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Q111Section C (Part C)111 of 120

Estimators for the scale of a Laplace distribution with location θ\theta

Let X1,X2,…,XnX_1,X_2,\dots,X_n (n≥2n\ge2) be a random sample from a continuous distribution with the probability density function

f(x∣θ)=12θe−∣x−θ∣θ,x∈R,f(x\mid\theta)=\frac1{2\theta}e^{-\frac{|x-\theta|}\theta},\quad x\in\mathbb R,

where θ(>0)\theta(>0) is an unknown parameter. Let Un=1n∑i=1nXiU_n=\tfrac1n\sum_{i=1}^nX_i, Vn=1n∑i=1nXi2V_n=\tfrac1n\sum_{i=1}^nX_i^2, and Sn2=1n−1∑i=1n(Xi−Un)2S_n^2=\tfrac1{n-1}\sum_{i=1}^n(X_i-U_n)^2. Then, which of the following statements are true?

Multiple select: one or more options are correct. Select all of them, then submit — marks need the exact set.

Options

Select every correct option, then submit.