Suppose that Y has a bin(n, ) distribution. For the model, logit() = , consider testing H

Suppose that Y has a bin(n,π ) distribution. For the model, logit(π) = α, consider testing H₀: α = 0 (i.e., π = 0.5). Let π̂ = y/n.

a. From Section 3.1.6, the asymptotic variance of α̂ = logit(π̂) is [nπ(l – π)]^–1. Compare the estimated SE for the Wald test and the SE using the null value of π, using test statistic [logit(π̂)/SE]². Show that the ratio of the Wald statistic to the statistic with null SE equals 4π̂(1 – π̂). What is the implication about performance of the Wald test if |α| is large and π̂ tends to be near 0 or 1?

b. Wald inference depends on the parameterization. How does the comparison of tests change with the scale [(π̂ – 0.5)/SE]², where SE is now the estimated or null SE of π̂?

c. Suppose that y = 0 or y = n. Show that the Wald test in part (a) cannot reject H₀: π = π₀ for any 0 < π₀ < 1, whereas the Wald test in part (b) rejects every such π₀.