Problem 85E

Let Y1, Y2, . . . , Yn denote a random sample from the density function given by

where α > 0 is known.

a Find the MLE of θ.

b Find the expected value and variance of .

c Show that is consistent for θ.

d What is the best (minimal) sufficient statistic for θ in this problem?

e Suppose that n = 5 and α = 2. Use the minimal sufficient statistic to construct a 90% confidence interval for θ. [Hint: Transform to a χ 2 distribution.]

Chapter Notes: Causal Inference ● Ultimate goal of scientific method ● Dependent Variable = Phenomenon we want to explain. (We denote with Y) ● Independent Variable = Variable that we hypothesize causes Dependent Variable to vary. (We denote with X) Fundamental Problem of Causal Inference ● Two States of the World: ○ 1) What we observe happen to Y given X=A ○ 2) Counterfactual: What would have happened to Y had X = B observed No effect Positive effect Negative counterfactual counterfactual effect counterfactual Conservative 90 90 80