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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14 views

Convergence of variance of sample median, pt. 2

Follow on question to this, answered negatively by Thomas Lumley. We reprint it here for convenience. In this SE question, it is stated that there is a central limit theorem for the sample median, ...
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Cental limit theorem, Chebyshev's inequality, and convergence of distributions through rescaling

I've been thinking about this issue for a few days and although read some of relevant questions on this site, still couldn't get it off my mind. Suppose we have $n$ i.i.d random variables $X_i$ with ...
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Limit of $\mathbb{P}(\sqrt n - 1 \leq \sqrt{X_1^2 + \cdots X_n^2} \leq \sqrt{n} + 1)$ wrt $n$ for standard multivariate normal $X$

Suppose $X$ is a standard multivariate normal distribution. Then what is the limit of the $c_n$ where the $c_n$ are $\mathbb{P}(\sqrt n - 1\leq \sqrt{X_1^2 + \cdots X_n^2} \leq \sqrt n + 1)$? I want ...
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Sample and population: definition for workplace statistics?

This might be a bit of a dunce's question, but I was wondering about the difference between a sample and a population. Obviously, if you have data relating to 200,000 people but you only look at 200 ...
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42 views

Convergence of variance of sample median

In this SE question, it is stated that there is a central limit theorem for the sample median, namely $$ \sqrt{n}(Y_n - m) \xrightarrow{d} N(0, [2f(m)]^{-2}), $$ as $n\to\infty$ where $Y_n$ is the ...
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Different regularity conditions for finite population CLT

I am having trouble understanding the different regularity conditions for different versions of the finite population central limit theorem. I would greatly appreciate any help or insight anyone has. ...
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1answer
60 views

Does a Binomial converge to Poisson or Normal?

I have read the answer here. Here the distinction is that If $n\to\infty$ and $p\to0$ while $np$ approaches some positive number $\lambda,$ then the binomial distribution approaches a Poisson ...
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Plain English explanation of Ito's integral?

I'm looking for a plain English explanation of Ito's integral. I don't need an exhaustive proof, derivation, etc. Just a simple ~this is effectively what it does and why it's better than a Riemann sum ...
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Is bootstrapping redundant for computing confidence intervals for the sample mean in large samples?

If we are just interested in computing confidence intervals for the population mean $\mu$ using a sample $X_1,X_2,\dots,X_n$ of $n$ iid random variables is bootstrapping redundant if $n$ is large? I ...
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CLT for non iid random variables

Assume $U_k$ are correlated standard normal random variables. Let $R_k := a_k U_k$. I'm looking for CLT of the sum $S_p := \sum_{k=1}^{p}\frac{R_k}{\sqrt{p}}$. Since $U_k$ are correlated, I'm looking ...
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Differences Between the Central Limit Theorem and Consistency

I have recently finished studying the central limit theorem and the idea of consistency. I am still a little fuzzy about them, so I was wondering what are some key similarities and differences of the ...
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Approximating $E[g(\overline X_n)]$ and want to bound the remainder using some form of CLT or Berry-Essen Theorem

If we have a set $X_1,\dots,X_n$ of iid random variables with finite mean $\mu$ and variance $\sigma$, the CLT says that $\sqrt{n}(\overline X_n - \mu) \stackrel{d}{\to} \mathcal{N}(0,\sigma^2)$. If ...
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1answer
70 views

Limiting distribution of $\sum_{j=1}^{p}\lambda_j U_j$

Assume $U_j$ are $\chi^2(1)$ random variables and $\lambda_1, \ldots, \lambda_p$ are the eigenvalues of a covariance matrix $\Sigma = (r^{|i-j|})_{ij}$ with a Toeplitz-type structure (for some fixed $|...
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Asymptotic properties of between-group estimates?

Suppose my data consists of the mean of several separate groups (sizes may vary) from an iid sample of an outcome variable Y and an independent variable X. The data generating process is $Y = \beta X +...
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what is the standard deviation of the geometric mean sample distribution?

I wrote a python script to take a population distribution of a random variable in the interval (0,1) to be uniform and make 2 sample distribution: The fist is the distribution of the arithmetic mean ...
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1answer
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confidence interval of $\beta$, where $X$'s are from exponential distribution

Suppose $X_i\overset{ind}{\sim}\mathcal{E}(\lambda_i)$, where $\lambda_i=(t_i\beta)^{-1}$, where $t_i$'s are positive known values and $\beta$ is positive unknown parameter. Here $i=1,\dots,n$. It can ...
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How to think about confidence intervals in the context of linear regression?

I think I understand confidence intervals for sampling distributions, but am trying to connect this understanding to the confidence bands I see around linear regression lines. According to the ...
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Limit distribution for a linear discrete-time stochastic process: limit of the sum of linearly transformed uniform distributions

I have posted this in math stack exchange, but I figured maybe this is a better forum for this kind of question. Suppose we have a stochastic linear process: $$x_{k+1} = Ax_{k} + Bw_{k} \qquad \text{...
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1answer
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Confidence interval for the mean of the uniform distribution

I can take samples of a random variable $X \sim U(a, b)$, where the length of $(a, b)$ is known. I am interested in its mean $E[X]$, estimated with $\hat{X_n} = \frac{1}{n} \sum_{i=1}X_n$, but I need ...
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OLS Regression: How can the Central Limit Thorem justify the assumption of normality of the error term?

Relative noob here. My understanding is that the CLT is used to derive the sampling distribution of the sample means. So from a population distribution, whether normal or not, if you take a sample and ...
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Central limit theorem for the function of an iid random variable

Given an iid random variable $X$, instead of the distribution $\sqrt{n}(n^{-1}\sum{X_{i}}-E[X])$ which is the result that the central limit theorem provides , I am interested in the distribution of $\...
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36 views

Mean and variance of the Gaussian resulting from Central Limit Theorem

Let $\{x_i\}$ be a set of iid random variables (not necessarily Gaussian distributed). The CLT states that $\frac{1}{n}\sum_{i=1}^n x_i$ is asymptotically normal. What do we know about the mean and ...
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Asymptotics of the maximum of k subsample means

Suppose we have $n$ i.i.d samples $X_1, X_2, \cdots, X_n$ from some real-valued distribution $P$. Let $\alpha \in [0,1]$ be a fixed constant. Select an uniformly random subset $S_1 \subset \{1,2,\...
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How many times do we need to sample to get the sampling distribution of means?

I was looking into the central limit theorem and noticed that every article talks about how big the sample size should be but they never indicate how many times should we be sampling to get the ...
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Does central limit theorem help in making inference only about the population mean and not other parameters?

As per the weak law of large numbers, if your sample size is large, your mean of the sample is likely to be closer to the population mean than in a smaller sample. Additionally the CLT tells us the ...
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54 views

Central Limit Theorem: Sample Size or Number of Samples?

The central limit theorem states that if we take a take a large enough sum of random variables, the sum will approach a normal distribution. I am confused about why we focus only on the sample size ...
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OLS assumption normallity of error term really needed?

As the title explains I was wondering whether the additional OLS assumption of having a normally distributed error term isn't redundant if the sample is large enough. I understand that we want the ...
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Finding cross-correlation p-value for non-normal time series with CLT

I have two non-normally distributed time series with 77 data points (each) which I am cross-correlating. The time series were pre-whitened to remove autocorrelation. Since I am not trying to use ...
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574 views

Why a sample of skewed normal distribution is not normal?

I was under the impression that if I randomly sample from a skewed normal distribution, the distribution of my sample would be normal based on central limit theorem, but the graph clearly shows that ...
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Stochastic Convergence - Radford Neal's Prior

There's question 1.3 in this set of questions already solved I found about stochastic convergence. I cannot understand where the expression $$\sum_{i} \phi_{i}^{2}(x)$$ came from and why is the last ...
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OLS vs LAD in cases with large n

Thanks to lots of helpful answers in the community, I figured that Least Absolute Deviations regression can give better estimations when the normality of residuals is violated (e.g. residuals ...
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Can this problem really be solved using central limit theorem?

My friend had this question on a test: Let $\{X_n\}_{n \in N}$ be a sequence of independent random variables with the same normal distribution $N(0, 2n)$. Check for the convergence of a sequence $\{...
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Network version of central limit theorem?

Is there a version of the central limit theorem for random variables that are connected through a network? More precisely, imagine I have $N$ vertices in a graph, and a random variable $X_{i}$ for ...
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Convergence of sum of (1) random variable that converges in distribution to Normal and (2) degenerate random variable that diverges to infinity?

Say that we have $\sqrt{n}(\hat{\mu} - \mu_0)$, which we can equivalently write as $\sqrt{n}(\hat{\mu} - \mu) + \sqrt{n}(\mu - \mu_0)$, where $\mu$ is the population mean, $\hat{\mu}$ is the sample ...
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105 views

Multinomial & Covariances

Assume we have $\mathbf{X} = (X_1,\ldots,X_k) \sim Multinomial(n,\mathbf{p}=(p_1,\ldots, p_k)).$ How can we find $\operatorname{Cov}(e^{X_i},e^{X_j})$? Tried to do CLT but it's led me nowhere.
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How to establish asymptotic normality of unbiased estimator and find asympotic variance?

Given a linear regression model with deterministic regressors $$y_{i} = x'_{i}\beta + \epsilon_i, \quad \epsilon \sim (0,\sigma^2) i.i.d., \quad i = 1, ...,n, \quad with \quad \mu_4 := \quad \mathbb{E}...
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Probability of averages

I have a random variable $Y$ and I am taking an independent sample of $n$ from this RV. I'll refer to this sample as $Y_n$, and I define the average of this sample as $\bar{Y}_n$. The maximum of this ...
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Which distributions originates from sampling near boundaries

I have a certain non-deterministic process which receives a parameter k that belongs to a fixed domain [a, b]. I am able to generate samples from the output of the process in the domain of k and by ...
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1answer
37 views

Central Limit Theorem Approximation and Relation to Law of Large Numbers

Assume the Linberg-Levy CLT to where we know $$\sqrt{n}\frac{\bar{X}_n-\mu}{\sigma}\xrightarrow{d}N(0,1).$$ I feel like I commonly see then that $$\bar{X}_n\approx N(\mu,\frac{\sigma^2}{n}),$$ but ...
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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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1answer
95 views

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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1answer
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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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1answer
37 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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60 views

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
107 views

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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112 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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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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88 views

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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