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首页入门与进阶:多元统计分析经典教材详解
入门与进阶:多元统计分析经典教材详解
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更新于2024-07-17
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《多元统计分析》是一本由Wolfgang Karl Härdle和Zdeněk Hávka合著的英文教材,专为那些希望深入了解多元统计分析的读者设计,无论他们是统计学的初学者还是已有一定基础的专业人士。本书第二版提供了丰富的实践内容,通过详尽的练习和解答,帮助读者掌握在实际应用中广泛使用的多元统计方法。 这本书涵盖了多元统计的核心概念,如多元正态分布、方差分析(ANOVA)、主成分分析(PCA)、因子分析、判别分析、回归分析的扩展形式(如多元线性回归、逻辑回归等),以及更高级的主题,如聚类分析、协方差矩阵的估计和模型选择。作者用清晰的语言阐述理论,同时辅以大量的实例和R语言中的Quantlets代码,以便读者能够在实践中理解和应用这些理论。 对于初学者而言,这本书不仅提供了基础知识的入门指导,还通过逐步深入的方式引导他们逐步掌握复杂的技术。而对于经验丰富的统计学家,它则可以作为参考书籍,补充他们在特定领域可能遗漏的细节或者寻求新的研究视角。 此外,该书的每个章节都配有详尽的练习题,旨在帮助读者巩固所学,提高技能。读者可以在实践中检验自己的理解,并通过解决这些问题来深化对多元统计理论的掌握。电子版本的链接可以从Springer官网获取,方便学习者随时随地查阅。 《多元统计分析》一书不仅是一本学术著作,也是一本实用的工具书,它将理论与实践相结合,适合不同层次的统计工作者使用,是提升统计分析能力的重要参考资料。版权信息表明,所有权利均受到保护,未经许可,复制或再利用部分或全部内容可能侵犯版权。
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Symbols and Notation
I can’t make bricks without clay.
Sherlock Holmes in “The Adventure of The Copper Beeches”
Basics
X; Y Random variables or vectors
X
1
; X
2
;:::;X
p
Random variables
X D .X
1
;:::;X
p
/
>
Random vector
X X has distribution
A; B Matrices
; Matrices
X ; Y Data matrices
˙ Covariance matrix
1
n
Vector of ones .1;:::;1
„
ƒ‚…
n-times
/
>
0
n
Vector of zeros .0;:::;0
„
ƒ‚…
n-times
/
>
I
p
Identity matrix
I.:/ Indicator function, for a set M is I D 1 on M,
I D 0 otherwise
i
p
1
) Implication
, Equivalence
Approximately equal
˝ Kronecker product
iff If and only if, equivalence
xv
xvi Symbols and Notation
Characteristics of Distribution
f .x/ pdf or density of X
f .x; y/ Joint density of X and Y
f
X
.x/; f
Y
.y/ Marginal densities of X and Y
f
X
1
.x
1
/;:::;f
X
p
.x
p
/ Marginal densities of X
1
;:::;X
p
O
f
h
.x/ Histogram or kernel estimator of f .x/
F.x/ cdf or distribution function of X
F.x; y/ Joint distribution function of X and Y
F
X
.x/; F
Y
.y/ Marginal distribution functions of X and Y
F
X
1
.x
1
/;:::;F
X
p
.x
p
/ Marginal distribution functions of X
1
;:::;X
p
f
YjXDx
.y/ Conditional density of Y given X D x
'
X
.t/ Characteristic function of X
m
k
kth moment of X
j
Cumulants or semi-invariants of X
Moments
E X; E Y Mean values of random variables or vectors X
and Y
E.YjX D x/ Conditional expectation of random variable or
vector Y given X D x
YjX
Conditional expectation of Y given X
Var.YjX D x/ Conditional variance of Y given X D x
2
YjX
Conditional variance of Y given X
XY
D Cov.X; Y/ Covariance between random variables X and Y
XX
D Var.X/ Variance of random variable X
XY
D
Cov.X; Y/
p
Var.X/ Var.Y/
Correlation between random variables X and Y
˙
XY
D Cov.X; Y/ Covariance between random vectors X and Y,
i.e.,
Cov.X; Y/ D E.X E X/.Y E Y/
>
˙
XX
D Var.X/ Covariance matrix of the random vector X
Symbols and Notation xvii
Samples
x; y Observations of X and Y
x
1
;:::;x
n
Dfx
i
g
n
iD1
Sample of n observations of X
X Dfx
ij
g
iD1;:::;nIjD1;:::;p
(n p) data matrix of observations of X
1
;:::;X
p
or of X D .X
1
;:::;X
p
/
T
x
.1/
;:::;x
.n/
The order statistic of x
1
;:::;x
n
H Centering matrix, H D I
n
n
1
1
n
1
>
n
Empirical Moments
x D
1
n
n
X
iD1
x
i
Average of X sampled by fx
i
g
iD1;:::;n
s
XY
D
1
n
n
X
iD1
.x
i
x/.y
i
y/ Empirical covariance of samples fx
i
g
iD1;:::;n
and
fy
i
g
iD1;:::;n
s
XX
D
1
n
n
X
iD1
.x
i
x/
2
Empirical variance of the sample fx
i
g
iD1;:::;n
r
XY
D
s
XY
p
s
XX
s
YY
Empirical correlation of X and Y
S Dfs
X
i
X
j
g Empirical covariance matrix of data matrix X
R Dfr
X
i
X
j
g Empirical correlation matrix of data matrix X
xviii Symbols and Notation
Distributions
'.x/ Density of the standard normal distribution
ˆ.x/ Distribution function of the standard normal
distribution
N.0; 1/ Standard normal or Gaussian distribution
N.;
2
/ Normal distribution with mean and variance
2
N
p
.; ˙/ p-Dimensional normal distribution with mean
and covariance matrix ˙
L
! Convergence in distribution
P
! Convergence in probability
CLT Central Limit Theorem
2
p
2
Distribution with p degrees of freedom
2
1˛Ip
1 ˛ Quantile of the
2
distribution with p
degrees of freedom
t
n
t-Distribution with n degrees of freedom
t
1˛=2In
1 ˛=2 Quantile of the t-distribution with n
degrees of freedom
F
n;m
F-Distribution with n and m degrees of freedom
F
1˛In;m
1 ˛ Quantile of the F-distribution with n and m
degrees of freedom
W
p
.˙; n/ Wishart distribution of a p-dimensional sample
covariance matrix with parameters ˙ and n
T
2
p;n
Hotelling T
2
-distribution with p and n degrees of
freedom
T
2
1˛Ip;n
1 ˛ Quantile of the Hotelling T
2
-distribution
with p and n degrees of freedom
Mathematical Abbreviations
tr.A/ Trace of matrix A
diag.A/ Diagonal of matrix A
diag.x/ Diagonal matrix with vector x on its diagonal, i.e.,
diagfdiag.x/gDx
rank.A/ Rank of matrix A
det.A/ or jAj Determinant of matrix A
abs.x/ Absolute value of x
hull.x
1
;:::;x
k
/ Convex hull of points fx
1
;:::;x
k
g
span.x
1
;:::;x
k
/ Linear space spanned by fx
1
;:::;x
k
g
Some Terminology
I consider that a man’s brain originally is like a little empty attic, and you have to stock it
with such furniture as you choose. A fool takes in all the lumber of every sort that he comes
across, so that the knowledge which might be useful to him gets crowded out, or at best is
jumbled up with a lot of other things so that he has a difficulty in laying his hands upon it.
Now the skilful workman is very careful indeed as to what he takes into his brain-attic. He
will have nothing but the tools which may help him in doing his work, but of these he has
a large assortment, and all in the most perfect order. It is a mistake to think that little room
has elastic walls and can distend to any extent. Depend upon it there comes a time when for
every addition of knowledge you forget something that you knew before. It is of the highest
importance, therefore, not to have useless facts elbowing out the useful ones.
Sherlock Holmes in “Study in Scarlet”
This section contains an overview of some terminology that is used throughout
the book. We thank David Harville, who kindly allowed us to use his TeX
files containing the definitions of terms concerning matrices and matrix algebra;
see Harville (2001). More detailed definitions and further explanations of the
statistical terms can be found, e.g., in Breiman (1973), Feller (1966), Härdle and
Simar (2015), Mardia, Kent, & Bibby (1979), or Serfling (2002).
Adjoint matrix The adjoint matrix of an n n matrix A Dfa
ij
g is the transpose
of the cofactor matrix of A (or equivalently is the nn matrix whose ijth element
is the cofactor of a
ji
).
Asymptotic normality A sequence X
1
; X
2
;::: of random variables is asymptot-
ically normal if there exist sequences of constants f
i
g
1
iD1
and f
i
g
1
iD1
such that
1
n
.X
n
n
/
L
! N.0; 1/. The asymptotic normality means that for sufficiently
large n, the random variable X
n
has approximately N.
n
;
2
n
/ distribution.
Bias Consider a random variable X that is parametrized by 2 . Suppose that
there is an estimator
O
of .Thebias is defined as the systematic difference
between
O
and ,
Ef
O
g. The estimator is unbiased if E
O
D .
xix
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