Exercise 5

Question 1.

The advertisement of the fast food chain of restaurants "FastBurger" claims that the average waiting time to get food in its branches is 30 seconds unlike 50 seconds for their competitors. Mr. Skeptic does not believe much in advertising and decided to test its truth by the following test: he will go to one of "FastBurger" branches, measure the waiting time, and if it is less than 40 seconds (the critical waiting time he fixed) he would believe in its advertisement. Otherwise, he would conclude that service in "FastBurger" is not faster than in other fast food companies. Mr. Skeptic also assumes that waiting time is exponentionally distributed.
  1. What are the hypotheses Mr. Skeptic tests? Calculate the probabilities of errors of both types for his test.
  2. Can you can suggest a better test to Mr. Skeptic with the same significance level?
  3. Mrs. Skeptic, Mr. Skeptic's wife, agreed in general with the test her husband has used but decided instead to fix a critical waiting time that will minimize the sum of probabilities of both error types. What will be her critical waiting time?
  4. Calculate the probabilities of the errors for Mrs. Skeptic's test and compare them with those of her husband. Comment the results.

Question 2.

Let Y1,...,Yn ~ U[0;θ].
  1. Derive the MP test at level α for testing two simple hypotheses H0:θ=θ0 vs. H1:θ=θ1, where θ1 > θ0.
  2. Calculate the power of the MP test.

Question 3.

Let Y1,...,Yn be a random sample from a distribution with the density function fθ=(y/θ)e-y2/(2θ), y >0, θ>0
  1. Find the MP test at level α for testing two simple hypotheses H0:θ=θ0 vs. H1:θ=θ1, where θ1 > θ0.
  2. Is there a UMP test at level α for testing the one-sided hypotheses H0:θ ≤ θ0 vs. H1:θ>θ0? What is its power function?
(Hint: show that Y2i ~ exp(1/(2θ)))

Question 4.

Suppose that the distribution of the data belongs to a one-dimensional exponential family with a parameter θ, and a natural parameter η=c(θ) is a strictly increasing function of θ.
  1. Show that the data has a monotone likelihood ratio in T(y) and derive a form of the UMP test for testing H0: θ ≤ θ0 vs. H1: θ > θ0.
  2. Apply the results of the previous paragraph to find UMP tests with the given significance level based on a sample of size n from a) exp(θ) and b) N(μ,θ) with the known μ.

Question 5.

  1. Given a random sample Y1,...,Yn from a normal distribution N(μ,σ2), where both parameters are unknown, derive the generalized likelihood ratio test (GLRT) for testing H0: σ2 ≤ σ02 vs. H1: σ2 > 02 with a given significance level α. Is it the UMP test?
  2. Repeat the previous paragraph for testing H0:σ2 = σ02 vs. H1: σ2 ≠ 02. Show that the GLRT accepts the null hypothesis when C1 < Σ(Yi-average(Y))2 < C2, where the critical values C1 and C2 satisfy: a)Χ2n-1(C2)-Χ2n-1(C1)=1-α ; b)C2-C1=n log(C2/ C1) and Χ2n-1 is the cdf of χ2n-1.