# Rice $\sim R(\nu,\sigma)$ to Noncentral $\chi^2$

Statistics beginner here.

I have a sample data set which is Rice distributed, $$R \sim R(\nu,\sigma)$$. However, I'm interested in fitting $$R^2$$. According to Wikipedia,

• If $$R ∼ Rice ⁡ ( ν , 1 )$$ then $$R^2$$ has a noncentral chi-squared distribution with two degrees of freedom and noncentrality parameter $$v^2$$.

I've also learnt that, given a data vector, it was possible to identify the population which generated it via MLE method.

Considering that the dataset has $$\sigma \neq 1$$

1. What strategy should I follow for parameter estimation in $$R \sim R(\nu,\sigma)$$?

2. Once I have the parameters, how can I make sure that above-mentioned transition to Noncentral $$\chi^2$$ applies to my dataset?

I apologize in advance if my questions are not well-structured due to my partial understanding.

1. Use MLE in $$R(\nu, \sigma)$$ to obtain $$(\overline{\nu}, \overline{\sigma})$$
2. Divide the data set by $$\sqrt{\overline{\sigma}}$$ ( Now it has unit variance $$R(\nu,1)$$ )
3. $$R^2(\nu,1)$$ has noncentral $$\chi^2$$ distribution with two degrees of freedom and noncentrality parameter $$v^2$$