bbefkr (version 4.2)

funopare.kernel: Functional Nadaraya-Watson estimator

Description

It implements the functional Nadaraya-Watson estimator to estimate the regression function. It depends on the type of semi-metric used as well as the optimal selection of bandwidth parameter

Usage

funopare.kernel(Response, CURVES, PRED, bandwidth, ..., kind.of.kernel = "quadratic", 
	semimetric = "deriv")

Arguments

Response

A real-valued scalar response of length n

CURVES

An (n by p) matrix of discretised data of functional curves

PRED

An (n by k) matrix of discretised data of functional curves. PRED can be the same as the CURVES or the discretised data points of a new functional curve

bandwidth

A real-valued bandwidth parameter

...

Other arguments

kind.of.kernel

Type of kernel function. By default, it is the Epanechnikov kernel

semimetric

Type of semi-metric. By default, it is the semi-metric based on the qth order derivative, where q is an integer

Value

NWweit

Estimated Nadaraya-Watson weights

Estimated.values

Estimated values of the regression function

Predicted.values

Predicted values of the regression function

band

Bandwidth of the functional NW estimator

Mse

In-sample mean squared error

Details

The functional NW estimator of the conditional mean can be expressed as a weighted average of response variable: $\sum^n_{i=1}K_h(d(x_i,x))y_i/\sum^n_{i=1}K_h(d(x_i,x))$, where $K(\cdot)$ is a kernel function which integrates to one, it has continuous derivative on the function support range. The semi-metric $d$ is used to measure distances among curves. For a set of smooth curves, the semi-metric based on derivative should be considered. For a set of rough curves, the semi-metric based on functional principal components should be used. The bandwidth $h$ controls the tradeoff between squared bias and variance in the mean squared error

References

H. L. Shang (2013) Bayesian bandwidth estimation for a semi-functional partial linear regression model with unknown error density, Computational Statistics, in press.

H. L. Shang (2013) Bayesian bandwidth estimation for a nonparametric functional regression model with unknown error density, Computational Statistics and Data Analysis, 67, 185-198.

X. Zhang and R. D. Brooks and M. L. King (2009) A Bayesian approach to bandwidth selection for multivariate kernel regression with an application to state-price density estimation, Journal of Econometrics, 153, 21-32.

F. Ferraty, I. Van Keilegom and P. Vieu (2010) On the validity of the bootstrap in non-parametric functional regression, Scandinavian Journal of Statistics, 37, 286-306.

F. Ferraty and P. Vieu (2006) Nonparametric Functional Data Analysis: Theory and Practice, Springer, New York.

F. Ferraty and P. Vieu (2002) The functional nonparametric model and application to spectrometric data, Computational Statistics, 17, 545-564.

See Also

bayMCMC_np_global, bayMCMC_np_local, bayMCMC_semi_global, bayMCMC_semi_local

Examples

funopare.kernel(Response = simresp_np_normerr, CURVES = simcurve_smooth_normerr, 
	PRED = simcurve_smooth_normerr, bandwidth = 2.0, range.grid=c(0,pi), q=2, nknot=20)