Can the log likelihood ratio for a simple vs simple hypothesis take a negative value?

Question

Can the log likelihood ratio for a simple vs simple hypothesis take a negative value? Since it approximately follows a chi-square distribution for large sample sizes and since a chi-square distribution takes only positive values do we reject or accept the null hypothesis if a negative value is obtained for the log likelihood ratio test under a particular simple vs simple hypothesis?

The log likelihood ratio test statistic in this case is $$-2Log\Lambda$$

Answer 1

Yes, the log likelihood ratio must take on both positive and negative values. If $\log \frac{f_1(x)}{f_0(x)}$ were positive for all possible values of the observed data $x$, then it would be true that $f_1(x) > f_0(x)$ for all $x$, and this is impossible. Remember that $f_1(x)$ and $f_0(x)$ are the probability density functions (or probability mass functions) of the observations under the two hypotheses, and if $f_1(x) > f_0(x)$ were to hold for all $x$, then it would be true that $$\int_{-\infty}^{\infty} f_1(x)\,\mathrm dx > \int_{-\infty}^{\infty} f_0(x)\,\mathrm dx$$ in contradiction of the fact that the integrals displayed above must both equal $1$. A similar argument applies (with sums instead of integrals) for probability mass functions.