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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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8 views

Upper bound of rarefaction curve

Rarefaction curve Rarefaction curve (see here for more background) is given by $$ S(m) = S_0 - \sum_{i=1}^{S_0}\left[ I(n-m\geq n_i)\prod_{l=0}^{m-1}\frac{n-n_i-l}{n-l}\right]=\\ S_0 - \sum_{i=1}^{S_0}...
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22 views

Formula for standard deviation when using non-parametric bootstrap

I'm using a bootstrap method to compute a confidence interval for the AUC of a logistic regression model. Let's say I'm performing 1000 resamples. When I find a result of 90.0 with a 95% confidence ...
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55 views

Bootstrap resampling from gaussian distribution.

I have a set of points $\{x_i,y_i\}$ with 20 elements on the scatter plot, the distribution looks like Gaussian. My goal is to estimate the Gaussian center $x_{center}=\mu$ of this distribution. I'm ...
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27 views

Proof of Sampling/Importance Resampling (Weighted Bootstrap) technique

From Casella Berger exercise 5.65: Let us have $X \sim f$. Then, assume we produce $m$ i.i.d. random variables $Y_1,...,Y_m$ from another distribution $g$. Let us have $$q_i = \frac{\frac{f(Y_i)}{g(...
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62 views

Sampling from a 2D to 1D distribution

I have one question/problem regarding sampling and I am not sure in which category this falls (it looks like bootstrapping) Let's say that I have some real surface $z = f(x, y)$ on some bounded domain ...
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23 views

Using hypothesis testing to check if a mean of a sample is bigger than the other sample.

Basically I am asked if people over age $35$ spend more money on average than people less or equal to age $35$ with significane level $0.05$. And I have a sample of the population that I can use for ...
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29 views

Choosing the test statistic for this bootstrap

Say I have a (sample) bin with balls colored red, green, and blue. I'm asked if the distribution of the colors is uniform with a required significance level of $0.01$, which means is there a $\dfrac{1}...
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34 views

Can we use Bootstrap to estimate the degrees of freedom for $K$-nearest-neighbors

Setting: We know that K-NN is a linear smoother model and then we can write $$ y = f(X) + \epsilon \implies \hat{y} = Sy \implies \hat{y}_i = \sum_{j=1}^{n}w(x_i,x_j)y_j, $$ where $w(x_i,x_j) = \frac{...
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44 views

What is the limit of $((x-1)/x)^x$ as $x$ goes to infinity?

I came across this limit when considering bootstrapping since $1 - ((x-1)/x)^x$ is the probability that observation $j$ from a sample of size $x$ does appears in a bootstrap sample also of size $x$. ...
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2answers
59 views

Probability of a median being $x_i$

I am trying to derive a probability function and here are the assumptions. Let $S = \{x_{(1)},x_{(2)},...,x_{(7)}\}$ be a set of distinct values that are ordered. Let $S*=\{x_1^*,x_2^*,...,x_7^*\}$ ...
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25 views

Relation between bootstrap resampling and $t$-Student test

After scouring the internet and several reference books for a couple of days, I couldn't really find an answer to the current problem I am trying to solve. I have some problem understanding why I can'...
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118 views

How to compute probability of a bootstrap sample

The Question Consider the samples $\{1, 3, 4, 6\}$ from some distribution. a) For one random bootstrap sample, find the probability that the mean is $1$. b) For one random bootstrap sample, find the ...
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80 views

Variance of a bootstrap estimator

Suppose we have a sample $X_1,X_2,...,X_n \sim F$, where the distribution $F$ is unknown. Let $T_n = g(X_1,X_2,...,X_n) = \bar{X}^2$, $\mu = \mathbb{E}[X_1]$, and define the following: $$\alpha_k = \...
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13 views

Aggregation of bootstraps on partitioned datasets

I am trying to evaluate the effect of a new machine learning model with regard to some metric $M$. So I am ABTesting the model and basically have two "real" datasets $A$ and $B$ where each ...
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6 views

Sampling from empirical distribution resambles drawing with replacement from sample.

When reading about bootstrap I've seen everywhere that sampling from the empirical distribution is the same as sampling values randomly with replacement from the observed data. Why is this true?
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17 views

Uncertainty handling for time series data

I am looking for information regarding uncertainty handling. Specifically, I am interested in finding information related to sampling techniques and scenario generation. I am very interested in ...
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57 views

Bootstrap and permutation resampling. What is the difference?

Recently while passing one course (on datacamp) I got acquainted with two methods of resampling ( I used them to simulate statistical hypothesis) : 1.Bootstrap resampling when we choose samples with ...
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59 views

Samples made with the bootstrap method and its means distribution

So,can we state that sample means of bootstrap samples are distributed normally? And if no , how we can find confidence interval for mean of general distribution? I know that we can calculate 2.5 and ...
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1answer
59 views

Statistics - Bootstrap Method

After scouring the internet and reference books for a couple of days I couldn't really find an answer to the current problem I am trying to solve. Lets say that I want to construct a confidence ...
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31 views

Apply function on two set to create a new (+ process validation: bootstrap iterations)

I am new to algebra and help would be more than welcome to tell me if the process I have built is OK, and if my attempt to apply formula on two sets to create a new one is also OK. Context I have a ...
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1answer
41 views

Apply formula on random sampling without replacement, but with replacement between each iterations

I am looking for an algebraic solution to explain that I apply a formula on a vector constituted of a random sampling of n elements in a population of size N without replacement. The formula is ...
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1answer
17 views

Consider why does bootstrap sampling distribution need to compute the estimate, If we knew the true parameters $\theta^∗$?

Page 192 of "Kevin Patrick Murphy. Machine Learning: A Probabilistic Perspective." says The bootstrap is a simple Monte Carlo technique to approximate the sampling distribution. This is ...
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23 views

Is it possible to cluster particles and then resample each cluster?

Is it possible to cluster particles (in the particle filter) and then resample each cluster separately? if yes, the resampling are done parallel?
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24 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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2answers
52 views

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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241 views

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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1answer
166 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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120 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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1answer
48 views

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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80 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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387 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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48 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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2answers
53 views

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
44 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
553 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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31 views

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
480 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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3answers
2k views

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
276 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
3k 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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30 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
848 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
196 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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67 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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1answer
192 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 ...