STB – 2019

• Let X1, X2, . . . , Xn be independent and identically distributed random variables with a common density function 

f(x,θ) = e−(x−θ)I(x ≥ θ), where θ ∈ R. 

1. Find the maximum likelihood estimator θ_n of θ based on X₁, . . . , X_n. 
2. Show that  is consistent for .
3. For a suitable normalizing factor (to be specified by you), find a non-degenerate limiting distribution of .