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1 vote

What is the intuition behind the single-pass algorithm (Welford's method) for the corrected sum of squares?

Ok, so when you update $S_{n-1}$ to $S_n$ there are two things that need fixing you need a term $(x_n-m_n)^2$ for the error in the $n$ observation you need to fix up $S_{n-1}$ because it was centered ...
Thomas Lumley's user avatar
0 votes
Accepted

Covariance of Best Linear Unbiased Estimators and arbitrary LUE

Consider the variance of $T_a=aT'+(1-a)T$, which (for any $a$) is another linear unbiased estimator $$\mathrm{var}[T_a]=a^2\mathrm{var}[T']+(1-a)^2\mathrm{var}[T]+2a(1-a)\mathrm{cov}[T,T']$$ or ...
Thomas Lumley's user avatar
0 votes

Covariance of Best Linear Unbiased Estimators and arbitrary LUE

Unbiasedness of $T$ and $T'$ imply $\sum_{i=1}^n \alpha_i = \sum_{i=1}^n \beta_i = 1$. $T$ having minimium variance means $\alpha_1, \ldots, \alpha_n$ minimize $\sum_{i=1}^n \alpha_i^2$ subject to $\...
angryavian's user avatar
  • 2,188
0 votes

K means clustering of image with k=1 vs mean of all pixels

Welcome to CrossValidated! If when you say "k-means 1" you mean a k-mean algorithm with only 1 cluster, then I think the centers of the cluster are represented by the multivariate mean ...
jmarkov's user avatar
  • 673
2 votes
Accepted

Calculating $E[(\sum X_i)^4]$

In pursuit of calculating the fourth power, it might be apt to have a look at the general form of expectation of $p$–th power of sum of iid random variables. $\rm [I]$ deduces $\mathbf E\left[S_n^p\...
User1865345's user avatar
  • 7,932
3 votes

Calculating $E[(\sum X_i)^4]$

I would start by setting $Y_i=X_i-\mu$, for $i=1\ldots,n$. You then have: $$ \mathrm{Var}(\bar{X}^2)=\mathrm{Var}((\bar{Y}+\mu)^2)= \mathrm{Var}(\bar{Y}^2+2\mu\bar{Y})\\ = \mathrm{Var}(\bar{Y}^2)+2\mu\...
Doctor Milt's user avatar
  • 2,672
6 votes

How is that possible that simple arithmetic mean works well even for strongly skewed distribution?

What is your definition about "good"? I assume you want to say the bias of sample mean is zero, it is consistent, if so, in 1947, Hsu and Robbins proved that the arithmetic mean converges ...
Tuobang Li's user avatar
0 votes

The probability of a random variable being larger than a sequence of random values

As mentioned by others you should notate it with capitals $X_{1}, X_{2}, \dots, X_{n}$. Let $A_{i}$ be the event $X_{i}$ is the largest. Then we have $1 = P(A_{1}) + P(A_{2}) + \dots + P(A_{n})$. ...
Elmex80s's user avatar
  • 101
0 votes

Counterexamples where Median is outside [Mode-Mean]

As the above answers suggest, any order is possible. However, the violation of the mean-median-mode inequality should generally only happen in a nearly symmetric distribution. I assume you are saying ...
Tuobang Li's user avatar
2 votes

Can ANOVAs or Student tests be conducted on any summary statistic at the individual level?

I wouldn't collapse any within-subject repeated measurements of RT into a mean or median, since they're correlated. The within-subject correlation among repeated measurements affects the overall ...
Leif Peterson's user avatar
0 votes
Accepted

How does a Random Sample relate to Random Processes and Random Variables?

An example of a random sample: $n$ i.i.d. draws $X_1, \ldots, X_n$ from a distribution $P$. Each $X_i$ is a random variable. You can also view $(X_1, \ldots, X_n)$ as a single draw from the product ...
angryavian's user avatar
  • 2,188
4 votes

In R, why are the results of summary(lm(y~-1))$sigma^2 and mean(y^2) same?

summary(lm(y~-1))$sigma^2 gives the variance of the residuals from your model, but your model is one that doesn't fit any parameters and just predicts $0$ every ...
Dave's user avatar
  • 60.9k
6 votes

In R, why are the results of summary(lm(y~-1))$sigma^2 and mean(y^2) same?

If you define the model as lm(y~1), you will get a value of 23.76667 for sigma^2. Your model does not have an intercept (that's ...
Doctor Milt's user avatar
  • 2,672
0 votes

When would we use tantiles and the medial, rather than quantiles and the median?

SAMPLE TANTILES Using order-statistics notation, let an ordered sample of size $n$ $$X_{(t)}\equiv \{x_{(1)},x_{(2)}, \dots, x_{(n)}\}.$$ Define the partial sums (in their usual sense) $$S_{(j)} \...
Alecos Papadopoulos's user avatar
-2 votes

Trimmed mean vs median

In statistics, there are three major errors: bias, variance, and contaminations. Sample mean is a consistent estimator, however, its variance and robustness is not desired in many scenarios, so ...
deq2's user avatar
  • 1
0 votes
Accepted

Subtracting the ideal-rank-mean in Wilcoxon rank-sum, what does it do

I had a further discussion on Datamethods. The resource I found most useful are Violation of proportional odds and MWU and Proportional odds. My current understanding is : The test reports the ...
aflip's user avatar
  • 21
1 vote

What do I do when I *want* to accept the usual null hypothesis?

Here you can employ theology instead of statistics If you want to accept a particular hypothesis, just accept it --- avoid wasting time by employing statistical analysis and pretending that you are ...
Ben's user avatar
  • 123k
4 votes

Box plots of monthly averaged water flow

This is up to you and what you want to find out. Are the outliers important? I could see how they could be (floods, droughts, etc) but maybe not. Or do you only want an average per month (maybe you ...
Peter Flom's user avatar
  • 117k
2 votes

Subtracting the ideal-rank-mean in Wilcoxon rank-sum, what does it do

This is my first answer on CV, so please make edits or let me know if it is wrong or missing anything! Now onto your question. Why not just the rank-sum? First the conclusion: comparing the rank-sum ...
Aliquid's user avatar
  • 21

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