# Questions tagged [independence]

Events (or random variables) are independent when information on some of them tells you nothing about the probability of occurrence (/ distribution) of the others. Please DO NOT use this tag for independent variable use [predictor] instead.

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### Non-independent explanatory variable in regression [closed]

I've been getting a refresher course on regression online, no teacher, and in basically every cases aside from Generalized Linear Mixed Model, the independence of explanatory variable is assumed (...
1 vote
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### Generating highly non-independent random samples

I'm testing performance of statistical tests in the face of non-independent data and I'd like to generate random data where I know the underlying statistical distribution. The easiest way to do it is ...
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### Regression with dependent observations of only one individual

Last week, I received a task to plan an analysis that my team wishes to perform. My objective is to measure if one physician agrees with the outputs that a certain tool generates for a set of N ...
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1 vote
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### Multivariable analysis for an ordinal dependent variable

I am investigating the factors associated with self-perception of oral health. My dependent variable is self-perception of oral health, which is an ordinal variable measured on a 5-point Likert scale. ...
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### Deriving the log-likelihood function for ACD (Autoregressive Conditional Duration) models

I am trying to understand the procedure that is shown in this survey to obtain the log-likelihood function to estimate the parameters for an ACD model: PACURAR, Maria. Autoregressive conditional ...
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### Statistical test for unequal sample sizes of repeated measures (non-parametric)

I'm looking to understand the differences between various sources (50) that evaluate an event based on 8 ordinal scales (varying from 2 to 4 values) associated with a discrete score between 0 and 10, ...
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### Borel-Cantelli lemma on conditional probabilities

In a probability space $\big( \Omega, \mathcal{F}, P \big)$, suppose $\{E_n\}_{n\in \mathbb{N}} \subseteq \mathcal{F}$ is a sequence of mutually independent events. By Borel-Cantelli Lemma, the ...
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### How to come up with an example that $E(\epsilon|z,\eta)=E(\epsilon|\eta)$ and $E(\epsilon)=0$ do not imply $E(\epsilon|z)=0$?

I'm trying to come up with an example showing that $E(\epsilon|z,\eta)=E(\epsilon|\eta)$ and $E(\epsilon)=0$ do not imply $E(\epsilon|z)=0$. The model is nonparametric IV model with the structural ...
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### Would a time series of sentiment valued between 1 and -1 be independent observations?

I am going through a course on linear regression and they say to check for independent observations: "based on the data collection process would the value of one data point impact the value of ...
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### Why do we make a time series stationary if the ARIMA, AR and other models are clearly working with the dependence of lags?

When we run a AR model, we are using a linear combination of its lags to predict the current value. So this means that the lags are related to each other (at least t-1, t-2, ..., t-n are related to t0)...
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### Test for independence of multivariate normality in R

I have 3 normal data matrices, i.e., that in each matrix, the rows are iid Multivariate Normal, but the rows of different matrices need not be identically distributed, but they have the same dimension....
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### Dependent variable as a ratio [duplicate]

I have a dependent variable that is a ratio of how many investments in a year were made in a certain industry. So e.g. the firm made 4 investments and two of those in the technology sector which would ...
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### Confidence intervals for proportions and independence

Suppose a company sells various products at different prices. I would like to compute confidence intervals for the contribution of each product to overal sales, preferably in terms of currency (e.g., ...
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1 vote
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### Can I fit distribution of pixel intensity?

I have an image and I want to plot the histogram of pixel intensity. The distribution seems a Poisson distribution (long tail). Can I fit this distribution using matlab function ...
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### Distribution of conditional independence

I have random variables W,A,Z,U and know that the following conditional independence holds: $W \perp (A,Z)|U$. Is it correct to then state that $W|ZA\sim W|U$? My reasoning is the following but am ...
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### Independence of real and imaginary part of the product of two independent normal variables

Let $X_1,X_2,Y_1,Y_2$ be iid standard normal variables $N(0,1).$ Let $X=X_1+iX_2,$ $Y=Y_1+iY_2$ and $Z=XY.$ We have : $Z=(X_1Y_1 - X_2Y_2) + i(X_1Y_2 + X_2Y_1).$ From https://en.wikipedia.org/wiki/...
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1 vote
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### Standard error calculation for difference in means

In the case of two independent samples, the formula for standard error of the difference in means is given by : $$\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}$$ Even though we are talking about a ...
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### Sampling: when is multivariate random variable interpretation dissimilar to repeated realization of single random variable interpretation

After reading many different posts on this site regarding the relationship between random variables and samples, I still have one lingering question (my apologies if I've missed any post explicitly ...
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### What is meant by the assumption of statistical independence among sources in Independent Component Analysis?

One of the underlying assumptions of independent component analysis (ICA) that I consistently see written is "statistical independence across the source signals". In the context of the ...
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### When are mean and variance estimates uncorrelated or independent

I know that in the case of the normal distribution, the MLE estimates of the mean and the variance are independent. My impression is that this is a rare property for a distribution to have. Are there ...
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### How to calculate the expectancy of the ratio of non-independent random variables?

How can I calculate this expectancy: $$E \left [ \frac{\sum_{t=1}^T{Z_tX_t}}{\sum_{t=1}^T{Z_t^2}} \right ]$$ where $Z_t \sim N(0,1)$ and $X_t \sim N(0,1)$ are independent? Any tricks? Is it ...
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### U -statistics for bi variate sample problem

Let $(X_1, Y_1), (X_2, Y_2),....,(X_n, Y_n)$ be iid random variables with joint distribution function $F(x, y)$ and $F(x), G(x)$ be the marginal distribution functions of $X_1$ and $Y_1$ respectively. ...
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### Does independence almost everywhere imply independence?

Let be $X$ and $Y$ two random variables such that, for any event $A$, $P( X \in A \mid Y) = P(X\in A)$ with probability 1. Can I conclude that $X$ and $Y$ are independent ?
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### PMF of the Independent Multivariate Bernoulli Distribution

I was reading this paper on the Multivariate Bernoulli Distribution, which provides the general form of the PMF in equation 3.1. The paper refers to this as the probability distribution function, but ...
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### Consequences of maintaining IID assumption for prediction model training, but relaxing it for model testing

Let's say you're developing a prediction model, and you are confident that your data are IID. For example, you have a dataset where each row represents a different patient, and you build a model to ...
In Wasserman's All of Statistics chapter 18, he defines the following undirected graph: Let $V$ be a set of random variables with distribution $\mathbb{P}$. Construct a graph with one vertex for each ...