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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questions
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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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28 views
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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0answers
40 views
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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1answer
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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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1answer
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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0answers
22 views
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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1answer
216 views
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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37 views
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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1answer
55 views
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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2answers
460 views
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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29 views
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
64 views
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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0answers
17 views
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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0answers
16 views
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{...
2
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1answer
47 views
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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0answers
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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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1answer
40 views
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 $\...
2
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1answer
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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1answer
25 views
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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1answer
42 views
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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2answers
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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2answers
93 views
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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0answers
16 views
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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2answers
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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32 views
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)$$ came from and why is the last ...
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1answer
25 views
OLS vs LAD in cases with large n
Thanks to lots of helpful answers in the community, I figured that Least Absolute Deviations regression can give better estimations when the normality of residuals is violated (e.g. residuals ...
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1answer
59 views
Can this problem really be solved using central limit theorem?
My friend had this question on a test:
Let $\{X_n\}_{n \in N}$ be a sequence of independent random variables with the same normal distribution $N(0, 2n)$. Check for the convergence of a sequence $\{...
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22 views
Network version of central limit theorem?
Is there a version of the central limit theorem for random variables that are connected through a network? More precisely, imagine I have $N$ vertices in a graph, and a random variable $X_{i}$ for ...
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Convergence of sum of (1) random variable that converges in distribution to Normal and (2) degenerate random variable that diverges to infinity?
Say that we have $\sqrt{n}(\hat{\mu} - \mu_0)$, which we can equivalently write as $\sqrt{n}(\hat{\mu} - \mu) + \sqrt{n}(\mu - \mu_0)$, where $\mu$ is the population mean, $\hat{\mu}$ is the sample ...
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1answer
105 views
Multinomial & Covariances
Assume we have $\mathbf{X} = (X_1,\ldots,X_k) \sim Multinomial(n,\mathbf{p}=(p_1,\ldots, p_k)).$
How can we find $\operatorname{Cov}(e^{X_i},e^{X_j})$?
Tried to do CLT but it's led me nowhere.
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1answer
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How to establish asymptotic normality of unbiased estimator and find asympotic variance?
Given a linear regression model with deterministic regressors $$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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0answers
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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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15 views
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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2answers
64 views
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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0answers
22 views
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
18 views
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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0answers
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 ...
5
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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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1answer
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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2answers
68 views
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 ...
2
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1answer
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 ...
