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Questions tagged [bootstrap-sampling]

Use this tag for questions related to a statistical test or metric that relies on random sampling with replacement.

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why bootstrap re-sampling size is same as length of the original sample

a non-parametric bootstrap: let $𝑥=(𝑥_1,…,𝑥_𝑛 )\overset{i.i.d}{\sim}𝐹$ and $F$ is unknown steps 1)Fix the number of bootstrap re-samples B. 2)Sample a new data set $𝑥^∗$ set of size $𝑛$ ...
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Bootstrap - percentage of items that are not selected [duplicate]

Why is percentage of items that are not selected in Bootstrap method $\frac{1}{e} \approx 37 \%$? It is related to Bootstrap-related issue . Thank you
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Bayesian bootstrap : is this correct?

So my problem is as follow : I have a given string of characters, and I would like to quantify the uncertainty linked to the probability of each letter types in the string, based on there observed ...
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19 views

Sampling distribution of a functional T

While studying the bootstrap method, I came across with the following definition of the sampling distribution of a functional T: Let's say $X_1,...,X_n$ are i.i.d with distribution function $F$, then ...
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Inequality regarding sample mean

I was looking at the book "Asymptotic Theory of statistics and probability, DasGupta A., 2008" and in one point of a proof they use an inequality which I have not been able to understand. Given that $...
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“Weak” conditions necessary for convergence of parameter being in bootstrap pivotal confidence interval?

My statistics text says that: In summary the $1-\alpha$ bootstrap pivotal confidence interval is: $$C_n = (2\hat{\theta}_n - \theta_{1-\frac{\alpha}{2}}^{*}, > 2\hat{\theta}_n - \theta_{\...
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Expected number of tries to choose x unique values

it's been a long time since I've dealt with probability so I thought I would ask here. I'm sampling elements independently and uniformly and with repetition from a population. Given that the ...
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10 views

Variance of (bootstrap) sample mean conditional on sample

I am having some trouble understanding the manipulation of the variance of the bootstrap mean, conditional on the sample. \begin{align*} Var(\bar{X}^B|X^S) & = Var({X}^B|X^S) \\ \\ & = \frac{E(...
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Is it the right way to use bootstrap on ANOVA test?

I've seen a model arguing that it is a bootstrap ANOVA test. The model is not what I knew. I tried to find if the model is a valid or not, but failed. The procedure is as follows: sample given data ...
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10 views

Consistency of bootstrapping asymptotic variance

Let $(X_n)_{n \in \mathbb{N}}$ be an i.i.d. sequence with distribution $F$ and let: \begin{align*} U_n = T(X_1,\dots,X_n) \end{align*} be an estimator for some quantity $\theta$. Suppose that: \begin{...
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Distribution of overlapped sampling

Given $n$ i.i.d. random variables $X_i$ for $1 \le i \le n$. Consider a moving block bootstrap sampling with block size $l$: $B_i = (X_i, \ldots, X_{i + l - 1})$ for $1 \le i \le n - l + 1$, there are ...
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How to create data out of sample?

I have a device, which measures events 1 out of R, where R is $O(10000)$. I started measurements at time $0$ up to time $T$, where $T$ is large. Now I have a sampled data set, recording event and its ...
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1answer
58 views

Why does bootstrapping approach the distribution of estimator, not mean of the estimator with normal distribution?

The wiki page (https://en.wikipedia.org/wiki/Bootstrapping_(statistics)) sated that bootstrapping allow one to compute and estimate the approximate distribution of the estimator. But why does ...
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21 views

sampling from an existing sample

I have an existing data set with 8 000 observations. I want to check an association between two parameters in the data set. I also want to see if maybe for this purpose I do not need 8 000 ...
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1answer
44 views

Probabilistic subsampling of an Erdős–Rényi graph

Suppose I have an Erdős–Rényi graph ${\cal G}(n,p)$, where $n$ is the total number of nodes and $p$ is the probability of an edge between any pair of nodes (edges are added independently). I subsample ...
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Calculating the unconditional distribution of a bootstrap replication

We have $(X_1,X_2,X_3...,X_n)$ iid $\mathcal{N}(\theta,\sigma^2)$ with $\sigma^2$ known. Let $\hat{\theta} = \frac{1}{n}\sum_{i=1}^n X_i$. We perform a parametric bootstrap procedure for $\hat{\theta}$...
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Nonparametric Boostrap Confidence Interval For $\text{Var }(\overline X)$

Let $\overline X$ denote the sample mean. If we want to find its variance, we have $\text{var } \overline X = \sigma^2/n$. Now, if we do not have $\sigma^2$ we can instead use: $$\hat \sigma^2 \ \ =...
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54 views

Bootstrap estimate of correlation

I have two sets of observations, the brain size $(x)$ and body size $(y)$ of various animals. I am constructing a bootstrap estimate of the correlation. In order to get the bootstrap estimates, I ...
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181 views

Given a data set of $n$ distinct values, show that the number of distinct bootstrap samples is $(2n-1)$ choose $n$.

Given a data set of $n$ distinct values, what is the proof that the number of distinct bootstrap samples is $(2n-1)$ choose $n$?
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1answer
37 views

Finding the expected value of the sum of $x_i^2$

I'm trying to figure out how to prove $$E\!\left(\bar{x} ^2\right) = \frac{1}{n} \sum_{i=0}^n \left(x_i ^2\right)$$ This is for a bootstrap, with no specification as to the distribution. I ...
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Probability that an element will not appear in a Sample of size N drawn from a Set of size N with replacement

The probability that an element of a Set A of size N will not appear in a Sample B of size N drawn with replacement from the aforementioned Set A is 1/3 if I am correct. But how can we prove this? I ...
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1answer
33 views

How many sets of N elements can we construct by sampling with replacement N elements?

Assuming we have a Set of N elements and we form samples of size N by sampling with replacement the original Set, how many such samples can we construct? Alternatively, the question is: How many ...
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2answers
235 views

Intrepretation of Bootstrap method in a simple example, with uniform population to infer.

In order to understand the functionality of bootstrap, i may use a population with uniform distribution to infer. We can generate a sample of 50 points from a uniform distribution $U(0, 1)$ with $\mu=...
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Test for difference of variances in populations with different means

I have two samples A and B, both of size 72, drawn from two different populations. I do not know anything about the population, only that they are non normal at all. I want to know the probability ...
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1answer
366 views

Why is a bootstrap method helping in some way?

The general methods with bootstrapping is always similar to that: We have a given sample $x_1,...,x_n$. Then we pick some elements of the sample randomly and put it then back to the sample; This ...
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The measure-theoretical definition of a bootstrap sample

I’m currently learning the bootstrap method, and I have two questions to ask about the definition of a bootstrap sample. Let $ (\Omega,\mathscr{S},\mathsf{P}) $ be a probability space. Let $ X_{1},\...
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2answers
185 views

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 ...
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1answer
2k views

How to estimate population mean by bootstrap sampling?

Let's say I have a population of size 1M, and I took a sample of 10k. For each individual of the 10k sample, I recorded an observation x. Afterwords I subsampled the 10k sample 1000 times with ...
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24 views

Parametric bootstrap (Edited)

The first question is quite easy, but I'm stuck with the second one. Any suggestion? http://imgur.com/a/hKmvB
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1answer
440 views

SVD: How to scale singular values after rotating U and V (Matlab)

I am very new to linear algebra... I construct a rectangular matrix A1 from some sampled data which is m x n where m > n [U1, S1, V1] = svd(A1) If I then ...
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1answer
124 views

What is meant by Adams Bashforth being a “boot strap” method?

People seem to say that the Adams-Bashforth method requires some "boot strapping" because it needs two initial conditions: $y_{n+1}=y_n+\frac{\Delta t}{2}[3f(t_n,y_n) - f(t_{n-1}, y_{n-1})]$ I ...
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54 views

Why is good to use bootstrapping samples?

I was researching a little about one algorithm called Random Forest, since it use bootstrapping samples, I mean it constructs several samples with replacement from a set, one question came to me, Why ...
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102 views

When is the bootstrap sampling method not applicable?

I have used once the bootstrap sampling method to obtain a confidence interval for the expected daily returns that I had calculated using some data given. As far as I have understood, this method can ...