modeest (version 1.06)

grenander: 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 mode estimate.

p

numeric. Paramater 'p' in Grenander 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 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*ny) - 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'
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))