Abstract
The coefficient of variation (CV) is a widely used scaleless measure of variability in many disciplines. However the inference for the CV is limited to parametric methods or standard bootstrap. In this paper we propose two nonparametric methods aiming to construct confidence intervals for the coefficient of variation. The first one is to apply the empirical likelihood after transforming the original data. The second one is a modified jackknife empirical likelihood method. We also propose bootstrap procedures for calibrating the test statistics. Results from our simulation studies suggest that the proposed methods, particularly the empirical likelihood method with bootstrap calibration, are comparable to existing methods for normal data and yield better coverage probabilities for nonnormal data. We illustrate our methods by applying them to two real-life datasets.
Acknowledgements
We are grateful to the constructive comments from the Associated Editor and the three anonymous Referees, which notably improved the quality of our manuscript.
Appendix
5.1 Proof of Theorem 2.2
Proof
Firstly we show that at
where
We then show that
Thus Theorem 2.2 follows directly from Theorem 2.1 in Jing et al. (2009).
5.2 R script for the proposed methods
[baselinestretch=0.75]if (F){ install.packages(“emplik”)}library(emplik)######################## EL functions######################cvhat4el.f = function(x){## CV estimate for EL n = length(x) m = floor(n/2) y = z = rep(NA,m) for (i in 1:m){ y[i] = (x[i]-x[m+i])^2/2 z[i] = (x[i]^2+x[m+i]^2)/2 } cvhat = sqrt(mean(y)/mean(z-y)) out = list(cvhat=cvhat,x=x, y=y, z=z) return(out)}el.f = function(y,z,tau){ zvals = y-tau^2*(z-y) tt = el.test(x=zvals,mu=0) ll = tt$“-2LLR” if(abs(sum(tt$wts)-length(y))>1) ll = 300 out = list(“-2LLR” = ll,zvals = zvals,tau = tau,n = n) return(out)}ci.el.f = function(x,avals = c(0.10,0.05),B = 1000,step = 0.01){ nalpha = length(avals) ci.alpha = array(NA,dim = c(nalpha,3,2)) cx.alpha = round(qchisq(1-avals,1),2) ## cut-off for chi-square n = length(x) m = floor(n/2) ttt = cvhat$el.f(x) cvhat = ttt$cvhat y = ttt$y z = ttt$z llboot = rep(NA,B) for (i in 1:B){ idx = sample(1:m,replace=T) ystar = y[idx] zstar = z[idx] llboot[i] = el.f(tau=cvhat, y=ystar, z=zstar)$“-2LLR” } cboot.alpha = round(quantile(llboot,prob=1-avals,na.rm=T),2) for (i in 1:nalpha){ if (el.f(tau = 1000,y = y, z = z)$“-2LLR” >= cx.alpha[i]){ ci = findUL(step = step, fun = el.f, MLE = cvhat, y = y, z = z, level = cx.alpha[i]) ci.alpha[i,1,] = c(ci$Low,ci$Up) } ci.alpha[i,3,] = ci.alpha[i,1,] if (el.f(tau = 1000,y = y, z = z)$“-2LLR” >=cboot.alpha[i]){ ci = findUL(step = step, fun = el.f, MLE = cvhat, y = y, z = z,level = cboot.alpha[i]) ci.alpha[i,3,] = c(ci$Low,ci$Up) } } out = list(ci.alpha = ci.alpha,x = x,avals = avals, B = B,cx.alpha = cx.alpha, cboot.alpha = cboot.alpha) return(out)}######################## JEL######################cvhat4jel.f=function(x){## CV estimate for JEL n=length(x) y=x^2 tt1 = mean(y) tt2 = sd(x)^2 cvhat = sqrt(tt2/(tt1-tt2)) out=list(cvhat=cvhat,x=x) return(out)}U_n.f=function(x,tau){ n=length(x) y=x^2 tt1 = mean(y) tt2 = sd(x)^2 tt=tt1*tau^2-(tau^2+1)*tt2 out=list(U=tt, tt1=tt1, tt2=tt2,x=x) return(out)}jkkf.f=function(x,tau){ n=length(x) Un=U_n.f(x,tau)$U Un1=vv=rep(NA,n) for (i in 1:n){ Un1[i]=U_n.f(x[-i],tau)$U } vjack=n*Un-(n-1)*Un1 out=list(vjack=vjack,n=n,tau=tau,x=x,Un=Un) return(out)}jel.f=function(tau,x){ n=length(x) vjack=jkkf.f(x=x,tau=tau)$vjack tt = el.test(x=vjack,mu=0) ll=tt$“-2LLR” if(abs(sum(tt$wts)-length(x))>1) ll=300 out=list(“-2LLR”=ll,vjack=vjack,tau=tau,n=n) return(out)}ci.jel.f=function(x,avals=c(0.10,0.05),B=1000,step=0.01){ n=length(x) cvhat=cvhat4jel.f(x)$cvhat nalpha=length(avals) ci.alpha=array(NA,dim=c(nalpha,3,2)) llboot=rep(NA,B) for (i in 1:B){ xstar=sample(x,n,replace=T) llboot[i]=jel.f(tau=cvhat,x=xstar)$“-2LLR” } cx.alpha=qchisq(1-avals,1) ## cut-off for chi-square cboot.alpha=round(quantile(llboot,prob=1-avals,na.rm=T),2) ## bootstrap cutoff for (i in 1:nalpha){ ci.alpha[i,1,]=rep(NA,2) if (jel.f(tau=1000,x=x)$“-2LLR”>=cx.alpha[i]){ ci=findUL(step=step, fun=jel.f, MLE=cvhat, x=x,level=cx.alpha[i]) ci.alpha[i,1,]=c(ci$Low,ci$Up) } ci.alpha[i,3,] = ci.alpha[i,1,] if (jel.f(tau=1000,x=x)$“-2LLR”>=cboot.alpha[i]){ ci=findUL(step=step, fun=jel.f, MLE=cvhat, x=x,level=cboot.alpha[i]) ci.alpha[i,3,]=c(ci$Low,ci$Up) } } out=list(ci.alpha=ci.alpha,x=x,avals=avals,B=B,cx.alpha=cx.alpha, cboot.alpha=cboot.alpha) return(out)}
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