## Statistical Inference

If you denote the *i*th measurement of the response by *y*_{i} and the
corresponding measurement of predictors by *x*_{i}, then

where *g* is the regression function and are independent
random errors with mean zero. If the errors are normally distributed
with constant variance, then you can obtain confidence intervals for
the predictions from PROC LOESS. You can also obtain confidence limits
in the case where is heteroscedastic but
has constant variance and *a*_{i} are
a priori weights that are specified using the WEIGHT statement
of PROC LOESS. You can do inference in the case in which the error
distribution is symmetric by using iterative reweighting.
Formulae for doing statistical inference under the preceding conditions
can be found in Cleveland and Grosse (1991) and Cleveland, Grosse, and Shyu (1992).
The main result of their
analysis is that a standardized residual for a loess model follows
a *t* distribution with degrees of freedom, where is called
the "lookup degrees of freedom." is a function of the smoothing
matrix *L*, which defines the linear relationship between the fitted and
observed dependent variable values of a loess model.

The determination of is computationally expensive and
is not done by default. It is computed if you specify the DFMETHOD=EXACT option
in the MODEL statement. It is also computed if you specify any of the
options CLM, STD, or T in the MODEL statement.

If you specify the CLM option in the MODEL statement, confidence limits are
added to the OutputStatistics table. By default, 95% limits are
computed, but you can change this by using the ALPHA= option in the MODEL statement.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.