New results on stochastic comparisons of finite mixtures for some families of distributions

Stochastic properties of generalized finite α-mixtures

“…Theorem 6.1 extends the result of Theorem 3.2 in Nadeb and Torabi [23] that compares two finite ordinary mixtures with respect to parameter .…”

“…and [19] and Nadeb and Torabi [23], where the authors make the similar statement for ordinary mixtures when 𝑛 > 2). In the following counterexample, we show that conditions ( 𝒑, 𝝀) ∈ U 2 and ( 𝒑, 𝜸) ∈ U 2 in Theorem 6.3 can not be dropped.…”

“…Also, we claim that B ≥ 0, because from condition (i), r(t|λ 1 ) ≤ (≥)r(t|λ 2 ), and condition (iii), α 1 ≤ α 2 , (α 1 r(t|λ 1 ) − α 2 r(t|λ 2 )) ≤ 0. On the other hand, from condition (ii), Nadeb and Torabi (2020), where the authors make the similar statement for ordinary mixtures when n > 2). Theorem 6.17 is illustrated by the following numerical example: Example 6.19.…”

“…Theorem 6.3 is proved for . We have not yet been able to prove whether the result holds for or not leaving it as an open problem for now (see also Section 4 of Hazra and Finkelstein [19] and Nadeb and Torabi [23], where the authors make the similar statement for ordinary mixtures when ).…”

“…Hazra and Finkelstein [19], using the multivariate majorization, have provided a stochastic comparison for two finite mixtures when the baseline distribution belongs to some semi-parametric families of distribution such as proportional hazards, accelerated lifetimes (scale models) and proportional reversed hazards models. Stochastic comparisons for two finite mixtures using majorization in the sense of the usual stochastic order, the hazard rate order and the reversed hazard rate order have been discussed by Nadeb and Torabi [23]. Albabtain et al [1] have considered a parametric family of weighted distributions and their mixtures and have provided some stochastic comparisons.…”