# np.gcv

0th

Percentile

##### Generalized cross-validation bandwidth selection in nonparametric regression models

From a sample ${(Y_i, t_i): i=1,...,n}$, this routine computes an optimal bandwidth for estimating $m$ in the regression model $$Y_i= m(t_i) + \epsilon_i.$$ The regression function, $m$, is a smooth but unknown function. The optimal bandwidth is selected by means of the generalized cross-validation procedure. Kernel smoothing is used.

Keywords
regression, time series, Statistical Inference, Nonparametric Statistics
##### Usage
np.gcv(data = data, h.seq=NULL, num.h = 50, estimator = "NW",
kernel = "quadratic")
##### Arguments
data
data[, 1] contains the values of the response variable, $Y$; data[, 2] contains the values of the explanatory variable, $t$.
h.seq
sequence of considered bandwidths in the GCV function. If NULL (the default), num.h equidistant values between zero and a quarter of the range of $t_i$ are considered.
num.h
number of values used to build the sequence of considered bandwidths. If h.seq is not NULL, num.h=length(h.seq). Otherwise, the default is 50.
estimator
allows us the choice between NW (Nadaraya-Watson) or LLP (Local Linear Polynomial). The default is NW.
kernel
allows us the choice between gaussian, quadratic (Epanechnikov kernel), triweight or uniform kernel. The default is quadratic.
##### Details

See Craven and Wahba (1979) and Rice (1984).

##### Value

• h.optselected value for the bandwidth.
• GCV.optminimum value of the GCV function.
• GCVvector containing the values of the GCV function for each considered bandwidth.
• h.seqsequence of considered bandwidths in the GCV function.

##### References

Craven, P. and Wahba, G. (1979) Smoothing noisy data with spline functions. Numer. Math. 31, 377-403. Rice, J. (1984) Bandwidth choice for nonparametric regression. Ann. Statist. 12, 1215-1230.

Other related functions are: np.est, np.cv, plrm.est, plrm.gcv and plrm.cv.

• np.gcv
##### Examples
# EXAMPLE 1: REAL DATA
data <- matrix(10,120,2)
data(barnacles1)
barnacles1 <- as.matrix(barnacles1)
data[,1] <- barnacles1[,1]
data <- diff(data, 12)
data[,2] <- 1:nrow(data)

aux <- np.gcv(data)
aux$h.opt plot(aux$h.seq, aux$GCV, xlab="h", ylab="GCV", type="l") # EXAMPLE 2: SIMULATED DATA ## Example 2a: independent data set.seed(1234) # We generate the data n <- 100 t <- ((1:n)-0.5)/n m <- function(t) {0.25*t*(1-t)} f <- m(t) epsilon <- rnorm(n, 0, 0.01) y <- f + epsilon data_ind <- matrix(c(y,t),nrow=100) # We apply the function a <-np.gcv(data_ind) a$GCV.opt

GCV <- a$GCV h <- a$h.seq
plot(h, GCV, type="l")
Documentation reproduced from package PLRModels, version 1.1, License: GPL

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