Prediction interval

A prediction interval is a random interval designed to contain a future observation with prescribed repeated-sampling probability. Unlike a confidence interval for a mean response, it includes both uncertainty in the fitted mean and the new observation's irreducible random variation.

For independent normal random variables with unknown common mean and variance, let ${\hat{\sigma }}^2=n^{-1}\sum (X_i-\bar{X}{)}^2$ and $n\ge 2$. The standardized future difference $X_0-\bar{X}$ is normal with variance ${\sigma }^2(1+1/n)$ and is independent of the residual sum of squares. Thus the ratio using the displayed standard error has Student's t-distribution with $n-1$ degrees of freedom. This is unconditional repeated-sampling prediction coverage, not a confidence interval for the mean.

For the full-rank normal linear model $Y=X\beta +\epsilon$ and an independent future response $Y^{\ast }=x^{\ast T}\beta +{\epsilon }^{\ast }$, putThen an exact $(1-\alpha )$ prediction interval isThe additional one inside the square root is the future observation's irreducible error variance; a confidence interval for its mean omits it.

When a full-rank normal regression with positive residual degrees of freedom happens to fit all supplied observations exactly, its residual estimate of variance is zero. The ordinary Studentized prediction-interval formula formally collapses, but the statistic divides by zero. Under positive true Gaussian variance such samples form a null set; exact rounded or constructed data must not be treated as proof that future randomness vanishes.

Suppose independent future log responses satisfy $Z_j=x_j^T\beta +{\epsilon }_j$, where the errors have variance ${\sigma }^2$. Conditional on a fitted normal linear model, the predicted log ratio $Z_1-Z_2$ has estimated mean $(x_1-x_2{)}^T\hat{\beta }$ and estimated varianceA Student t interval on this logarithmic scale can be exponentiated to obtain a prediction interval for the positive ratio.