# Questions tagged [theory]

For questions about statistical theory. Always include a more specific tag as well.

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### How do we call a more extreme case of fat tails than a power law?

According to Wikipedia the most extreme case of a fat tail follows a power law: The most extreme case of a fat tail is given by a distribution whose tail decays like a power law. That is, if ...
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### Conceptually, what happens when you take (A - B) and divide it by (1 - B)? What is it used for?

I've come across this transformation a lot in my advanced stats course and I'm curious as to what it accomplishes. The basic structure is: A - B --------- 1 - B Here's an example of its use when ...
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### Statistical Theory of Power Sampling (within R code)

I am curious about the statistical theory behind power.prop.test in R. I have dived into the code behind it and detailed it in Latex format. My question here is: ...
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### Bounding the distributional error introduced by bootstrapping

Suppose I have some data $x_N$ of `size' $N$ which were drawn according to some measure $P_N(x_N)$. I'm imagining $x_N$ as being any of: an exchangeable / iid sequence of length $N$ a stationary ...
13 views

### Sum of pure errors

We know that in a simple linear regression model that the sum of all the residuals is 0 but why is it that the sum of all the pure errors is also 0? Is there a relation between them?
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### Compare predicted versus actual outcomes in a GLM

I read somewhere that you could compute a "residual value" for a GLM by taking the actual values of your response variable divided by the predicted value of that response variable. For example, ...
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### What is the terminology for the length of time an individual data point represents?

I'm having trouble conceptualizing this or putting it into word(s). It has been bothering me off and on for months now. As an example (financial, as that's all I know), let's say that Sarah wants to ...
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### Existence of an optimal learning algorithm for a given distribution

I am working on Exercise 3.8 from Understanding Machine Learning by Shai Shalev-Shwartz and Shai Ben-David. In it we consider the binary classification problem where $\mathcal{X}$ is a set of data ...
41 views

### The effect on expectation of biasing a distribution by a monotonic function

Let $$x \sim g(x)$$ where $g(x)$ denotes the pdf of $x$. Let the pdf of another variable $x^*$ be denoted by $f(x^*)$ and let $$f(x^*) \propto g(x^*) z(x^*)$$ where $z(x^*)$ is a monotonic ...
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### Large difference in contingency table level affecting Cramer's V?

The Wikipedia article for Cramer's V states the following without citation: Note that as chi-squared values tend to increase with the number of cells, the greater the difference between r (rows) ...
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### Optimization equivalence

Can someone help me with the step by step demonstration of the following equivalence used in SVM: $$maximize: m = \frac{1} {\|w\|} \equiv minimize: m =\frac{1} {2}\|w\|^2$$ I would be most grateful ...
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### minimum replicate number for comparative metagenome

I am in the process of designing an experiment that will look for differences in the gut bacteria of sick and healthy animals. The question that has been troubling us is how many animals should we ...
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### Who first invented/proposed/formulated the IQM (Interquartile Mean) and the Truncated/Trimmed Mean?

I've looked this up on google for quite some time but couldn't find the answer! Does any of you know who or how can I find who first proposed these statistics? How can access their respective ...
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### Theory: When to use one multi-factorial ANOVA vs. many one-factorial ANOVAs

I was recently peer reviewing a paper that used statistical tests in an odd way that didn't seem right to me. However, I'm not an expert so just wanted to talk theoretically about some of the things I ...
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### Understanding Spectral Density of a time series in terms of Regression on Sinusoids

This may be a very trivial question but I would like to understand spectrum density of a time series in terms of regression. This is what I know from reading on the net: A time series can be ...
27 views

### A good literature for statistical learning theory

any recommendations for good literature on statistical learning theory? I mean, something what goes into more details than Elements of Statistical Learning, in terms of losses, empirical error ...
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### Maximum sample size for one-way ANOVA?

Lists of requirements for one-way ANOVA include the following: Samples should be mutually independent Samples should be from a population with a normal distribution Samples should have the same ...
62 views

### Can the AUC (logistic regression) be used as a measure of quality of the model? [duplicate]

If I have a good measure of the AUC, what does it tell me in relation to the model and its variables?
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### How do you prove that the following 2 hypothesis classes' VC-dimensions are equal?

Given a hypothesis class $H=\{h:X\to\{0,1\}\}$. Let $c\in H$ be the correct predictor. Denote $H^c = \{c\Delta h:h\in H\}$, where $c\Delta h=(h\backslash c)\cup (c\backslash h)$. Please prove that ...
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### Learning from multiple very varied data sets?

Suppose we have a set of objects $X$ (e.g. individual humans). Suppose also that humans can be described by a set of (potentially very high-dimensional) variables $V_i$, (e.g. $V_1$ is a picture of ...
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### Why isn't the least squares predictor $\Phi(\Phi^\top\Phi)^{-1}\Phi^\top$ simply the identity matrix? [duplicate]

Given target vector $y$. Want to predict it using linear regression $h(w) = w^Tx$ Let $\Phi$ be the least squares matrix, i.e., $\Phi = \begin{bmatrix} x_1^\top \\ \vdots\\ x_n^\top \end{bmatrix}$ ...
I created a multivariate regression following the scheme $$y = \beta_0 + \sum^n_{i=1}\beta_i*x_i$$ and got an average deviation ofaround 5%. When I tried the regression without the $\beta_0$ I got a ...