Version 6 April 2004
These notes work through a simple example to show how one can programRto do both jackknife and bootstrap sampling. We start with bootstrapping.
Bootstrap Calculations
Rhas a number of nice features for easy calculation of bootstrap estimates and confidence intervals. To see how to use these features, consider the following 25 observations:
8.26 6.33 10.4 5.27 5.35 5.61 6.12 6.19 5.2 7.01 8.74 7.78 7.02 6 6.5 5.8 5.12 7.41 6.52 6.21 12.28 5.6 5.38 6.6 8.74
Suppose we wish to estimate the coefficient of variation, CV =√
Var/ x. Let’s do this with a bootstrap estimator.
First, let’s put the data into a vector, which we will callx,
> x <-c(8.26, 6.33, 10.4, 5.27, 5.35, 5.61, 6.12, 6.19, 5.2, 7.01, 8.74, 7.78, 7.02, 6, 6.5, 5.8, 5.12, 7.41, 6.52, 6.21, 12.28, 5.6, 5.38, 6.6, 8.74)
Now let’s define a functon in R, which we will call CV, to compute the coefficient of variation,
> CV <- function(x) sqrt(var(x))/mean(x) So, let’s compute the CV
> CV(x) [1] 0.2524712
To generate a single bootstrap sample from this data vector, we use the command
>sample(x,replace=T)
which generates a bootstrap sample of the data vectorxby sampling with replacement.
Hence, to compute the CV using a single bootstrap sample,
> CV(sample(x,replace=T)) [1] 0.2242572
The particular value thatRreturns for you will be different as the sample is random.
Some other useful commands:
> sum(x)returns the sum of the elements inx
> mean(x)returns the mean of the elements inx
> var(x)returns the sample variance, i.e.,P
i(x− x)2/(n− 1)
> length(x)returns the number of items inx(i.e., the sample size n)
Note that thesumcommand is fairly general, for example
> sum((x-mean(x))^ 2) computesP
i(x− x)2
So, lets now generate 1000 bootstrap samples. We first need to specify a vector of real values of lenght 1000, which we will callboot
> boot <-numeric(1000)
We now generate 1000 samples, and assign the CV for bootstrap sample i as the ith element in the vectorboot, using aforloop
for (i in 1:1000) boot[i] <- CV(sample(x,replace=T))
The mean and variance of this collection of bootstrap samples are easily obtained using themeanandvarcommands (again, your values may differ),
> mean(boot) [1] 0.2404653
> var(boot) [1] 0.00193073
A plot of the histogram of these values follows using hist(boot)
Likewise, the value corresponding to the (say) upper 97.5
> quantile(boot,0.975) [1] 0.3176385
while the value corresponding to the lower 2.5% follows from
> quantile(boot,0.025) [1] 0.153469
Recall from the notes that the estimate of the bias is given by the difference between the mean of the bootstrap values and the initial estimate,
> bias <- mean(boot) - CV(x)
and an bootstrap-corrected estimate of the CV is just the original estimate minus the bias,
> CV(x) - bias [1] 0.2644771
Assuming normality, the approximate 95% confidence interval is given by CVd± 1.96p
Var(bootstrap) (or adjusting for the bias an lower and upper values of
> CV(x) - bias - 1.96*sqrt(var(boot)) [1] 0.1783546
> CV(x) - bias + 1.96*sqrt(var(boot)) [1] 0.3505997
Efron’s confident limit (Equation 11 on resampling notes) has an upper and lower value of
> quantile(boot,0.975) [1] 0.3176385
and
> quantile(boot,0.025) [1] 0.153469
While Hall’s confidence limits (Equation 12) has an upper and lower value of
> 2*CV(x) - quantile(boot,0.025) [1] 0.3514734
and
> 2*CV(x) - quantile(boot,0.975) [1] 0.1873039
Jackknife Calculations
We now turn to jackknifing the sample. Recall from the randomization notes that this involves two steps. First, we generate a jackknife sample which has value xiremoved and then compute the ith partial estimate of the test statistic using this sample,
bθi(x1· · · xi−1, xi,· · · xn)
We then turn this ith partial estimate into the ith pseudovalue bθi∗using (Equation 5c in random notes)
bθ∗i = nbθ− (n − 1)bθi
where bθis the estimate using the full data.
Let’s see how to code this inRusing the previous vectorxof data with our test statistic again being the coefficient of variation (and hence our functionCVpreviously defined).
We first focus on generating the ith partial estimate and ith pseudovalue. We need to take the original data vectorxand turn it into a vector (which we denotejack) of lenght n− 1 as follows. First, we need to specify toRthat we are creating the jackknife sample vector of the n− 1 sampled points
jack <- numeric(length(x)-1)
As before, we will use the commandlenght(x)in place of n. We also need to specify to Rthat we will be generating a vectorpseudoof the n pseudovalues
pseudo <- numeric(length(x))
Next, we need to fill in the elements of thejacksample vector as follows. For j < i, the jth element ofjackis the same as the jth element ofx; for j = i we exclude the value of x, while for j > i, the j− 1th element ofjackis the jth element ofx. We can state all this using a logicalif .. elsestatement within aforloop,
for (j in 1:length(x)) if(j < i) jack[j] <- x[j]
else if(j > i) jack[j-1] <- x[j]
We can then compute the ith pseudovalue (for the CV) as follows:
pseudo[i] <- lenght(x)*CV(x) -(lenght(x)-1)*CV(jack)
Finally, we top this all off by looping through the n possible i values, giving the final code as
jack <- numeric(length(x)-1) pseudo <- numeric(length(x)) for (i in 1:length(x)) { for (j in 1:length(x))
{if(j < i) jack[j] <- x[j] else if(j > i) jack[j-1] <- x[j]} pseudo[i] <- length(x)*CV(x) -(length(x)-1)*CV(jack)}
Note the use of the parenthesis ({, }) to delimit the appropriate elements in each loop. The mean and variance of the pseudovalues are easily found using
> mean(pseudo) [1] 0.2617376
> var(pseudo) [1] 0.07262871
Likewise, a histogram of the pseudovalues is generated using hist(pseudo)
Recall that the mean of the pseudovalues is the bootstrap estimator, while var(pseudo)/n is the variance of this estimator,
>var(pseudo)/length(x) [1] 0.002905148
An approximate 95% confidence interval is given by mean(pseudo)±t0.975,n−1p
var(pseudo)/n UsingR, the upper and lower limits become
> mean(pseudo) + qt(0.975,length(x)-1)*sqrt(var(pseudo)/length(x)) [1] 0.3729806
> mean(pseudo) - qt(0.975,length(x)-1)*sqrt(var(pseudo)/length(x)) [1] 0.1504947
Giving the approximate 95% jackknife confidence interval as 0.150 to 0.372.
Here’s a summary of the various estimated values, variances, and confidence intervals
Method Estimated CV Variance 95% interval
Original Estimate 0.252
Jackknife 0.262 0.0029 0.150 - 0.373
Bootstrap 0.264 0.0019
Bootstrap (normality) 0.178 - 0.351
Bootstrap (Efron) 0.153 - 0.318
Bootstrap (Hall) 0.187 - 0.351
