Questions tagged [mathematical-statistics]

Mathematical theory of statistics, concerned with formal definitions and general results.

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4 votes
2 answers
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What are good resources/criteria for judging human bias in data collection?

I've just been given a stack of polling data to analyse. Some of the questions are obviously leading or present subtle incentives (for the poller or polled) for specific answers. Of other questions I'...
12 votes
5 answers
4k views

Standard reference for classical mathematical statistics?

Can anyone recommend some books that are considered to be standard references for classical (frequentist) statistics? IE, fairly comprehensive, and also, been around for a while so that typos and ...
3 votes
2 answers
339 views

How to prove the exponential expansion recursive form?

Consider the following exponential expansion form: $$ \exp\left[\sum_{k=1}^\infty \gamma_k x^k\right] = \sum_{j=0}^\infty\delta_j x^j $$ where $\gamma_k$'s are known, and $\delta_0=1$, $$\delta_{j+1}...
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18 votes
2 answers
4k views

Why does the supremum of the Brownian bridge have the Kolmogorov–Smirnov distribution?

The Kolmogorov–Smirnov distribution is known from the Kolmogorov–Smirnov test. However, it is also the distribution of the supremum of the Brownian bridge. Since this is far from obvious (to me), I ...
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14 votes
1 answer
934 views

Intuition for higher moments in circular statistics

In circular statistics, the expectation value of a random variable $Z$ with values on the circle $S$ is defined as $$ m_1(Z)=\int_S z P^Z(\theta)\textrm{d}\theta $$ (see wikipedia). This is a very ...
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7 votes
3 answers
562 views

Non-trivial bound for $E[\exp(Z^2)]$ when $Z \sim {\rm Bin}(n, n^{-\beta})$ with $\beta \in (0,1)$

How to find a non-trivial upper bound on $E[\exp(Z^2)]$ when $Z \sim {\rm Bin}(n, n^{-\beta})$ with $\beta \in (0,1)$? A trivial bound is obtained for substituting $Z$ with $n$. A background on this ...
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104 votes
13 answers
80k views

Simple algorithm for online outlier detection of a generic time series

I am working with a large amount of time series. These time series are basically network measurements coming every 10 minutes, and some of them are periodic (i.e. the bandwidth), while some other aren'...
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2 votes
2 answers
504 views

What types of data analysis do not count as statistics?

When does data analysis cease to be statistics ? Are the following examples all applications of statistics ?: computer vision, face recognition, compressed sensing, lossy data compression, signal ...
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-4 votes
1 answer
423 views

Lies, Damn Lies and Statistics [closed]

Is there something about statistics that lends itself to this sort of saying, or is it just that people will say anything to support their case, and this includes citing irrelevant or incomplete ...
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37 votes
8 answers
15k views

What is Bayes' theorem all about?

What are the main ideas, that is, concepts related to Bayes' theorem? I am not asking for any derivations of complex mathematical notation.
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142 votes
21 answers
102k views

What's the difference between probability and statistics?

What's the difference between probability and statistics, and why are they studied together?
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34 votes
7 answers
4k views

How do you convey the beauty of the Central Limit Theorem to a non-statistician?

My father is a math enthusiast, but not interested in statistics much. It would be neat to try to illustrate some of the wonderful bits of statistics, and the CLT is a prime candidate. How would you ...
55 votes
19 answers
9k views

Mathematical Statistics Videos

A question previously sought recommendations for textbooks on mathematical statistics Does anyone know of any good online video lectures on mathematical statistics? The closest that I've found are: ...
50 votes
6 answers
14k views

Motivation for Kolmogorov distance between distributions

There are many ways to measure how similar two probability distributions are. Among methods which are popular (in different circles) are: the Kolmogorov distance: the sup-distance between the ...
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7 votes
0 answers
482 views

Testing (and proving) the randomness of numbers [duplicate]

Possible Duplicate: Testing random variate generation algorithms What's a good way to test a series of numbers to see if they're random (or at least psuedo-random)? Is there a good statistical ...
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11 votes
1 answer
480 views

What are some good frameworks for method selection?

I have been looking into theoretical frameworks for method selection (note: not model selection) and have found very little systematic, mathematically-motivated work. By 'method selection', I mean a ...
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93 votes
10 answers
21k views

What is meant by a "random variable"?

What do they mean when they say "random variable"?
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