Confidence Intervals for Common Signal-to-Noise Ratio of Several Log-Normal Distributions

Abstract

Log-normal distribution is used widely in application fields such as economics and finance. This paper considers confidence interval estimates for common signal-to-noise ratio of log-normal distributions based on generalized confidence interval (GCI), adjusted method of variance estimates recovery, and computational approaches. A simulation study is conducted to compare the performance of these confidence intervals. A Monte Carlo simulation is applied to report coverage probability and average length of the confidence intervals. Based on the simulation study, for $k=$ 3, the GCI can be used. For $k=$ 6, the results of GCI approach perform similarly to the results of computational approach. For $k=$ 10, the computational approach can be considered as an alternative to estimate the confidence interval. A numerical example based on real data is presented to illustrate the proposed approaches.

Appendix

R code for outputs in Tables 1, 2, and 3

CI.SNRLN $=$ function(M,m,k,n(k),mean,sqrt.var(k)){

ni $=$ rep(0,k); s $=$ rep(0,k); xbar $=$ rep(0,k); mu.hat.RML $=$ rep(0,k)

sigma.hat.RML $=$ rep(0,k); Rtheta.GCI $=$ rep(0,m)

theta.CA $=$ rep(0,m); CP.GCI $=$ rep(0,M); CP.AM $=$ rep(0,M)

CP.CA $=$ rep(0,M); Length.GCI $=$ rep(0,M); Length.AM $=$ rep(0,M)

Length.CA $=$ rep(0,M); alpha $=$ 0.05; z.alpha $=$ qnorm(1-(alpha/2))

sqrt.var $=$ c(sqrt.var(k)); theta(k) $=$ 1/sqrt(exp(sqrt.var(k)$^\wedge 2$)-1)

var.theta(k) $=$ ((sqrt.var(k)$^\wedge$4)$*$(exp(2$*$(sqrt.var(k)$^\wedge 2$))))/(2$*$(n(k)-1)

$*$(((exp(sqrt.var(k)$^\wedge 2$))-1)$^\wedge 3$))

theta.p $=$ c(theta(k)); var.theta.p $=$ c(var.theta(k))

theta $=$ (sum(theta.p/var.theta.p))/(sum(1/var.theta.p))

for(i in 1:M){

x(k) $=$ rnorm(n(k),mean,sqrt.var(k)); xbar.(k) $=$ mean(x(k))

s.(k) $=$ sd(x(k)); ni $=$ c(n(k)); xbar $=$ c(xbar.(k)); s $=$ c(s.(k))

thetahat $=$ 1/sqrt((exp($\hbox {s}^\wedge 2$))-1)

var.thetahat $=$ (($\hbox {s}^\wedge 4$)$*$(exp($2*$($\hbox {s}^\wedge 2$))))/($2*$(ni-1)$*$(((exp($\hbox {s}^\wedge 2$))-1)$^\wedge 3$))

frac1 $=$ sum(thetahat/var.thetahat); frac2 $=$ sum(1/var.thetahat)

thetahat.large $=$ frac1/frac2

for(j in 1:m){

V $=$ rchisq(k,ni-1); Rsig.sqrt $=$ ((ni-1)$*$($\hbox {s}^\wedge 2$))/V

Rvar $=$ (($\hbox {Rsig.sqrt}^\wedge 2$)$*$(exp($2*$(Rsig.sqrt))))/($2*$(ni-1)$*$(((exp(Rsig.sqrt))-1)$^\wedge 3$)); Rtheta $=$ 1/sqrt((exp(Rsig.sqrt))-1)

Rtheta.GCI[j] $=$ sum(Rtheta/Rvar)/sum(1/Rvar)}

L.CI1 $=$ quantile(Rtheta.GCI,0.025,type=8)

U.CI1 $=$ quantile(Rtheta.GCI,0.975,type=8)

CP.GCI[i] $=$ ifelse(L.CI1<theta&&theta<U.CI1,1,0)

Length.GCI[i] $=$ U.CI1-L.CI1

t.x $=$ qt((1-alpha/2),(ni-1)); l1 $=$ (1/sqrt((exp($\hbox {s}^\wedge 2$))-1))-(t.x$*$sqrt(var.thetahat))

u1 $=$ (1/sqrt((exp($\hbox {s}^\wedge 2$))-1))+(t.x$*$sqrt(var.thetahat)); z $=$ qnorm(alpha/2)

var.l $=$ ((thetahat-l1)$^\wedge 2$)/($\hbox {z}^\wedge 2$); var.u $=$ ((u1-thetahat)$^\wedge 2$)/($\hbox {z}^\wedge 2$)

var.t $=$ (var.l+var.u)/2; thetahat.w $=$ (sum(thetahat/(var.t)))/(sum(1/(var.t)))

L.CI2 $=$ thetahat.w-(z.alpha$*$sqrt(1/(sum(1/var.l))))

U.CI2 $=$ thetahat.w+(z.alpha$*$sqrt(1/(sum(1/var.u))))

CP.AM[i] $=$ ifelse(L.CI2<theta&&theta<U.CI2,1,0)

Length.AM[i] $=$ U.CI2-L.CI2

mu.hat.RML $=$ xbar; mu.hat.RML(k) $=$ mu.hat.RML[[k]]

sigma.hat.RML $=$ s; sigma.hat.RML(k) $=$ sigma.hat.RML[[k]]

for(j in 1:m){

x.RML(k) $=$ rnorm(n(k),mu.hat.RML(k),sigma.hat.RML(k))

xbar.RML.(k) $=$ mean(x.RML(k)); s.RML.(k) $=$ sd(x.RML(k))

xbar.RML $=$ c(xbar.RML.(k)); s.RML $=$ c(s.RML.(k))

thetahat.RML $=$ 1/sqrt((exp(s.$\hbox {RML}^\wedge$2))-1)

var.RML $=$ (($\hbox {s.RML}^\wedge 4$)$*$(exp($2*$($\hbox {s.RML}^\wedge 2$))))/($2*$(ni-1)$*$(((exp($\hbox {s.RML}^\wedge 2$))-1)$^\wedge 3$))

frac1.RML $=$ sum(thetahat.RML/var.RML); frac2.RML $=$ sum(1/var.RML)

theta.CA[j] $=$ frac1.RML/frac2.RML }

L.CI3 $=$ quantile(theta.CA,0.025,type=8)

U.CI3 $=$ quantile(theta.CA,0.975,type=8)

CP.CA[i] $=$ ifelse(L.CI3<theta&&theta<U.CI3,1,0)

Length.CA[i] $=$ U.CI3-L.CI3}

cat(“CP|GCI$=$”, mean(CP.GCI),“Length|GCI$=$”, mean(Length.GCI),“$\backslash$ n”)

cat(“CP|AM$=$”, mean(CP.AM),“Length|AM$=$”, mean(Length.AM),“$\backslash$ n”)

cat(“CP|CA$=$”, mean(CP.CA),“Length|CA$=$”, mean(Length.CA),“$\backslash$ n”)}