# 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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### Is the Berry-Esseen theorem useful for justifying normality?

The Kolmogorov-Smirnov (KS) test tells one how confident they can be that a sample comes from a hypothesized distribution. It is my understanding that this test can be used to justify whether or not ...
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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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### Alternative distribution of $T^2$ statistic without Gaussian assumption

Background Let $p(x)$ be an arbitrary distribution defined on $\mathbb{R}^d$. Define $\mu = \mathbb{E}[x]$. Given an i.i.d. sample $x_1, \ldots, x_n \sim p(x)$, consider the following $T^2$ statistic ...
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### Can CLT be invoked to perform T-tests and other parametric tests on samples from a population that has non-normal distribution?

I have data (about 1150 data points) and it is not normally distributed. Within this sample I want to compare means of two groups and see if they are significantly different. Can I use T-test for it, ...
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### Central limit theorem backwards

Does the fact that some quantity in nature is normally distributed necessarily imply that the quantity can be meaningfully expressed as a sum of smaller iid components (e.g. IQ is a sum of small ...
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### For which parameters does the Central Limit Theorem work?

I have a question about the Central Limit Theorem in the context of estimating a population parameter through a sample statistic. The most known case is that the CLT asserts that a sampling ...
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### Central limit theorem in Bagging

I am in process of trying to understand the statistical theory behind Machine learning. I came across the fact that central limit theorem plays a key role in the Bagging algorithm (in ML). I searched ...
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### (When) does integrals of stochastic process follow any central limit theorem (converge to a normal distribution)?

I am trying to understand the central limit theorem established for integrals. Specifically, let $\left( X_n \right)$ be a sequence of random variables. I understand that under a set of conditions, ...
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### Convergence of the Bootstrap to the true mean

Suppose that $X_i,$ $i= 1, 2 \dots$ are iid with finite positive variance, then let $X_{i,n}^*$ $(i = 1, \dots, m)$ be a bootstrap sample of size m from $\lbrace X_1, \dots, X_n\rbrace$, then \...