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I am trying to find the Maximum Likelihood Estimator of an i.i.d. sample $X_1, \ldots, X_n$ arising from the model $N(\theta, 1)$, where $\theta \in [0,\infty)$. I have done this problem previously where the mean was not restricted to be non-negative, and found the MLE to be equal to the sample mean (as you would expect).
Please could you explain why this situation is different and how it should be approached, as this is confusing me quite a bit!
Many thanks.
The maximum likelihood estimate is that value in the parameter space such that the likelihood is maximized. In the unrestricted case the MLE should be the sample mean. But in the restricted case the MLE cannot be negative as it will not belong to the parameter space then. Also the likelihood will be maximized at the sample mean. However if the sample mean is negative then the value in $[0,\infty)$ where the likelihood is maximized is $0$. hence the MLE is the sample mean if the sample mean is positive, otherwise the MLE is $0$.
