MLE & Sufficient Statistics for Uniform(-θ/2, θ/2) | GATE ST 2025 | Problem-52 Part-1| RitwikMath

Master **parameter estimation** for uniform distribution \(X_1, \dots, X_n \sim \text{Uniform}(-\theta/2, \theta/2)\), θ 0 unknown!

**Key results:**
- **Density**: \(f(x|\theta) = \frac{1}{\theta} I(|x| \leq \theta/2)\)
- **Likelihood**: \(L(\theta) = \theta^{-n} I(\max_i |X_i| \leq \theta/2)\)
- **MLE**: \(\hat{\theta}_{MLE} = 2 \max_i |X_i|\) ✓ (Option A incorrect - missing absolute value)
- **Sufficient statistic**: \((X_{(1)}, X_{(n)}) = (\min X_i, \max X_i)\) ✓ (By factorization theorem)

**Why absolute value matters**: If most extreme value is large negative, \(2 \max X_i\) underestimates true θ needed to cover the range.

Perfect for GATE aspirants tackling **order statistics**, **MLE derivation**, and **sufficiency**!

#GATE2025 #GATEStatistics #UniformDistribution #MaximumLikelihood #SufficientStatistic #OrderStatistics #GATESTPYQs #StatisticalInference #PointEstimation

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