rayleigh

1. Does Sqrt of confidence intervals require a correction factor?

I have the following formula for confidence intervals on samples from a Rayleigh process: \frac{2(n-1)\overline{r^2}}{\chi_2^2} \leq \widehat{\sigma^2} \leq \frac{2(n-1)\overline{r^2}}{\chi_1^2} I want to give confidence intervals in terms of \sigma, not \sigma^2. If [x, y] are the 95%...
2. Distribution of Extreme Spread for (n, sigma)

Given a Rayleigh process R(s) generating samples X_i -- which is equivalent to a bivariate normal process with 0 correlation and both sigmas = s -- what is the distribution of the Extreme Spread of n samples? Extreme spread ES\{X\}_n \equiv max_{i, j}|X_i - X_j| E[ES_n(\sigma)], and...
3. How is this a MLE?

In all the literature I can find it is stated (and "proven" trivially) that for i.i.d. samples r with Rayleigh distribution \sigma the MLE is \widehat{\sigma} = \frac{\sum r_i^2}{2n}, and it is an unbiased estimator for \sigma. But any Monte Carlo test shows that's not true: Only the square...
4. Rayleigh estimator and correction factor

We're using the Rayleigh distribution for some real-world scenarios. We often need to estimate its parameter (sigma) from samples R of size N where N is very small. The estimator we're using for sigma, \widehat{\sigma} = \sqrt{\frac{\sum r_i^2}{2n}}, is biased. Using Monte Carlo analysis...