Questions tagged [iid]
iid is an acronym for independent and identically distributed. Many statistical methods assume that the data are iid; that is, that each observation comes from the same distribution and is independent of other observations.
255 questions
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How to test i.i.d. assumption?
Given a sample $X_1,\ldots X_n$, how can I test the hypothesis that these are i.i.d. samples from a fixed (unknown) distribution?
To add context, assume this is a time series and I want evidence ...
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Is Copula Modeling Suitable for Accounting for Temporal Dynamics in Olive Plantation Data?
I am working on a project analyzing olive plantation data, where I aim to simulate the relationship between investment costs (Costs), revenues (...
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Product of iid random variables contradiction [duplicate]
I was working through some econometrics problems, and my professor used the result that the product of iid random variables is iid, but I've seen some posts that this might not always be the case.
My ...
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Are $X_i \varepsilon_i$ iid?
Take:
$X_i , \ i = 1, ... , n $ iid.
$\varepsilon_i , \ i = 1, ... ,n$ also iid.
$X_i \not \perp \varepsilon_j$ (they are not necessarily independent)
Are $X_i \varepsilon_i$ iid ?
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Why GLM doesn't have an error term and why shouldn't residuals be i.i.d?
I've read dozens on post on the subject but I cannot figure this out. From what I've gathered, GLMS don't include an error term in their formulation unlike linear models (LM). I was wondering why (or ...
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Method-of-moment of n IID random variables
The method-of-moment of $\sigma$ for the following pdf is
$$
\text{pdf}(x,\sigma) = \frac{x}{\sigma^2}\exp(-\frac{1}{2}\frac{x^2}{\sigma^2})
$$
$$
E[x] = \int_{0}^{\infty}\frac{x^2}{\sigma^2}\exp(-\...
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1
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Is the Distribution of Survival Times always IID?
I was reading about an approach to Survival Analysis called "First Hitting Time Models" (threshold regression): https://www.jstatsoft.org/article/view/v066i08 , Can Survival Models model the ...
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If $X_1, \dots, X_n$ iid, are $f(X_1), \dots, f(X_n)$, also iid? [duplicate]
If I have independent and identically distributed random variables $X_1, \dots, X_n$, then are $f(X_1), \dots, f(X_n)$ themselves independent and identically distributed?
I think the answer is yes, ...
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How to find the MGF of the max of a set of i.i.d. exponential random variables
As the title suggests, I would like to find the MGF of the max of iid exponential random variables. Assume $Z=\max(x_{1},...,x_{n})$, where $x_{i}$ is distributed as exponential($\beta$) and has pdf $\...
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Are these trials identically distributed? These trials from a panel data are all Bernoulli experiments; do they have same probability of success p?
Scenario: I have an ordered/indexed sample that supposedly comes from a binomial distribution with fixed probability of success p. (By ordered/indexed I mean that not only I care about the percentage ...
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IID assumption in proportion hyp test
I am asked to test a hypothesis that a manufacturing line makes p% faulty parts in a month, it's assumed that the p% is independent of the month. My approach is as simple as it gets, take a random ...
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1
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If a strictly stationary process is also independent, does this imply i.i.d.?
Suppose I have a time series process $\{X_t\}$ that is strictly stationary in the sense that the joint distribution of $[X_{t_1},...,X_{t_k}]$ and $[X_{t_1+a},...,X_{t_k+a}]$ are the same for any set ...
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Permutations of iid Random Variables [duplicate]
Suppose that $X_1, X_2, X_3$ are iid random variables. I have seen this fact many times that $$\mathbb{P}(X_1<X_2<X_3)=\frac{1}{6}$$ but I want to know that why every permutation of $X_1, X_2, ...
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How to: Bootstrap Prediction Intervals for Regression Models with non IID noise?
Question:
Consider a regression model $Y=m(X)+\epsilon$ for which $\epsilon$ is neither independent of $x$ nor identically distributed.
How would we go about generating prediction intervals in such a ...
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A constant $c$ minus the iid random variables still iid? [duplicate]
Let $x_0,x_1,...x_n$ be iid (independenta and identically distributed) random variables. Then, $m_0,m_1,...m_n$ be defined as $c-x_0,c-x_1,....c-x_n$, where c is a constant greater than $x_i$ ($i \in \...
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Are there examples of ML or stats approaches that are valid for IID data, but not exchangeable data?
Lots of supervised learning theory is motivated using the IID assumption. Do most of these methods apply equally well if data is only exchangeable, and not IID? Can you provide an example where this ...
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Central limit theorem for asymptotically i.i.d. random variables
I observe a sequence of r.v. $X_1, X_2, \dots$ where each $X_i$ is a function of the sample size $n$.
When $n \rightarrow \infty$ I have the following result: $X_1 \rightarrow^d E_1, X_2 \rightarrow^d ...
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Data collection after the model is built and deployed
I have built a machine learning model which predicts whether a customer will buy a product or not. The model performs well on cross validation tests. Now, I will deploy it in production to recommend ...
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$E(SN)$ for aggregate claim amount $S$, $S=X_{1}+...+X_{n}, X_{i}$ are iid [duplicate]
Consider the following model for aggregate claim amounts $S$:
$S=X_{1}+X_{2}+...+X_{N}$
where the $X_{i}$ are independent, identically distributed random variables representing individual claim ...
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Test for distribution equality
This question touches Kolmogorov-Smirnov testing, but asks actually something different.
Consider independent random variables $X_1, \dots, X_n$. I want to test the following hypothesis:
$$
H_0: X_i\ ...
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If we remove half the samples from an IID dataset, is the remaining half still IID?
I am generating 10,000 pairs of X and Y such that both X and ...
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Bernoulli distribution with random means
Let $S = \frac{X_1 + \cdots + X_n}{n}$ where the $X_i$ are IID Bernoulli distributed with mean $p$, then $E[S] = p$ and $Var(S) = \frac{p(1-p)}{n}$.
Now consider the slightly more complex setup where $...
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References on data partitioning (cross-validation, train/val/test set construction) when data are non-IID
Consider a prediction setting in which we are interested in training a regression or classification function $f$ with inputs $X \in \mathbb{R}^k$ and target $Y$, and assessing its expected ...
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Confusion about independent and identically distributed?
Say that I wish to measure the height of male within the population (so gender=Male is the only factor I am accounting for). Say I collect 100 observations of male height from an elderly population. ...
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Central limit theorem for dependent binary-related variable
Let $Y\sim N(\mu, \sigma^2)$ and given sample size $n$, we have an iid sample $\{Y_1, ..., Y_n\}$. We sample $X$ (size $n$) from Bernoulli with probability $\pi$. Denote $Z_i=X_iY_i$. Then, when $X_i=...
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