Local polynomial regression fitting with Epanechnikov weights

Description

Fast and stable algorithm for nonparametric estimation of regression functions and their derivatives via local polynomials with Epanechnikov weight function.

Usage

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lpepa(x, y, bandwidth, deriv = 0, n.out = 200, x.out = NULL,
      order = deriv+1, mnew = 100, var = FALSE)

Arguments

x

vector of design points, not necessarily ordered.

y

vector of observations of the same length as x.

bandwidth

bandwidth(s) for nonparametric estimation. Either a number or a vector of the same length as x.out.

deriv

order of derivative of the regression function to be estimated; defaults to deriv = 0.

n.out

number of output design points where the function has to be estimated. The default is n.out=200.

x.out

vector of output design points where the function has to be estimated. The default value is an equidistant grid of n.out points from min(x) to max(x).

order

integer, order of the polynomial used for local polynomials. Must be <= 10 and defaults to order = deriv+1.

mnew

integer forcing to restart the algorithm after mnew updating steps. The default is mnew = 100. For mnew = 1 you get a numerically “super-stable” algorithm (see reference SBE\&G below).

var

logical flag: if TRUE, the variance of the estimator proportional to the residual variance is computed (see details).

Details

More details are described in the first reference SBE\&G (1994) below. In S\&G, a bad finite sample behaviour of local polynomials for random designs was found. For practical use, we therefore propose local polynomial regression fitting with ridging, as implemented in the function lpridge. In lpepa, several parameters described in SBE\&G are fixed either in the fortran routine or in the R-function. There, you find comments how to change them.

For var=TRUE, the variance of the estimator proportional to the residual variance is computed, i.e., the exact finite sample variance of the regression estimator is var(est) = est.var * sigma^2.

Value

a list including used parameters and estimator.

x

vector of ordered design points.

y

vector of observations ordered according to x.

bandwidth

vector of bandwidths actually used for nonparametric estimation.

deriv

order of derivative of the regression function estimated.

x.out

vector of ordered output design points.

order

order of the polynomial used for local polynomials.

mnew

force to restart the algorithm after mnew updating steps.

var

logical flag: whether the variance of the estimator was computed.

est

estimator of the derivative of order deriv of the regression function.

est.var

estimator of the variance of est (proportional to residual variance).

References

See also http://www.unizh.ch/biostat/ under ‘Manuscripts’ etc.

- Numerical stability and computational speed:
B. Seifert, M. Brockmann, J. Engel and T. Gasser (1994) Fast algorithms for nonparametric curve estimation. J. Computational and Graphical Statistics 3, 192–213.

- Statistical properties:
Seifert, B. and Gasser, T. (1996) Finite sample variance of local polynomials: Analysis and solutions. J. American Statistical Association 91(433), 267–275.

Seifert, B. and Gasser, T. (2000) Data adaptive ridging in local polynomial regression. J. Computational and Graphical Statistics 9, 338–360.

Seifert, B. and Gasser, T. (1998) Ridging Methods in Local Polynomial Regression. in: S. Weisberg (ed), Dimension Reduction, Computational Complexity, and Information, Vol.30 of Computing Science \& Statistics, Interface Foundation of North America, 467–476.

Seifert, B. and Gasser, T. (1998) Local polynomial smoothing. in: Encyclopedia of Statistical Sciences, Update Vol.2, Wiley, 367–372.

Seifert, B., and Gasser, T. (1996) Variance properties of local polynomials and ensuing modifications. in: Statistical Theory and Computational Aspects of Smoothing, W. Härdle, M. G. Schimek (eds), Physica, 50–127.

See Also

lpridge, and also lowess and loess which do local linear and quadratic regression quite a bit differently.

Examples

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data(cars)
attach(cars)

epa.sd <- lpepa(speed,dist, bandw=5)		# local polynomials

plot(speed, dist, main = "data(cars) & lp epanechnikov regression")
lines(epa.sd$x.out, epa.sd$est,  col="red")
lines(lowess(speed,dist, f= .5), col="orange")
detach()

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