modeest (version 1.06)

Chernoff: The Chernoff Distribution

Description

Density, distribution function, quantile function and random generation for the Chernoff distribution.

Usage

dchern(x, log = FALSE)
pchern(q, lower.tail = TRUE, log.p = FALSE)
qchern(p, lower.tail = TRUE, log.p = FALSE)
rchern(n)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations.

log, log.p

logical. If TRUE, probabilities p are given as log(p).

lower.tail

logical. If TRUE (default), probabilities are $P[X \le x]$, otherwise, $P[X > x]$.

Value

• dchern gives the density, pchern gives the distribution function, qchern gives the quantile function and rchern generates random deviates.

Warning

Presently, pchern, qchern, and rchern are *very* slow.

source

dchern, pchern and qchern are calculated from the C code due to Groeneboom and Wellner (2001).

Details

The Chernoff distribution is the law of the random variable $$Z = \arg\max_{t} (B(t) - t^2)$$ where $B(t)$ is two-sided Brownian motion starting from zero. Up to a location and a scale parameter, this distribution is the limit distribution of the Chernoff mode estimator (also called the naive mode estimator), see Chernoff (1964).

References

• Chernoff H. (1964). Estimation of the mode.Ann. Inst. Statist. Math.,16:31-41.
• Groeneboom P.and Wellner J.A. (2001). Computing Chernoff's distribution.J. Comput. Graph. Statist.,10:388-400.

See Also

mlv for general mode estimation; naive for the Chernoff mode estimate; distribMode for computation of the mode of the classical distributions

Examples

dchern(seq(0.1,0.9,0.1))
curve(dchern(x), xlim = c(-2,2))