modeest (version 2.4.0)

grenander: The Grenander mode estimator

Description

This function computes the Grenander mode estimator.

Usage

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

Value

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

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.

Author

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

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

# 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))