Foralargenumber n of independentPoissonrandomvariables {Yi}, with = E(Yi), consider testing H0: = 0. (a) Showthatthescoreteststatisticis

Foralargenumber n of independentPoissonrandomvariables {Yi}, with μ = E(Yi), consider testing H0: μ = μ0.

(a) Showthatthescoreteststatisticis Z =

º

n( ¯ Y − μ0)~º

μ0.

(b) Showthatthe Waldteststatisticis Z =

º

n( ¯ Y − μ0)~

º

¯ Y . Under H0, why wouldyou expectthenormaldistributionapproximationtobebetterforthescoreteststatistic?

(c) Showthatthevalueofthelikelihood-ratioteststatisticforobserveddata {yi} is 2(L1 − L0) = 2[n(μ0 − ¯y) + n¯y log(¯y~μ0)].

(d) Explainhowthesetestscanbeusedtoconstructconfidenceintervals.Illustrateforthe Waldandscoreconfidenceintervals,andexplainwhyyouwouldexpectthescoremethod to performbetter.