Bounding the mean vs bounding the next draw

Question

You collect n i.i.d. observations (say daily returns), with sample mean \bar x and sample standard deviation s. No regression, just one number and its scatter. Distinguish a confidence interval for the population mean \mu from a prediction interval for the next single observation. Which is wider, and what happens to each as n 	o \infty?

Accepted Answer

The two answer different questions. Confidence interval for the mean \mu. Its only uncertainty is how well \bar x pins down \mu. The standard error of the mean is s/\sqrt{n}, giving \bar x \pm t{n-1}\,\frac{s}{\sqrt{n}}. As n 	o \infty this width 	o 0: with enough data you know the average return arbitrarily precisely. Prediction interval for the next observation x{	ext{new}}. A future draw is \mu plus its own noise term of variance \sigma^2. On top of the estimation error in \bar x you must swallow that fresh \sigma^2: \operatorname{Var}(x{	ext{new}} - \bar x) = \sigma^2 + \frac{\sigma^2…

Bounding the mean vs bounding the next draw

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