Chi-Squared distributions, non-parametric tolerance intervals, and solving the inverse regularized gamma function...for a? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T19:21:43Z http://mathoverflow.net/feeds/question/78094 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78094/chi-squared-distributions-non-parametric-tolerance-intervals-and-solving-the-in Chi-Squared distributions, non-parametric tolerance intervals, and solving the inverse regularized gamma function...for a? Lucas 2011-10-14T02:58:41Z 2011-10-14T02:58:41Z <p>Conover's book on non-parametric statistics gives as equation 1 in section 3.3 a formula for finding the sample size required to yield a given tolerance interval, namely <code>$n\approx\frac{1}{4}x_{1-\alpha}\frac{1+q}{1-q}+\frac{1}{2}\left(r+m-1\right)$</code> where $n$ is the sample size, $\alpha$ is the confidence level, $q$ is the proportion of the population encompassed by the tolerance interval, $r$ is the index of the lower order statistic, <code>$x_{1-\alpha}$</code> is the $1-\alpha$'th quantile from a chi-squared distribution with $2\left(r+m\right)$ degrees of freedom, and $m$ governs the width of the interval measured in terms of the number of order statistics involved ($s=n+1-m$ where $s$ is the index of the upper order statistic in the interval). Trying to solve this for the size of the tolerance interval (roughly put that is, we are actually solving for $r+m$ here) yields <code>$r+m=2n-\frac{1}{2}x_{1-\alpha}\frac{1+q}{1-q}+1$</code>. From here, we can apply the fact that the $p$'th quantile of the chi-squared distribution with $v$ degrees of freedom is <code>$x_p=2Q^{-1}\left(\frac{v}{2},0,p\right)$</code> where <code>$Q^{-1}\left(a,0,x\right)$</code> is the inverse regularized gamma function of $a$ and $x$ (thank you, Wolfram Alpha, for the table that allowed me to figure that out); however, I have no idea how one would solve that for $v$. Has this even been attempted before, or am I insane?</p>