# A question about pdfs with likelihood ratio order

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. 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?

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