I have a problem with a likelihood interval; Here on the page $74$ in the example $4.1$ they write:
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The likelihood interval at 15% cutoff (computed numerically) is $(0.50,0.96)$; since the likelihood is reasonably regular this is an approximate $95\%$ CI for $\theta$.
I do not follow what do they mean by this sentence: how did they come from 15% to 95% ?
On
This is just an ad hoc approximation in relating likelihood intervals to traditional confidence intervals. It is has been observed in many "typical" situations that the 15% likelihood interval and the 95% confidence interval are roughly the same. Of course in relating them it brings up the questions "why would we use likelihood intervals to approximate confidence intervals!? Why not just use confidence intervals from the get go if that is what we want!?". The truth is that likelihood intervals are philosophically a different way of quantifying the uncertainty in an estimate. Historically they were a more popular choice, but over time the use of confidence intervals has become a much more common.
This is derived in Section $2.6$ of the book, “Likelihood-based intervals” (p. $35$ of the $2013$ paperback edition) under a normality assumption. The derivation uses a $\chi_1^2$ distribution, but you can actually get the same result using a Gaussian: $95\%$ of the probability lies within $\sqrt2\operatorname{erf}^{-1}(0.95)\approx1.96$ standard deviations of the mean, and at this point the probability density function has a value $\exp\left(-1.96^2/2\right)\approx0.15$ of its maximum value.