grenander: The Grenander mode estimator

Description

This function computes the Grenander mode estimator.

Usage

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

Arguments

numeric. Vector of observations.

numeric. The bandwidth to be used. Should belong to (0, 1].

numeric. Paramater 'k' in Grenander's mode estimate, see below.

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.

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.

Examples

Run this code

# 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'
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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