Questions tagged [variance]
The expected squared deviation of a random variable from its mean; or, the average squared deviation of data about their mean.
3,443
questions
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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 ...
0
votes
1answer
16 views
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 ...
0
votes
1answer
22 views
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}$$\sum_{i=1}^{n} X_i$ , I ...
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0answers
6 views
How to measure ML classification model stability?
It's well known that decision trees are unstable models, i.e. they have high variance. It can be easily shown by adding Gaussian noise to variable(s) and checking that completely different tree is ...
0
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0answers
48 views
Covariance of two subsample means [closed]
A SRS of size $m = m_1 + m_2$ with mean $\bar{x}$ is drawn
from a finite population and a simple random subsample of size $m_1$ is
drawn from it with mean $\bar{x}_1$ and $\bar{x}_2$ is the mean of ...
0
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0answers
36 views
How can I compute the variance of $aX^2+bX$?
Let $X \sim \mathcal{N}(0,1)$. How can I compute the variance of:
$$
Y=aX^2+bX
$$
1
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0answers
10 views
How can I compare two response ratios in meta-analysis?
For example, there are several studies about how males and females response to certain condition.
Here's the data I got:
Xmc, the mean data of male control group, Xfc, the mean data of female control ...
1
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0answers
45 views
Compute a Monte Carlo estimate. Which of the variances (of $\hat{\theta}$ and $\hat{\theta}^{*}$) is smaller, and why?
Compute a Monte Carlo estimate $\hat{\theta}$ of $$ \theta = \int_{0}^{0.5} e^{-x} dx $$ by sampling from Uniform$(0, 0.5)$, and estimate the variance of $\hat{\theta}$. Find another Monte Carlo ...
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13 views
How to analyze small dataset? [closed]
I have a small dataset with continues variables , dependent variable is normally distributed and linearly associated with the independent variable. I tried doing linear regression but the model doesn'...
2
votes
0answers
25 views
prove the difference between mean and median is less than the variance [duplicate]
Suppose $X$ is a random variable with finite variance. Let $m$ denote
the median of $X$ and $\mu$ the mean of $X$, i.e. $\mu=\mathbb{E}(X)$.
Show $$(m-\mu)^2\leq\text{var}(X)$$
Intuitively this is ...
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0answers
9 views
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,
$\...
1
vote
1answer
21 views
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 ...
1
vote
0answers
23 views
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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0answers
8 views
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).
...
1
vote
1answer
26 views
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 ...
1
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0answers
24 views
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 ...
0
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0answers
40 views
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 ...
1
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0answers
14 views
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 ...
7
votes
2answers
107 views
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 ...
0
votes
1answer
68 views
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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0answers
5 views
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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0answers
51 views
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 ...
1
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0answers
25 views
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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23 views
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)
$$
I have 2 datasets $X=(X_1,...,X_n)$ with $Y=(Y_1,...,Y_n)$ that I need to compare, and estimate the ...
1
vote
1answer
34 views
Sample Variance and Population Variance for Ungrouped data
A study of the effect of smoking on sleep patterns is conducted. The measure observed is the time, in minutes, that it takes to fall asleep. These data are obtained:
Smokers: $69.3, 56.0, 22.1, 47.6, ...
1
vote
1answer
33 views
Different formulas for $\hat{\sigma}^2$
My book introduces $\hat{\sigma}^2 = \frac{1}{n} \sum_{i=1}^n \hat{e}_i^2$, which makes sense to me since to my understanding $\hat{\sigma}^2$ is the error variance estimator. However, I have also ...
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0answers
25 views
Variance as Covariance [duplicate]
Is there a useful interpretation of variance as a special case of covariance, Var(X) = Cov(X,X)?
-1
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0answers
54 views
Distribution of estimated variance of residuals in simple regression model [duplicate]
We have our simple regression estimated model:
$$
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
$$
But how can we prove ...
0
votes
0answers
25 views
Distribution of hat sigma in simple linear regression model [duplicate]
How can I prove that the distribution of estimated σ^2 (the distribution of the estimated variance of errors εi, in case of homoschedasticity) is equal to a chi-squared with n-2 df * σ^2/(n-2) ?
2
votes
2answers
99 views
Expression of $E[(X-a)^3]$ as a function of $\operatorname{Var}(X)$ and/or $\sigma_x$
Just a question:
I would be able to express $\mathrm{E}\left[(X-a)^{3}\right]$ as a function of $\sigma_x$
and/or $\sigma_x^2$, with $a$ a constant (surely $\mathrm{E}\left[X\right]$ terms should ...
0
votes
0answers
12 views
Variance of DFT of filtered noise
I am struggling with the following question:
Let v(t) be a stationary stochastic process with Gaussian probability distribution and power spectral density $S(\omega)$. Let the DFT of $v(t)$ be $V(k)=\...
4
votes
1answer
45 views
Variance involving two independent variables
hello I have two independent variable P and Q. They are both non-negative. Now I define two new variables on them:
The first variable $$R_1=\alpha P+(1-\alpha)Q.$$ Since P and Q are independent, so $$...
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0answers
4 views
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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0answers
12 views
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 ...
0
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0answers
29 views
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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22 views
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 ...
0
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0answers
17 views
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{...
0
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0answers
20 views
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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0answers
62 views
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 ...
0
votes
1answer
42 views
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 ...
2
votes
1answer
24 views
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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0answers
31 views
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:
$$...
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0answers
37 views
Is the inverse of the sample variance integrable?
Is the inverse of the sample variance integrable? That is, does it hold that
$$
E\bigg[\bigg(\frac{1}{n}\sum_{i=1}^n X_i^2 - \overline{X}_n^2\bigg)^{-1}\ \bigg] < \infty.
$$
0
votes
0answers
20 views
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 ...
0
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0answers
3 views
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 ...
0
votes
0answers
7 views
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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0answers
39 views
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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vote
2answers
24 views
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 ...
9
votes
2answers
669 views
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. ...
0
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0answers
9 views
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 ...
