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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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  • $\begingroup$ you may want to define $F_i^{-1}$ --- inverse function, reciprocal, cdf ? $\endgroup$
    – Carlo Beenakker
    Commented Apr 7, 2019 at 10:11
  • $\begingroup$ Thanks for pointing this out. It was forgotten. The question is now updated. $\endgroup$
    – Ozzy
    Commented Apr 7, 2019 at 18:29
  • $\begingroup$ updated; I mean inverse function. $\endgroup$
    – Ozzy
    Commented Apr 7, 2019 at 20:13
  • $\begingroup$ I assume that pdfs are always positive, and the random variables are continuous. $\endgroup$
    – Ozzy
    Commented Apr 7, 2019 at 23:18
  • $\begingroup$ Does $t_i= \arg\max_p F^{-1}_{i+1}(p) - F^{-1}_i(p)$ mean $t_i= \arg\max_p (F^{-1}_{i+1}(p) - F^{-1}_i(p))$? $\endgroup$
    – Iosif Pinelis
    Commented Apr 8, 2019 at 13:10

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