All Questions
1,569 questions with no upvoted or accepted answers
2
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290
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Distribution of Sample Variances for Half Normal Distributions
If $X$ is distributed according to a normal distribution with zero-mean $\mathcal{N}(0, \sigma_N^2)$, $Y:=\vert X\vert$ is said to be distributed according to a half-normal distribution, cf. 2.
I am ...
- 71
2
votes
0
answers
414
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Expectation and Variance of Sum of dependent discrete variables
Q. 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 ...
- 21
2
votes
2
answers
155
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Estimate population mean from "best of N" samples
If I have a data set for which I know all measurements represent the largest of N observations, is there a good method for estimating the mean of all observations? So for example if N=10 and I have 3 ...
- 121
2
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0
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309
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Calculating a variance term for AUC using data points extracted from a figure
I am currently performing a meta-analysis where I am pooling area under the curve values.
For some papers I only have access to the figure. For example:
Using WebPlotDigitizer I can extract the mean ...
2
votes
0
answers
38
views
Autocovariance of a Stochastic Process
Let us define a stochastic process $X_t$ as follows:
$$X_t = \sum_{i=0}^p \alpha_i \epsilon_{t-i} + \sum_{i=0}^q \beta_i \delta_{t-i}$$
where {$ε_t$} and {$δ_t$} are mutually independent normally ...
- 71
2
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0
answers
336
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How to estimate the phenotypic variation explained by top SNPs from a GWAS study?
I have conducted a large-scale GWAS study and got a few significantly associated SNPs. I used GEMMA with -lmm 1 options to run ...
- 21
2
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2
answers
779
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What does variation in X mean when interpreting $R^2$ in linear regression?
I was reading a document about the coefficient of determination and saw that $R^2$ shows how much of the variation in y is represented by the variation in x. What I couldn't understand was the part ...
- 515
2
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28
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Portion of variance explained by individual properties with ANOVA
I have used the ANOVA test to compare the expected values of three datasets. The data sets contain data about the lifespans of three brands of frying oils. I have used Excel to perform the test and I ...
- 123
2
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27
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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,
$\...
- 3,242
2
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answers
2k
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How to interpret Hill estimate of tail index
I'm seeking a non-technical explanation of how to interpret the Hill estimate of the tail index for fat-tailed data, and, if possible, some explanation of seemingly contradictory results that ...
- 21
2
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answers
870
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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 $L - l$.
- 143
2
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150
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A non statistical/mathematical analogy to max vs argmax
I recently had a discussion on the topic 'usefulness/awareness of the function argmax() in non descriptive analysis'. That means areas, where you do not want to ...
- 1,658
2
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309
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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 ...
user226073
2
votes
0
answers
77
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How to calculate the Influence function for the half variance
So we know that the influence function $IF$ for a functional $v$ at a point $y$ is roughly defined as:
$$IF(v,F,y)=lim_{e\rightarrow 0} \frac{v(Y,G_y)-v(Y,F_y)}{e}$$
where
$$G_y(y)=1(Y>y)*e + (1-e)...
- 762
2
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0
answers
344
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What are the mean and variance of the square of a chi square?
Let $x$ be a random gaussian variable with mean=0 and sd=1, which is then squared (thus a chi-squared variable), so $y=x^2$. I understand that the expected value of $y^2$ is actually the variance of $...
- 688
2
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24
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How to construct GLMM with differing random effect variance structure by group?
I have a longitudinal dataset with a normally distributed outcome variable, a normally distributed predictor variable, and a binary grouping variable. I am trying to construct a GLMM with differing ...
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2
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0
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424
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Does sample variance has a Chi-square distribution?
Let $X_1, X_2, \ldots, X_n$ be a random sample from $N(\mu, \sigma^2)$. Does
$S^2=\frac{\sum^n_{i=1}(X_i-\bar X)^2}{n-1}$ has a Chi-square distribution?
I know that $\frac{(n-1)S^2}{\sigma^2}=\frac{\...
- 21
2
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0
answers
42
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What family of full support probability distributions satisfy that the density of any point in the domain vanishes as the variance goes to infinity?
Let $f(x,\sigma^2)$ be a representative element of a family of PDF's with full support over the reals that is indexed by their variance $\sigma^2$. Under what general conditions of the family of ...
- 141
2
votes
1
answer
280
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Ways to shrink standard errors in models for discrete dependent variables
Consider a simple Probit model
$$
Y_i=1\{X_i\beta+\epsilon_i\geq 0\}
$$
where $\epsilon_i$ is standard normal independent of $X_i$.
(1) Cardinality of the support of $X_i$
Is it true that (and, if yes,...
- 935
2
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0
answers
324
views
PCA with low variance ratio
I am trying to conduct PCA on a dataset with 17 features (which includes dummy variables; I converted two categorical variables into their corresponding dummy variables), and the first two principal ...
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2
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0
answers
101
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Variance of the simple regression coefficients $\text{Var}[\hat\beta]=\sigma^2 E[(X^T X)^{-1}]$ seems invalid? It seems to always blow up to $\infty$?
Suppose for simplicity we consider simple linear regression with $n=2$ i.i.d. observations.
The conditional variance of the estimated least squares regression coefficients will be
$$
\text{Var}[\hat \...
- 805
2
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0
answers
71
views
F test with inverted hypotheses
When using the F test to see whether pooled variance is appropriate, would it not be more useful to assume that the population variances are not equal and let the alternative hypothesis be that they ...
- 128
2
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0
answers
31
views
How to test that a sequence of variances rank ascendingly?
I am investigating forecast optimality. Diebold (2017, p. 334, list item d) indicates that one of the desirable properties of a good forecast is
Optimal forecasts have $h$-step-ahead errors with ...
- 69.5k
2
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0
answers
218
views
Mean and variance of the Beta distribution using identities of exponential families
I was studying the part of exponential families from Statistical Inference (George Casella, Roger L. Berger) and they give the following definition of an exponential family:
$$
f(x|\pmb{\theta}) = h(x)...
- 31
2
votes
0
answers
190
views
Variance of the method of moments estimator for $\mu$ of log-normal distribution
Suppose the data has originated from a log-normal distribution with parameters $\mu$ and $\sigma$ (i.e. the mean and standard deviation of the underlying normal distribution). I have only the sample ...
- 21
2
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0
answers
18
views
Analyzing the variance of an outcome variable: modelling standard deviation/sigma itself
Is the following a correct approach for sigma modelling?
Let’s assume we have a Y variable named hours in lognormal scale. We would like to know how these hours changed in time (variable named year).
...
- 2,277
2
votes
1
answer
47
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Does Variance Reduction really help the Nonparametric Test?
In Online Experiment, Working on Variance Reductions could help a lot for the traditional parametric tests like test two-proportion (CTR) or two-mean t-test, as it significantly improves the power and ...
- 177
2
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0
answers
49
views
MLE for the number of samples given $k$ largest values
I have the views on the top 100 videos using a tag in TikTok and want to estimate the total number of videos in that tag. I know the distribution for other tags so I can make a guess as to what it is ...
- 2,608
2
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0
answers
187
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Derivation of skewness and kurtosis algebra of random variables
In algebra of random variables, the symbolic rule for computing variance of random variable $X\in\mathbb{R}^{n\times p}$ multiplied by a coefficent vector, $a\in\mathbb{R}^p$, is
$$\text{Var}(X\cdot a)...
- 4,049
2
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answers
57
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Is it Sufficient to Truncate a Left Censored Distribution?
A colleague explained their approach to dealing with left censored data in an analysis, and while I don't think it is the best approach, I am not sure if it is insufficient or not.
My colleague has ...
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2
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70
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OLS: Show that $\sum_{i=1}^{n}\hat{x}_{i1}\hat{r}_{i1}=0$ and $\sum_{i=1}^{n}\hat{x}_{i1}\hat{u}_{i}=0$
The following is stated in appendix 3A in Jeffrey Wooldridge book (5 edition page 114): Introductory Econometrics - A Modern Approach:
$$
\sum_{i=1}^{n}\hat{x}_{i1}\hat{r}_{i1}=0
$$
$$
\sum_{i=1}^{n}\...
- 21
2
votes
0
answers
260
views
How to calculate Variance after Convolution process
I am trying to understand is it possible to say what variance I'll have if apply convolution to image. In my case I have matrix(image) and I am applying Gaussian Blur(with known parameters $\sigma_X = ...
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2
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0
answers
948
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Stochastic dominance and mean preserving spread
I need someones help on understanding the concepts of stochastic dominance and mean preserving spread. I have an exercise which could be used for explanation.
Consider the following lotteries:
L1 ={...
- 21
2
votes
1
answer
44
views
Expected Values, Variance and Covariance
I have come across this theorem on bayesian updating.
Suppose there is a random variable r of law N (rbar, 1/a).
Also, $$x_i = r+\epsilon_i $$ where the error term is distributed N(0,1/b).
Then:
...
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2
votes
0
answers
209
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Prove variance of locally weighted regression increases with degree
I am interested in proving the following fact for locally weighted polynomial regression from The Elements of Statistical Learning by Hastie et. al.
It can be shown that $||l(x_0)||$ increases with ...
- 556
2
votes
1
answer
51
views
Does Covariance Improve Variance of Random Variable?
I read a while back that we use covariance to improve the variance (conditional variance) of some random variable (i believe this was in context to Kalman filter). This concept is bit intriguing me ...
- 900
2
votes
0
answers
853
views
Is the variance of the residuals in linear regression constant (assuming constant-variance noise)?
In a linear regression model where $Y = X\beta+\epsilon$, with a residual vector $e=y-X\hat\beta$ where $\hat\beta=(X^TX)^{-1}X^Ty$ is the optimal regression coefficients given the data $X$ and $y$. ...
- 1,209
2
votes
0
answers
90
views
Asymptotic variance of Metropolis-Hastings estimates on a disjoint subdivision of the state space
I'm running the Metropolis-Hastings algorithm on a state space $(E,\mathcal E)$ which can be disjointly subdivided into regions $E_1,\ldots,E_k$, $k\in\mathbb N$ ($k\approx1e5$). On $E$, I have a ...
- 213
2
votes
1
answer
239
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Will change in standard deviation impact covariance?
If we increase the degree of standard deviation of one variable, does it affect covariance of two variables?
Example, two variables are there, A & B, the covariance of A & B is 100, and the ...
2
votes
0
answers
71
views
Proportion of Variance for Variational Autoencoders
For example for PCA, the proportion of variance explained is proportional to the eigenvalue of the respective feature.
Now for VAEs, is there a way to estimate the amount of variance that is ...
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2
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0
answers
201
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Variance of Linear Regression Coefficients in Terms of VIF
I am trying to understand how to rewrite the variance of regression coefficients in terms of the variance inflation factor. To be specific If we have a regression with design matrix X, I know that we ...
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2
votes
1
answer
302
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The approximation of the variance of MLE (Cramer-Rao Lower Bound)
This is in In Casella's Statistical Inference，page 473, the approximation of the variance of MLE (Cramer-Rao Lower Bound). I really confused with the conclusion:
$Var_{\hat{\theta}}h(\hat{\theta})$ ...
- 1,325
2
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answers
92
views
Estimating the asymptotic variance of a specific Metropolis-Hastings estimator
Remark:
I've added a detailed description of the actual setting of the application to the end of the question.
I'm running the Metropolis-Hastings algorithm with target distribution $\hat\mu$ (see ...
- 213
2
votes
1
answer
55
views
Variances of normal distribution
So my girlfriend posed me this question, from an exam that she had to take:
Assume that the length of adult men and women is normally distributed with means of 182 cm and 168 cm respectively. ...
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2
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716
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Is there an alternative to Box's M test for data that is not multivariate normally distributed?
My research question is to test whether two groups have a difference in variance-covariance across multiple measures. However, the data do not follow the multivariate normality assumption required for ...
- 333
2
votes
0
answers
52
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Calculating a variance
I want to calculate a variance of an estimator $Z$. $Z$ is the derivative of another estimator $Y$ with respect to some parameter $\theta$. I have the variance of $Y$ and now want to compute the ...
- 91
2
votes
0
answers
46
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Which is higher? - variance of a real-valued normal random variable or variance of integer rounding of that real-valued normal random variable?
Let A_r be a real-valued normal random variable whose mean is an integer A.
Let A_i be the rounding of A_r such that A_r = A_i + e, where e is an uniformly distributed random variable taking the ...
2
votes
0
answers
871
views
Why is Half-Cauchy, Half-Student-t as prior for variance parameters better than a normal distribution?
Gelman often refers to using half-cauchy or half-student-t distributions for variance parameters. Why is it better than using a vague normal distribution such as N(0,10)? Can somebody explain me the ...
- 164
2
votes
0
answers
48
views
Exponential Inequality For Probability of Being Close to Maximum
Given $n$ independent identically distributed random variables $X_1, X_2, \ldots, X_n$ that have $|X_i| < \lambda$ for all $i$. Let $\max(X)$ be the maximum of these $n$ variables.
Is there a ...
- 103
2
votes
0
answers
200
views
Paired non-parametric test for a difference in variance
I have $N = 391$ paired data points $\mathbf a = a_1, \ldots a_N$ and $\mathbf b = b_1, \ldots b_N$.
Both $\mathbf a$ and $\mathbf b$ are approximately normally distributed with a mean of zero.
I ...
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