Return to Answer

If you know it: Exact CI. However, if the population is known to be exponential, then we know that $\frac{\bar X}{\mu} \sim\mathsf{Gamma}(\mathrm{shape}=1/n, \mathrm{rate}=1/n).$ We can 'pivot' this relationship to make an exact 95% CI for $\mu$ of the form $\left(\frac{\bar X}{U}, \frac{\bar X}{L}\right),$ where $L$ and $U$ cut probability $0.025$ from the lower and upper tails of $\mathsf{Gamma}(1/50, 1/50).$ This exact 95% CI for $\mu$ is $(13.57, 23.69).$

If you know it: Exact CI. However, if the population is known to be exponential, then one can show that $\frac{\bar X}{\mu} \sim\mathsf{Gamma}(\mathrm{shape}=1/n, \mathrm{rate}=1/n).$ and 'pivot' this relationship to make an exact 95% CI for $\mu$ of the form $\left(\frac{\bar X}{U}, \frac{\bar X}{L}\right),$ where $L$ and $U$ cut probability $0.025$ from the lower and upper tails of $\mathsf{Gamma}(1/50, 1/50).$ This exact 95% CI for $\mu$ is $(13.57, 23.69).$

Parametric bootstrap CI. Now suppose that we know that the population is exponentially distributed, so that $X_i \stackrel{iid}{\sim}\mathsf{EXP = \mathrm{rate}}).$

Parametric bootstrap CI. Now suppose that we know that the population is exponentially distributed, with $X_i \stackrel{iid}{\sim}\mathsf{EXP}( \mathrm{rate}=1/\mu).$

set.seed(2021) 
# non-parametric bootstrap, re-sample from sample
d.re = replicate(3000, mean(sample(x, 50, rep=T))-a.obs)
UL = quantile(d.re,c(.975,.025))
a.obs - UL
   97.5%     2.5% 
 9.12105 23.81885 

For my fictitious data x the resulting 95% parametric bootstrap CI is $(12.44, 22.13).$ This interval is narrower than the nonparametric bootstrap CI because it is based on the additional information that the population is exponential. [I did more re-samples here because parametric bootstrap CIs with larger numbers of resamples may be noticably more accurate.]

set.seed(2021)
# parametric bootstrap, sample 50 from EXP(rate=1/a)
d.re = replicate(10000, mean(rexp(50,1/obs.a))-a.obs)
UL = quantile(d.re,c(.975,.025))
a.obs - UL
   97.5%     2.5% 
12.44381 22.13479 

set.seed(2021)
# parametric bootstrap of ratios, sample 50 from EXP(rate=1/a)
r.re = replicate(3000, mean(rexp(50,1/obs.a))/a.obs)
UL = quantile(r.re,c(.975,.025))
a.obs / UL
   97.5%     2.5% 
13.66134 23.76732 

set.seed(2021) 
# non-parametric bootstrap, re-sample from sample
a.obs = mean(x); a.obs
[1] 17.584

d.re = replicate(3000, mean(sample(x, 50, rep=T))-a.obs)
UL = quantile(d.re,c(.975,.025))
a.obs - UL
   97.5%     2.5% 
 9.12105 23.81885 

For my fictitious data x the resulting 95% parametric bootstrap CI is $(12.44, 22.13).$ This interval is narrower than the nonparametric bootstrap CI because it is based on the additional information that the population is exponential. [I did more re-samples here because parametric bootstrap CIs with larger numbers of resamples may be noticeably more accurate.]

set.seed(2021)
# parametric bootstrap, sample 50 from EXP(rate=1/a.obs)
a.obs = mean(x); a.obs
[1] 17.584
d.re = replicate(10000, mean(rexp(50,1/a.obs))-a.obs)
UL = quantile(d.re,c(.975,.025))
a.obs - UL
   97.5%     2.5% 
12.44381 22.13479 

set.seed(2021)
# parametric bootstrap of ratios, sample 50 from EXP(rate=1/a.obs)
r.re = replicate(3000, mean(rexp(50,1/obs.a))/a.obs)
UL = quantile(r.re,c(.975,.025))
a.obs / UL
   97.5%     2.5% 
13.66134 23.76732