Exercise 4

Question 1.

Let Y1,..., Y2 ~ U[0;θ].
  1. Verify that Ymax/θ is a pivotal quantity.
    (hint: the distribution of Ymax was derived, in particular, in Exercise 3, Question 3)
  2. Show that [Ymax;α-1/nYmax] is a 100(1-α)% confidence interval for θ.

Question 2.

  1. Suppose [T1(y),T2(y)] is a 100(1-α)% confidence interval for a parameter θ. Suppose also that a function g(θ) is increasing withing the range of θ. Show that [g(T1(y)),g(T2(y))] is a 100(1-α)% confidence interval for g(θ).
  2. What is a 100(1-α)% confidence interval for g(θ) if g(θ) is decreasing?

Question 3.

  1. Let Y be a continuous random variable with c.d.f. F(y). Define a random variable Z=F(Y). Show that Z ~ U[0,1] for any F.
  2. Let y1,...,yn be a sample from a distribution F(y;θ), where θ is an unknown parameter. Show that h(y;θ)=-(ln F(y1;θ) + ... + ln F(yn;θ)) is a pivotal quantity.
  3. Find the distribution of h(y;θ)
    (hint: recall a distribution of a function of a random variable from a basic probability course to find first the distribution of ln F(yi;θ) and use various properties of χ2 distributions)

Question 4.

Let y1 and y2 be independent observations from U[θ-1,θ+1].
  1. Show that y-θ is a pivotal quantity and use this fact to derive a confidence interval of the form [(y1+y2)/2 - δ, (y1+y2)/2 + δ] for θ.
    (hint: again, you'll probably need to recall the distribution of a sum of two independent random variables from a basic probability course and to apply the general result for your specific case)
  2. Find the 80% confidnce interval for θ from a sample 4.0, 5.9. Are the results confusing? (do not feel ashamed to share your confusion!)

Question 5.

Given a sample y1,...,yn from a p-dimensional multinormal distribution with an uknown mean vector μ and a known variance-covariance matrix V, find
  1. a conservative confidence region for μj, j=1,...,p using Bonferroni approach
  2. an exact confidence region for μj, j=1,...,p
    (hint: recall that if Y ~ N(μ,V), then (Y-μ)'V-1(Y-μ) ~ χ2p).