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Let X1,…, Xn be a sample of iid random variables with pdf f (x ∶ ?)...

Let X1,…, Xn be a sample of iid random variables with pdf f (x ∶ ?) = 1/? for x ∈ {1, 2,…, ?} and Θ = ℕ. Determine the MLE of ?.

Math   Statistics-And-Probability   
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Answer #1

Note- Proceeding in general approach (differentiating and equating to zero), we cannot find MLE for this problem. We have to approach analytically to obtain MLE.

Joint probability distribution function is given by

We have to maximize L.

So, we have to minimize i.e. to minimize .

Using notion of order statistic we have

Clearly, minimum possible value of is .

So, MLE of is .

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