. Consider testing the hypotheses H 0 : λ = 2 versus Ha : λ > 2 using 25 observations from a Poisson( λ ) model. Rote application of the central limit theorem would suggest rejecting H 0 at α = 0 . 05...

.
Consider testing the hypotheses
H0 :λ
= 2 versus
Ha
:λ >
2 using 25 observations from a Poisson(λ) model. Rote application of the central limit theorem would suggest rejecting
H0 at
α
= 0.05 when
Z
≥ 1.645, where
Z
= (¯X
− 2)/
√ 2/25.

a.
Estimate the size of this test (i.e., the type I error rate) using five Monte Carlo approaches: standard, antithetic, importance sampling with unstandardized and standardized weights, and importance sampling with a control variate as in Example 6.12. Provide a confidence interval for each estimate. Discuss the relative merits of each variance reduction technique, and compare the importance sampling strategies with each other. For the importance sampling approaches, use a Poisson envelope with mean equal to the
H0 rejection threshold, namely
λ
= 2.4653.

b.
Draw the power curve for this test for
λ
∈ [2.2,
4], using the same five techniques. Provide point wise confidence bands in each case. Discuss the relative merits of each technique in this setting. Compare the performances of the importance sampling strategies with their performance in part (a).