quantilesLogConDens {logcondens} | R Documentation |
Function to compute p_0-quantile of
widehat F_m(t) = int_{x_1}^t widehat f_m(t) dt,
where widehat f_m is the log-concave density estimator, received via activeSetLogCon
. The
formula to compute a quantile at u in [widehat F_m(x_j), widehat F_m(x_{j+1})] for
j = 1, ..., n-1 is:
widehat F_m^{-1}(u) = x_j + (x_{j+1}-x_j) G^{-1}_{(x_{j+1}-x_j)(widehat varphi_{j+1}-widehat varphi_j)} Big( frac{u - widehat F_m(x_j)}{ widehat F_m(x_{j+1}) - widehat F_m(x_j)}Big),
where G^{-1}_theta is described in qloglin
.
quantilesLogConDens(p0, x, phi, Fhat)
p0 |
Real number where quantil should be computed. |
x |
Sorted vector of original observations {x} = (x_1, ..., x_m). |
phi |
Vector (widehat varphi_m(x_1), ..., widehat varphi_m(x_m)), representing the function widehat varphi_m, as computed by activeSetLogCon . |
Fhat |
Vector (widehat F_{m,i})_{i=1}^m with entries widehat F_{m,i} = int_{x_1}^{x_i} exp(widehat varphi_m(t)) dt, as computed by |
Returns the real number q_0 = inf_{x}{widehat F_m(x) >= p_0}.
Kaspar Rufibach, kaspar.rufibach@gmail.com
Lutz Duembgen, duembgen@stat.unibe.ch,
http://www.staff.unibe.ch/duembgen
## estimate gamma density set.seed(1977) x <- sort(rgamma(200, 2, 1)) res <- activeSetLogCon(x, w = NA, print = FALSE) ## compute 0.95 quantile of Fhat q <- quantilesLogConDens(0.95, x, res$phi, res$Fhat) plot(x, res$Fhat, type = 'l'); rug(x) abline(h = 0.95, lty = 3); abline(v = q, lty = 3)