统计常用定义

1. MSE
   The mean squared error between an estimator A and the corresponding population parameter θ is
   The mean squared error can be decomposed into a bias term and a variance term. The bias term is the difference between the parameter of interest and the expected value of the estimator. The variance term corresponds to the variation of the estimator around its expected value.
2. Bias-variance tradeoff
   For any estimator A of a population parameter μ,
   其中左边的括号代表variance，右边的括号代表bias。

IfE (Xi) =μfor all the random variables in a sequence thenE( ̄Xn)=μ.

(Variance of sample mean).IfX1,...,Xnis a sequence of uncorrelated ran-dom variables with the same meanμand varianceσ2, thenVar( ̄Xn)=σ2n.

(Sample variance).LetX1,X2,...be a sequence of random variables belong-ing to the same probability space. The sample variance is defined asS2:=1n−1n∑i=1(Xi− ̄X)2

IfX1,X2,...are uncorrelated and share the same meanμand varianceσ2thenE(S2)=σ2

(Central Limit Theorem).IfX1,X2,...is an iid sequence with boundedvariance, such thatE (Xi) =μandVar (Xi) =σ2, the sample mean ̄Xn:=1n∑ni=1Xiconverges in distribution to a Gaussian random variable with meanμand varianceσ2/n.