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I have been solving some exercises on Maximum Likelihood estimation and I got across this one-

It is known that the proportion of smokers (p) in a population lies in the interval $[1/3, 2/3]$. In a random sample of N individuals selected from the population, it was found that M were smokers. The maximum likelihood estimate of p based on the above data is

(A) $max\left\{1/3, M/N\right\}$

(B) $min\left\{M/N, 2/3\right\}$

(C) $M/N$

(D) none of the above.

I am completely new to this kind of problem where an interval for the parameter is given. My approach is to take MLE as $M/N$ if $M/N$ lies in the given interval, but it does not match with any of the options. I don't know what to take MLE if $M/N$ lies outside of the interval. Please tell me how to proceed in this kind of situation.

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  • $\begingroup$ When the parameter space is restricted, MLE takes on the boundary values when $M/N$ lies outside of the interval $[1/3,2/3]$. $\endgroup$ Apr 25, 2019 at 12:31
  • $\begingroup$ @StubbornAtom Ok. So here should it be option A,B or something else? $\endgroup$
    – Ankit Seth
    Apr 25, 2019 at 14:35
  • $\begingroup$ The MLE takes all three values $M/N, 1/3$ and $2/3$. Based on this what do you think will be the correct option? $\endgroup$ Apr 25, 2019 at 14:39
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    $\begingroup$ I think it should be $max\left\{1/3,M/N\right\}$ if $M/N < 2/3$ and $min\left\{M/N,2/3\right\}$ if $M/N > 1/3$. $\endgroup$
    – Ankit Seth
    Apr 26, 2019 at 5:22
  • $\begingroup$ Yes, that is one way to write down the MLE. $\endgroup$ Apr 26, 2019 at 7:18

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