Greetings,
I have the followingÂ
Problem:
Given k (=10) discrete independent random variables X_i with n_i (= 5 to 20)
values each,compute quantiles of the distribution of the sum X = X_1+...+X_k.
Here X has n=n_1 x n_2 ... n_k distinct values which is too large to list them
all together with
their probabilities.
I tried several approaches:
(A) Convolution:
each X_j is approximated with Y_j=X_j+Z, where Z is
an N(0,sigma) variable with small sigma. Then Y_j is a probability mixture of
the normal
variables N(x_j,sigma), where the x_j runs over all values of X_j, and has a
highly oscillatory density.
The density of Y=\sum Y_j is the convolution of the densities of the Y_j.
I need this density at a sizeable number of points and this turns out to be too
slow.
The issue seems to be the convergence of the convolution integrals slowed down
by the oscillatory nature
of the densities of the Y_j. When the densities are behaved better (e.g. normal
RVs), the computation
of such a convolution is quite fast.
(B) Characteristic function:
X will be approximated with Y=X+Z, where Z is normal N(0,sigma) with small
sigma.
Y has a density (which it is impossible to compute directly) but the
characteristic function
(_continuous_ Fourier transform) cf_Y of Y can easily be computed analytically
(without knowing
the density of Y)
Now let s be a numeric vector. I want to get the density f_Y(s) of Y evaluated
along s.
The proper way of doing this would be to apply the inverse continuous Fourier
transform to the function cf_Y at each point in s.
This is far too slow. That's why I tried to apply the inverse _discrete_
Fourier transform to the vector of values cf_Y(s)
and that does not yield anything reasonable.
This baffles me since I was under the impression that the discrete Fourier
transform is an approximation to the continuous
Fourier transform and so should yield the values of the latter, if the value
grid s is fine enough.
Should this work?
Note that this inversion would definitely work if I could compute the
_discrete_ Fourier transform of the density f_Y
along s, but regrettably this is not possible since
(a) the density of f_Y is far too complicated, and
(b) the discrete Fourier transform of a sum Y = Y_1+Y_2+...+Y_k of independent
random variables Y_j is not the product of the discrete Fourier transforms of
the Y_j.
Any ideas how I could approach this problem with the tools of R?
Thanks in advance for all replies,
Michael Meyer
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