Help! Reading Material for Set Theory & Non-Parametric

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Hi Guys,

This is just a copypasta of a post I made to Kaggle.com/fourms. It got no responses in 6 days! Maybe it was just a bad question? Anyway, any material you can recommend would be appreciated...

 

First post, so please be gentle. Also, keep in mind I only went for a math minor (finance major).

OK, so up until last week I had no idea non-parametric statistics even existed. As far as my preliminary research has indicated, unlike parametric statistics, non-parametric statistics do not use distributions to make inference about a total population from a sample (and also predominately use medians in lieu of means).

After taking abstract algebra and some discrete mathematics, I am fairly competent with set theory and cantor notation. I also took an advanced logic class covering methods of proof ("reductio ad absurdum" and all that).

All the texts I have found refer to "assumptions" that must be true in order to apply parametric statistics to a sample, but they don't go over what these are or how to test for them!

Can you guys recommend a book(s) that...

  • Enumerates the assumptions made by parametric models (Gauss, F, T, Z, etc), describes how to test for these with set theory (preferably cantor notation), and does a good job of delineating these proofs.
  • A book which acts as a road-map for different statistical techniques, which assumes absolute familiarity with basic parametric statistics. What I have found so far has been so rudimentary, I spend all my time re-reading verbose descriptions of basic stat.

Thanks Guys!

Source

Perhaps incorrectly, I am basing much of my current understanding off these two finds during my googling: 

23 1 Parametric vs non parametric statistics 10 22

Choosing Between a Nonparametric Test and a Parametric Test

Any input would be appreciated! 

Cheers,

TJ

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Hi TJ,

you are asking a number of different statistics related questions, plus mixing it up with set theory... I will do my best to point you in the right direction, provided I have understood what you are looking for.

1) Parametric vs. non-Parametric methods: to use parametric methods a number of assumptions need to be verified. In general the assumptions are different depending on the distribution your data come from, so first you have to verify the fact that your data come from a specific distribution. You should be looking into Goodness-of-fit tests. This can get your started:

https://en.wikipedia.org/wiki/Goodness_of_fit

2) To use non-parametric methods on the other hand you make no assumptions on the underlying distribution of your data, hence there is no need to test for any particular distribution. Still some assumptions apply, for example that of independence of observations. On the flip side non-parametric test are in general less powerful, exactly because they make to assumptions on the underlying distribution, and may require more data to provide you with a reliable conclusion.

https://en.wikipedia.org/wiki/Nonparametric_statistics

3) Your are asking for a sort of "guide" on how to chose which statistics to apply to a specific task/situation. Have a look at this diagram (also available in PDF from the same page) which summarizes it very concisely: 

http://abacus.bates.edu/~ganderso/biology/resources/statistics.html#whichtest

Hope this helps. 

Cheers,
Marco.

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Amazing Thanks!