Questions tagged [variance]

The expected squared deviation of a random variable from its mean; or, the average squared deviation of data about their mean.

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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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Variance of projected data

What's the expression for the variance of projection of $N$ data points onto a vector $\mathbf{v}$ which is not necessarily a unit vector? I could only find an expression when $\mathbf{v}$ is taken to ...
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Cramer-Rao Lower Bound for Poisson Adjusted MLE

For X$_i$ iid Poisson($\theta$), let y($\theta$)=$\theta$e$^{-\theta}$ be the function when X=1. Since, normally, the MLE of $\theta$ for a Poisson is $\bar{x}$ = $\frac{1}{n}$$\sum_{i=1}^{n} X_i$ , I ...
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How to measure ML classification model stability?

It's well known that decision trees are unstable models, i.e. they have high variance. It can be easily shown by adding Gaussian noise to variable(s) and checking that completely different tree is ...
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Covariance of two subsample means [closed]

A SRS of size $m = m_1 + m_2$ with mean $\bar{x}$ is drawn from a finite population and a simple random subsample of size $m_1$ is drawn from it with mean $\bar{x}_1$ and $\bar{x}_2$ is the mean of ...
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How can I compute the variance of $aX^2+bX$?

Let $X \sim \mathcal{N}(0,1)$. How can I compute the variance of: $$ Y=aX^2+bX $$
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How can I compare two response ratios in meta-analysis?

For example, there are several studies about how males and females response to certain condition. Here's the data I got: Xmc, the mean data of male control group, Xfc, the mean data of female control ...
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Compute a Monte Carlo estimate. Which of the variances (of $\hat{\theta}$ and $\hat{\theta}^{*}$) is smaller, and why?

Compute a Monte Carlo estimate $\hat{\theta}$ of $$ \theta = \int_{0}^{0.5} e^{-x} dx $$ by sampling from Uniform$(0, 0.5)$, and estimate the variance of $\hat{\theta}$. Find another Monte Carlo ...
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How to analyze small dataset? [closed]

I have a small dataset with continues variables , dependent variable is normally distributed and linearly associated with the independent variable. I tried doing linear regression but the model doesn'...
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prove the difference between mean and median is less than the variance [duplicate]

Suppose $X$ is a random variable with finite variance. Let $m$ denote the median of $X$ and $\mu$ the mean of $X$, i.e. $\mu=\mathbb{E}(X)$. Show $$(m-\mu)^2\leq\text{var}(X)$$ Intuitively this is ...
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bias–variance decomposition related to median?

In evaluating or designing an estimator $\hat\theta$ of a population parameter $\theta$, the most common approach is to look at its bias, $\operatorname{E} \hat\theta - \theta$, its variance, $\...
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Is the bias-variance tradeoff consistent across all x values in a linear model?

I'm having trouble making the transition from the concept of bias of a single estimator to bias in a linear model. The clearest explanation I've found is this detailed simulation of bias and variance ...
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Matrix Covariance Algebra

In the structural equation modeling (SEM) context, one of the modeling frameworks is called the reticular action model (RAM). In RAM, the observed variables (y) and latent variables (η) are combined ...
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How to compute the covariance error term in an astrophysics context?

I have posted initially on physics exchange but don't get any answer, so I try hopefully here which seeems to be a more appropriate forum (I am going to delete the initial post on physics exchange). ...
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Theoretical variance of the (binary) accuracy score for a random guesser

Suppose we want to compute accuracy for a binary classifier (assuming balanced classes): Acc = (TP+TN)/N Where N = TP + TN + FP + FN. For the case of a pure random guesser where each (actual) positive ...
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How to compute % contribution to total variance

I would like to determine the percentage of variance each group contributes to an observed total variance. I would like to know the simplest way to calculate this, and to gain some intuition as to ...
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Question about confidence intervals and prediction intervals

Considering following linear multiple regression model: \begin{equation} y=X\beta + e, \end{equation} where observations $y\in\Re^n$, coefficents $\beta\in\Re^p$ and $e\sim N(0,\sigma I)$ is a white ...
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Multivariate datasets comparison in R

I would like to compare the variance of two multivariate datasets describing the same population between them (e.g. Covariance) but also the specificity of one dataset regarding the total variance of ...
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variance estimation using order statistics

I have four largest samples drawn from a distribution of N i.i.d Gaussian R.V. with standard deviation (Sigma) where sigma is unknown. N is known to be between 50-200. Mean is given to be 0. How do ...
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Upper bound for variance of $\hat{\beta}$ in multiple linear regression

The variance of the beta estimator in an ordinary-least-squares multiple linear regression to express $Y$ as a (linear) function of $X$, $\hat{\beta}$, can be expressed as (knowing $X$ and $\sigma^2$ ...
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How to interpret SD from output of anovaVCA model?

I'm using "VCAdata1" data from VCA Library containing device measurements from 3 devices on each day (1-21) and on each lot (1-3) and subsetting the data to include only sample=5 . So for ...
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Bias and variance of an estimator of a model mean

I have a binary classification model and I need to use its output to estimate the means of groups of observations. I have two questions: A. Can I compute the the bias and variance of the estimator of ...
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Change in standard deviation when a value is removed

Let's say a list of numbers $L$ has standard deviation $S$. Is there a formula for finding $S$ if I remove an element $l$ from $L$? Assume we know the mean of both $L$ and $L - l$.
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Variance of ratio of sums

How do I compute the variance of a ratio of sums ? $$ Var\big(\frac{\sum_i X_i}{\sum_j Y_j}\big) $$ I have 2 datasets $X=(X_1,...,X_n)$ with $Y=(Y_1,...,Y_n)$ that I need to compare, and estimate the ...
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Sample Variance and Population Variance for Ungrouped data

A study of the effect of smoking on sleep patterns is conducted. The measure observed is the time, in minutes, that it takes to fall asleep. These data are obtained: Smokers: $69.3, 56.0, 22.1, 47.6, ...
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Different formulas for $\hat{\sigma}^2$

My book introduces $\hat{\sigma}^2 = \frac{1}{n} \sum_{i=1}^n \hat{e}_i^2$, which makes sense to me since to my understanding $\hat{\sigma}^2$ is the error variance estimator. However, I have also ...
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Variance as Covariance [duplicate]

Is there a useful interpretation of variance as a special case of covariance, Var(X) = Cov(X,X)?
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Distribution of estimated variance of residuals in simple regression model [duplicate]

We have our simple regression estimated model: $$ Yi=b_{0}+b_{1}Xi $$ We know that the estimator for $\sigma^2$ is: $$ \hat\sigma^2=\frac{1}{n-2}\sum_{i=1}^{n}\varepsilon_i^2 $$ But how can we prove ...
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Distribution of hat sigma in simple linear regression model [duplicate]

How can I prove that the distribution of estimated σ^2 (the distribution of the estimated variance of errors εi, in case of homoschedasticity) is equal to a chi-squared with n-2 df * σ^2/(n-2) ?
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Expression of $E[(X-a)^3]$ as a function of $\operatorname{Var}(X)$ and/or $\sigma_x$

Just a question: I would be able to express $\mathrm{E}\left[(X-a)^{3}\right]$ as a function of $\sigma_x$ and/or $\sigma_x^2$, with $a$ a constant (surely $\mathrm{E}\left[X\right]$ terms should ...
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Variance of DFT of filtered noise

I am struggling with the following question: Let v(t) be a stationary stochastic process with Gaussian probability distribution and power spectral density $S(\omega)$. Let the DFT of $v(t)$ be $V(k)=\...
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Variance involving two independent variables

hello I have two independent variable P and Q. They are both non-negative. Now I define two new variables on them: The first variable $$R_1=\alpha P+(1-\alpha)Q.$$ Since P and Q are independent, so $$...
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Scaling measurement error (noise) in Gaussian process regression

I noticed that Gaussian process regression (GPR) performs poorly when the target values are too small (see here). One solution for this is to previously normalize the targets to zero mean and unit ...
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Estimating variance of slope & MSE Without Full Data Set for Regression

For a simple linear regression model, you have taken a sample of size 8 and made the following computations: 𝑋 bar= 8.6 𝑌 bar = 8.0 𝑏0= 8.7 𝑠x2= 1.1 𝐶𝑜𝑣(𝑏0,𝑏1)≈ -3.5 Estimate the variance of ...
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How to calculate var(var) in time series

I try to calculate var(var) with Markov Process in time series by Bartlett(1946, P28) https://www.jstor.org/stable/2983611?seq=2#metadata_info_tab_contents , I use quadratic form calculate,but I didn'...
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Within- and between-group variance does not add up to the total variance

The variance is additively decomposable (Shorrocks 1982). That is, when decomposing it into within- and between-group components, the within and between-group component should add up to the total ...
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unbiased estimation of the variance of $p$ (proportion) of a random sample without replacement

Given a random sample without replacement of size $n$ from population of size $N$ and $p$ is the estimator of the proportion $P$. How could one show that: \begin{equation*} \frac{N-n}{N(N-1)}pq \end{...
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Variance approximation for large scale least squares problems

The least squares problem I am solving has the objective function $S=\sum_i (y_i-f_i(\textbf{x},\boldsymbol{\beta}))^2$. The variance for this can be approximated as $\text{var}(\beta_j)\approx\frac{S}...
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Second order with Delta method on a ratio to improve variance estimation accuracy

Following a previous post on math exchange without success, I have applied the "Delta method" that says : Delta method : There are alternative formulations of this expression which may be ...
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Portfolio theory: confusion about variance-covariance matrix

I am taking an introductory course to finance in my Master's, and wanted to go further in the topic of portfolio theory (I am an engineering bachelor graduate, but as I just hinted, I am new to ...
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Variance of predicted value in a linear regression when $n \to \infty$

The following question is from Kutner's Applied Linear Statistical Models - Ch 2 - 2.12 To answer the question a few pieces of information are needed, provided below: What I gather the question is ...
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Is the inverse of the sample variance uniformly integrable?

Let $X_1,X_2,\dots,X_n$ be a sample of $n$ independent and identically distributed observations of a continuous population random variable $X$. Define $Z_n$ to be the inverse of the sample variance: $$...
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Is the inverse of the sample variance integrable?

Is the inverse of the sample variance integrable? That is, does it hold that $$ E\bigg[\bigg(\frac{1}{n}\sum_{i=1}^n X_i^2 - \overline{X}_n^2\bigg)^{-1}\ \bigg] < \infty. $$
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Explaining and Addressing the Bias-Variance Tradeoff

Let's say we have a model whose training error is 7% and the validation error is 10%. What does this mean in terms of the bias-variance tradeoff? I know that high validation error and low training ...
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Interpreting divergence between standard error and observed standard deviation of population

I have a simple dataframe. Users come to my website and sign up for a trial. I consider each sign up as a conversion. But my conversion rate is fluctuating wildly every day even though I have about ...
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Associate uncertainities to variance's estimate of a distribution with heavy tails

I have a set of $M$ variables $\{ \lambda_N^{(i)} \}_{i=0}^{M(N)}$. The distribution of $\lambda$ depends on the $N$ parameter; for $N \rightarrow \infty$ the distribution shrinks around the mean ...
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Object cloning expected value and variance

Suppose there exist special marbles that can clone themselves while keeping their special cloning ability. The probability that any special marble independently clones itself is 0.01/sec (i.e. there ...
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Estimate how many values fall below a specific deviation using the empirical rule

I'm trying to estimate how many values fall within a portion of the standard deviation Lets say I have: A sample size of 100 and Average of 50. and a start deviation of 10. Using the Empirical Rule ...
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Looking for a measure of variance of variance?

I was hoping you would be able to help me identify the statistic I am looking for, or point me in the right direction. Is there a statistical measure that will represent variance of variance (or s.d. ...
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Can we derive the Variance of the least squares slope WITHOUT assuming that $X_i$s are fixed or deterministic? [duplicate]

Everywhere in the literature, I have seen that while deriving the variance of the least squares slope estimate $Var(\hat \beta_1) = \dfrac{\sigma ^2}{SS_{xx}}$, we always assume that $X_i$s are fixed ...

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