)
Since
has a kink in the case
,
, higher-order
interpolations are futile in non-adaptive methods and
is a suitable upper bound for the
interpolation error. By experimentation,
suffices to keep the
interpolation error comparatively small.
evaluations of
(
) suffice to ensure an accuracy
of 1 digit in the approximation of the 1%-quantile over a sample of one- and
two-factor cases.
function evaluations are needed for two digits
accuracy.
The
XploRe
implementation of the Fourier inversion is split up as follows:
- z=
VaRcharfDGF2
(t,par)implements
the function
for the complex argument t and the
parameter list par.
- z=
VaRcorrfDGF2
(x,par)implements
the correction term
for the argument x
and the parameter list par.
- vec=
-
gFourierInversion
(N,K,dt,t0,x0,charf,par)
implements a generic Fourier inversion like in
(1.39). charf is a string naming the function to be
substituted for in (1.39). par is the
parameter list passed to charf.
|
gFourierInversion
can be applied to
VaRcharfDG
, giving the
density, or to
VaRcharfDGF2
, giving the cdf minus the Gaussian
approximation. The three auxiliary functions are used by
- l=
VaRcdfDG
(par,N,K,dt)
to approximate the cumulative distribution function (cdf) of the
distribution from the class of quadratic forms of Gaussian vectors with
parameter list par. The output is a list of two vectors
x and y, containing the cdf-approximation on a grid
given by x.
- q=
cdf2quant
(a,l)
approximates the a-quantile from the list l, as returned
from
VaRcdfDG
.
- q=
VaRqDG
(a,par,N,K,dt)calls
VaRcdfDG
and
cdf2quant
to approximate an
a-quantile for the distribution of the class of quadratic forms
of Gaussian vectors that is defined by the parameter list par.
|
The following example plots the 1%-quantile for a one-parametric family of
the class of quadratic forms of one- and two-dimensional Gaussian vectors: