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 be the standard normal c.d.f. and let  = ϕ. Show P{tn ≤ t} = (t) − 1 4n

(t + t 3

)ϕ(t) + O(n−2

) (13.63)

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

1/2

]} .

By Taylor expansion inside the expectation, along with moments of S2 n , one can deduce (13.63).