How to find a critical value of a distribution ?

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Discussion Overview

The discussion revolves around finding an analytical method to extract critical values from generic distributions, specifically focusing on Gaussian and Chi-Square distributions. Participants explore the relationship between critical values and the challenges associated with deriving them without relying on distribution tables.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant seeks an analytical method for determining critical values from distributions, expressing frustration over reliance on distribution tables.
  • Another participant explains that computing critical values involves solving integrals, noting that if no analytical solution exists for the integral, then an analytical method for finding the critical value is also not possible.
  • Some participants question whether closed-form solutions exist for Gaussian and Chi-Square distributions, with one expressing uncertainty about their own attempts to find such solutions.
  • There is mention of the anti-derivative of the functions involved, with a suggestion that it has not been found and may not exist in a finite expression of standard functions.
  • One participant clarifies their earlier statement about seeking an analytic way to solve for critical values, acknowledging the complexity of such problems, especially for higher-degree polynomials.

Areas of Agreement / Disagreement

Participants generally agree that finding closed-form solutions for critical values in Gaussian and Chi-Square distributions is challenging, but there is no consensus on whether such solutions definitively do not exist.

Contextual Notes

Participants express uncertainty regarding the existence of closed-form solutions and the nature of the integrals involved. The discussion highlights the limitations of current methods and the dependence on numerical approximations for critical value extraction.

szandara
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I am trying to find an ANALYTICAL way to extract a critical value from a generic distribution. I need a way to find it analytically because I am trying to find a relationship between the critical values.

I always find methods that depends on the distribution tables, but I would like to know how these tables are built.

I did not find anything on the internet?

any advice? ( in particular I am working with Gaussian and ChiSquare distributions)

thanks

Simone
 
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Computing critical values for a statistical test from distributions involves solving equations such as \int_c^\infty f(x) dx = 0.05 for the value c. If there is no "analytical" solution for writing the integral then there is no "analytical" way to compute the critical value. I am assuming that by "analytical", you mean a relatively simple "closed form" formula. Is that what you mean? I know of no "analytical" way to solve the intergrals involved in the normal and chi-squared distributions. The tables are built by using numerical approximations to solve the equations.
 
Last edited:
yes that was actually what I wanted to know.

So, I must suppose that there's no closed form for Gaussian and Chi Square distributions?
I haven't tried myself yet.

S
 
szandara said:
yes that was actually what I wanted to know.

So, I must suppose that there's no closed form for Gaussian and Chi Square distributions?
I haven't tried myself yet.

S

For those problems, that is equivalent to solving the anti-derivative of those functions, which so far as Stephen Tashi has said, has not been found (yet).

If you find it, be sure to let us know!
 
chiro said:
For those problems, that is equivalent to solving the anti-derivative of those functions, which so far as Stephen Tashi has said, has not been found (yet).

If you find it, be sure to let us know!

I believe they have been proven not to exist using only a finite expression of standard functions.

However, just like sin(x) is just a conventional name for a function, we also have \Phi(x) which represents the anti-derivative of the standard normal distribution function.
 
I like Serena said:
I believe they have been proven not to exist using only a finite expression of standard functions.

However, just like sin(x) is just a conventional name for a function, we also have \Phi(x) which represents the anti-derivative of the standard normal distribution function.

Yeah I said the wrong thing, thanks for pointing that out.

I meant to say an analytic way of solving for c, which for many cases (even polynomials with degree higher than say 4 or 5) present a challenge.
 

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