## Pages

### An R function for bootstrap confidence intervals

Here I introduce a R function made by myself to calculate several confidence intervals of mean/median including exact CI, basic bootstrap CI, percentile bootstrap CI & studentized bootstrap CI.

#### Arguments

There are five arguments in this R function bootCI():

• x — a numeric vector specifying $x_i$
• alpha = 0.05 — a numeric specifying $\alpha$ value
• alternative = c("t", "l", "g") — a single character string specifying two-sided, less or greater tail
• B = 1999 — a integer specifying number of bootstrap samples
• quantileAlgorithm = 7 — a integer specifying the argument type passed to quantile()

#### Examples

We can define a numeric vector $x = [5\, 2\, 3\, 6\, 8\, 19\, 1.5]$ including $N=7$ items as num:

> num <- c(5, 2, 3, 6, 8, 19, 1.5)

After loading the above function, we can call the function to calculate CIs:

> bootCI(num)
Summary of x
Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
1.500   2.500   5.000   6.357   7.000  19.000
CIs of mu
2.5%     97.5%
$CI.exact 0.7778728 11.936413$CI.basic       1.7125000  9.714286
$CI.percentile 3.0000000 11.001786$CI.studentized 2.6785623 17.669412

The above results show that

\begin{aligned} \text{exact CI} & = ( \hat{\mu} + t_{\frac{\alpha}{2}} \cdot \hat{se}_{\hat{\mu}} \;,\; \hat{\mu} + t_{1-\frac{\alpha}{2}} \cdot \hat{se}_{\hat{\mu}} ) \\ \text{basic CI} & = ( 2\hat{\mu} - \hat{\mu}_{1-\frac{\alpha}{2}}^{*} \;,\; 2\hat{\mu} - \hat{\mu}_{\frac{\alpha}{2}}^{*} ) \\ \text{percentile CI} & = ( \hat{\mu}_{\frac{\alpha}{2}}^* \;,\; \hat{\mu}_{1-\frac{\alpha}{2}}^* ) \\ \text{studentized CI} & = ( \hat{\mu} + t^*_{\frac{\alpha}{2}} \cdot \hat{se}_{\hat{\mu}} \;,\; \hat{\mu} + t^*_{1-\frac{\alpha}{2}} \cdot \hat{se}_{\hat{\mu}} ) \\ \end{aligned}

where $\hat{\mu}$ is mean of $x$, $t_{\frac{\alpha}{2}}$ is lower $\alpha/2$ critical value for the $t$ distribution given $\text{df} = N-1$, $\hat{se}_{\hat{\mu}}$ is the standard error of the mean of $x$, $\hat{\mu}_{\frac{\alpha}{2}}^*$ is $\alpha/2$ percentile of the mean of bootstrapped $x$, and $t^*_{\frac{\alpha}{2}} = \frac{\hat{\mu^*}-\hat{\mu}}{\hat{se}^*_{\hat{\mu^*}}}$ is t-value based on bootstrapped $x$. See

for more details.

We can assign a different $\alpha$, direction or number of bootstrap samples:

> bootCI(num, alpha=0.01)
Summary of x
Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
1.500   2.500   5.000   6.357   7.000  19.000
CIs of mu
0.5%    99.5%
$CI.exact -2.0962642 14.81055$CI.basic       -0.1428571 10.28571
$CI.percentile 2.4285714 12.85714$CI.studentized  1.1394952 25.22030

> bootCI(num, alternative="g")
Summary of x
Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
1.500   2.500   5.000   6.357   7.000  19.000
CIs of mu
5% 100%
$CI.exact 1.926445 Inf$CI.basic       2.714286  Inf
$CI.percentile 3.285714 Inf$CI.studentized 3.332740  Inf

> bootCI(num, B=99)
Summary of x
Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
1.500   2.500   5.000   6.357   7.000  19.000
CIs of mu
2.5%     97.5%
$CI.exact 0.7778728 11.936413$CI.basic       1.6053571  9.330357
$CI.percentile 3.3839286 11.108929$CI.studentized 2.5532298 15.532786

The returned list may be helpful for someone:

> myCI <- bootCI(num)
> myCI$CI.percentile 2.5% 97.5% 3.00000 10.85714 > str(myCI) List of 8$ x             : num [1:7] 5 2 3 6 8 19 1.5
$alpha : num 0.05$ alternative   : chr "t"
$B : num 1999$ CI.exact      : Named num [1:2] 0.778 11.936
..- attr(*, "names")= chr [1:2] "2.5%" "97.5%"
$CI.basic : Named num [1:2] 1.86 9.71 ..- attr(*, "names")= chr [1:2] "2.5%" "97.5%"$ CI.percentile : Named num [1:2] 3 10.9
..- attr(*, "names")= chr [1:2] "2.5%" "97.5%"