Description Usage Arguments Value References Examples

Wrapper function which does some preparatory calculations and then calls the actual “workhorse” functions which do the main computations for kernel adaptive regression estimation of Eichner & Stute (2012). Finally, it structures and returns the obtained results. Summarizing information and technical details can be found in Eichner (2017).

1 2 |

`x.points` |
Vector of location(s) at which the regression estimate is to be computed. |

`data` |
Data frame or list with one component named |

`kernel` |
A character string naming the kernel to be used for the adaptive estimator. This must partially match one of "gaussian", "rectangular" or "epanechnikov", with default "gaussian", and may be abbreviated to a unique prefix. (Currently, this kernel is also used for the initial, non-adaptive Nadaraya-Watson regression estimator which enters into the estimators of bias and variance as described in the references.) |

`Sigma` |
Vector of value(s) of the scale parameter |

`h` |
Numeric scalar for bandwidth |

`theta` |
Numeric scalar for value of location parameter |

If `length(x.points)`

= 1, a list of eight components with the
following names and meanings:

`x` | Scalar x-value in `x.points` at which the
regression estimator was computed. |

`y` | Estimated scalar value of m(x) at point in
`x.points` . |

`sigma.adap` | The found scalar minimizer of the MSE-estimator, i.e., the adaptive smoothing parameter value. |

`msehat.min` | The found scalar minimum of the MSE-estimator. |

`Sigma` | Vector with the σ-grid on which the
minimization process was performed. |

`Bn` | Vector with the estimator of bias on that
σ-grid. |

`Vn2` | Ditto for the variance. |

`MSE` | Ditto for the MSE. |

If `length(x.points)`

> 1, a list with the same component names as
above, but then

`x` | Vector `x.points` with x-values at which the regression
estimator was computed. |

`y` | Vector of estimated values of m(x) at the x-values in
`x.points` . |

`sigma.adap` | Vector of the found minimizers of the MSE-estimator, i.e., the adaptive smoothing parameter values. |

`msehat.min` | Vector of the found minima of the MSE-estimator. |

`Sigma` | Vector with the σ-grid on which the
minimization process was performed. |

`Bn` | (`length(Sigma)` by `length(x.points)` )-matrix
with the estimated values of the bias on the
σ-grid in their columns (which correspond to the
x-values). |

`Vn2` | Ditto for the variance. |

`MSE` | Ditto for the MSE. |

Eichner & Stute (2012) and Eichner (2017): see `kader`

.

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require(stats)
# Regression function:
m <- function(x, x1 = 0, x2 = 8, a = 0.01, b = 0) {
a * (x - x1) * (x - x2)^3 + b
}
# Note: For a few details on m() see examples in ?nadwat.
x0 <- 5 # The point x_0 at which the MSE-optimal kernel adjusted
# nonparametric estimation of m should take place. (Recall: for m's
# default values a minimum is at 2, a point of inflection at 4, and
# a saddle point an 8; an "arbitrary" point would, e.g., be at 5.)
n <- 100 # Sample size.
sdeps <- 1 # Std. dev. of the \epsilon_i: \sqrt(Var(Y|X=x))
# (here: constant in x).
design.ctr <- x0 + 0.5 # "centre" and "scale" of the design, i.e.,
design.scl <- 1 # in the normal scenario below, expected value and
# std. dev. of the distribution of the x_i's.
set.seed(42) # To guarantee reproducibility.
x <- rnorm(n, mean = design.ctr, sd = design.scl) # x_1, ..., x_n
Y <- m(x) + rnorm(length(x), sd = sdeps) # Y_1, ..., Y_n
data <- data.frame(x = x, y = Y)
# Computing the kernel adaptive regression estimator values
#**********************************************************
x.points <- seq(-3.3 * design.scl, 3.3 * design.scl, length = 101) +
design.ctr # x-grid on which to draw and estimate the regr. fct. m.
Sigma <- seq(0.01, 10, length = 51) # \sigma-grid for minimization.
fit <- kare(x.points = x0, data = data, Sigma = Sigma)
## Not run:
# Grafical display for the current data set
#******************************************
# Storing the curent settings of the graphics device
# and changing its layout for the three plots to come:
op <- par(mfrow = c(3, 1), mar = c(3, 3, 2, 0.1)+0.1,
mgp = c(1.5, 0.5, 0), tcl = -0.3, cex.main = 2)
# The scatter plot of the "raw data":
plot(y ~ x, data = data, xlim = range(data$x, x.points),
ylim = range(data$y, fit$y, na.rm = TRUE),
main = bquote(n == .(n)), xlab = "x", ylab = "y")
# The "true" regression function m:
lines(x.points, m(x.points), lty = 2)
# The MSE-optimal kernel adjusted nonparametric regression estimator
# at x_0, i.e., the point (x_0, \hat m_n(x_0)):
points(fit$x, fit$y, col = "red", pch = 4, cex = 2)
# The legend for the "true" regression function m and for the point
# (x_0, \hat m_n(x_0)):
legend("topleft", lty = c(2, NA), pch = c(NA, 4),
col = c("black", "red"), bty = "n", cex = 1.2,
legend = c(as.expression(bquote(paste("m with ",
sigma(paste(Y, "|", X == x))
== .(sdeps)))),
as.expression(bquote(paste(hat(m)[n](x[0]), " at ",
x[0] == .(x0))))))
# Visualizing the estimators of (Bias_n(sigma))^2 and
# Var_n(sigma) at x0 on the sigma-grid:
with(fit,
matplot(Sigma, cbind(Bn*Bn, Vn2), type = "l", lty = 1:2,
col = c("black", "red"), xlab = expression(sigma), ylab = ""))
# The legend for (Bias_n(sigma))^2 and Var_n(sigma):
legend("topleft", lty = 1:2, col = c("black", "red"), bty = "n",
legend = c(expression(paste(widehat(plain(Bias))[n]^2, (sigma))),
expression(widehat(plain(Var))[n](sigma))),
cex = 1.2)
# Visualizing the estimator of MSE_n(sigma) at x0 on the sigma-grid
# together with the point indicating the detected minimum, and a legend:
plot(fit$Sigma, fit$MSE, type = "l",
xlab = expression(sigma), ylab = "")
points(fit$sigma.adap, fit$msehat.min, pch = 4, col = "red", cex = 2)
legend("topleft", lty = c(1, NA), pch = c(NA, 4),
col = c("black", "red"), bty = "n", cex = 1.2,
legend = c(expression(widehat(plain(MSE))[n](sigma)),
substitute(group("(", list(plain(Minimizer),
plain(Minimum)), ")")
== group("(", list(x, y), ")"),
list(x = signif(fit$sigma.adap, 4),
y = signif(fit$msehat.min, 4)))))
par(op) # Restoring the previous settings of the graphics device.
## End(Not run)
``` |

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