Hi,
My first question is what is your n when you say fixed n. I assume the
lambda is the mean of the poisson distribution that you want to take sample
from.
Another question is about the sample size. It does not make too much
sense to make a sample of size 1.
Let's assume that you want to fix the sample size to be 100 and change
lambda from 0.1 to 5 with an increment of 0.1. For each lambda, plan to
run, say, 1000 times. Then the following will be my approach. Recall that
the function cover returns 1 when lambda is in the confidence interval and 0
otherwise. resulting_matrix is created with size 50 x 2 with 0 populated.
The matrix is to store lambda and the proportion of samples with lambda
inside the confidence interval calculated from samples. With the resulting
matrix, one can see that lambdas are in the first column with values of 0.1
to 5 with increment of 0.1. The corresponding proportions are in the second
column. All of the proportions are from 0.917 to 0.969 as the last line
shows.
Hope this helps.
> cover <- function(lambda, sample.size, significance.level) {
+ x <- rpois(sample.size,lambda)
+ estimate <- mean(x)
+ lower <- estimate - qnorm(1 - significance.level/2) *
sqrt(estimate/sample.size)
+ upper <- estimate + qnorm(1 - significance.level/2) *
sqrt(estimate/sample.size)
+ if (lambda > lower & lambda < upper){1}else{0}
+ }
> resulting.matrix <- matrix(0, nrow=50,ncol=2)
> for (i in 1:50)
+ {
+ resulting.matrix[i,1] <- 0.1 * i
+ resulting.matrix[i,2] <- mean(sapply(1:1000,function(x)
cover(0.1*i,100,0.05)))
+ }
> resulting.matrix
[,1] [,2]
[1,] 0.1 0.917
[2,] 0.2 0.949
[3,] 0.3 0.928
[4,] 0.4 0.939
[5,] 0.5 0.943
[6,] 0.6 0.949
[7,] 0.7 0.942
[8,] 0.8 0.939
[9,] 0.9 0.945
[10,] 1.0 0.943
[11,] 1.1 0.962
[12,] 1.2 0.933
[13,] 1.3 0.947
[14,] 1.4 0.951
[15,] 1.5 0.946
[16,] 1.6 0.939
[17,] 1.7 0.946
[18,] 1.8 0.953
[19,] 1.9 0.964
[20,] 2.0 0.943
[21,] 2.1 0.937
[22,] 2.2 0.944
[23,] 2.3 0.945
[24,] 2.4 0.950
[25,] 2.5 0.954
[26,] 2.6 0.946
[27,] 2.7 0.945
[28,] 2.8 0.949
[29,] 2.9 0.956
[30,] 3.0 0.953
[31,] 3.1 0.941
[32,] 3.2 0.949
[33,] 3.3 0.943
[34,] 3.4 0.956
[35,] 3.5 0.950
[36,] 3.6 0.944
[37,] 3.7 0.952
[38,] 3.8 0.958
[39,] 3.9 0.938
[40,] 4.0 0.944
[41,] 4.1 0.950
[42,] 4.2 0.945
[43,] 4.3 0.948
[44,] 4.4 0.962
[45,] 4.5 0.969
[46,] 4.6 0.956
[47,] 4.7 0.950
[48,] 4.8 0.955
[49,] 4.9 0.946
[50,] 5.0 0.945
> range(resulting.matrix[,2])
[1] 0.917 0.969
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