Re: Stochastic Taylor series
- From: hrubin@xxxxxxxxxxxxxxxxxxxx (Herman Rubin)
- Date: 6 Jan 2009 12:42:03 -0500
In article <96335dfc-8f88-40be-9eae-d5026f0e7e39@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Peter <pwolynes@xxxxxxxxx> wrote:
Hi,
I am wondering if the following type of Taylor series is known .
I am working with the following quantity:
z = ArcTan[ (sin(a) + x)/(cos(a) + y) ]
where x,y i.i.d. N(0,s^2] where s<1 (we'll say s<<1 to simplify the
problem)
I want to study the statistical properties of z. Now, the distribution
of z can be worked out (since the distribution of the ratio of two
independent normal rvs is a known one), but in my case, the
distribution is too complicated to work with very easily.
As a consequence, I want to do a Taylor series expansion of z wrt x
and y (about x=0 and y=0).
The form for the Taylor series has a nice simple form (esp for the
first few orders):
z = a + cos(a)* x - sin(a)* y - sin(2a) *(x^2/2) - sin(2a)*(y^2/2) -
cos(2a) * x*y.
Since x and y are statistically independent, computing approximations
to the mean, variance, etc of z can then be performed rather easily.
Technically, however, I am wondering how valid this is. x and y will
almost surely have a magnitude less than unity, but cases may arise in
which the magnitude of x relative to cos(a) may not be small, and it
is not obvious to me that that wouldn't cause problems.
Thanks in advance for any help/insight on whether people perform this
type of Taylor series expansions and what conditions are required for
them to be well-defined.
Peter
The Taylor series here does not converge for all x and
y, as the ArcTan has singularities at +-i. For small s,
the approximation does have value, and the moments can
be approximated from the series. Large values of either
x or y can produce problems. Since ArcTan is bounded
for real arguments, these approximations MIGHT be good.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.
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