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

→ V1/2 , where V is Poisson with mean δ2, and so if z1−α is not an integer, PFn {tn > tn−1,1−α} → P{V1/2 > z1−α} .

(ii) Show, for α < 1/2, the limiting power of the t-test against Fn satisfies P{V1/2 > z1−α} ≤ 1 − P{V = 0} = exp(−δ2

) .

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

(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.

Section13.5