grenander function

The Grenander mode estimator

This function computes the Grenander mode estimator.

grenander(x, bw = NULL, k, p, ...)

Arguments

• 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 p = Inf, the function venter is used.
• ...: Additional arguments to be passed to venter.

Returns

A numeric value is returned, the mode estimate. If p = Inf, the venter mode estimator is returned.

Details

The Grenander estimate is defined by

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.

Note

The user may call grenander through mlv(x, method = "grenander", bw, k, p, ...).

Examples

# 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))

References

• 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.

See Also

mlv for general mode estimation; venter for the Venter mode estimate.

Author(s)

D.R. Bickel for the original code, P. Poncet for the slight modifications introduced.