Solving Overdetermined Problems: X2 Distribution Requirements

In summary, the experts are discussing the relationship between the observed chi square (X2obs) and the number of degrees of freedom (m-n) in an overdetermined problem with m data points and n parameters. The book states that X2obs follows a chi-squared distribution if the data points are normally distributed. However, one expert believes that the number of degrees of freedom is always m-n, regardless of the data distribution. Another expert suggests that the Central Limit Theorem may be used to justify the assumption of normality. Finally, a source is provided that discusses the requirement of normally distributed data for the chi-squared distribution to be applicable.
  • #1
Niles
1,866
0
Hi

I'm not sure this is the right place to post, but I'll go ahead. In my book it says that if I am dealing with an overdetermined problem with m data points and n parameters (so m>n), then my observed chi square X2obs follows a X2 distribution with m-n degrees of freedom if the data points are normally distributed.

I thought that the number of degrees of freedom was always m-n, regardless of what distribution my data follows. Am I right or is it correct what the book is stating?
 
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  • #2
Niles said:
Am I right or is it correct what the book is stating?

I think no one has answered this because you haven't given a clear statement of what the book said. For example, what kind of parameters is the book talking about? Means? Covariances? Any old parameter? What kind of data are the "data points"?

Do you have a source or link that supports your own opinion that the random variables need not be normally distributed?
 
  • #3
Niles said:
Hi

I'm not sure this is the right place to post, but I'll go ahead. In my book it says that if I am dealing with an overdetermined problem with m data points and n parameters (so m>n), then my observed chi square X2obs follows a X2 distribution with m-n degrees of freedom if the data points are normally distributed.

I thought that the number of degrees of freedom was always m-n, regardless of what distribution my data follows. Am I right or is it correct what the book is stating?

The Chi-squared distribution has an essential parameter called number of degrees of freedom. So, the bolded and red text in your quote is all part of the name.
 
  • #4
By "parameters" I mean parameters used to make a fit to the data. And data points are physically measured data, which is why I believe the book is so keen on always dealing with normally distributed data (cf. Central Limit Theorem).

I have no source for my statement. In fact I believe I might be wrong. But I still think it is an interesting question: If I am dealing with data that isn't Gaussianly distributed, then how would I go about and make a goodness-of-fit estimate, considering I can't use X2?

Thanks.
 
  • #5
Niles said:
By "parameters" I mean parameters used to make a fit to the data.

And by parameters I meant coefficients that characterize the probability density function, just like the expectation a and the standard deviation [itex]\sigma[/itex] in the normal distribution [itex]\mathcal{N}(a, \sigma)[/itex], or the endpoints a and b in the uniform distribution [itex]\mathcal{U}(a, b)[/itex] or the parameter [itex]\lambda[/itex] in the Poisson distribution [itex]\mathcal{P}(\lambda)[/itex].
 
  • #6
The "if the data points are normally distributed." part may be invoked by using the Central Limit Theorem. If the data points are sums or averages of many RVs then one may assume it is "close to" normally distributed and thus the statistic is "close to" chi-squared.

(BTW: one should say "regardless of" or "irrespective of" or even "irregarding" but not "irregardless".)
 
  • #8
  • #9
http://en.wikipedia.org/wiki/Cochran%27s_theorem" [Broken] gives the precise conditions when the distribution is chi-square and what the number of degrees of freedom is.
 
Last edited by a moderator:

1. What are overdetermined problems?

Overdetermined problems are mathematical problems in which there are more equations than unknown variables. This means that there is no single solution that satisfies all of the equations simultaneously.

2. What is the X2 distribution?

The X2 distribution, also known as the chi-squared distribution, is a probability distribution used in statistics to model the distribution of certain types of data. It is often used in hypothesis testing and confidence interval calculations.

3. How do you solve overdetermined problems using the X2 distribution?

To solve overdetermined problems using the X2 distribution, you can use a method called the method of least squares. This involves finding the line or curve that minimizes the sum of the squared differences between the observed data and the predicted values. The value of X2 is then calculated to determine the goodness of fit of the solution.

4. What are the requirements for using the X2 distribution in solving overdetermined problems?

There are a few requirements for using the X2 distribution in solving overdetermined problems. First, the errors in the data should be normally distributed. Additionally, the data should be independent and the sample size should be large enough. Finally, the values of the independent variable should be known and the relationship between the variables should be linear.

5. Can the X2 distribution be used for nonlinear relationships in overdetermined problems?

The X2 distribution is typically used for linear relationships in overdetermined problems. However, it can also be used for nonlinear relationships if the data is transformed to make it linear. This can be done by taking logarithms or using other mathematical transformations to make the data fit a linear relationship.

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