Originally Posted by
eigenvector11 No, there is not a typo in regards to the way the density function is defined. That is how it is defined in the question. The question asks me to derive the form of a large sample 95% confidence interval based upon the maximum likelihood estimator of theta hat. So from what I understand from the notes in class, this requires me to find the maximum likelihood estimator (which I've done), and then find the Fisher information for this distribution (which is where I need help). The Fisher information involves the Cramer-Rao lower bound (i.e. taking the natural log of the density, finding two derivatives, taking the negative expected value of the second derivative, taking the inverse of this, multiplying by n, and taking the square root, etc.).
I will put it more forcefully then. What you have does not agree with how the main online sources think the Rayleigh distribution is defined. Not a show stopper in itself since there is more than one definition out there. What is a show stopper is that it does not integrate up to 1.
Numerical evidence:
Code:
>dx=0.0001;
>x=dx/2:dx:20;
>theta=1;
>f=x^2/(theta^2)*exp((-x^2)/(2*theta^2));
>
>sum(f)*dx
1.25331
>
Symbolic evidence: see attachment
CB