Marginal Density for Vasicek process MLE

Hi,

I am trying to perform the test statistics Ait Sahalia proposes in his paper “Testing Continuous Time Models of the Spot interest Rate” in order to test the correctness of the parameterisation of the Vasicek process.

I have calculated the kernel density approximation π(ri) and all I need to do now is find the parameter’s vector w=[θ,k,sigma] that minimizes the distance between the parametric marginal density π(ri,w) and the kernel density approximation π(ri).

 w = argmin(1/n)Sum(π(ri,w)-π(ri))^2     (1)

The parametric marginal density for a Vasicek process is a normal density given by the formula:

π(ri,w)=(1/sqrt(2*pi*sigma^2/2*k))*exp[-.5((r-θ)/sqrt(sigma^2/2k))^2]

I am thinking to implement maximum likelihood estimation in order to find the optimal set of parameters in 1.

However I am struggling a bit with the maximum likelihood function of (2) and I was wondering if anyone has solved a maximum likelihood problem for a normal distribution with 3 parameters – and furthermore if using MLE for a 3-parameter normal distribution would give good estimates.

Many thanks in advance Best Regards Ele