Suppose $f_1,f_2,\dots$ are pdfs of absolutely continuous random variables with the same support (say an interval). Assume that $\{f_i\}$ are strictly positive in their support. Furthermore, $\frac{f_i(x)}{f_j(x)}$ is increasing in $x$ for any $i<j$. This is sometimes called the likelihood ratio order.

Let $t_i= \arg\max_p F^{-1}_{i+1}(p) - F^{-1}_i(p)$. Here $F^{-1}_i$ is the inverse function of the CDF associated with random variable i.

Is $\{t_i\}$ an increasing sequence? Is it a decreasing sequence? If not, can you provide counterexamples?

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