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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How to find the variance for $\frac{\sum Y}{\sum X}$ when X and Y are both independent and normal random variables
The original problem is:
Given $Y_i = \beta X_i + \epsilon_i$, $i=1,2,...,n$, where $X \sim N(\mu, \tau^2)$ iid and $\epsilon \sim N(0, \sigma^2)$ iid, $X$ and $\epsilon$ are independent. What is the ...
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How to analyze variance from repeating a test with the same unit vs variance across units?
Suppose I want to buy a CNC machine to automate guitar-making, for instance. I have access to a warehouse of CNC machines that are all reported to be the same model and advertised as having equal ...
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Calculating a (combined) variance of two samples with unequal variances
Before I state my question, I would like to point out that I do not have a mathematics/statistics background, so please excuse my lack of rigor in the terminology I use.
I have two datasets of unequal ...
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What is the intuition behind the single-pass algorithm (Welford's method) for the corrected sum of squares?
The corrected sum of squares is the sum of squares of the deviations of a set of values about its mean.
$$
S = \sum_{i=1}^k\space\space(x_i - \bar x)^2
$$
We can calculate the mean in a streaming ...
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Effective degrees of freedom for residual variance in ridge regression
The definition of the effective degrees of freedom (dof) in Ridge Regression via the trace of the "hat matrix" is well known (see e.g. Hastie and Tibshirani's Generalized Additive Models).
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Expectation and variance of bivariate skew normal distribution
I am fitting a bivariate skew normal distribution to a 2D data through the sn package in R. I get a $2 \times 1$ vector of ...
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How do I combine the error when multiplying a number by a proportion [closed]
I wish to know how many seals are in an area. Seals have been counted in a portion of the area, once each month over multiple years. Separately, several seals in the area have been fitted with GPS ...
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when focus on the pre-test for the normality test and ignore the pre-test for equality of variances?
When should a pre-test for normality only without a pre-test for equality of variances be performed before location tests like this paper did ((Rochon, J., Gondan, M. & Kieser, M. To test or not ...
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Variance of Multimodal Generalized von Mises Distribution?
How do you calculate the variance of a Multimodal Generalized von Mises (MGvM) distribution? Given its complexity with multiple modes and asymmetry, I'm looking for:
Any formula or method to calculate ...
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How would one describe such irregular data?
The situation is as follows (physics based): I have an array (7) of pixel sensors (imagine phone cameras) and a ton (millions) of particles crossing them (very large N). Each particle crossing a ...
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Understanding proof of variance of mean of random effects model and relationship to E[MS_treatments]
My textbook makes the following proof:
But I cannot understand how it arrives.
The variance of $y_{ij} = \sigma^2 + \sigma_t^2$ by assumption
this leads to:
$MS_e = \sigma^2$
$MS_{treatments} = \...
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Calculating $E[(\sum X_i)^4]$
Trying to figure where I'm going wrong with the following. My goal is to calculate var$(\bar X_n^2)$ using $E[(\bar X_n)^4]=\frac{1}{n^4}E[(\sum X_i)^4]$ given that $X_1,...X_n$ are iid with $EX_1=\mu,...
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Mean Squared Error for point estimation
I am attempting to understand Mean Squared Error when evaluating point estimators for particular parameters of interest. The book we are reading for class states the following:
The mean squared error (...
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Is the Between-Groups Variance a Covariance?
I am currently working through a book/class in quantitative genetics, and in Falconer and Mackay's Introduction to Quantitative Genetics, the following line stumped me:
"The between-group ...
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In stratified sampling, why is the stratum population variance obtained by dividing by 1 less than the stratum size
I am aware that flavors of this question get asked a lot, for e.g., here. I am fine with the sample variance being divided by $n-1$ and that is what makes it an unbiased estimator of the population ...
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In R, why are the results of summary(lm(y~-1))$sigma^2 and mean(y^2) same?
Consider the following data and the model:
y <- c(9,3,-2,7,6,12)
lm(y~-1)
summary(lm(y~-1))$sigma^2
[1] 53.83333
I expected ...
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lme output interpretation when including a varIdent(~ 1 | subject)
I am working on a project where I want to determine the sources of variability. I built the following model using the nlme package and lme function:
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How to interpret the differences in estimated variances?
I estimated the variance of Bitcoin in several ways using the var command in R, and within a GARCH model. I get series that look a bit similar, but the y-axis gives ...
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Zero variance but non-zero skewness
I was thinking of a hypothetical distribution where the mean(first cumulant) is non-zero, second cumulant(variance) is zero, and the third cumulant(skewness) is non-zero. The higher order cumulants ...
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Optimal three parameter variable stabilizing transformation of a Poisson
In the paper: "On the classical choice of variance stabilizing transformations and an application for a Poisson variate", Shaul K. Bar-Lev and Peter Enis give an optimal two parameter ...
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Variances of observed values and corresponding beta hats in Linear Regression [closed]
Is there any relation between the variance of the observed values of a predictive variable and the variance of the estimator of the corresponding beta parameter?
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Width of Confidence Intervals for Variance Estimates in Contrast to Point Estimates
We are conducting a variance decomposition using a hierarchical linear random effects Bayesian model to investigate the variance in a DV that is affected by three nested layers. We estimate credible (...
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Independent Component Analysis (ICA): Why rotate whitened data by principal components instead of right singular vectors?
I have a data matrix $ X $ that is $n \times m$, where $n$ is the number of features and $m$ is the number of samples and $ n < m$. Let the Singular Value Decomposition (SVD) of $X$ be $$ X = U \...
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Construct an estimator for the asymptotic variance of the MLE and and its asymptotic distribution
This is a question I have come across while learning about statistical inference and data analysis.
I think I have been able to solve question (a) and (b) already.
My solutions are:
$$
\hat{p}_{MLE} = ...
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When do variance of MLE and variance of posterior match?
Assuming a Gaussian likelihood, $y \mid x, w \sim \mathcal{N}(w^\top x, \sigma^2)$, the variance of the least squares estimate $\hat{w} = \mathrm{argmax}_w p(y \mid X, w)$ is $\mathbb{V}[\hat{w} \mid ...
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How to find $\mathbb{E} \left[\frac{\bar{\mu}}{\bar{\sigma}^2}\right]$?
I asked the same question on math stacks: MathStacks:, and some user suggest to ask it here for better insight. So this question has found interest in many research problems, but there have been no ...
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Question regarding probability and maximum possible variance
I have the following question:
Suppose we have a set of 10 numbers (1, 2, ... , 10), each with a certain probability tagged to it.
Is it true that the highest possible variance is achieved when 1 and ...
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Unusual variance measure
I explore centrality measures for graphs and discretized regions.
The data array (for given point) is all distances from this point to the boundary of region.
At the beginning I used standard ...
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Estimating variance by regressing squared residuals
For a linear regression, one way to estimate the range of a prediction for a new observation is to calculate the prediction interval for the new observation.
What if, instead, the squared residuals ...
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Derive Cramer-Rao lower bound for $Var(\hat{\theta})$ given that $\mathbb{E}[\hat{\theta}U]=1$
I am trying to derive the Cramer-Rao lower bound for $Var(\hat{\theta})$ given that we already know $\mathbb{E}[U]=0$, $Var(U)=I(\theta)$ and $\mathbb{E}[\hat{\theta}U]=1$. I am struggling with using ...
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Standard deviation squared is variance. What is standard error of the mean squared? [closed]
Is it correct to say that there is a sample variance and a mean variance, and their square roots are the standard deviation and the standard error of the mean, respectively?
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Why the MSE of the fitted data is not equal to the sum of the bias and the variance in R?
I use simple linear regression and I want to find the decomposition of MSE, that is as a sum of the bias, the variance and the variance of the error terms. I have the following code:
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How to look at between-group differences for a single gene using RNA seq data
I have an RNA seq dataset, but I am only interested in the expression of a single pre-specified gene and to compare it between 2 groups (patient phenotypes). Some have suggested (without a reference) ...
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How to establish what distribution to compare my statistic to
Apologies for this poorly titled question, but I've been taking some statistics courses and sometimes when you try to learn too many things in too little time you want to take a step back to check if ...
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What is this hybrid(mixed) random variable’s variance?
X ∼ Uniform(a,b), a<b (Discrete) where f(x)=1/n where n=b-a+1 and Y ∼ Uniform(c,d), c<d (Continuous) where g(y)=1/d-c. X and Y are independent. Let z = x - y. I was able to find the E(Z), ...
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AR(n) process error variance in R
I was wondering where does the "sigma^2 estimated as 12.2" in the following example came from. I tried to compute by myself the variance of the residuals, however it seems I probably forgot ...
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What is the variance decomposition method?
For $i = 1, \ldots, m$ and $j = 1, \ldots , n$ we have observations $x_{ij}$. We can assume that
$$
x_{ij} = y_{i} + z_{ij}, \qquad y_{i} \sim \mathcal{N}(\mu_{y},\sigma_{y}^{2}), \quad z_{ij} \sim \...
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Variance for Serial Meditation (Model 6)
I have this SPSS output and I'm wondering how I report variance for the model.
Additionally, how much of the variance was explained by exercise both directly and indirectly (through both mediators) in ...
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mean and variance
Let X be a discrete random variable such that X = 0 with probability 0.5 and X = 1 with probability 0.5. Let Y be a discrete random variable such that Y = 1 when X = 1 and Y = 0 when X = 0. What is ...
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Why is the variance smaller for the same coefficient in a reduced regression model vs. full regression model? [duplicate]
Let's say we have two estimators for $\beta$.
$\beta$ denotes all a full set of coefficients, one for each covariate in a dataframe.
$\beta$ can be split into $\beta_p$ and $\beta_r$, where $p$ ...
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Expectation and Variance of two sets
In genomics, you have an input control (I) and a treatment (T) where then you determine the ratio T/I. You perform multiple replicates for each but the number of replicates is not always the same. ...
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Variance in set of different random variables
Imagine we have sample of values a_1, a_2,...a_n, with a_i value originating from a normal distribution with mean mu_i_a and variance var_i_a, thus each value is single realization of different random ...
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Unbiased estimator of $\sigma^4$
In the post [here], the user asked the question
$\{X_i\}_1^n$ is random sample from $N(\mu, \sigma^2)$ with unknown parameters. Find an unbiased estimator of $\sigma^4$.
The solution uses a property ...
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Which type of t-test is best to use for my data
I'm looking for some guidance on which type of t-test is most suitable to use for my data.
I want to compare the means of the variable 'Rate' between two groups in my data. I have 6 years of data and ...
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Misunderstanding on the use of Popoviciu and von Szokefalvi Nagy's inequalities on the variance of a unbiased estimator
Let $X_1,\cdots,X_n$ be (discrete in my case) i.i.d. and bounded between $m$ and $M$. I'm interested in bounding the variance of an unbiased estimator:
$$\mathbb{V}\left[\frac1n\sum_{i=1}^nX_i\right]$$...
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Standard deviation of standard deviation under non-normality
In this post, an unbiased estimator for the standard deviation of the standard deviation under normality is provided.
I would be interested in such an estimator without the normality assumption, i.e., ...
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Difference between F-test and confidence intervals on variance estimates
Given n samples from a normally-distributed variable X, we estimate variance as $s^2=\frac{1}{n-1}\sum{(x_i - \bar{x})^2}$. We can also get a confidence interval for such a variance estimate as:
$$...
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Hypothesis testing of normal distribution, unknown mean unknown variance
Suppose that we make measurements of an effect, and we know that the values that we obtain follow a normal distribution. But we don't know the mean nor the variance. The hypothesis is that 95% of the ...
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How to quantify the similarity between three sets of complex numbers? [closed]
I have multiple groups of measurements, each containing three sets of complex numbers (impedances of the same thing measured under three conditions). The Nyquist plots belows shows two of such groups.
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Covariance matrix for data
Assume $n*p$ data matrix $X$, where n is the number of observations and p is the number of features. We are interested in the covariance among features.
I have seen notations where covariance matrix ...
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