I am using the code below to get familiar with Maximum Likelihood Estimation. I am fitting to a normal distribution. Following the MLE estimate, comparison is made with (i) mean() & std(); (ii) normfit(); and (iii) MLE estimate of the error of the estimates.
I'd like to get the feedback of more experienced MLE users. For parameter 2 (sigma, or standard deviation), the estimate is 0.4684, with 95% confidence intervals at 0.3311 and 1.0192. The estimate is much closer to the lower end of the confidence interval. I understand that that PDF for the estimate of sigma may be heavily skewed. Would this be the explanation? The reason I ask is because a paper I am studying uses the MLE estimate of the error on sigma, which leads to a +/- delta_sigma, which basically pretends that the confidence intervals are symmetric around the estimate. The above skewing of estimated sigma relative to the confidence intervals means that I have to be careful and ensure that this assumption wasn't carried through to all the calculations that depend on confidence intervals.