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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185 views

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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118 views

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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65 views

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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596 views

Square roots of sums absolute values of i.i.d. random variables with zero mean

In an earlier question, I asked about the limiting distribution of the square root of the absolute value of the sum of $n$ i.i.d. random variables each with finite non-zero mean $\mu$ and variance $\...
5
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259 views

Central Limit Theorem when the dimension size increases with the sample size

Let $X_1, X_2,\ldots, X_n \in \mathcal{R}^d$ and be zero-mean, unit variance random variables. Here the dimension ($d$) is a function of the sample size($n$) i.e, $d=f(n)$. For example $d = \sqrt{n}$. ...
4
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1answer
99 views

Convergence in Distribution for i.i.d. data

Let $X_1,X_2,\ldots,X_n$ be i.i.d. RVs with $E(X_{i})=\mu$ and $V(X_{i})=\sigma^2$, $\sigma <\infty$.Is it possible to find real sequences $a_{n}$ and $b_{n}$ such that $a_{n}(\bar{X}^3_{n}-b_{n})$ ...
4
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187 views

CLT confidence intervals

I have come across the following statement: $(*)$ The width of CLT-based 99% confidence intervals is $6\sigma n^{-1/2}$. How does one derive this? Is there a general formula? I tried to ...
4
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782 views

Central limit theorem: applicability for assumptions of different tests

Since many statistical procedures (e.g. t-test, ANOVA, Pearson’s r (for efficient estimates)) require the normal distribution of the tested variables ('normality-assumption') one may ask if (at least) ...
3
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29 views

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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17 views

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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117 views

How are the variance of an estimate to $\int_Bf\:{\rm }d\mu$ and the deviation of $f$ from the mean $\frac1{\mu(B)}\int_Bf\:{\rm d}\mu$ related?

Let $(E,\mathcal E,\mu)$ be a probability space, $(X_n)_{n\in\mathbb N_0}$ be an $(E,\mathcal E)$-valued ergodic time-homogeneous Markov chain with stationary distribution $\mu$ and $$A_nf:=\frac1n\...
3
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151 views

Terminology: “Central Limit Theorem” for Delta Method

This is a question about when is it appropriate to call an asymptotic normality statement, the "Central Limit Theorem" (CLT). More specifically, suppose I have $X_1, X_2, \dots X_n$ iid from a ...
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42 views

Inverse Gaussian Distribution and the Central Limit Theorem

Let the random variables $Y_1,\ldots,Y_n$ be independent and identically distributed (i.i.d.) (standard) Inverse Gaussian random variables with parameters $\mu$ and $\lambda$. Then, let the random ...
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100 views

Stationary processes that do not satisfy Gordin's central limit theorem

We are doing an assignment for our Advanced Econometrics course for which we are trying to illustrate Gordin's Central Limit Theorem by simulation. We used an AR(1) process to show that if the ...
3
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66 views

Can the Berry-Esseen theorem tell us whether acceptable inference may be achieved by parametric tests?

I refer in particular to such choices: 1) t-test (or its generalizations: ANOVA or Hotelling's $t^2$) vs its non-parametric alternatives (e.g., U Mann-Whitney test and its generalizations); 2) Pearson'...
3
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85 views

confusion about the link between residuals, error terms, sample size and CLT in ANOVA

I feel a little confused about the assumption of the ANOVA and what it ensures mathematically the errors have to be iid and normally distributed N(0,1). independance of observation. Is it not a ...
3
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1answer
85 views

Are there any results on the distribution of the posterior mean across data sets?

I know that this question borderlines on Bayesian and frequentist philosophy, somewhat related to this question. Bayesian point estimation sometimes uses the mean of the posterior distribution. That ...
3
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53 views

Why to care about local uniform convergence of characteristic function in proving multivariate CLT?

Let $X_{1},X_{2}, \dots \sim_{i.i.d.}$ some Borel probability distribution on $\mathbb{R}^{k}$; let $\mathbb{E}X_{1} \equiv m$; let $\text{var} X_{1} \equiv V$. Then it can be shown that $\sqrt{n}(\...
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69 views

statistical inference - what about population size?

I'll try to explain my question since it's very theoretical. In general, using statistical inference we try to estimate parameters of the population based on a sample. Several theorems show that as ...
3
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140 views

Weakest assumptions for central limit theorem to hold?

Let $X_i$, $i=1,2,...$ be any sequence of random variables (i.e. not necessarily independent or identically distributed). What are some weak conditions you know of on $X_i$ that makes $\sum_i X_i$, ...
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115 views

Sample mean vs population mean

I was provided with the following: $X_1,\ldots,X_5$ are independent random samples from a distribution with mean $5$ and standard deviation $3$. Then I was asked to find the mean and standard ...
3
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165 views

Local Version of Bernstein Von-Mises Theorem?

The Bernstein-Von Mises theorem says that, under reasonable conditions, the posterior distribution $p(\theta | x_{1},\ldots,x_{n})$ converges weakly to the normal distribution after suitable rescaling....
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141 views

bound on expectation of a two-variable function under an independent distribution

Consider a probability distribution $P(x)$, a set observed samples $S = \{x_1,\cdots, x_n\}$ where $x_i \stackrel{iid}{\sim} P(x)$ for $i \leq n$, and a symmetric function $h(x,y)$. How can one ...
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64 views

Total probability distribution of multiple random lotteries

My question: Imagine $d$ identical lotteries. Each individual lottery picks a cost $c_{i}$ between $0$ and $1$. Picking a costs occurs with probability distribution $f(c)$. The total cost of these $d$...
3
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111 views

How to find an appropriate sample size given information about the CV?

You are planning to collect a (simple random) sample to estimate the mean of a non-negative random variable. It is known that the population coefficient of variation (CV) is 1.2. Use the central limit ...
3
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75 views

Shouldn't a function of data from a PDF repeated over and over on new data eventually yield a Gaussian PDF?

I got into an interesting discussion with a co-worker today and we are not sure what the answer is: We have $N=1000$ samples from a Rayleigh PDF. We take those $N$ samples, and compute their (...
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196 views

Central Limit Theorem - interval estimation

I'm rolling a regular dodecahedron (12-side die) 1200 times. I need to find an interval, in which the total count of prime-number results will lie with the probability of 95%. I have to use the ...
2
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49 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 ...
2
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64 views

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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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 ...
2
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1answer
60 views

CLT for Medians for Distributions Without Means

Is there a version of the CLT for distributions that don't have a mean, but have a median, and finite moments of order > 1? (if such distributions exist)
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35 views

CLT - Adding small samples to one big sample

According to CLT, the SE is the SD of the distribution of several samples means. This SE depends on each sample mean, the SD of each sample and N (the size of each sample which I test). Since there ...
2
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0answers
83 views

Why does the Normal Distribution have inflection points at +-1 standard deviations?

Supposedly this was Laplace's first error curve: Small errors occur more frequently, large errors less frequently; the shape of the Laplace Error Curve above roughly makes sense to me. Looking at ...
2
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39 views

Asking for feedback on the application of a Central Limit Theorem

Let $\{X_{n,i}:1\leq i \leq d_n\}$ be a triangular array of mean zero random variables where $d_n$ is a positive increasing sequence ($d_n\leq n$). Under some conditions, a Central Limit Theorem ...
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0answers
47 views

Asymptotics of 2 x 2 precision matrix

Edited to give the answer... but I still don't understand where it came from! Suppose we have $$X_1, X_2,..., X_n \overset{i.i.d.}{\sim} N(0, \Omega^{-1})$$ where $\Omega \in \mathbb{R}^{2 \times 2}$ ...
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100 views

Estimation and hypothesis testing for the difference in squared bias for two random variables

My Question: Let $X_t$ and $Y_t$ denote two time-series random variables, both of which are estimates of the random variable $\theta_t$. Let $U_t = X_t - \theta_t$, and $V_t = Y_t - \theta_t$. The ...
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45 views

What is the limiting posterior in the generalized Bayesian central limit theorem?

The central limit theorem characterizes the limiting distribution of the sum of increasingly many finite-variance independent random variables: the limit is Gaussian. The generalized central limit ...
2
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61 views

Convergence of series of dependent random variable, central limit theorem

My friend and I have a problem on central limit theorem. Given $X_1,X_2......$ are i.i.d random variables with mean $\mu$=0, variance $\sigma^2=1$(may or may not be normally distributed). If we ...
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0answers
58 views

Beyond the CLT: guarantees on the shape of the sample mean distribution?

If the L.L.N. tell us where our sample mean is going, and the C.L.T. "extend it" telling us how fast the variance is decaying, do we have an other tool telling us how fast the shape (or the further ...
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0answers
18 views

Conditions for Central Limit Theorem

I want to apply this version of Central limit theorem for triangular arrays. I'm interested only on condition $2$. Let $\{X_{n,i}:1\leq i \leq d_n\}$ be a triangular array of mean zero random ...
2
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0answers
121 views

Asymptotic normality of quadratic form?

Let $X$ be a $p$-dimensional vector that is asymptotically normal such that $$\sqrt{n}(X - \mu_X) \stackrel{d}\longrightarrow N(0, \Sigma)$$, and let $H$ be a random $p\times p$ symmetric matrix, ...
2
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0answers
46 views

CLT and convergence of Variance

I am looking at a problem where the sum of the individual $X_i$ is $S_n=X_1+\dotsm+X_n$. The probability is given as, $P(X_i=i)=P(X_i=-i)=\frac{i^{-\alpha}}{4}$ and $P(X_i=0)=1-\frac{i^{-\alpha}}{2}$. ...
2
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37 views

Good book on characteristic functions that includes the CF-proof of the CLT

The title basically says it all. I would like to learn about CF in order to understand the proof of the CLT that makes use of CF. Ideally I would like to read a book that does not only give proves of ...
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73 views

Applying CLT to difference in sample proportions?

I am reading a derivation that has the following statement about a two-sample proportion test: $$ \frac{\hat p_1 - \hat p_2}{\sqrt{(\frac{1}{n_1}+\frac{1}{n_2})\hat p(1-\hat p)}} \stackrel{d}{\...
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0answers
282 views

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, ...
2
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0answers
222 views

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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413 views

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 ...
2
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0answers
156 views

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 ...
2
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0answers
113 views

(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, ...
2
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0answers
443 views

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 \...