chausenwho pulled himself out of swamp by his own bootstraps as an examplethe bootstrap estimate of the expectation of is *ey = which is simply the sample mean of {yisimilarlythe bootstrap estimate for vary is the sample variance vary (yi ) ( = pk bootstrap estimators for the bias and mse are and = (yi ) respectively note that for these estimators the unknown quantity is replaced with its original estimator confidence intervals can be constructed in the same fashion we mention two variantsthe normal method and the percentile method in the normal methoda confidence interval for is given by ( + - / )normal method percentile method where is the bootstrap estimate of the standard deviation of ythat isthe square root of ( in the percentile methodthe upper and lower bounds of the confidence interval for are given by the / and / quantiles of ywhich in turn are estimated via the corresponding sample quantiles of the bootstrap sample {yithe following example illustrates the usefulness of the bootstrap method for ratio estimation and also introduces the renewal reward process model for data example (bootstrapping the ratio estimatora common scenario in stochastic simulation is that the output of the simulation consists of independent pairs of data ( )( )where each is interpreted as the length of period of time - socalled cycle -and is the reward obtained during that cycle such collection of random variables {(ci ri )is called renewal reward process typicallythe reward ri depends on pnt the cycle length ci let at be the average reward earned by time tthat isat = ri /twhere nt max{ cn tcounts the number of complete cycles at time it can be shownsee exercise that if the expectations of the cycle length and reward are finitethen at converges to the constant er/ec this ratio can thus be interpreted as the long-run average reward estimation of the ratio er/ec from data ( )(cn rn is easytake the ratio estimator ac howeverthis estimator is not unbiased and it is not obvious how to derive confidence intervals fortunatelythe bootstrap method can come to the rescuesimply resample the pairs {(ci ri )}obtain ratio estimators * * and from these compute quantities of interest such as confidence intervals as concrete examplelet us return to the markov chain in example recall that the chain starts at state at time after certain amount of time the process returns to state the time steps form natural "cyclefor this processas from time onwards the process behaves probabilistically exactly the same as when it startedrenewal reward process long-run average reward ratio estimator