Statistics/Statistical Inference (6) 썸네일형 리스트형 Concentration Inequalities 오늘의 글의 목차는 다음과 같습니다. 1. Basic concentration inequalities 2. Azuma-Hoeffding inequality 3. McDiarmid's inequality 1. Basic concentration inequalities Theorem 1.1 Markov Inequality $P(X\geq t)\leq\frac{E[X]}{t}=O(\frac{1}{t})$ $X$ should be non-negative proof $E[X]=\int_0^{\infty}xp(x)dx\geq\int^{\infty}_{t}xp(x)dx\geq t\int^{\infty}_tp(x)dx=tP(X\geq t)$ Remark 1.2 Generalization of Markov Inequal.. Sample Space Definition 1.1 (Random Experiment) (a) All outcomes of the experiment should be known in advance (b) any performance of the experiment results in an outcome that is not known in advance, and (c) the experiment can be repeated under identical conditions Definition 1.2 (Sample Space) sample space of a statistical experiment is a pair $(\Omega,S)$, where (a) $\Omega$ is the set of all possible outc.. Location and Scale Family Theorem 3.5.1 Let $f(x)$ be any pdf and $\mu>0$ and $\sigma>0$. Then $g(x|\mu,\sigma)=\frac{1}{\sigma}f(\frac{x-\mu}{\sigma})$는 pdf 이다. Definition 3.5.2 $f(x)$ 를 pdf라고 할 때, family of pdfs $f(x-\mu)$, $-\infty 지수족 Definition(exponential families) A family of pdfs or pmfs is called an exponential family if it can be expressed as $f(x|\theta)=h(x)c(\theta)exp(\Sigma^k_1w_i(\theta)t_i(x))$ h(x)와 t(x)는 observation x에 대한 real-valued functions들이고 $c(\theta)$아 $w(\theta)$는 parameter에 대한 real-valued functions 들이다. Example 3.4.1 : binomial exponential family 이항분포 (n,p)도 exponential family 라고 할 수 있는데, $f(x|p)=\bino.. Ancillary and Complete Statistics Definition 6.2.16 A statistics $S(X)$ whose distribution doesn't depend on the parameter $\theta$ is called on ancillary statistics. location family와 scale family에서 ancillary statistic을 생각해보자 Example 6.2.17 (Uniform ancillary statistic) 예제 6.2.15를 이어서 생각해보면, $R=X_{(n)}-X_{(1)}$ 은 ancillary statistic이다. $R$은 $\theta$ 에 dependent 하지 않다! $X_{(n)},X_{(1)}$의 joint distribution은 $g(x_{(1)},x_{(n)}|\th.. Sufficient Statistics : Factorization Theorem casella 6절 내용 중 일부를 정리한 내용입니다. Sufficiency Principle parameter $\theta$에 대한 충분통계량이란, Sample들이 주어졌을 때, $\theta$에 대한 모든 정보를 담고 있는 통계량을 의미한다. 따라서 $T(\mathbf{X})$가 $\theta$에 대한 충분통계량이면, $\theta$는 오직 $T(\mathbf{X})$에만 의존한다. If $T(X)$ is sufficient statistic for $\theta$ , then any inference about $theta$ should depend on sample $X$ only through the value $T(X)$ . That is if $x\;and\;y$ are two sample.. 이전 1 다음
