# Maximum likelihood estimation versus a given interval

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.

• When the parameter space is restricted, MLE takes on the boundary values when $M/N$ lies outside of the interval $[1/3,2/3]$. Apr 25, 2019 at 12:31
• @StubbornAtom Ok. So here should it be option A,B or something else? Apr 25, 2019 at 14:35
• 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? Apr 25, 2019 at 14:39
• 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 $M/N > 1/3$. Apr 26, 2019 at 5:22
• Yes, that is one way to write down the MLE. Apr 26, 2019 at 7:18