When sampling from a normal distribution, one can derive an Edgeworth expansion for the t-statistic as follows. Suppose X1,...,Xn are i.i.d.

N(µ, σ2) and let tn = n1/2(X¯n − µ)/Sn, where S2 n is the usual unbiased estimate of σ2. Let Φ be the standard normal c.d.f. and let Φ
= ϕ. Show P{tn ≤ t} = Φ(t) − 1 4n(t + t 3

)ϕ(t) + O(n−2

) (11.91)

as follows. It suffices to let µ = 0 and σ = 1. By conditioning on Sn, we can write P{tn ≤ t} = E{Φ[t(1 + S2 n − 1)1/2

]} .

By Taylor expansion inside the expectation, along with moments of S2 n, one can deduce (11.91).