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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2
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1answer
34 views

Which error is displayed in an error ellipse?

I have some bivariate data and I have calculated the error ellipse in the following way: I have first calculated the covariance matrix and then to obtain the radii of the ellipse I have taken the ...
0
votes
0answers
11 views

Robust variance estimators

Given $N$ data points in $\mathbb{R}^p$ - some of which are outliers (drawn from a different distribution from the inliers) - what sorts of algorithms have been designed to estimate the robust ...
1
vote
0answers
24 views

How remove variations in a time series X caused by another time series Y?

I have a time series on a monthly basis (a commodity) of which much variation is caused by the weather. I want to adjust this commodity for weather changes. I use Heating degree day as a proxy for ...
2
votes
0answers
32 views

Which observation has the largest variance of the residual?

a) For which of the following observations (obs1, obs2, obs3) is the variance of the residual the largest? Which observation has the highest leverage? And which one the smallest? Explain why. ...
1
vote
1answer
42 views

Variance and covariance notation: $\sigma^2 V_1$, $\sigma^2 V_2$

I am reading an internal paper that says: Let $\sigma^2 V_1$ equal the variance of $\sum_{m\in M}Z_m - Z_0$ and $\sigma^2V_2$ equal the covariance of $||M||^{-1}\sum_{m\in M}Z_m - Z_0$ and $Z_m ...
2
votes
0answers
25 views

$\sigma^2 \le (\mu-a)(b-\mu)$ for all probability distributions bounded on $[a,b]$? [duplicate]

Let $\mu$ be the mean and $\sigma$ the standard deviation of a probability distribution defined on the bounded interval $[a,b]$ (that is, the probability that the random variable lies outside $[a,b]$ ...
1
vote
0answers
20 views

Sample variance and error using Monte Carlo

Asked to compute estimator for the following function, $\theta = \int_0^\infty e^{-x^2}$ which can be solved by transforming the limits to 0 to 1 and solving the following expectation using Monte ...
4
votes
0answers
36 views

Variance inflation factor for generalized additive models

In the usual VIF calculation for a linear regression, each independent/explanatory variable $X_j$ is treated as the dependent variable in an ordinary least squares regression. i.e. $$ X_j = \beta_0 + ...
3
votes
0answers
72 views

covariance term in simple linear regression

I am trying to derive the expression for variance of $\hat{\beta_0}$ in simple linear regression. I substitute $\bar{y} - \hat{\beta_1} \bar{x}$ for $\hat \beta_0$ but in the intermediate steps the ...
3
votes
2answers
185 views

Why does sphericity diagnosed by Bartlett's Test mean a PCA is inappropriate?

I understand that Bartlett's Test is concerned with determining if your samples are from populations with equal variances. If the samples are from populations with equal variances, then we fail to ...
1
vote
1answer
43 views

How to make sense of 6mo worth of weight loss data?

-I have ~6mo of data that contains daily weight/caloric intake & #carbs/fats/proteins. I'm trying to figure out a way to make sense of the data. It's not a perfect experiment so there are other ...
0
votes
1answer
33 views

pairwise variance ratio in Stata

Suppose, I want to compute the pairwise correlation in Stata, then I do the following: sysuse auto pwcorr Now, I want to compute the variance for each variables ...
1
vote
0answers
20 views

Posterior variance reduction

As detailed on its Wikipedia page, Mutual information, $I(X,Y)$, can be bounded by the Jensen inequality to show that it is always positive. Also, one can show that $$ I(X,Y) = H(X) - H(X|Y). $$ ...
2
votes
0answers
35 views

Variance of the sum of correlated random variables

i'm trying to compute the variance of the random variable $$X = \frac{1}{N}\sum_{i=1}^N x_i$$ where $x_i$ are correlated identical random variables (mean and variance defined) obtained from a ...
3
votes
2answers
137 views

Overdispersion in logistic regression

I'm trying to get a handle on the concept of overdispersion in logistic regression. I've read that overdispersion is when observed variance of a response variable is greater than would be expected ...
11
votes
3answers
434 views

Deduce variance from boxplot

I was wondering how to deduce the variance of a variable using a boxplot. Is it at least possible to deduce if two variables have the same variance observing their boxplot?
0
votes
1answer
96 views

$E(\overline{x})$ vs $E(x)$?

If I have a problem down to $$k(E(x^2) - E(x)^2) = E(\overline{x}^2) - E(\overline{x})^2$$ ...can I say $k = 1$? $X$ has an unbiased mean and a variance. Each $x$ in $X$ is independent. Or is it ...
6
votes
1answer
521 views

How can a distribution have infinite mean and variance?

It would be appreciated if the following examples could be given: A distribution with infinite mean and infinite variance. A distribution with infinite mean and finite variance. A distribution with ...
1
vote
0answers
15 views

Why did fixing the covariance value cause AMOS to mis-estimate variance?

I have 100 data points on two variables, a and b. The correlation between the two is .3 and the SD is 1. When I run the ...
4
votes
0answers
57 views

Avoiding large variances when taking the logs of small values

I have two random variables $(X$ and $Y)$ that are always positive. The assumption I'm making is that their logs follow normal distributions (i.e., $N(\overline{\log(X)},s^2_{\log(X)})$ and ...
1
vote
1answer
63 views

P-value of F-test to compare two variances (var.test in R)

I am trying to understand where the p-value of a F-test comparing two variances comes from. More specifically, the p-value given by R's var.test function does not ...
0
votes
0answers
24 views

Forecast mean and variance for group data

Apologies if this is a bit of a simple question, but I haven't been able to find any answer to this over the past week and it's driving me crazy. Background Info: I have a dataset that tracks the ...
0
votes
1answer
25 views

How to compare distributions of two groups where each group has multiple observations from a small group?

I have data on continuous measurements (length of time of a behaviour) from two groups of individuals which differ in their phenotype (phenotype A or B, the variable used to group them). Each group is ...
7
votes
1answer
103 views

Do the mean and the variance always exist for exponential family distributions?

Assume a scalar random variable $X$ belongs to a vector-parameter exponential family with p.d.f. $$ f_X(x|\boldsymbol \theta) = h(x) \exp\left(\sum_{i=1}^s \eta_i({\boldsymbol \theta}) T_i(x) - ...
0
votes
0answers
10 views

How do I test for equal variances in a nested design?

I've got a dataset where stations (3) are nested within beaches (5). Every station is replicated 6 times per beach, so that I have a total of 90 observations. I now want to test for equal variances, ...
2
votes
1answer
28 views

Dealing with poorly estimated/missing explanatory variable values in GLMs

Context I am using generalised linear models to analyse some ecological data looking at the relationship between the population density of moth larvae and the prevalence (%) of viral mortality in the ...
0
votes
0answers
9 views

Estimating variance of prediction error in bootstrapped training sets with clustered data

I have C clusters with m elements each. I split the C clusters into a large training set D and a test set T. Hence, each element in D and T has m related elements, so its a cluster. I want to ...
1
vote
1answer
42 views

delta method with higher order terms to improve variance estimation accuracy

I need to apply the delta method principle using a Taylor expansion that retains higher order terms (i.e. to second or third order) in order to improve the accuracy of variance estimation. The ...
0
votes
0answers
9 views

comparing variation of a blended product from populations of different sizes (dilutions)

Original question: If population 1 consists of blended piles of size X_1 and population 2 consists of blended piles of size 4*X_1. Should I expect the standard deviation of X_1 to be equal to 4*the ...
3
votes
1answer
37 views

Variance of absolute value of a rv

Suppose that $X \backsim iid (\mu, \sigma^2)$. We are interested in $E (|X|)$ and ${\rm Var}(|X|)$. Can you suggest a way to proceed? I thought of rewriting $|X|$ as : $|X| = Xd - X(1-d)$, ...
1
vote
1answer
58 views

Basic statistical equation

How would I go about showing that: $$\sum_{i=1}^n (X_i-\bar{X})^2=\sum_{i=1}^n X_i^2-n\bar{X}^2\,,$$ (where $\bar{X}=\frac{1}{n}\sum_{i=1}^n X_i$)? Thank you for your help!
4
votes
1answer
115 views

Variance of Z for Z = X + Y, when X and Y correlated

So I'm trying to show that ${\rm Var}(Z) \le 2({\rm Var}(X)+{\rm Var}(Y))$ for $Z = X + Y$. This seems to be pretty easy to show given that $X$ and $Y$ are uncorrelated. But I'm running into trouble ...
0
votes
1answer
62 views

Expectation / minimizing variance of weighted sum of means

(I'm assuming the second $\bar x$ should be a $\bar y$), but I'm mostly confused how to solve this problem because it seems like since $\bar x$ and $\bar y$ are values, not random variables, $W$ is ...
1
vote
1answer
33 views

Is variance replacing square of standard deviation in a t test?

I am writing a t test using the code here : How to perform two-sample t-tests in R by inputting sample statistics rather than the raw data? But I would like to use the root of the variance as an ...
-3
votes
1answer
48 views

Self-study question

I'm currently working on a self study worksheet. I understand most parts of the solution for part III, but I can't seem to make out how this comes about: QUESTION: ANSWER:
2
votes
0answers
60 views

Estimating variance for identically non independent data

Let $X_{ij}$ with $1\leq i<j\leq n$ (that are $X_{12},\dots, X_{1n},\dots,X_{(n-1)n}$) be ${n \choose 2}$ identically normal distributed $N(0,\sigma^2)$ such that $ \text{corr}(X_{ij},X_{rs})=\rho ...
3
votes
3answers
104 views

Looking for a proof that overfitting a model leads to greater variance estimates (under OLS)

So I've been trying to algebraically prove that overfitting a model leads to greater variance values for the parameter estimates. I've gotten close (reduced the problem to showing a certain matrix is ...
1
vote
0answers
15 views

Calculating variation of one variable within one experiment

I am conducting a preliminary study, where people assess (7-point likert scala) different pictures. Besides asking for my 2 Independent variables (which are the manipulations for my main study, an ...
4
votes
1answer
127 views

Derive Variance of regression coefficient in simple linear regression

In simple linear regression, we have $y = \beta_0 + \beta_1 x + u$, where $u \sim iid\;\mathcal N(0,\sigma^2)$. I derived the estimator: $$ \hat{\beta_1} = \frac{\sum_i (x_i - \bar{x})(y_i - ...
2
votes
1answer
50 views

Taylor expansion to contain sample mean, sample variance, sample skewness, and sample kurtosis

I have the following expression: $$\frac{1}{p} \ln\left(1+\frac{p^1}{1!n}\sum_{i=1}^n x_i + \frac{p^2}{2!n} \sum_{i=1}^n x_i^2 + \frac{p^3}{3!n} \sum_{i=1}^n x_i^3 + \frac{p^4}{4!n} \sum_{i=1}^n ...
8
votes
2answers
317 views

Is variation the same as variance?

This is my first question on Cross Validated here, so please help me out even if it seems trivial :-) First of all, the question might be an outcome of language differences or perhaps me having real ...
2
votes
1answer
18 views

Variance of an autocorrelated random variable two periods in the future with Bayesian updating

I observe draws of some random variable $Y$ over time where $Y_{t} = aY_{t-1} + \epsilon_{t}$. $\epsilon \sim N(0, 1/\rho_\epsilon)$ and $a$ is an unknown parameter with prior distribution $a \sim ...
0
votes
0answers
8 views

How to create a correction factor based on measures from a known material?

I work with medical imaging and have the following issue: -- every time I scan a patient I obtain images of a calibration phantom alone, followed by images of subject + calibration phantom in the ...
1
vote
1answer
34 views

Adjustment of medians in the Siegel-Tukey test vs. the Wilcoxon test

I have two (unpaired) distributions. The distributions are far from being normal, so I'm using the Wilcoxon test to compare their medians: the Wilcoxon test does not reject the null-hypothesis ...
0
votes
0answers
23 views

Testing homogeneity of variances among samples across replicates

I would like to receive any advice on the following question: If one wants to test that variances are homogeneous among a certain number of population samples a possibility is to use the Levene’s ...
2
votes
0answers
31 views

Correlation of fixed effects with multiple response variables in MCMCglmm

I'm working with a mixed model for which I have several response measurements for every individual. One goal is to determine the sampling variance/covariance of the fixed effect estimates for a ...
2
votes
0answers
42 views

Calculating standard deviation after spatially and temporally averaging data

I have a spatio-temporal data set with (n,m) spatial and k temporal dimensions. My initial analysis consisted of spatially averaging the data and looking at the time dependent behavior. This resulted ...
0
votes
1answer
22 views

$\operatorname{var}[\frac{b}{a} B(a-b)-b B(b)]$ with $b\leq a$ and $b\geq 0$; $B=$brownian motion

I want to calculate: $\operatorname{var}[\frac{b}{a} B(a-b)-b B(b)]$ with $b\leq a$ and $b\geq 0$; $B=$brownian motion. I started like this: $(\frac{b}{a})^2 \operatorname{var}[ B(a-b)]+-b ^2 ...
2
votes
1answer
29 views

Require some help with biological replications with one extreme outlier

My problem is i have 3 biological replications (reps) withe each having 4-5 technical reps with two of the biological reps having comparable results for the treatment and control. These values ...
2
votes
2answers
111 views

Bias and Variance - Errors in R example

I am trying to better understand the bias and variance trade-off, and tried to create a R example. It attempts to calculate the bias and variance of smoothing splines with different parameters. ...