Delta Method /Taylor's Theorem/Fisher Transformation

最近學習時遇到的三個工具

基本來自維基百科。

1. Delta Method

來源：https://en.wikipedia.org/wiki/Delta_method
下面只列出了Univariate的部分，以及貼出證明部分；至於Multivariate的部分因爲暫未涉及到，所以這裏沒有貼出，其實還是相似的。

In statistics, the delta method is a result concerning the approximate probability distribution for a function of an asymptotically normal statistical estimator from knowledge of the limiting variance of that estimator. It was first described in 1938 by Robert Dorfman^[1]

Univariate delta method

While the delta method generalizes easily to a multivariate setting, careful motivation of the technique is more easily demonstrated in univariate terms. Roughly, if there is a sequence of random variables X_n satisfying

where θ and σ² are finite valued constants and D denotes convergence in distribution, then

for any function g satisfying the property that g′(θ) exists and is non-zero valued.

證明：注意到主要運用了泰勒展開。

2. Taylor’s Theorem

鏈接：https://en.wikipedia.org/wiki/Taylor's_theorem
注意到這邊也只是貼出最基礎的理論part和應用時最常用的兩個式子方便我自己查閱。更多的可以轉去鏈接部分查看原文。

In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a k-th order Taylor polynomial. For analytic functions the Taylor polynomials at a given point are finite order truncations of its Taylor series, which completely determines the function in some neighborhood of the point. The exact content of "Taylor's theorem" is not universally agreed upon. Indeed, there are several versions of it applicable in different situations, and some of them contain explicit estimates on the approximation error of the function by its Taylor polynomial.

Taylor's theorem is taught in introductory level calculus courses and it is one of the central elementary tools in mathematical analysis. Within pure mathematics it is the starting point of more advanced asymptotic analysis, and it is commonly used in more applied fields of numerics as well as in mathematical physics. Taylor's theorem also generalizes to multivariate and vector valued functions on any dimensions n and m. This generalization of Taylor's theorem is the basis for the definition of so-called jets which appear in differential geometry and partial differential equations.

Theorem Part:

Application Part:

3. Fisher Transformation

…of the sample correlation coefficient.
鏈接：https://en.wikipedia.org/wiki/Fisher_transformation

簡單的版本：
$\rho$是相關係數，$r$是相關係數的估計(sample correlation coefficient)。有如下性質：
$\sqrt{n} [\frac{1}{2}log(\frac{1+r}{1-r})-\frac{1}{2}log(\frac{1+\rho}{1-\rho}){\xrightarrow {D}} N(0,1)]$

複雜的版本：