GATE 2024 ST – Question 24
An iid sample n≥2 has density θ(2x)^(θ−1) on 0<x≤1/2 and θ(2−2x)^(θ−1) on 1/2<x≤1,zero otherwise,θ>0. Put L=Σ_(X_i≤1/2)ln(2X_i)+Σ_(X_i>1/2)ln(2−2X_i), and K=Σ_all ln(2X_i)+Σ_all ln(2−2X_i). Which is the MLE?
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: (B) −n/L
Explanation
The log likelihood is n lnθ+(θ−1)L. Differentiation gives n/θ+L=0, so θ̂=−n/L. Almost surely L<0, and the second derivative −n/θ²<0 confirms the maximum. **B**.