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Let $X$ be a chi squared variable with $121$ degrees of freedom. So the density $f_X$ of $X$ is defined by

$$ f_X(x)=\frac{\big(\frac{x}{2}\big)^{\frac{121}{2}-1}}{\Gamma(\frac{121}{2})}{{e}^{-\frac{x}{2}}} $$

I would like to compute $P(X>126)$ with an accuracy of $10^{-2}$. I know that a standard method is to approximate the distribution of $X$ by a normal distribution ( ${\cal N}(121,\sqrt{242})$ here), but I do not know of any control on the error made in this approximation. In theory this is just a problem of computing a definite integral with a good enough precision, but it seems to exceed the capacity of my formal calculator (indeed, the value $\Gamma(\frac{121}{2})$ is very large, its integer part has 81 digits.)

Is there a rigorous (and working!) method to solve this ?

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1 Answer 1

up vote 1 down vote accepted

You could use Cornish-Fisher asymptotic expansion formula to take into account higher moments, specifically the skewness of $\mathcal{s}$ and kurtosis $\mathcal{k}$: $$ \mathcal{s}\left(\chi^2_{121}\right) = \frac{2\sqrt{2}}{11} \quad \mathcal{k}\left(\chi^2_{121}\right) = 3 + \frac{12}{121} $$

The Cornish-Fisher expansion would approximate the inverse CDF function $Q(q)$ as a polynomial in the inverse CDF $\tilde{Q}(q)$ of the standard normal distribution. It reads: $$ Q(q) = 121 + \left( -\frac{2}{3} + \frac{2171}{99 \sqrt{2}} \tilde{Q}(q) + \frac{2}{3} \tilde{Q}^2(q) + \frac{1}{99 \sqrt{2}} \tilde{Q}^3(q)\right) $$ Now using the method of binary splitting on the calculator it is not hard to find that $Q(q) > 126$ implies $\tilde{Q}(q) > 0.359853$, that is the requested probability.

Indeed, checking with Mathematica, you see that this comes pretty close:

In[27]:= N[
 Probability[x > 126, x \[Distributed] ChiSquareDistribution[121]]]

Out[27]= 0.359493

Compare with the pure normal approximation:

In[29]:= Block[{ch2d = ChiSquareDistribution[121]},
 N[Probability[x > 126, 
   x \[Distributed] 
    NormalDistribution[Mean[ch2d], StandardDeviation[ch2d]]]]
 ]

Out[29]= 0.373949
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Thanks! But the wolfram link is not very helpful for a newbie like me on this subject. Is there a standard reference containing a more complete treatment on the Cornish-Fisher expansion ? –  Ewan Delanoy Oct 23 '12 at 14:12
    
You could use riskgloggary website on the subject. Here is another site with the derivation. I picked my knowledge on the subject from "Kendall's advanced course of statistics" book. There plenty of books on Amazon covering it though. Search for "Cornish-Fisher" expansion. –  Sasha Oct 23 '12 at 14:35
    
Nowhere in all those sites is there any mention of an effective control on the error made in the approximation, which is what the OP asks about. What do you know about the quality of your approximation of $Q$ by the polynomial in $\tilda{Q}$ ? –  Ewan Delanoy Oct 23 '12 at 14:56
    
Let $X$ be a $\chi^2$ random variables and let $Z$ be a standard normal random variable. What is happening is that we approximate $F^{(-1)}_X( F_Z(q) )$ as a polynomial in $q$. Assuming $X$ is "nearly" normal, it is reasonable to assume that such an approximation is accurate at least when $q$ is sufficiently distant from extremities $q=0$ and $q=1$. Alternatively, you could use Gram-Charlier A series or Edgeworth series. –  Sasha Oct 23 '12 at 16:05

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