Bootstrap Confidence Interval For Nonparametric Regression

Sri Haryatmi Kartiko
Math Dept, Gadjahmada University,
Yogyakarta, Indonesia

Abstract
In regression, pairs of data (X1, Y1), . . . , (Xn, Yn) are observed, and the objective is to estimate m(x) = E[Y |X = x]. Nonparametric techniques allow m to be estimated without any assumptions on its underlying form. Unfortunately, this flexibility can make it difficult to ascertain whether observed features are physically meaningful or the result of random variability in the data. Confidence intervals gives a mechanism to assess the impact of this variability on the estimate.

The bootstrap approach to constructing confidence intervals for kernel regression estimates is proposed. First, kernel Nadaraya Watson regression estimator is used to calculate the residuals, which approximate the true unobserved errors.Second, the residuals are resampled, to create ”new” errors. Third, the resampled errors are added to an estimate of the regression function, creating a
new bootstrap data set (X1, Y_1 ), . . . , (Xn, Y_n ). Finnaly, the bootstrap data set is smoothed, taking into account the bias and used for constructing quantile bootstrap confidence interval. Simulation study is conducted to see that the coverage probability approximates its level of significance.

Key words : kernel function, bias, bootstrap, quantile, coverage probability.

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