# Rule of thumb for bandwidth selection. Implements Fan and Gijbels(1996)'s Rule of thumb for bandwidth selection ```r thumbBw(x, y, deg, kernel, weig = rep(1, length(y))) compDerEst(x, y, p, weig = rep(1, length(y))) ``` ## Arguments - `x`: x covariate data values. - `y`: y response data values. - `p`: order of local polynomial estimator. - `deg`: Local polynomial estimation degree($p$). - `kernel`: Kernel used to perform the estimation. - `weig`: weights if they are required. ## Details See Fan and Gijbels(1996) book, Section 4.2. This implementation is also considering weights. `compDerEst` computes the $p+1$ derivative of the regression function in a simple manner, assuming it is a polynomial in $x$. `thumbBw` gives a bandwidth selector by means of pilot estimator given by `compDerEst` and the mean of residuals. ## Returns `thumbBw` returns a single numeric value, while `compDerEst` returns a data frame whose components are: - **x**: x values. - **y**: y values. - **res**: residuals for the parametric estimation. - **der**: derivative estimation at x values. ## References Fan, J. and Gijbels, I. **Local polynomial modelling and its applications\/**. Chapman & Hall, London (1996). Wand, M.~P. and Jones, M.~C. **Kernel smoothing\/**. Chapman and Hall Ltd., London (1995). ## Author(s) Jorge Luis Ojeda Cabrera. ## See Also `regCVBwSelC`, `pluginBw`. ## Examples ```r size <- 200 sigma <- 0.25 deg <- 1 kernel <- EpaK xeval <- 0:100/100 regFun <- function(x) x^3 x <- runif(size) y <- regFun(x) + rnorm(x, sd = sigma) d <- data.frame(x, y) cvBwSel <- regCVBwSelC(d$x,d$y, deg, kernel, interval = c(0, 0.25)) thBwSel <- thumbBw(d$x, d$y, deg, kernel) piBwSel <- pluginBw(d$x, d$y, deg, kernel) est <- function(bw, dat, x) return(locPolSmootherC(dat$x,dat$y, x, bw, deg, kernel)$beta0) ise <- function(val, est) return(sum((val - est)^2 * xeval[[2]])) plot(d$x, d$y) trueVal <- regFun(xeval) lines(xeval, trueVal, col = "red") xevalRes <- est(cvBwSel, d, xeval) cvIse <- ise(trueVal, xevalRes) lines(xeval, xevalRes, col = "blue") xevalRes <- est(thBwSel, d, xeval) thIse <- ise(trueVal, xevalRes) xevalRes <- est(piBwSel, d, xeval) piIse <- ise(trueVal, xevalRes) lines(xeval, xevalRes, col = "blue", lty = "dashed") res <- rbind( bw = c(cvBwSel, thBwSel, piBwSel), ise = c(cvIse, thIse, piIse) ) colnames(res) <- c("CV", "th", "PI") res ```

thumbBw function