Confidence interval Concept Check
1 point possible (graded)
As in the previous section, let X1,…,Xn∼iidexp(λ) . Let
λˆn:=n∑ni=1Xi
denote an estimator for λ . We know by now that λˆn is a consistent and asymptotically normal estimator for λ .
Recall qα/2 denote the 1−α/2 quantile of a standard Gaussian. By the Delta method:
λ∈[λˆn−qα/2λn−−√,λˆn+qα/2λn−−√]=:I
with probability 1−α . However, I is still not a confidence interval for λ .
Why is this the case?
This is the case because the interval I is based on estimating the standard deviation of lambda, denoted by λn. However, λn itself is also an estimator and thus subject to sampling variability. Therefore, substituting λn for its estimate λˆn introduces an additional source of uncertainty, which affects the accuracy of the interval I. This means that the interval I may not always contain the true value of λ with the desired confidence level of 1-α. Therefore, I cannot be considered a confidence interval for λ.
To construct a valid confidence interval for λ, we need to account for the uncertainty in estimating λn. This can be achieved by using a pivotal quantity, which is a function of the data and the unknown parameter that has a known distribution. By using a pivotal quantity, we can derive a confidence interval that properly captures the uncertainty in estimating λ and provides the desired confidence level.