Exercise2.1 Let Y1, ...,Yn be arandomsamplefromthefollowingdistributionswiththeunknown parameter(s). Estimatethembymaximumlikelihoodandbythemethodofmoments. 1. doubleexponentialdistribution f (y) = 2 e|y|,

Exercise2.1 Let Y1, ...,Yn be arandomsamplefromthefollowingdistributionswiththeunknown parameter(s). Estimatethembymaximumlikelihoodandbythemethodofmoments.

1. doubleexponentialdistribution fθ (y) = θ

2 e−θ|y|, −∞ < y < ∞, θ > 0 2. shifted exponentialdistribution fθ,τ (y) = θe−θ(y−τ), y ≥ τ, θ > 0, where

(a) τ is known

(b) both θ and τ are unknown (Hint: tocalculatethemomentsofashiftedexponentialdistributiononecaneasilyshowthat Y = T +τ, where T ∼ exp(θ))
3. Paretodistribution fα,β (y) = αβ

β
y α+1 , y ≥ β, α,β > 0 4. fθ (y) = θyθ−1, 0 ≤ y ≤ 1, θ > 0