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My statistics class makes a big deal out of the following fact: for any random sample $X_1, \ldots, X_n$ with continuous, invertible cdf $F(x;\theta)$, $$-2\sum_i\ln F(X_i;\theta) \sim \chi^2(2n).$$ That surely would have been very useful in the days when most researchers didn't have the computational capacity to find quantiles of an arbitrary distribution, and were forced to rely on massive lookup tables of $\chi^2$ quantiles. But today I can write a simple script to find quantiles of any distribution I like. This makes me less inclined to massage a $\chi^2$ variable out of my data every time I want to design a hypothesis test or confidence interval.

Am I missing some reason why this trick is still relevant in the computer age?

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  • $\begingroup$ That is an asymptotic result, so an approximation in finite samples (with some assumptions on $F$). And the $F$ probably should be $f$, that is, the corresponding density ... $\endgroup$
    – kjetil b halvorsen ♦
    Commented Nov 11, 2022 at 18:38
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    $\begingroup$ The result is true as stated. Each $F(X_i;\theta) \sim unif(0, 1)$, which means that their negative logs are $exp(1)$-distributed. The fact that the statistic is $\chi^2$ can be verified with moment generating functions. $\endgroup$
    – Frank Seidl
    Commented Nov 11, 2022 at 19:04
  • $\begingroup$ OK, my bad ... but it assumes that $\theta$ is the true parameter value, so must be replaced by an estimator and then becomes approximate. $\endgroup$
    – kjetil b halvorsen ♦
    Commented Nov 11, 2022 at 19:10
  • $\begingroup$ The point is for the pivotal quantity to depend on $\theta$. For instance, I might design a test which assumes the null hypothesis $\theta=\theta_0$, and rejects if the statistic exceeds a $\chi^2$ quantile. The idea is that under the null hypothesis, the statistic with $\theta=\theta_0$ should be $\chi^2(2n)$, and if the value I compute is too large to have come from that distribution, I can reject the null hypothesis. $\endgroup$
    – Frank Seidl
    Commented Nov 11, 2022 at 19:16
  • $\begingroup$ Good, I have a bad day ... pivots are always useful, maybe you can point to an application of this where you have a better solution? $\endgroup$
    – kjetil b halvorsen ♦
    Commented Nov 11, 2022 at 19:31

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