Exercise5.9 Let Y1, ...,Yn be arandomsamplefromashiftedexponentialdistributionwithadensity f (y) = e(y), y . 1. Find theMLE

Exercise5.9 Let Y1, ...,Yn be arandomsamplefromashiftedexponentialdistributionwithadensity fτ (y) = e−(y−τ), y ≥ τ.

1. Find theMLE ˆ τn for τ (see alsoExercise2.1).

2. Is ˆ τn consistent inMSE?

3. Showthat n( ˆ τn−τ) ∼ exp(1).

4. Find theasymptoticdistributionof

√

n( ˆ τn−τ).

5. WhydoestheasymptoticnormalityofMLEnotholdinthiscase?