“convergence in distribution”的概念、定义、翻译、参考文献-科学参考

convergence in distribution

Mathematics

A sequence of random variables X_n converges in distribution to a random variable X if F_n(x) → F(x) for all real x, where F_n and F denote the respective cumulative distribution functions. This is also known as weak convergence and is written $X_{n} \overset{d}{\to} X .$

Economics

A sequence of random variables x₁,…, x_n,… with corresponding distribution functions F₁(x),…F_n(x),… converges in distribution (or weakly) to the random variable x with distribution function F(x) if the sequence of the corresponding distribution functions converges to F at all continuity points of F

$x_{n} \overset{d}{\to} x \Leftrightarrow lim_{n \to \infty} | F_{n} (x) - F (x) | = 0 \Leftrightarrow | F_{n} (x) - F (x) | < ε$

for every ε‎ > 0 starting from some n. The distribution given by F(x) is called the limiting or the asymptotic distribution of x_n. This concept allows the approximation of the unknown distribution F_n(x) of an estimator or a test statistic with a known asymptotic distribution F(x). If x is a constant the limiting distribution is degenerate, i.e. collapses to a single point. In this case (but not in general) convergence in distribution also implies convergence in probability.

单词	convergence in distribution
释义	convergence in distribution Mathematics A sequence of random variables X_n converges in distribution to a random variable X if F_n(x) → F(x) for all real x, where F_n and F denote the respective cumulative distribution functions. This is also known as weak convergence and is written $X_{n} \overset{d}{\to} X .$ Economics A sequence of random variables x₁,…, x_n,… with corresponding distribution functions F₁(x),…F_n(x),… converges in distribution (or weakly) to the random variable x with distribution function F(x) if the sequence of the corresponding distribution functions converges to F at all continuity points of F $x_{n} \overset{d}{\to} x \Leftrightarrow lim_{n \to \infty} \| F_{n} (x) - F (x) \| = 0 \Leftrightarrow \| F_{n} (x) - F (x) \| < ε$ for every ε‎ > 0 starting from some n. The distribution given by F(x) is called the limiting or the asymptotic distribution of x_n. This concept allows the approximation of the unknown distribution F_n(x) of an estimator or a test statistic with a known asymptotic distribution F(x). If x is a constant the limiting distribution is degenerate, i.e. collapses to a single point. In this case (but not in general) convergence in distribution also implies convergence in probability.
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