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Erschienen in:
2010 | OriginalPaper | Buchkapitel
Bootstrap Calibration in Functional Linear Regression Models with Applications
verfasst von : Wenceslao González-Manteiga, Adela Martínez-Calvo
Erschienen in: Proceedings of COMPSTAT'2010
Verlag: Physica-Verlag HD
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Our work focuses on the functional linear model given by
$$Y=\langle\theta,X\rangle+\epsilon,$$
where
Y
and
ε
are real random variables,
X
is a zero-mean random variable valued in a Hilbert space
$$(\mathcal{H},\langle\cdot,\cdot\rangle)$$
, and
$$\theta\in\mathcal{H}$$
is the fixed model parameter. Using an initial sample
$$\{(X_i,Y_i)\}_{i=1}^n$$
, a bootstrap resampling
$$Y_i^{*}=\langle\hat{\theta},X_i\rangle+\hat{\epsilon}_i^{*}$$
,
$$i=1,\ldots,n$$
, is proposed, where
$$\hat{\theta}$$
is a general pilot estimator, and
$$\hat{\epsilon}_i^{*}$$
is a naive or wild bootstrap error. The obtained consistency of bootstrap allows us to calibrate distributions as
$$P_X\{\sqrt{n}(\langle\hat{\theta},x\rangle-\langle\theta,x\rangle)\leq y\}$$
for a fixed
x
, where
P
X
is the probability conditionally on
$$\{X_i\}_{i=1}^n$$
. Different applications illustrate the usefulness of bootstrap for testing different hypotheses related with
θ
, and a brief simulation study is also presented.