Recent content by dbooksta

I generated a quantile function \hat X using Monte Carlo simulation. The random variable I simulate is the mean value of 5 draws from an i.i.d. range statistic Y. I.e., I have Y(\sigma) \sim \sigma F(), and I simulated the value of X(\sigma=1) \sim \sum_{1}^{5} y(1)_i / 5. Is it valid for...

I derive two different statistics for characterizing the dispersion in a random variable. One, call it E, is an average of ranges. The other, call it R, is an unbiased estimator of the random variable's parameter. But E is far more popular than R because it's easier to calculate and observe...

I'm looking at bivariate Gaussian variables centered on zero. The chi distribution provides moments for a normalized multivariate Gaussian -- i.e., a random variable Y = \sqrt{\sum (\frac{X_i}{\sigma_i}})^2. But I want the variance for the unnormalized vectors Y' = \sqrt{\sum X_i^2}, so I...

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%...

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...

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...

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...