# Maximum likelihood estimator based on 1 datum for non-canonical discrete distribution

One observation is taken from a discrete distribution with a parameter $$\theta$$. There are 3 possible values of $$\theta$$: 1, 2, and 3. The PMF is given below. What is the MLE of $$\theta$$?

I suspect that $$X = 3$$ has something to do with the answer, because it has non-zero probabilities for all 3 values of $$\theta$$.

I also know that the MLE cannot be terribly complicated, because there is only one observation to work with.

• This looks like the right answer! Thank you! Would you be so kind to write it as an answer, so that I can vote it as a solution? – MSE Oct 26 '18 at 18:18

Let $$\hat \theta$$ be the maximum likelihood estimate of $$\theta$$.

$$\hat \theta = \begin{cases} 1, & \text{for } X = 0 \text{ or } 1\\ 2, & \text{for } X = 3\\ 3, & \text{for } X=4\\ 2 \text{ with prob 0.5 and 3 with prob 0.5}&\text{for } X=2 \end{cases}$$

• This may just be convention, but I'm not used to seeing a stochastic maximum likelihood estimate. – Frank Oct 26 '18 at 19:46
• @a_statistician If $X=2$, then is it correct to say that $\hat{\theta} = 2.5$? I say this because the average value of the 2 values of $\theta$ for that particular value of $X$ is 2.5. – MSE Oct 28 '18 at 15:50
• $\hat \theta = 2.5$ is not the answer for this problem. If $X=2$, find a fair coin, specify head to 2, throw coin, if head $\hat \theta = 2$, otherwise $\hat \theta = 3$ – user158565 Oct 28 '18 at 15:56

It depends on the sample value x you have drawn. MLE maximizes the likelihood of observing the data with regard to the parameter. There is no MLE without data.

• This is really more of a comment than an answer. – jbowman Oct 26 '18 at 17:35
• In this question, the MLE is considered to be a function defined on the sample space, with image in the parameter space. The question provides enough information to define an MLE. – whuber Oct 26 '18 at 18:48
• can I convert this to a comment? – bellmaneqn Oct 26 '18 at 19:21