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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 )`

instead of simulating

`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.