Re: MLE of a restricted RV

On Aug 22, 8:49 am, Scott Seidman <namdiestt...@xxxxxxxxxxxxxx> wrote:
Outlier <MTBrenne...@xxxxxxxxx> wrote in news:1187742184.834976.297520
@q3g2000prf.googlegroups.com:

I want to find the MLE for an angle, a. The sine of the angle is
normally distributed s.t.
X ~ N( sin(a0), s^2)

Let me try to recast this a bit, and see if that offers any insight. I'm
assuming your data is not circular.

Instead of thinking about the angle itself, try thinking about this as a
vector of length 1 and angle theta in the complex plane (you can cast the
problem this way with Euler's formula). Now, the Imaginary component of
this vector is sin(theta), and is distributed normally (as defined by
your problem). The real part, when the problem is constrained this way,
is a simple trig relationship to the imaginary part).

I'm dealing with a problem much like this in neuroscience-- except the
magnitude of the vector is also free to change-- and I think I have a
valid approach, which I'm writing up soon for Neuroscience Methods. I'm
doing my stats on the slope of a zero-intercept orthogonal least squares
fit to the data. The slope is related to the angle, of course, by an
arctan.

I have a prelim version of this "in press" in appedix form in
Experimental Brain Research,http://www.springerlink.com/content/100473/?Content+Status=Accepted
doi number: 10.1007/s00221-007-1072-3 Author: Seidman
This approach uses an ordinary least squares approach, but the orthogonal
least squares is much more correct. Feel free to contact me if you need
more detail-- just remember to reverse the first part of my email addy.

--
Scott
Reverse name to reply

Dear Dr. Seidman:

Thank you for the reply. I will look at your paper, and see if your
ideas could be fruitfully applied to my particular problem.

I have a feeling that the solution to my problem might be easier than
I originally had anticipated though. I can sketch out what I think the
correct solution should be rather easily.

Instead of the pdf, consider the cdf. Here is an example of how I
could compute the conditional cdf.
Let our stat model be:
X = sin(a0) + e where e~N(0,s^2), s known
Suppose I want to compute the conditional probability:
P(X<x| a=a0) when X<-1 (for the case of a "bad" rv value)
but this is:
P(sin(a0)+e<x | a=a0) = P(e < x-sin(a0) | a=a0)
since I know the pdf, I can compute this quantity (which I believe
works out to {Erf[(x-sin(a0))/(s*sqrt(2)]+1}/2). By taking the
derivative of the conditional cdf, I will have the conditional pdf,
whose maximum I can now find.

When I think about it, my use of the word "censor" was perhaps not
technically correct. When I looked over the references about censoring
rvs, they dealt with cases where someone was trying to apply a
distribution, say like the normal distribution, to model an rv like
income. In this case, it is never possible to have a negative income
*with such a model*. I emphasize the last phrase because the idea of
censoring here is saying that the rv itself (not just the
deterministic part) cannot be negative. In my case, this is not true,
the rv can have negative values, and the model can account for them.

Now of course, the task at hand is to work it out and make sure it
makes sense and is computationally manageable. (As I look back over
the argument, it almost seems *too* simple, although I can't see
anything wrong with it. My main concern is that it seems like taking
the derivative of the cdf will simply give me back the same pdf with
the change of variables formula, so I obviously need to think this
over a bit more.) I appreciate your reference as soon I'll be looking
at multi-parameter versions of this problem, and my first instinct is
that a least-squares approach may be the best way to proceed in such a
case.

Best Regards,

Matt Brenneman

.

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