Confidence intervals based on the CLT: ever useful?

Question

Suppose, for concreteness, that I am trying to estimate the mean of a population using a random sample of size $N$.

Many elementary books discuss forming a confidence interval for the population mean by using the central limit theorem to argue that the sample mean is approximately normally distributed.

However, the central limit theorem is a theorem about the limit as $N \to \infty$. But if $N$ were really large then the width of the confidence interval would be really small and giving just a point estimate would be good enough.

So it seems that we are implicitly assuming that there is a range of $N$ for which the CLT is not too bad an assumption but for which $N$ is not so large as to shrink the confidence interval to practically a point.

My question is: is there any basis for making this implicit assumption? Is there any way to judge what this range of useful $N$ is for a particular application?

Answer 1

The speed of convergence towards Gaussian depends on the exact laws, and in particular on the values of the cumulants:

• if there are no cumulant of order 1 or 2, i.e. no mean or no variance, you cannnot expect convergence to the normal;
• if cumulants of order 1 and 2 exist, it all depends on the cumulants of higher order. The smaller they are, the faster the sum converges to the normal, because the starting law is already close to the normal.

For instance, it is generally considered that a $Poi(20)$ law can be well approximated by a $N(20,20)$. However, $\sqrt{20}$ is still non-negligible compared to $20$.

Bounds for the difference between normal and exact are given by the Berry - Esseen theorem, but I am not sure they would be applicable in practice because you would need to know the third moment, which is difficult to estimate from the data.

In practice, judgment is required, and the approximation must be avoided whenever laws show a strong difference with normality: very non-centered or heavy tails (see for instance a qq plot).

Bootstrapping might also help: if your empirical data is close enough to normal, then bootstrapping and reestimating should give something close again to the approximate formula.