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STAT 5102 HW12

Question 1 (large-sample likelihood ratio test for comparing Poisson samples): Let X1 , · · · , Xm ∼ Poisson(λ), Y1 , · · · , Yn ∼ Poisson(µ), and X1 , · · · , Xm , Y1 , · · · , Yn be jointly inde- pendent. Consider the testing problem

H0 : λ = µ against H1 : λ µ .

In this question, you are asked to derive the large sample likelihood ratio test by answering the following questions. For most sub-questions below, the correct answer only requires one or two lines of derivations/explanations. You may refer to the lecture note 10, page 39 – 43 for the large sample theory of likelihood ratio test and an example of applying the theory.

1. What is the maximum likelihood estimator, denoted by ( , ), for (λ, µ)? (Hint: if we only look at X1 , · · · , Xm , the MLE for λ is = )

2. What is the restricted MLE, denoted by ( 0 , 0 ), of (λ, µ) if we restrict that λ = µ so that H0 holds?

3. What is the number of free parameters under H0? (i.e., dim(Ω0)) What is the number of free parameters for the entire parameter space? (i.e., dim(Ω)).

4. Write down the formula for the likelihood ratio statistic Λ( x), and determine the test statis- tic and the rejection region according to the Wilk’s theorem so that the size of the test is approximately α (assuming m and n are large so the theorem applies).

Question 2 (invert conﬁdence intervals) A pharmaceutical company develops an oral antivi- ral medicine (molnupiravir) for COVID-19. The following results are part of a scientiﬁc report in a study on the eﬀectiveness of the medicine: 1

In the analysis of all participants who had undergone randomization, the percentage of participants who were hospitalized or died through day 29 was lower in the molnupiravir group than in the placebo group (6.8% [48 of 709] vs. 9.7% [68 of 699]; diﬀerence, −3.0 percentage points; 95% CI, −5.9 to −0.1).

Based on the above results, answer the following questions:

1. Can we conclude that molnupiravir (when compared with the placebo) reduced the percentage of hospitalization or death (through day 29), at the signiﬁcance level α = 0.05?

2. Can we conclude that molnupiravir (when compared with the placebo) reduced the percentage of hospitalization or death (through day 29), at the signiﬁcance level α = 0.01?