【 以下文字转载自 Military 讨论区 】 发信人: nashsimon (Nash), 信区: Military 标 题: Poisson variables comparisons in log-concavity, dispersive order 发信站: BBS 未名空间站 (Fri Oct 4 21:40:33 2019, 美东)
Let X be a Poisson variable with parameter λ with a probability mass function, f(k), where k = 0, 1, 2 … We know the index of log-concavity is the function rf(k) = f(k)^2/(f(k-1)f(k+1) = (k+1)/k>1. So, Poisson is log- concave.
Many books have: "The random variable Y is dispersive if, and only if, Y has a logconcave density."
Poisson variable is discrete. Do we have: A Poisson variable X is dispersive?
Or, more specifically, let X be a Poisson variable with parameter λ1 with a probability mass function. Let Y be a Poisson variable with parameter λ2 with a probability mass function. Let λ2 > λ1. Is Y more dispersive than X?
Dispersive order: Let X and Y be random variables with quantile functions F-1 and G-1 respectively. If F-1(b)- F-1(a) <= G-1(b)- G-1(a) whenever 0 Any reference?
【 以下文字转载自 Military 讨论区 】
发信人: nashsimon (Nash), 信区: Military
标 题: Poisson variables comparisons in log-concavity, dispersive order
发信站: BBS 未名空间站 (Fri Oct 4 21:40:33 2019, 美东)
Let X be a Poisson variable with parameter λ with a probability mass
function, f(k), where k = 0, 1, 2 … We know the index of log-concavity is
the function rf(k) = f(k)^2/(f(k-1)f(k+1) = (k+1)/k>1. So, Poisson is log-
concave.
Many books have: "The random variable Y is dispersive if, and only if, Y has a logconcave density."
Poisson variable is discrete. Do we have: A Poisson variable X is dispersive?
Or, more specifically, let X be a Poisson variable with parameter λ1 with a probability mass function. Let Y be a Poisson variable with parameter λ2
with a probability mass function. Let λ2 > λ1. Is Y more dispersive than X?
Dispersive order: Let X and Y be random variables with quantile functions F-1 and G-1 respectively. If F-1(b)- F-1(a) <= G-1(b)- G-1(a) whenever 0
Any reference?