The Grenander mode estimator
This function computes the Grenander mode estimator.
grenander(x, bw = NULL, k, p, ...)
x |
numeric. Vector of observations. |
bw |
numeric. The bandwidth to be used. Should belong to (0, 1]. |
k |
numeric. Paramater 'k' in Grenander's mode estimate, see below. |
p |
numeric. Paramater 'p' in Grenander's mode estimate, see below.
If |
... |
Additional arguments to be passed to |
The Grenander estimate is defined by
( sum_{j=1}^{n-k} (x_{j+k} + x_{j})/(2(x_{j+k} - x_{j})^p) ) / ( sum_{j=1}^{n-k} 1/((x_{j+k} - x_{j})^p) )
If p tends to infinity, this estimate tends to the Venter mode estimate;
this justifies to call venter if p = Inf.
The user should either give the bandwidth bw or the argument k,
k being taken equal to ceiling(bw*n) - 1 if missing.
A numeric value is returned, the mode estimate.
If p = Inf, the venter mode estimator is returned.
The user may call grenander through
mlv(x, method = "grenander", bw, k, p, ...).
D.R. Bickel for the original code, P. Poncet for the slight modifications introduced.
Grenander U. (1965). Some direct estimates of the mode. Ann. Math. Statist., 36:131-138.
Dalenius T. (1965). The Mode - A Negleted Statistical Parameter. J. Royal Statist. Soc. A, 128:110-117.
Adriano K.N., Gentle J.E. and Sposito V.A. (1977). On the asymptotic bias of Grenander's mode estimator. Commun. Statist.-Theor. Meth. A, 6:773-776.
Hall P. (1982). Asymptotic Theory of Grenander's Mode Estimator. Z. Wahrsch. Verw. Gebiete, 60:315-334.
# Unimodal distribution x <- rnorm(1000, mean = 23, sd = 0.5) ## True mode normMode(mean = 23, sd = 0.5) # (!) ## Parameter 'k' k <- 5 ## Many values of parameter 'p' ps <- seq(0.1, 4, 0.01) ## Estimate of the mode with these parameters M <- sapply(ps, function(p) grenander(x, p = p, k = k)) ## Distribution obtained plot(density(M), xlim = c(22.5, 23.5))
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