Re: MLE Problem

In article <1157557340.128270.18910@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Brenneman <brennemt@xxxxxxxxxx> wrote:
Hi,

I am having a bugger of a time with a problem that doesn't seem too
hard.

I want to find an MLE for the non-centrality parameter of a non-central
chi-squared distbtn.

Normally, this would be a bear, but in this case, the problem is not
too bad, because the distribution has only 2 degrees of freedom, so my
pdf is:

(i) f(z) = z * exp[-(z^2 + c^2)/2] * I_0(c*z)
where c is my non-centrality param and I_0 is modified Bessel function
of zero order.

The log function is:
(ii) Ln(f) = ln(z) - (z^2 + c^2)/2 + ln[ I_0(cz) ]

and using the fact that:
(iii) d/dx I_0(x) = I_1(x)

gives us the derivative of the log liklihood function (I'll call F) as:
(iv) F(z) = 1/z - z + c * I_1(cz)/I_0(cz)

The problem is the eqn that the critical value, c, must satisfy, which
is:
(v) (z^2 - 1)* I_0(cz) = (cz)* I_1(cz)

I thought to try to solve this by using the power series expansions of
both modified Bessel functions and then, by equating terms with like
powers of z, determine what the parameter c must be. The problem is
that this method does not seem to work no matter how I work it: the LHS
of eqn (v) has a constant in it, which the RHS doesn't. If you try to
choose c to produce a constant value on the RHS, the only possible soln
is c=1/z, this still will not work.

Of course this cannot work; a power series expansion
starts at 0, and since I_j(w) > 0 for all non-zero
real w, z must exceed 1.

I considered also the asymptotic expression for I0(c*z) when x is
large, and it suggests that the value c=z should work, but I am still
bothered by the fact that the MLE method does not seem to work here.

The asymptotic expansion to be used is for I_1/I_0.

Any suggestions?

--
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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