Re: MLE Problem


David Jones wrote:
Brenneman 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)


? Why are you taking the derivative with respect to z, and not c?

David Jones

Because I did something _very_ stupid: sorry.
But, even if you take the derivative wrt c, the problem still does not
work out, since the derivative of the Log liklelihood function is now:
(v.a) 0 = -c + z * I_1(cz)/I_0(cz)
or
(v.b) c*I_0(cz) = z*I_1(cz)
The power series expansion still will not work using this eqn since the
first few terms in the p.s. expansion of I_0(cz) are:
(vii.a) 1 + (cz)^2/4 + (cz)^4/36 + ...
and that of I_1(cz) is:
(vii.b) (cz)/2 + (cz)^3/16 + (cz)^5/384 + ...
and subtituting the last two expressions into (v.b) still gives an eqn
for which I cannot see a soln.

Matt

.

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