Apologies again. Pasting error on 1 st posting, the
content was truncated....
Hi.
Should've posted this long ago but I guess it's
better late than never.
Cor Stolk and David Jones pointed out that Ted
Harding's proof (see 1st summary) had implicitly
assumed that the functions were not only
differentiable but also monotonic. See below:
>>>>Ted Harding:
Now suppose x = g(y), and let dx corrspond to dy, so
that dP, the probability that X is in dx, is also the
probability that Y is in dy. <== assumption of
monotonicity.
In this case, Cor proposes a proof which does not rely
on differentiability but only on monotonicity. Also
below proposal for constructing the desired non-linear
functions from Cor, David Jones and David McNulty.
>>>>Cor Stolk:
==> Assume the function is monotonic.
my way of attacking the problem is to look at the
cumulative distribution
function.
If the cdf for X is a normal cdf, than the cdf for Y
(a nonlinear
transformation of X) is non-normal.
How to prove this intuition?
Y(X) is a continuous nonlinear function, so there must
exist three distinct
values of X (say x1, x2 and x3, transformed into y1,
y2 and y3; both
{x1,x2,x3} and {y1,y2,y3} correspond to cdf values P1,
P2 and P3) such that
the intervals between them are streched in unequal
degree:
(y2-y1)/(x2-x1) differs from (y3-y2)/(x3-x2).
(If no such triplet exists than the transformation is
linear).
It follows that (x3-x2)/(x2-x1) differs from
(y3-y2)/(y2-y1).
In a normal distribution for given P1, P2 and P3 the
ratio
(x3-x2)/(x2-x1) is fixed (you don't require me to
prove this, do you?).
X has a normal distribution and fixes the value of
this ratio.
So it follows that Y does not have the proper ratio
and hence cannot have a
normal distribution.
He also proposes a way to construct a non-linear
continuous funtion:
We start from an X with distribution N(0,1). Now let
us first consider the
lower half of the graph (X<0; 0<=CDF<0.5). Let us
stretch this part of the
X-axis in such a way that this part of the transformed
curve has the shape
of a N(0,1) distribution on a reduced vertical scale.
This transformation
can be a monotonic increasing mapping (-inf,0) on
(-inf,+inf).
For the upper half, we first put X' = -X, and then we
apply the same
transformation.
The single point X=0 can be treated as a limiting
case; the procedure
guarantees that aproaching it from above or from below
gives the same
result.
In this way we have generated a mapping X to X"; this
mapping is continuous
and nonlinear, and the distribution of X" is normal:
every X" <> 0
corresponds to two X-values each with half the
required probability density.
>>>>>David Jones:
Most of the relies are implicitly assuming a monotonic
transformation function. The following approach yields
an non-monotonic function.
First, assume that the original rv is standard Normal.
Then transforming this using the standard Normal
distribution function gives a uniform rv.
Second, transform the uniform rv to another uniform
rv. For example using y = 2x-1 if x is less than one
half and y=2-2x otherwise.
Third, transform the new uniform rv back to standard
Normal using the inverse of the standard Normal
distribution function.
Of course this transform does send values close to
zero out to plus infinity, but it is possible to find
a better uniform-to-uniform transformation which will
avoid this effect. See attached bitmap file for a
suggestion using a sectionally linear function which
has starts and finishes with sections with a slope of
one.
>>>>David McNulty:
Hi,
This is an algorithmic approach.
To start with lets look at the easy case and consider
whether there is a
discontinuous function:
Step 1: Use a suitable funtion of the random
variable as a seed within a
random number generator. The output ought to be
roughly uniform[0, 1].
Step 2: Use the probability integral transform to
"Re-Normalise" the
output.
Optional on a case by case basis.
Step 3: Adjust the scales as necessary using
strictly monic
transformations to touch up the shape of the
distribution.
Since both steps are deterministic the algorithm
creates a discontinuous
non-linear transformation of the original variable to
a normally
distributed variable.
To get a continuous function replace step 1 with a
continuous function with
high variation e.g. a fast sawtooth.
Don't ask what the finished distribution will look
like I am running on
intuition!
Cheers,
Hendra I. Nurdin
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