MLE Problem

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.

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.

Any suggestions?

TIA (VM!)

Matt Brenneman

.

Relevant Pages

  • Probability Theory Problem
    ... I want to find an MLE for the non-centrality parameter of a non-central ... where c is my non-centrality param and I_0 is modified Bessel function ... of eqn has a constant in it, which the RHS doesn't. ...
    (sci.math)
  • Re: MLE Problem
    ... I am having a bugger of a time with a problem that doesn't seem too ... I want to find an MLE for the non-centrality parameter of a ... where c is my non-centrality param and I_0 is modified Bessel ... gives us the derivative of the log liklihood function ...
    (sci.stat.edu)
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