Re: MLE of a restricted RV

Outlier wrote:
On Aug 22, 2:44 am, Ray Koopman <koop...@xxxxxx> wrote:
On Aug 21, 5:23 pm, Outlier <MTBrenne...@xxxxxxxxx> wrote:



Hi,

I am curious if anyone can suggest a solution to this problem.

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)

Of course, I am interested in the distribution of phi. Making the
normal change of variables requires I restrict the pdf so |X|<1.
Letting:

FX(x|a0) = CDF for X given a0

then I can define the "mixed" pdf:

Case 1 : F(-1|a0) for X=-1
f (y) = Case 2 : Normal PDF for -1<X<1 (using
transformation of rv rule)
where pdf is scaled by
F(1|a0)-F(-1|a0) Case 3 : 1 - F(1|a0) for X=1
Case 4 : 0 |X| > 1

Cases 1 and 3 could be thought of as being equivalent to assigning
MLE = -pi/2 and pi/2 resp.

When you try to find the MLE for a value of X s.t. |X|<1, the
problem you run into is that the scale factor in the denominator of
the pdf is dependent on the parameter of interest a0. When I do
calculations with Mathematica, where I fix a value of X and
evaluate the pdf for various values of a0, the maximum is not
typically at a value even remotely near ArcSin(X).

Is there a way to remedy this problem or a better estimation method
that would work in such a situation?

TIA,

Matt

Look up the von Mises distribution.

Hi Dr. Koopman:

I appreciate the suggestion and I know that the von Mises distribution
is "like" a wrapped normal distribution, but I have the feeling this
may not really solve my problem in the end. First, the von Mises is
like a normal rv, but it isn't a normal distribution, which is one
problem (although perhaps not a terribly big one : the variance is
less than one, which would give a kappa (concentration parameter) that
was relatively large and might make the normal approximation a decent
one). Maybe I am missing something in your suggestion or I am missing
a connection, but I don't think the problem here though here is due to
an underlying periodicity. The problem I am having is that I cannot
make the change of rvs I want without having to "censor" the rv
values, and I am trying to find out if there is a standard method for
performing MLE with a censored rv. I did find one reference that deals
with this problem (once I figured out that the keyword was censor),
but I am curious if anyone has had experience with this sort of
problem and what comments/suggestions they might make.

Thanks again,

Matt B.

(i) if von Mises won't do, you should consider some of the other "circular distributions".
(ii) "once I figured out that the keyword was censor" may indicate that even for what you are trying to do, you are not heading quite correctly .... you may actually want a truncated version of the Normal distribution, rather than a censored one.

David Jones
.

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