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(Solved): Problem 3. Comparing two Poisson CIs. Let Y_(1),dots,Y_(n)^( iid )Pois(\lambda ), and let hat(\l ...

Problem 3. Comparing two Poisson CIs. Let

Y_(1),dots,Y_(n)∼^( iid )Pois(\lambda )

, and let

hat(\lambda )()/(b)=ar (Y)

. In class, we discussed two possible

1-\alpha

asymptotic CIs for

\lambda

. Interval 1. Using a normal approximation for

hat(\lambda )

, namely

\sqrt(n)(hat(\lambda )-\lambda )->dN(0,\lambda )

, we get the interval

[(hat(\lambda ))-z((\alpha )/(2))\sqrt(((hat(\lambda )))/(n)),(hat(\lambda ))+z((\alpha )/(2))\sqrt(((hat(\lambda )))/(n))].

If the left endpoint is negative, we can replace it by 0 without affecting the coverage probability, since the true

\lambda

is never negative. Interval 2. Using a normal approximation for

\sqrt(h)at(\lambda )

, namely

\sqrt(n)(\sqrt(h)at(\lambda )-\sqrt(\lambda ))->dN(0,(1)/(4))

, we get the following interval for

\sqrt(\lambda )

:

[\sqrt(h)at(\lambda )-z((\alpha )/(2))\sqrt((1)/(4n)),\sqrt(h)at(\lambda )+z((\alpha )/(2))\sqrt((1)/(4n))].

Again, if the left endpoint is negative, we can replace it by 0 . We can then transform both endpoints to get an interval for

\lambda

:

max(0,\sqrt(h)at(\lambda )-z((\alpha )/(2))\sqrt((1)/(4n)))^(2),(\sqrt(h)at(\lambda )+z((\alpha )/(2))\sqrt((1)/(4n)))^(2)

While both intervals are asymptotically valid, they have different coverage probabilities in finite samples. Conduct a simulation study to compare the coverage of the two intervals for the 9 com- binations of sample size

n=10,30,100

and true parameter values

\lambda =0.1,0.5,1

. You can take the target coverage probability to be

1-\alpha =0.95

. For each combination, perform at least 100,000 simulations. In each simulation, you can choose to simulate

hat(\lambda )

directly from

(1)/(n)Pois(n\lambda )

Y_(1),dots,Y_(n)

. (In R, this would be lambdahat

=

rpois(1,

n*

lambda)/n.) Report the simulated coverage probabilities in two tables, and briefly comment on your findings.

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