lientz() R function from [modeest]

The empirical Lientz function and the Lientz mode estimator

The Lientz mode estimator is nothing but the value minimizing the empirical Lientz function. A 'plot' and a 'print' methods are provided.


lientz(x, bw = NULL)

## S3 method for class 'lientz'
plot(x, zoom = FALSE, ...)

## S3 method for class 'lientz'
print(x, digits = NULL, ...)

## S3 method for class 'lientz'
mlv(x, bw = NULL, abc = FALSE, par = shorth(x), optim.method = "BFGS", ...)

Arguments

x: numeric (vector of observations) or an object of class "lientz".
bw: numeric. The smoothing bandwidth to be used. Should belong to (0, 1). Parameter 'beta' in Lientz (1970) function.
zoom: logical. If TRUE, one can zoom on the graph created.
...: if abc = FALSE, further arguments to be passed to optim, or further arguments to be passed to plot.
digits: numeric. Number of digits to be printed.
abc: logical. If FALSE (the default), the Lientz empirical function is minimised using optim.
par: numeric. The initial value used in optim.
optim.method: character. If abc = FALSE, the method used in optim.

Returns

lientz returns an object of class c("lientz", "function"); this is a function with additional attributes:

xthe x argument
bwthe bw argument
callthe call which produced the result

mlv.lientz returns a numeric value, the mode estimate. If abc = TRUE, the x value minimizing the Lientz empirical function is returned. Otherwise, the optim method is used to perform minimization, and the attributes: 'value', 'counts', 'convergence' and 'message', coming from the optim

method, are added to the result.

Details

The Lientz function is the smallest non-negative quantity $S(x,b)$ , where $b$ = bw, such that

F(x+S(x,\beta)) - F(x-S(x,\beta)) \geq \beta.F(x+S(x,b)) - F(x-S(x,b)) >= b.

Lientz (1970) provided a way to estimate $S(x,b)$ ; this estimate is what we call the empirical Lientz function.

Note

The user may call mlv.lientz through mlv(x, method = "lientz", ...).

Examples


# Unimodal distribution
x <- rbeta(1000,23,4)

## True mode
betaMode(23, 4)

## Lientz object
f <- lientz(x, 0.2)
print(f)
plot(f)

## Estimate of the mode
mlv(f)              # optim(shorth(x), fn = f)
mlv(f, abc = TRUE)  # x[which.min(f(x))]
mlv(x, method = "lientz", bw = 0.2)

# Bimodal distribution
x <- c(rnorm(1000,5,1), rnorm(1500, 22, 3))
f <- lientz(x, 0.1)
plot(f)

References

Lientz B.P. (1969). On estimating points of local maxima and minima of density functions. Nonparametric Techniques in Statistical Inference (ed. M.L. Puri, Cambridge University Press, p.275-282.
Lientz B.P. (1970). Results on nonparametric modal intervals. SIAM J. Appl. Math., 19 :356-366.
Lientz B.P. (1972). Properties of modal intervals. SIAM J. Appl. Math., 23 :1-5.

lientz function