Question

1. Let X1, . . . , Xn be i.i.d. continuous RVs with density pθ(x) = e−(x−θ), x ≥ θ for some unknown θ > 0. Be sure to notice that x ≥ θ. (This is an example of a shifted Exponential distribution.) (a) Set up the integral you would solve for find the population mean (in terms of θ); be sure to specify d[blank]. (You should set up the integral by hand, but you can use software to evaluate it.) (b) You could find the population mean by evaluating the integral in the previous part, but here is another way. It can be shown that if X ∼ pθ then (X − θ) ∼ Exponential(1), hence the name “shifted Exponential”. Use this fact to find the population mean. (c) Use the previous part to suggest an unbiased estimator of θ, and find its variance. (Hint: when finding variance use the fact about shifted Exponential from the previous part; what affect does shifting have on variance?) (d) Find the likelihood function L(θ) and find the MLE of θ. (Hint: don’t try calculus; plot the likelihood, similar to MLE of Uniform(θ, 1). Remember: pθ(x) > 0 only if x ≥ θ.)

Answer 1

We will use the info above