# Bootstrap method & Confidence Interval

I'm trying to figure out how this method works. My data:

• 1000 samples from unknown distribution.
• I need to create 40 vectors from those 1000 samples (each vector with 20 samples)
• For every one of the 40 vectors, I need to do the bootstrap method for:
• Finding the confidence interval ($\alpha$ = 0.05) in three methods: t, quantiles & normal.
• We need the confidence interval for the standard deviation.

(R langauge)

My way until now:

• I've created this 40 vectors (each one with 20 samples)
• Let's say that the bootstrap constant is 1000.
• What is actually the process of "doing bootstrap" for each vector with 20 samples? How can we create a confidence interval for this vector in each one of these methods I've mentioned?

I will be glad for any help.

• Your post has been edited and re-structured. I checked against the original and the edit seems OK. But your question is long, and I would urge you to verify if the 'bullet' structure of the edit is what you intended. // Are you interested only in confidence intervals for the population standard deviation $\sigma?$ What you mean by the 3 methods (t, quantiles, normal) is not exactly clear; bootstrapping is only a few decades old and the terminology is still not totally standard. Can you provide R code for at least one of the methods? – BruceET Apr 7 '17 at 20:48
• Please edit the information in your 'Comment as Answer' into your Question. Do you mean standard deviation ('Standardabweichung') instead of standard mean? Or do you mean standard error of the mean? At the very least, in your title, you should be able to say what you are estimating--mean ($\mu$), SD ($\sigma$), SE ($\sigma/\sqrt{n}$). (Then delete your non-Answer, before it attracts down-votes.) My answer uses a quantile method. I don't see how a method with t or normal would work for SD. Maybe my code will give you a start. – BruceET Apr 7 '17 at 22:20

## 2 Answers

@BruceET

I cannot comment because i've needed to reset my account. Anyway, I'm interested only in CI for the population standard mean. By 3 methods, I've meant that there are 3 options to calculate the CI (one with quantiles, one with t distribution, and one with normal distribution).

I wanna provide R code, but that's my challange to understand what is the code :)

• yes, standard deviation. – bootstrapper0 Apr 7 '17 at 22:11
• Then my Answer is relevant. Does it seem similar to any types of bootstrap procedures you have encountered? In particular, does it match quantile, t, or normal? // BTW, if you want someone to see a comment (quickly), you have to put it at the end of their Answer (or Question) or you have to use @name so it will show up on their home page. [You should see this because it's at the end of your Answer.] – BruceET Apr 9 '17 at 2:10

Here is one example of finding a 95% nonparametric bootstrap confidence interval for the population standard deviation (SD) $\sigma,$ based on a sample x of size $n = 20$ from an unknown population distribution.

 x
 240 314 354 183 321 325 271 273 272 255
 276 250 261 303 348 294 274 254 258 421
s.obs = sd(x);  s.obs
50.67365


If we knew the population distribution, we could find the distribution of the ratio $R = S/\sigma$ based on the population distribution. Then we could find values $L$ and $U$ that cut 2.5% from the lower tails, respectively, of the distribution of $R$ so that

$$0.95 = P(L \le R \le U) = P\left(\frac{S}{U} \le \sigma \le \frac{S}{L}\right),$$

where $S$ is the sample SD of the sample of $n = 20.$ Then the desired 95% CI would be $(S/U,\,S/L).$

However, we do not know the distribution of $R$ and we seek to estimate $L$ and $U$ by using a bootstrap method.

Enering the so-called bootstrap world, we take $B = 1000$ re-samples from x, each of them a re-sample of size $n = 20$ taken with replacement from x. Temporarily, we take the observed SD ($S_{obs} = 50.67365$) as a proxy for the unknown population SD $\sigma;$ that is $\sigma^* = 50.67365.$ Then, for each of the $B$ re-samples, we find $R^* = S^*/\sigma^*.$ We find quantiles .025 and .975 of the $B$ values $R^*$ as estimates $L^*$ of $L$ and $U^*$ of $U,$ respectively. [Notice that quantities referring to re-sampling are denoted by $*$'s.

Back in the real world, we find the 95% nonparametric bootstrap CI of $\sigma$ as $(S_{obs}/U^*, S_{obs}/L^*).$ [Here $S_{obs}$ returns to its original role as the observed SD of our sample x.]

The R code for this procedure follows. In the code we use -re instead of $*$.

B = 1000;  n = length(x);  sg.re = s.obs;  r.re = numeric(B)
for (i in 1:B) {
x.re = sample(x, n, repl=T);  s.re = sd(x.re)
r.re[i] = s.re/sg.re  }
L.re = quantile(r.re, .025);  U.re = quantile(r.re, .975)
LCL = s.obs/U.re;  UCL = s.obs/L.re
c(LCL, UCL)
97.5%     2.5%
37.33546 88.26224


So the 95% nonparametric bootstrap CI for $\sigma$ is $(37.3,\,88.3).$ Because this is a simulation procedure, subsequent runs may give slightly different results. My second run of the program above gave slightly different results that still round to $(37.3,\,88.3).$

Now it is time for a confession: I generated x from a normal distribution as follows:

set.seed(1234); x = round(rnorm(20, 300, 50))


So I know that the data are normal. The standard 95% CI for $\sigma$ of a normal population is $\left(\sqrt{\frac{(n-1)S^2}{U_q}}, \sqrt{\frac{(n-1)S^2}{L_q}}\right),$ where $L_q$ and $R_q$ are quantiles .025 and .975, respectively, of $\mathsf{Chisq}(\nu = n-1).$ So the traditional parametric 95% CI for $\sigma$ is $(38.5, 70.0)$. Because knowing the population distribution introduces new and useful information into the process of estimation, we cannot expect the normal-based CI to be the same as the bootstrap CI, but they are not much different for practical purposes.

sqrt((n-1)*var(x) / qchisq(c(.975,.025), n-1))
 38.53682 74.01249