Question:medium

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from $U(\theta, \theta+1)$ then the maximum likelihood estimate of $\theta$

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For $U(\theta, \theta+k)$, the MLE is any value in $[X_{(n)}-k, X_{(1)}]$. If $X_{(1)} - (X_{(n)}-k) > 0$, there are infinitely many solutions, hence non-unique.
Updated On: Jun 6, 2026
  • is unique and is equal to $\min(X_{1}, X_{2}, \dots, X_{n})$
  • is unique and is equal to $(\max(X_{1}, X_{2}, \dots, X_{n})-1)$
  • is not unique
  • does not exist
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The Correct Option is C

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