=+ij = , where is an unknown parameter, and ni = l, 1 i

=+ijβ = μ, where μ is an unknown parameter, and ni = l, 1 ≤ i ≤ m. Show that in this case, the left side of

(12.46) is equal to g(μ) = σ−2 l k=0 φ(k, μ)P(y1· = k), where φ(t, μ) is the unique solution u to the equation σ−2u + lh(μ + u) = t and P(y1· = k) = l k



E

 {exp(μ + η)}k

{1 + exp(μ + η)}l



with the expectation taken with respect to η ∼ N(0, σ2).

(v) Take σ2 = 1. Make a plot of g(μ) against μ and show that g(μ) is not identical to zero (function). [Hint: You make use numerical integration or the Monte Carlo method to evaluate the expectations involved in (iv).]