Maximum Smoothed Likelihood Estimation of the Centre of a Symmetric Distribution

Abstract

Estimating the centre of a symmetric distribution is one of the basic and important problems in statistics. Given a random sample from the symmetric distribution, natural estimators of the centre are the sample mean and sample median. However, these two estimators are either not robust or inefficient. Other estimators, such as Hodges-Lehmann estimator (Hodges and Lehmann, Ann Math Stat 34:598–611, 1963), the location M-estimator (Huber, Ann Math Stat 35:73–101, 1964) and Bondell (Commun Stat Theory Methods 37:318–327, 2008)’s estimator, were proposed to achieve high robustness and efficiency. In this paper, we propose an estimator by maximizing a smoothed likelihood. Simulation studies show that the proposed estimator has much smaller mean square errors than the existing methods under uniform distribution, t-distribution with one degree of freedom, and mixtures of normal distributions on the mean parameter, and is comparable to the existing methods under other symmetric distributions. A real example is used to illustrate the proposed method. The R code for implementing the proposed method is also provided.

Appendix: R code for calculating \(\hat{\mu }_{Smo}\)

library("KernSmooth")

library("ICSNP")

norm.kern=function(x,data,h)

{

out=mean( dnorm( (x-data)/h ) )/h

out

}

dkern=function(x,data)

{

h=dpik(data,kernel="normal")

out=lapply(x,norm.kern,data=data,h=h)

as.numeric(out)

}

dfint=function(x,data,mu)

{

p1=dkern(x,data)

p2=log(0.5*dkern(x,data)+0.5*dkern(2*mu-x,data)+1e-100)

p1*p2

}

pln=function(mu,data)

{

h=dpik(data,kernel="normal")

out=integrate(dfint, lower=min(data)-10*h,

                  upper=max(data)+10*h, data=data, mu=mu)

-out$value

}

hatmu.smooth=function(data)

{

##Input: data set

##Output: maximum smoothed likelihood estimate

hl.est=hl.loc(data)

est=optim(hl.est,pln,data=data,method="BFGS")$par

est

}

##Here is an example

set.seed(1221)

data=rnorm(100,0,1)

hatmu.smooth(data)

##Result: 0.1798129