Stochastic Comparisons of Random Extremes from non-identical Random Variables

Abstract

We propose some new results on the comparison of the minimum or maximum order statistic from a random number of non-identical random variables. Under the non-identical set-up, with certain conditions, we prove that random minimum (maximum) of one system dominates the other in hazard rate (reversed hazard rate) order. Further, we prove variation diminishing property (Karlin [8]) for all possible restrictions to derive the new results.

Figures (6)

• Figure 1: Graph of $w_1(x)$.
• Figure 2: Graph of $w_2(x)$.
• Figure 3: Graph of $\frac{\bar{F}_{1:N}(-\ln y)}{\bar{G}_{1:N}(-\ln y)}$
• Figure 4: Graph of $\frac{g_{1:n}(y)}{f_{1:n}(y)}$
• Figure 5: Graph of $\frac{f_{1:n_1}(y)}{f_{1:n_2}(y).}$

Theorems & Definitions (18)

• Definition 2.1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Proposition 3.4
• Theorem 3.1
• Remark 3.1