Let Yn,1,...,Yn,n be i.i.d. bernoulli variables with success probability pn, where npn = and 1/2 =

Let Yn,1,...,Yn,n be i.i.d. bernoulli variables with success probability pn, where npn = λ and λ1/2 = δ. Let Un,1,...,Un,n be i.i.d. uniform variables on (−τn, τn), where τ 2 n = 3p2 n. Then, let Xn,i = Yn,i + Ui, so that Fn is the distribution of Xn,i. (Note that n1/2µ(Fn)/σ(Fn) = δ.)

(i) If tn is the t-statistic, show that, under Fn, tn d

→ V 1/2 , where V is Poisson with mean δ2, and so if z1−α is not an integer, PFn {tn > tn−1,1−α} → P{V 1/2 > z1−α} .

(ii) Show, for α < 1/2, the limiting power of the t-test against Fn satisfies P{V 1/2 > z1−α} ≤ 1 − P{V = 0} = exp(−δ2

) .

This is strictly smaller than 1 − Φ(z1−α − δ) if and only if

Φ(z1−α − δ) < exp(−δ2

) .

Certainly, for small δ, this inequality holds, since the left hand side tends to 1−α

as δ → 0 while the right hand side tends to 1.