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It seems like there is a discrepancy between these two authors on what a LRT is.

Casella and Berger state on pg. 375. That the LRT statistic is:

$\lambda(x)=\frac{L(\hat{\theta}_0|x)}{L(\hat{\theta}|x)}$

While Wasserman in "All of Statistics" states the likelihood ratio statistic is:

$\lambda = 2\log\left(\frac{L(\hat{\theta})}{L(\hat{\theta}_0)}\right)$

Are these the same? If so is one more common than the other?

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  • $\begingroup$ Does L in Casella and Berger refer to the likelihood or the log-likelihood? Same question for Wasserman. $\endgroup$ May 3, 2016 at 18:37
  • $\begingroup$ Both refer to the likelihood. $\endgroup$
    – Alex
    May 5, 2016 at 15:28

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There does not seem to be any reasonable response to this question, therefore I offer a late explanation.

Casella's is more standard, however, Wasserman's gives you the asymptotic null distribution. That is, if we denote the Casella's statistic by $T_C$ and Wasserman's by $T_W$, then of course $T_W=-2\log(T_C)$ and under some regularity conditions you can show that under the null, $T_W\to \chi^2_k$ weakly where $k$ is the difference in dimensions of the null and full parameter space.

So Wasserman gives you the ready statistic whose asymptotic distribution is well known.

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