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Questions tagged [sampling-distribution]

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Why do we assume samples have the same variance when deriving standard error? [closed]

In all derivations I've seen of the standard error formula $\sigma/\sqrt{n}$, it is assumed all the samples in the sampling distribution have the same variance ($\sigma^2$). Why is it assumed they all ...
statataka's user avatar
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0 answers
39 views

T Distribution and CLT

By definition, the T distribution is the ratio of standard normal variable and sqrt of scaled $\chi^2$ variable. The "popularized" version of (one sample) t statistic goes like this: $\frac{\...
Kaiwen Wang's user avatar
2 votes
0 answers
43 views

Exact sampling distribution of Kendall's tau under independence (no ties case)

I am looking for an R package for calculating the CDF and the inverse-CDF of the sample Kendall's tau (*) under the null assumption of independence, in the case of no ties. (*) Just to be clear, I am ...
Vicent's user avatar
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0 answers
22 views

Distribution of the relative range ($R/\sigma$) [duplicate]

Let us assume a $n$-sized random sample $\{ X_1 , \dots, X_n \}$ from a random variable $X \sim N(\mu, \sigma)$. Let $X_\mathrm{min} = \mathrm{min}\{ X_1 , \dots, X_n \}$ $X_\mathrm{max} = \mathrm{...
Vicent's user avatar
  • 789
1 vote
1 answer
147 views

Linear Regression - Proof that coefficients estimated via OLS follow a normal distribution [duplicate]

The aim of my question can be better illustrated by this quote extracted from the third chapter of Elements of Statistical Learning (link to book): I'm trying to understand why, given that the error ...
Frederico Portela's user avatar
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0 answers
26 views

Sample covariance of t distribution and degree of freedom

If $X$ is a P by N size matrix, $X_{ij} \sim N(0,\sigma_i^2)$ if I standardize this X matrix with sample mean and sample variance (assuming I don't have access to the population mean and variance) I ...
maddy's user avatar
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12 votes
3 answers
3k views

Do the 2.5th and 97.5th percentile of the theoretical sampling distribution of a statistic always contain the true population parameter?

I am trying to understand the validity of bootstrap percentile confidence intervals and I have stumbled on the following from these slides: Suppose we want to set a 95% confidence interval on $θ$, ...
ado sar's user avatar
  • 477
0 votes
1 answer
168 views

Calculating the probability my observation, $Y_i$, is drawn from a random variable $X$?

If I sample a population distribution 2,000 times and get an estimator for the population mean, $\mu$, and the standard deviation, $\sigma$, how can I use these to get the probability that an ...
Connor's user avatar
  • 635
1 vote
2 answers
71 views

Interpretation of distribution that appears when calculating CI for population mean

Let $X \sim \mathcal{N}(\mu, \sigma)$ be the model for a normally distributed population, described by the probability density function $f_{X}(x; \mu, \sigma)$. We can denote $\mathbf{X} = (X_1, X_2, \...
ivan's user avatar
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0 answers
31 views

Is it possible that a confidence interval for a sampling distribution of the sample proportion to be < 0 or > 1? [duplicate]

A proportion or percentage must be between 0 and 1, therefore, a confidence interval for a proportion should also be between 0 and 1. But is it possible that if the "10-successes-and-10-failures&...
Maxime Dupré's user avatar
2 votes
1 answer
125 views

Why use the bootstrap for a skewed statistic when you can use a transform?

Let's say you are working with a statistic (say, the mean of the population) of a skewed distribution with a long, long tail such that confidence intervals must be very skewed to achieve reasonable ...
Estimate the estimators's user avatar
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0 answers
26 views

Sampling distribution, bias and variance of cross-validation methods (particularly LOOCV)

(TL;DR version below) If my understanding is correct, bias/variance are measures of goodness of fit of a statistical estimator w.r.t. the sampling distribution. So if I have a statistic $t(X)$ that ...
statkun's user avatar
  • 63
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1 answer
25 views

Proper way of constructing sampling distribution

While the proper way of constructing the sampling distribution would be to repeatedly sample from population with unknown parameters, I'm not convinced it is the case in practice. Surveys performed in ...
Student's user avatar
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2 votes
1 answer
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Is true that the sampling distribution of $\ln \left(\chi^{2}\right)$ converges to normality much faster than the sampling distribution of $\chi^{2}$?

If true is the consequence true that $X \sim \chi^{2}(k)$ then $\sqrt{2 X}$ is approximately normally distributed with mean $\sqrt{2 k-1}$ and unit variance? Also true that If $X \sim \chi^{2}(k)$ ...
user avatar
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102 views

sampling distribution of a test statistic when doing permutation test

In a permutation test, we have a test statistic $T$ (for example t-test, f-test, etc.) that is applied to the original data to obtain an observed value $o_{obs}$ for the test statistic. Next, the data ...
userOnion's user avatar
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50 views

Standard error of sampling distribution mean vs standard deviation of the sampling distribution?

In this resource, towards the bottom, the authors write: The next step is to estimate the standard error of the mean. If we knew the population variance, we could use the following formula, $sigma_M = ...
jbuddy_13's user avatar
  • 3,372
2 votes
1 answer
603 views

Sampling distribution of GBM Maximum-Likelihood estimator

Given the geometric Brownian diffusion $$ X_t = \mu X_t \, dt + \sigma X_t \, d W_t$$ I learnt that its maximum likelihood estimators are the following as this web article suggests $$\hat \mu = \frac{\...
student's user avatar
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1 vote
0 answers
27 views

Calculating the Standard Deviation of the Sampling Distribution when using Bernoulli Sampling

I am trying to calculate the standard deviation of a sampling distribution when using Bernoulli sampling (where each item is given a separate, equal, non-zero probability of being included in the ...
ElectricRocket's user avatar
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0 answers
45 views

Is there a good graphical representation for the sampling distribution of relative risk?

For pedagogical reasons, I am interested in communicating graphically the sampling distribution of relative risks (histogram and CDF plot), specifically in the context of a matched-pair study design: ...
Alexis's user avatar
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1 answer
104 views

Random samples within the Central Limit Theorem - why select with permutation with repetition?

"The central limit theorem states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the ...
Peiran Yu's user avatar
  • 119
4 votes
1 answer
128 views

Sampling distribution of sample variance

Let $(X_1, X_2, ..., X_n)$ be a random sample from Bernoulli($p$). Find the sampling distribution of $S_n^2=\frac{\sum_{i=1}^n(X_i -\bar X)^2}{n-1}$. I've just started learning about sampling ...
Tapi's user avatar
  • 311
8 votes
3 answers
3k views

Confidence band for simple linear regression - why the curve? [duplicate]

I am really struggling to understand why confidence bands for regression lines have a curve to them. A few example plots showing curved CIs, taken from this post: Shape of confidence interval for ...
dataphile's user avatar
1 vote
1 answer
199 views

Computing probability distributions over bootstrap samples for two statistics

I have a data set $x= c(0.9575,0.4950,0.1080,0.9359,0.6326)$ and I'm trying to compute the probability distributions for the statistics $\bar X^* - \bar X$ and $\sqrt n(\bar X^* - \bar X)/s^*$, over ...
Novice's user avatar
  • 581
4 votes
1 answer
673 views

Two Sample Z-test with Non - normal distributions (A/B testing)

I am doing my first steps in statistics and your help will be much appreciated. Sorry for not providing any data, since this is just a made-up exercise that I could not quite find the answer for. Let'...
vvan's user avatar
  • 43
0 votes
1 answer
117 views

Can bootstrapping be used to find the standard error of variance?

I am a bit new to the field and had a question regarding bootstrapping. As bootstrapping gives an idea of the standard deviation of the sampling distribution of the mean, does it also give an idea of ...
Andrew Dane's user avatar
-1 votes
1 answer
391 views

What is the meaning behind a sample distribution if you only have one sample?

I'm trying to understand the meaning behind the central limit theory and the importance of CLT for inferential statistics. The problem that I encountered has to do with sample distributions. ​ I do ...
IBI's user avatar
  • 3
1 vote
2 answers
1k views

Sampling distribution: sampling with replacement or without?

The following paragraph occurs in Essentials of Statistics for the Behavioral Sciences(10th edition): If you actually wanted to construct the distribution of sample means, you would first select a ...
Mainul Islam's user avatar
3 votes
1 answer
5k views

Binomial vs hypergeometric finite sampling distribution

I was reading these notes:Finite Population Sampling with Application to the Hypergeometric Distribution and I have a question just about the first two pages. The first page, they derive the variance ...
Steve's user avatar
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1 vote
1 answer
347 views

Estimating population parameter from bootstrapped sample distribution, other than mean value

I'm a rather novice to statistics, so my terminology here might not be totally correct. But I will do my best to explain my question clearly. My question is about understanding whether it is making ...
Bicycle-riding Dog's user avatar
3 votes
0 answers
800 views

Sampling/Asymptotic Distribution of Estimated Coefficients of Logistic Regression

If I understand correctly, in a logistic regression, we have that $Y_i \mid X \sim Bern(S(X\beta))$ where $S(x)$ is the sigmoid function. Suppose we estimate $\beta$ using MLE and get $\hat \beta$. ...
Dayne's user avatar
  • 2,661
1 vote
0 answers
1k views

Sampling distribution of Pearson correlation coefficient

Suppose I have draw a random sample of points $(x_i,y_i)$ iid from some distribution, then I compute the Pearson correlation coefficient $\overline{\rho}$ of the points in the sample. Is $\overline{\...
D.W.'s user avatar
  • 6,688
0 votes
0 answers
31 views

Which field of science applies the log-normal sample median distribution?

Which field of science applies the log-normal sample median distribution (i.e. the sample median distribution with samples from a log-normal parent)? I noticed that incubation periods are assumed log-...
LLT's user avatar
  • 11
1 vote
0 answers
47 views

How to use a p-value sampling distribution?

As part of my research I have been looking into the sampling distribution of p-values. Even though p-values are on their way out, they are still used ubiquitously with the normal $\alpha$ =.05 cutoff (...
Matthew Ferrell's user avatar
2 votes
2 answers
1k views

How to interpret confidence interval and prediction interval in simple regression "in/with the context of sampling distribution"?

With the context of sampling distribution, in regression analysis, is the following an appropriate interpretation? Assumptions : X & Y have a linear relationship sample size is large enough for ...
rahul-ahuja's user avatar
0 votes
0 answers
51 views

Likelihood function and sampling distribution symbol

I read a introductory stat book that the likelihood function has symbol like , e.g., L(theta, x); while the sampling distribution has the symbol like f(x,theta). May I ask does anyone know whether the ...
Soother's user avatar
1 vote
0 answers
64 views

Asymptotic Distribution Using CLT

I have random variables $X_1, X_2, ... , X_n \sim \text{IID } f_X$ using the density function: $$f_X(x) = \frac{2x}{\theta^2} \cdot \mathbb{I}(0 \leqslant x \leqslant \theta).$$ I have to use the ...
user avatar
4 votes
1 answer
141 views

degrees of freedom in a t-test in Kruschke's BEST paper

This question is based on a comment by John Kruschke in his BEST paper, pages 589-590. Kruschke, John K. "Bayesian estimation supersedes the t test." Journal of Experimental Psychology: ...
Dave's user avatar
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5 votes
1 answer
99 views

Question about the conceptual sampling distribution for one time events

in this paper: Do Political Protests Matter? Evidence from the Tea Party Movement*, the authors use rainfall on the day of the tea party protests as a source of plausibly exogenous variation in rally ...
Steve's user avatar
  • 681
2 votes
2 answers
127 views

right skewed sample does not lead to normally shaped sampling distributions

I will provide R code for a reproducible example. I am calculating the difference in means for two groups. I get a sampling distribution by permutations but instead of a normally shaped distribution ...
tomas hujo's user avatar
3 votes
2 answers
711 views

Sampling distribution of the mean of the discrete-power law distribution

For a certain problem I wish to generate random integers $k$ so that their distribution follows $p_k \sim k^{-\alpha}$ for $k \geq k_{\text{min}}$, $k_{\text{min}} > 0$. I am following the ...
Peaceful's user avatar
  • 613
10 votes
2 answers
2k views

Sampling distribution of the mean of a Beta

Say we have $X \sim \text{Beta}(\alpha, \beta)$. What's the sampling distribution of its sample mean? In other words, what distribution does the sample mean $\bar{X}$ of a Beta follow?
Josh's user avatar
  • 4,518
4 votes
1 answer
9k views

Proof that the sampling distribution of the sample variance from $N(0,1)\sim \chi_{n-1}^2$

Is this true? How to verify it? From the definition of chi square I can not judge whether it is chi square.
Richard Lin's user avatar
3 votes
2 answers
4k views

Finding sampling distribution of normal MLE and likelihood

I'm reviewing old exams in preparation for a statistics final, and I'm stuck on a particular question: Suppose that you have n independent random variables $Y_i$, with each distributed normal with ...
Florian D'Souza's user avatar