modeest (version 2.1)

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

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.

Details

The Lientz function is the smallest non-negative quantity \(S(x,\beta)\), where \(\beta\) = bw, such that $$F(x+S(x,\beta)) - F(x-S(x,\beta)) \geq \beta.$$ Lientz (1970) provided a way to estimate \(S(x,\beta)\); this estimate is what we call the empirical Lientz function.

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.

See Also

mlv for general mode estimation; shorth for the shorth estimate of the mode

Examples

# NOT RUN {
# 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)
# }