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Sufficient conditions for some transform orders based on the quantile density ratio

A. Arriaza, F. Belzunce, C. Martínez-Riquelme

Abstract

In this paper we focus on providing sufficient conditions for some transform orders for which the quantile densities ratio is non-monotone and, therefore, the convex transform order does not hold. These results are interesting for comparing random variables with a non-explicit expression of their quantile functions or they are computationally complex. In addition, the main results are applied to compare two Tukey generalized distributed random variables and to establish new relationships among non-monotone and positive aging notions.

Sufficient conditions for some transform orders based on the quantile density ratio

Abstract

In this paper we focus on providing sufficient conditions for some transform orders for which the quantile densities ratio is non-monotone and, therefore, the convex transform order does not hold. These results are interesting for comparing random variables with a non-explicit expression of their quantile functions or they are computationally complex. In addition, the main results are applied to compare two Tukey generalized distributed random variables and to establish new relationships among non-monotone and positive aging notions.
Paper Structure (6 sections, 12 theorems, 60 equations, 5 figures)

This paper contains 6 sections, 12 theorems, 60 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Sufficient conditions for the transform orders
  3. Applications
  4. A summarized result to compare random variables in the transform orders
  5. Some implications in reliability and aging notions
  6. Conclusions

Key Result

Theorem 2.1

Let $X$ and $Y$ be two non-negative random variables with differentiable quantile functions $F^{-1}$ and $G^{-1}$ and density functions $f$ and $g$, respectively. Let us assume that the ratio $f(F^{-1}(p))/g(G^{-1}(p))$ has n modes at $0<p_1 < p_2 < \ldots < p_n < 1$, such that the ratio is initiall for all $p\in(0,1)$. Then, the number of modes of $G^{-1}(p)/F^{-1}(p)$ is less or equal that $n+1$

Figures (5)

  • Figure 1: Plots of the empirical transform $G_m^{-1}(F_n)$ for guinea pigs (on the left) and SPF Fisher 344 male rats (on the right).
  • Figure 2: Domain for $\alpha_1$ and $\alpha_2$ such that the ratio of the quantile density functions is unimodal (grey area).
  • Figure 3: Plots of the ratio of the quantile density functions (left), the $\delta$ function (upper right) and the function that appears in \ref{['lim_cond']} and \ref{['lim_conddmrl']} (lower right), for $X \sim T(4,1,2.5)$ and $Y \sim T(1.5,1,1.5)$.
  • Figure 4: Plots of the ratio of the quantile functions (left), the ratio of the mean inactivity quantile functions (upper right) and the ratio of the mean residual quantile funcions (lower right), for $X \sim T(4,1,2.5)$ and $Y \sim T(1.5,1,1.5)$.
  • Figure 5: Plots of the hazard rate function (upper left plot), the hazard rate on average function (upper right plot), the function given in \ref{['ifraclas']} (lower left plot) and the hazard rate weighted on average function (lower right plot), for $X \sim G(0,2,2)$.

Theorems & Definitions (29)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Theorem 2.5
  • ...and 19 more