R: The Empirical Lientz Function and The Lientz Mode Estimator

lientz {modeest}

R Documentation

The Empirical Lientz Function and The Lientz Mode Estimator

Description

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

Usage

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

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.
`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`.
`zoom`	logical. If `TRUE`, one can zoom on the graph created.
`digits`	numeric. Number of digits to be printed.
`...`	if `abc = FALSE`, further arguments to be passed to `optim`, or further arguments to be passed to `plot.default`.

Details

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

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.

Value

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

`x`	the `x` argument
`bw`	the `bw` argument
`call`	the 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.

Note

The user should preferentially call mlv.lientz through mlv(x, method = "lientz", ...). This returns an object of class mlv.

Author(s)

P. Poncet

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.

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))]
M <- mlv(x, method = "lientz", bw = 0.2)
print(M)
plot(M)

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

[Package modeest version 2.1 Index]