Homework 10. Review of point, interval and Bayesianestimation for Midterm 2
Due Thursday, 11/17, by 11 am
(1) Review lecture notes, homework 6, 7, 8, 9, relevant sections of chapters 10
and 11.
(2) Confidence Intervals
The length of the skulls of 10 fossil skeletons of an extinct species of bird
has a mean of 5.68 cm and a standard deviation of 0.29 cm. Assuming that
such measurements are normally distributed, find a 98% confidence interval
for the mean length of the skulls of this species of bird. You don’t have to
simplify your answer or explicitly find the score that you are using.
(3) Method of maximal likelihood and method of moments
Xi has probability density θ1 · e−(x−δ)/θ for x ≥ δ, and 0 otherwise. I.i.d.
sample of size 12 has sample mean 4, variance 16, minimum 1.
(a) Find the maximal likelihood estimates for θ and δ.
(b) Find the method of moments estimates for θ and δ.
(4) Properties of point estimators
X1 , X2 , . . . , Xn are i.i.d. random variables, Xi has probability density
e−(x−θ) for θ ≤ x < ∞, θb := min{X1 , X2 , . . . , Xn }.
(a) Argue without computation that θb is a biased estimator of θ.
(b) Find cn such that cn + θb is an unbiased estimator of θ.
(c) Is θb sufficient? Justify your answer.
(d) Is θb consistent? Justify your answer.
(5) Bayesian estimation
(X1 , X2 , . . . , X10 ) are i.i.d. random variables with Xi ∼ U [0, θ]. θ has prior
distribution U [0, 10]. Sample maximum is 5. Find mean of the posterior
distribution.
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