Question

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

• A. is unique and is equal to $\min(X_{1}, X_{2}, \dots, X_{n})$
• B. is unique and is equal to $(\max(X_{1}, X_{2}, \dots, X_{n})-1)$
• C. is not unique
• D. does not exist

Hint:

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.

Answer 1

The Correct Option is C

This problem concerns the MLE of a Uniform distribution where both bounds depend on the same parameter.

Step 1: \color{redWrite the Likelihood Function
The PDF is $f(x; \theta) = 1$ for $\theta \le x \le \theta+1$.
The Likelihood function for $n$ observations is $L(\theta) = 1$, provided all $x_i$ satisfy $\theta \le x_i \le \theta+1$.

Step 2: \color{redAnalyze the Constraints on $\theta$
The condition $\theta \le x_i$ for all $i$ implies $\theta \le \min(x_i) = X_{(1)}$.
The condition $x_i \le \theta+1$ for all $i$ implies $\max(x_i) \le \theta+1$, which means $\theta \ge \max(x_i) - 1 = X_{(n)} - 1$.

Step 3: \color{redDetermine the Range of MLEs
Combining these, the likelihood $L(\theta)$ is a constant ($1$) for any $\theta$ in the interval:
$X_{(n)} - 1 \le \theta \le X_{(1)}$.
Since the likelihood is constant and at its maximum over this entire interval, any value within this range is a valid Maximum Likelihood Estimator.

Step 4: \color{redConclusion
Since there is an entire interval of values that maximize the likelihood, the MLE is not unique.
Final Answer: (3).