Questions tagged [central-limit-theorem]

For questions about the central limit theorem, which states: "Given certain conditions, the mean of a sufficiently large number of iterates of independent random variables, each with a well-defined mean and well-defined variance, will be approximately normally distributed." (Wikipedia)

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Why does bootstraping not seem to produce a normal distribution for this data?

I am trying to calculate the 95% confidence interval of the mean value of the population. I have this data: ...
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Sample mean of a geometric distribution

Let $D$ be a distribution with finite mean $\mu$ and finite variance $\sigma^2$. Consider the distribution $S_n$ of the sample mean of $n$ i.i.d. values from $D$. I understand that the Central Limit ...
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Mann-Whitney Normal Approximation process help

Let $X_{1}, X_{2}, ..., X_{n}$ is i.i.d sample from $X$ and $Y_{1}, Y_{2}, ..., Y_{m}$ is i.i.d sample from $Y$. And both samples are independent each other. Trying Mann-Whitney U-test then, $U =$ $\...
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Implications of zero limiting variance

Assume that I have a sequence of random variables $X_1, X_2, \dots$ with means $\mu_1, \mu_2, \dots$ such that $\lim_{n \to \infty} \operatorname{Var}(X_n) = 0$. Can I claim that for large enough $n$ ...
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28 views

confidence interval for sample that is not normal distributed

I have one sample of only 86 values, which is not normally distributed according to the Shapiro-Wilk normality test. Can I still use this formula/code (sorry R code) to estimate the 95% confidence ...
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Confidence Interval for Estimator using Delta method

The statement I am given the following discrete distribution with $\theta>0$ $$p(x) = \left(\frac{\theta}{1+\theta}\right) ^{2-x}\left(\frac{1}{1+\theta}\right)^{x-1} \hspace{1cm} x=1,2$$ I need to ...
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1answer
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theoretical confidence interval depending on sample size [closed]

I am using R and plain English to express my question. Let us say I have a "true"/made up population, which is normally distributed with a mean of 500000 and a standard deviation of 13000: <...
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1answer
43 views

possible use case of central limit theorem for analysts

This is a bit of a long shot but I would appreciate any help please. I have to do a basic stats course for our analysts, which I try to make as applicable and useful as possible using our data (e.g. ...
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49 views

Does Normality of a Time Series imply Stationarity and Viceversa?

I have a theory question which never became completely clear to me. Reading Hamilton (1995) I understod that the stationarity requirement for time series data stands as the normality requirement for ...
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2answers
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Does the Central Limit Theorem imply that $(\hat{X}_n - \bar{x}) = o_p(1)$ at rate $O_p(1/\sqrt{n})$?

Let $\left\{\hat{X}_n\right\}$ be a sequence of estimators that converges in probability to the constant $\bar{x}$, i.e., $\left(\hat{X}_n - \bar{x}\right) = o_p(1)$. Then say that, by some applicable ...
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Rate of convergence of $\hat Q_{xx}^{-1} = \left(\frac{\mathbf{X}^T \mathbf{X}}{n}\right)^{-1}$ to the probability limit?

Consider the simple linear regression model. $$ y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \quad \quad \quad \quad i = 1,2,\dots,n. $$ Let $\mu_x$ and $\sigma_x^2$ represent the mean and variance of ...
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Understanding rate of convergence for realized estimators

I'm a Econ student currently taking a small course on realized measures/estimators. I'm a bit confused about the meaning behind rate of convergence for each type of estimator. I'll give some ...
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ICA: a question about the non-gaussian requirement

I'm new in the ICA processing and I'm trying to understand the non-gaussian requirement. I read that the problem is that, if the composed data is $\mathbf{x}=\mathbf{As}$ with $\mathbf{A}$ (unknown) ...
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CLT-based confidence intervals not working in code

I wrote code to draw $n$ samples from a categorically distributed random variable $C$ with probabilities $p_i$ for each value $i$ and to use those samples to compute an approximation $q_i = n_i/n$ of ...
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Does a vast amount of probability and statistical literature make a mistake when they make use of CLT/asymptotic normality?

Suppose we toss a fair coin $N$ times and we are interested in the probability that we get at least $cN$ heads for $c\in [0,1]$. We can model this situation by letting $S_N = \sum_{i=1}^N X_i$ where $...
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Central Limit Theorem and Skewed Distribution

I'm looking for a simple answer to this question relating the central limit theorem and Gaussian and skewed distributions, if one exists. I used the binomial function to generate calculations of the ...
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1answer
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Central Limit Theorem for Truncated observations

Consider a random variable $X$ with values in $\left[0,\infty\right)$ such that $E\left[X\right]=\infty$. Given $M > 0$ I want to estimate the expected value of $X$ truncated at $M$. That is I want ...
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1answer
61 views

variance of difference of two means $\bar X-\bar Y$

For the mean of sample $\bar X$, $\frac {\bar X -\mu}{\sigma_X/\sqrt{N}}$ has normal distribution. According to CLT, $\bar X$ has a variance of $\sigma_x/\sqrt{N}$. For two means $\bar X, \bar Y$ of ...
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Whats the difference between sample size and sampling size?

I am currently studying a r code that's used for verifying the CLT and in that function, there are three inputs. First is n which is the number of observations second is a parameter for a geometric ...
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1answer
42 views

How to use CLT on statistical inference?

I have an issue for how a sample of the population can be used to infer about the population parameters. For example, see the following questions: The average weekly earnings for female social ...
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Finding the p-values for one sample t-test in different cases

Calculate and interpret the $p-$ value in the following situations: (a) one-sample $t$ - test for testing (i) $H_{0}: \mu=\mu_{0}$ vs $H_{1}: \mu \neq \mu_{0}$ (ii) $H_{0}: \mu \leq \mu_{0}$ vs $H_{1}:...
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Central Limit Theorem - does the number of times the samples are taken matter in terms of the CLT?

Let's say that in R, we generate $n$ random variables $Y_1, \dots, Y_n$ which all follow an exponential distribution. We then construct the mean $\bar{Y_n}$. The process is repeated $m$ times, so we ...
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1answer
45 views

Why is it that when you add normally distributed random variables the variance gets larger but in the Central Limit Theorem it gets smaller?

When you add two independent normal distributions the resulting distributions' variance is the sum of the variances i.e. it gets larger. However, the Central Limit Theorem states that when ...
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1answer
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Central Limit Theorem formula transformation with iid variables

I was looking into Central Limit Theorems and how a CLT is derived and I found this source quite helpful. The only thing I am having trouble to comprehend is the transformation of the formula $$\frac{\...
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31 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{\...
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1answer
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Do I need to sample the distribution in order to apply the central limit theorem?

My scenario is the following: I have a sample set of 1000+ rows, of which my variable of interest is a non-normal distribution. I haven't tested throughtly to check what kind of distribution it is, ...
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1answer
52 views

Central Limit Theorem and Normal Distribution

I have the following questions as homework where I have to decide whether they are true or false. a) The standard deviation of the distribution of the sum of independent random variables is also ...
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Convergence of Normalised Sum of IID Random Variables

I have a Markov chain X that starts from the stationary distribution. Let define $S_n = X_1 + \cdots + X_n.$, where $X_i$ is the state of the Markov chain. Let's have 3 states. I wanted to prove the ...
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2answers
51 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 ...
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How is the Central Limit Theorem related to Maximum Likelihood Estimation?

While studying the Maximum Likelihood Estimation, I often hear that the Central Limit Theorem kicks in do to the Confidence Interval in relation to the Maximum Likelihood Estimation. As far as I know, ...
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Is there a law or concept related to unchanging average?

I have a survey comprising 30 questions with 50k respondents where respondents mark each answer on a 5 point scale. An average of all the questions for all the respondents is metric we track monthly. ...
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Why does MLE tend to normal distribution

We have $X_1,\dots, X_n$ are iid (the distribution can be of any type, e.g. Bernoulli (p), normal ($\mu, \sigma^2$), Poisson ($\lambda$). If we use MLE $\hat \theta$ to estimate any parameter $\theta$ ...
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1answer
123 views

Confidence interval true/false questions

A sprinkler system is being installed in a large office complex. Based on a series of test runs, a 99% confidence interval for the population mean $\mu$, the average activation time of the sprinkler ...
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Weak Law of Large Number to Central Limit Theorem [closed]

Consider a distribution with unknown mean μ and population standard deviation σ=30. Using the Weak Law of Large Numbers, what is the minimum sample size in order to attain a probability of at least 99%...
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Confidence interval for average of different binomial trials - can I apply central limit theorem?

I have person-level data, and for each person, I have the racial breakdown of the geographic area in which they live. For example, person A might have 100 people in their area, and 80 (80%) of them ...
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33 views

Convergence in distribution and Slutsky's theorem

It is known that from the CLT, if $X_i \stackrel{\text{iid}}{\sim} F$ for some distribution $F$ with finite variance, then $$\frac{1}{\sqrt{n}} \sum_{i=1}^n (X_i - \text{E}[X]) \stackrel{d}{\to} N(0,\...
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1answer
135 views

Proof of multivariate central limit theorem

$\newcommand{\phi}{\varphi}$ $\newcommand{\eps}{\epsilon}$ I'm using the book called 'A Course in Large Sample Theory' from Thomas S. Ferguson. During studying the proof of the central limit theory in ...
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1answer
68 views

How is confidence interval related to central limit theorem?

I am currently looking into Confidence Interval and sees that Confidence Interval is possible based on Central Limit Theorem. So, I have been looking for how Central Limit Theorem is related to ...
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Interpretation of odd Central Limit Theorem (i.i.d) condition

My class was taught a third sufficient condition for the CLT to hold in the i.i.d. case that can replace the Lindeberg or Lyapunov conditions. I have never seen this condition before and am wondering ...
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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 ...
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Asymptotic distribution of mme of coefficient of variation

For random sample from unif(0,1) distribution, method of moments estimator for coefficient of variation is sample mean divided by sample standard deviation. Here, coefficient of variation theta is mu/...
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140 views

Central limit theorem and standard normal distribution

In wiki you can find the next text: If n is large enough then a variance should be around 0 but not 1?
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1answer
43 views

CLT with inconsistent estimator

So I have the OLS estimator that is inconsistent due to the mean independence assumption being violated. I'm asked whether $\sqrt{n}(\hat{\beta}-\beta)$ converges when the sample size $n$ goes to ...
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Distribution of bootstrap and central limit theorem

Let's take a simple example: we have 100000 observations, and we want to estimate the mean. In theory, the distribution of the estimator is a normal distribution according to the Central limit theorem....
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15 views

What is the relationship between Central Limit Theorem and Confidence Interval and Hypothesis Testing?

While studying Central Limit Theorem, Confidence Interval, and Hypothesis Testing, I have come across a question that how central limit theorem applied to confidence interval and hypothesis testing. I ...
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37 views

Solving the limit of the standard normal CDF $\Phi$

I am trying to solve what the Survival function for a std. normal, so $1-\Phi(z) = P(Z>z)$, converges to as $\lim_{z \to +\infty}$. I think this may involve the error function but I'm getting stuck ...
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21 views

Convergence of function of central limit theorem

I have an estimator of the form: $$\frac{\hat{\mu}-c}{\sqrt{n\hat{\sigma}^2}}$$ Where $c$ is constant, $\hat{\mu}$, is an estimate, $n$ is number of observations and $\hat{\sigma}^2$ is a variance ...
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1answer
15 views

What is the justification for using the sample mean in confidence intervals?

In the set-up for the classical CLT we have that $$\frac{\sqrt{n}}{\sigma}(\bar{X}_n-\mu)\to^d N(0,1)$$ as $n\to \infty$, which gives rise to the $1-\alpha$ asymptotic confidence interval formula for $...
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0answers
32 views

In practice how well does asymptotic normality of the MLE hold?

There is a lot of theory about asymptotic normality of the MLE and many people use the result to generate confidence intervals given finite sample data. But a key question here is how large a sample ...
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1answer
26 views

Combined variance estimate for samples of varying sizes

I'm working on my master thesis, and something's come up where I don't know if I'm "allowed" to do this and call it good science, I've scoured the internet to no avail of finding my answer, ...

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