Prediction interval

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In statistics, a prediction interval bears the same relationship to a future observation that a confidence interval bears to an unobservable population parameter. Prediction intervals predict the distribution of individual points, whereas confidence intervals estimate the true population mean or other quantity of interest that cannot be observed.

[edit] Example

Suppose one has drawn a sample from a normally distributed population. The mean and standard deviation of the population are unknown except insofar as they can be estimated based on the sample. It is desired to predict the next observation. Let n be the sample size; let μ and σ be respectively the unobservable mean and standard deviation of the population. Let X1, ..., Xn, be the sample; let Xn+1 be the future observation to be predicted. Let

\overline{X}_n=(X_1+\cdots+X_n)/n

and

S_n^2={1 \over n-1}\sum_{i=1}^n (X_i-\overline{X}_n)^2.

Then it is fairly routine to show that

{X_{n+1}-\overline{X}_n \over \sqrt{S_n^2+S_n^2/n}}={X_{n+1}-\overline{X}_n \over S_n\sqrt{1+1/n}}

has a Student's t-distribution with n − 1 degrees of freedom. Consequently we have

\Pr\left(\overline{X}_n-T_a S_n\sqrt{1+(1/n)}\leq X_{n+1} \leq\overline{X}_n+T_a S_n\sqrt{1+(1/n)}\,\right)=p

where Ta is the 100((1 + p)/2)th percentile of Student's t-distribution with n − 1 degrees of freedom. Therefore the numbers

\overline{X}_n\pm T_a {S}_n\sqrt{1+(1/n)}

are the endpoints of a 100p% prediction interval for Xn + 1.

[edit] See also

[edit] References

  • Chatfield, C. (1993) "Calculating Interval Forecasts," Journal of Business and Economic Statistics, 11 121-135.
  • Meade, N. and T. Islam (1995) "Prediction Intervals for Growth Curve Forecasts," Journal of Forecasting, 14 413-430.
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