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删失数据的估计方程AI 解读

General estimating equations based on independent identically distributed random variables which may be right-censored are considered. The equations which appear to be natural analogues of the uncensored equations require full distributional assumptions, and a distribution-free modification based on the product-limit estimator is suggested. It follows from self-consistency that the resulting estimating functions are simply means obtained from the product-limit estimator. The…

通过大小变量与形状变量的独立性刻画分布族AI 解读

Let $X_1, \cdots, X_n$ be $n \geqslant 2$ positive random variables and $G(\mathbf{X})$ a positive variable satisfying $G(a\mathbf{X}) = aG(\mathbf{X})$ for all $a > 0$. Then $G$ is a size variable, and $\mathbf{X}/G$ is a shape vector. If $X_1, \cdots, X_n$ are independent, then the independence of shape and the size variable $G(\mathbf{X})$ characterizes (i) the lognormal distribution if $G(\mathbf{X}) = \Pi X^{1/n}_i$, (ii) the generalized gamma distribution if $G(\mathbf{X}) = (\sum X^b_i)^{1/b}$…

大上界下非负有界总和变量的F独立性AI 解读

Consider r ≥ 2 random variables X 1 (t), …, Xτ(t), each taking values in [0, t], but subject to the constraint Ω r j=1 Xj (t) ≤ t. The concept of F-independence for such variables is a modification of the concept of independence, which takes the constraint into account, and is applicable when (X 1 (t), … Xr (t) ∈ {(X 1 (t'), …, X τ (t')); t' ∈ I} for some index set I [2, 3]. We consider F-independence properties as t → ∞. We show that F independence properties become…

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具有贝塔条件分布的多元分布AI 解读

Distributions having beta conditional distributions arise in connection with the generation of distributions for random proportions which do not necessarily possess neutrality properties (see [3]). The family of bivariate distributions for which both conditional distributions are beta is evaluated, and some multivariate extensions are briefly discussed. The bivariate Dirichlet distribution is characterized by the assumptions that both conditional distributions and one…