The quantity \(-l''(\hat{\theta};x)\) is called the “observed information,” and \(1/ \sqrt{-l''(\hat{\theta};x)}\) is an approximate standard error for \(\hat{\theta}\). As the loglikelihood becomes more sharply peaked about the MLE, the second derivative drops and the standard error goes down.

When calculating asymptotic confidence intervals, statisticians often replace the second derivative of the loglikelihood by its expectation; that is, replace \(-l''(\hat{\theta};x)\) by the function

\(I(\theta)=-E[l''(\hat{\theta};x)]\),

which is called the expected information or the Fisher information. In that case, the 95% confidence interval would become

\(\hat{\theta}\pm 1.96 \dfrac{1}{\sqrt{I(\hat{\theta})}}\) (3)

When the sample size is large, the two confidence intervals (2) and (3) tend to be very close. In some problems, the two are identical.

Next, what follows are a few examples of asymptotic confidence intervals.