np.est {PLRModels}

Nonparametric estimate of the regression function

Description

This routine computes estimates for m(newt_j) (j=1,...,J) from a sample {(Y_i, t_i): i=1,...,n}, where:

Y_i= m(t_i) + \epsilon_i.

The regression function, m, is a smooth but unknown function, and the random errors, {\epsilon_i}, are allowed to be time series. Kernel smoothing is used.

Usage

np.est(data = data, h.seq = NULL, newt = NULL,
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

the considered bandwidths. If NULL (the default), only one bandwidth, selected by means of the cross-validation procedure, is used.

newt

values of the explanatory variable where the estimates are obtained. If NULL (the default), the considered values will be the values of data[,2].

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 Fan and Gijbels (1996) and Francisco-Fernandez and Vilar-Fernandez (2001).

Value

YHAT: a length(newt) x length(h.seq) matrix containing the estimates for m(newt_j)

(j=1,...,length(newt)) using the different bandwidths in h.seq.

Author(s)

German Aneiros Perez [e-mail]

Ana Lopez Cheda [e-mail]

References

Fan, J. and Gijbels, I. (1996) Local Polynomial Modelling and its Applications. Chapman and Hall, London.

Francisco-Fernandez, M. and Vilar-Fernandez, J. M. (2001) Local polynomial regression estimation with correlated errors. Comm. Statist. Theory Methods 30, 1271-1293.

See Also

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

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)
h <- aux$h.opt
ajuste <- np.est(data=data, h=h)
plot(data[,2], ajuste, type="l", xlab="t", ylab="m(t)")
plot(data[,1], ajuste, xlab="y", ylab="y.hat", main="y.hat vs y")
abline(0,1)
residuos <- data[,1] - ajuste
mean(residuos^2)/var(data[,1])



# 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 estimate the nonparametric component of the PLR model
# (CV bandwidth)
est <- np.est(data_ind)
plot(t, est, type="l", lty=2, ylab="")
points(t, 0.25*t*(1-t), type="l")
legend(x="topleft", legend = c("m", "m hat"), col=c("black", "black"), lty=c(1,2))