## On-line predictive linear regression.(English)Zbl 1160.62065

Summary: We consider the on-line predictive version of the standard problem of linear regression; the goal is to predict each consecutive response given the corresponding explanatory variables and all the previous observations. The standard treatment of prediction in linear regression analysis has two drawbacks: (1) the classical prediction intervals guarantee that the probability of error is equal to the nominal significance level $$\varepsilon$$, but this property per se does not imply that the long-run frequency of error is close to $$\varepsilon$$; (2) it is not suitable for prediction of complex systems as it assumes that the number of observations exceeds the number of parameters.
We state a general result showing that in the on-line protocol the frequency of error for the classical prediction intervals does equal the nominal significance level, up to statistical fluctuations. We also describe alternative regression models in which informative prediction intervals can be found before the number of observations exceeds the number of parameters. One of these models, which only assumes that the observations are independent and identically distributed, is popular in machine learning but greatly underused in the statistical theory of regression.

### MSC:

 62J05 Linear regression; mixed models 62G08 Nonparametric regression and quantile regression 62M20 Inference from stochastic processes and prediction 68T05 Learning and adaptive systems in artificial intelligence

### Software:

PredictiveRegression
Full Text:

### References:

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