Bootstrap-t Confidence Interval on Local Polynomial Regression Prediction

In local polynomial regression, prediction
confidence interval estimation using standard theory will
give coverage probability close to exact coverage
probability. However, if the normality assumption is not
met, the bootstrap method makes it possible to apply it. The
working principle of the bootstrap method uses the
resampling method where the sample data becomes a
population and there is no need to know the distribution of
the sample data is normal or not. Indiscriminate selection
of smoothing parameters allows scatterplot results from
local polynomial regressions to be rough and can even lead
to misleading statistical conclusions. It is necessary to
consider the optimal smoothing parameters to get local
polynomial regression predictions that are not overfitting
or underfitting. We offer two new algorithms based on the
nested bootstrap resampling method to determine the
bootstrap-t confidence interval in predicting local
polynomial regression. Both algorithms consider the
search for optimal smoothing parameters. The first
algorithm performs paired and residual bootstrap samples,
and the second algorithm performs based on residuals with
residuals. The first algorithm provides a scatterplot and
reasonable coverage probability on relatively large sample
data. In contrast, the second algorithm is more powerful for
each data size, including for relatively small sample data
sizes. The mean of the bootstrap-t confidence interval
coverage probability shows that the second algorithm for
second-degree local polynomial regression is better than
the other three. However, the larger the sample data size
gives, the closer the closer the average coverage
probability of the two algorithms is to the nominal
coverage probability