MLE of a Uniform Distribution28/2/2023 I noticed something surprising about the Maximum Likelihood Estimator (MLE) for a uniform distribution yesterday. Suppose we’re given sample $X’ = {x_1, x_2, … x_n}$ from a uniform distribution $X$ with parameters $a,b$. Then the MLE estimator for $a = min(X’)$, and $b = max(X’)$. [1] All straight forward so far. However, examining the estimators, we can also say with probability = 1 that $a < min(X’)$, and similarly that $b > max(X’)$. Isn't it strange that the MLE estimators are clearly less/more than the true values? So what can we do instead? (Since Gauss did a lot of the early work on MLE, here's a portrait of him as a young man. )
Source: https://commons.wikimedia.org/wiki/File:Bendixen__Carl_Friedrich_Gau%C3%9F,_1828.jpg 
AuthorI work as an actuary and underwriter at a global reinsurer in London. Categories
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