modeest (version 2.1)

grenander: The Grenander Mode Estimator

Description

This function computes the Grenander mode estimator.

Usage

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.

p

numeric. Paramater 'p' in Grenander's mode estimate. If p = Inf, function venter is used.

...

further arguments to be passed to link{venter}

Value

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

Details

The Grenander estimate is defined by $$ \frac{ \sum_{j=1}^{n-k} \frac{(x_{j+k} + x_{j})}{2(x_{j+k} - x_{j})^p} }{ \sum_{j=1}^{n-k} \frac{1}{(x_{j+k} - x_{j})^p} } $$

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.

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

Examples

# NOT RUN {
# 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'
p <- seq(0.1, 4, 0.01)

## Estimate of the mode with these parameters
M <- sapply(p, function(pp) grenander(x, p = pp, k = k))

## Distribution obtained
plot(density(M), xlim = c(22.5, 23.5))
# }