Stat

Stat/Junk

Contraction mapping theorem

Definition contraction mapping Let (X,d)(X,d) be a complete metric space. Then a map T:XXT:XX is called a contraction mapping on XX if there exists c[0,1)c[0,1) such that d(T(x),T(y))c d(x,y)d(T(x),T(y))c d(x,y) for all x,yX.x,yX. Definition fixed point If T:XXT:XX, then a point xXxX such that T(x)=xT(x)=x is called a fixed point of T.T. Theorem Contraction mapping theorem (also know..

Stat/Junk

Orlicz norm

Definition Let ΨΨ be an increasing convex function from [0,)[0,) onto [0,)[0,) and XX be a random variable. The Orlicz norm XΨXΨ is defined by XΨ=inf{c>0:EΨ(|X|c)1},XΨ=inf{c>0:EΨ(|X|c)1}, where the infimum over the empty set is . Examples of Ψ Ψ(x)=exp(xp)1Ψp(x) for $p \geq..

Stat/Junk

Riesz representation theorem

Preliminaries and notation Linear and antilinear maps By definition, an antilinear map (also called a conjugate-linear map) f:HY is a map between vector spaces that is additive : f(x+y)=f(x)+f(y)   for all x,yH, and antilinear (also called conjugate-linear or conjugate-homogeneous) : $$ f(cx) = \bar{c}f(x) ~~~ \text{for all } x ∈ \mathcal{H} \..

Stat/Junk

Covering number

Definition Let (T,d) be a semi-metric space. For ε>0, an ε-net of T is a subset Tε of T such that for every tT there exists a tεTε with d(t,tε)ε. The ε-covering number N(T,d,ε) of T is the infimum of the cardinality of ε-nets of T, that is, $$ N(T, d, \..

Stat/Junk

Lipschitz continuity

Definition Given two metric spaces (X,dX) and (Y,dY), where dX denotes the metric on the set X and dY is the metric on set Y, a function f:XY is called Lipschitz continuous if there exists a real constant K0 such that, for all x1 and x2 in X, dY(f(x1),f(x2))KdX(x1,x2). In particular, a real-valued function $f : \mathbb{R} → \mathbb{..

Stat/Empirical Process

1. Symmetrization

본 내용은 아래의 노트 (Lecture notes in empirical process theory, Kengo Kato, 2019)를 참조하여 작성하였습니다. https://drive.google.com/file/d/0B7C_CufYq6j6QU5rblF2Yl85d3c/view?resourcekey=0-ItZa4Z1yrAGhUa7scVo_aw empirical_process_v3.pdf drive.google.com The main object is to study probability estimates of the random quantity PnPF:=supfF|PnfPf|, and lim..

Stat/Empirical Process

0. Notation and Setting

본 내용은 아래의 노트 (Lecture notes in empirical process theory, Kengo Kato, 2019)를 참조하여 작성하였습니다. https://drive.google.com/file/d/0B7C_CufYq6j6QU5rblF2Yl85d3c/view?resourcekey=0-ItZa4Z1yrAGhUa7scVo_aw empirical_process_v3.pdf drive.google.com A sample space Ω is the set of all possible outcomes of a random element. A collection of subsets of a sample space Ω denoted by A. A col..

Stat/Spatial Stat

3. Some theory for point-referenced data models

본 내용은 아래의 책을 참조하여 작성하였습니다 https://www.researchgate.net/publication/224839687_Hierarchical_Modeling_and_Analysis_of_Spatial_Data 3.1 Formal modeling theory for spatial processes ... 3.1.1 Some basic stochastic process theory for spatial processes ...

Stat/Spatial Stat

2. Basics of point-referenced data models

본 내용은 아래의 책을 참조하여 작성하였습니다 https://www.researchgate.net/publication/224839687_Hierarchical_Modeling_and_Analysis_of_Spatial_Data Chap 1의 내용에서 알 수 있듯, 해당 주제의 fundamental concept은 stochastic process {Y(s):sD}이며 여기서 Dr-dimensional Euclidean space이다. r=1이면 단순한 time series이다. 이 spatial model에서는 주로 r>2이며 이러한 경우를 spatial process라 부른다. 2.1 Elements of point-referenced modeling 2...

Stat/Spatial Stat

1. Overview of spatial data problems

본 내용은 아래의 책을 참조하여 작성하였습니다 https://www.researchgate.net/publication/224839687_Hierarchical_Modeling_and_Analysis_of_Spatial_Data 1.1 Introduction to spatial data and models Spatial data는 크게 3가지 type으로 구분된다. point-referenced data, where Y(s) is a random vector at a location sRr, where s varies continuously over D, a fixed subset of Rr that contains an r-dimen..