数理统计教学视频38集
" t/ h9 _/ n7 z. O8 o. k# u主讲人：邵军
2 f# x. e1 S4 B& h: X5 [美国威斯康星大学麦迪逊分校，统计学教授。
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) x/ q2 e5 x4 q+ O, L) u邵军，男，1957年出生，1982年毕业于华东师范大学数学系后留校任教.1983年进入University of Wisconsin-Madison攻读博士学位。1987年，邵军教授从WU-Madison毕业，先后执教于Purdue University 以及 University of Ottawa。并于1994年，回到UW-Madison，现为UW-Madison统计系教授，兼系主任。除此以外，邵军教授还曾是美国国家统计局以及著名生物制药咨询公司Covance的Seniro Research Fellow (1996-1997, 1998)。
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在学术界，邵军教授曾是JASA的Associate Editor(1993-1996,1999-2005)，是J. of Multivariate Analysis的Co-editor(2002-2005)。现在，邵军教授还是Statistica Sinica 的Assocaite Editor 以及Sankhya的Co-editor。除此以外，邵军教授还曾是. y$ W' x. S; n6 U- U% _
国际泛华统计学会（International Chinese Statistical Association）的President-elect以及Board of Directors。自1996年来已培养了多名统计学博士生。
; i( Z. R, Z+ K8 W! v( @# M, ~6 r 在匿名评审的英文学术刊物上，邵军教授有着逾百篇的论文发表，并著有多部学术专著。具体地说，邵军教授的, \" H) m+ d- t
主要研究兴趣如下：$ H- }: B. S4 E* D2 a
(1) The jackknife, bootstrap and other resampling methods5 N5 ~4 n! @1 I5 U& X! g7 _: v
(2) Asymptotic theory in linear and nonlinear models0 G" n8 l3 P5 g( [
(3) Model selection
I4 e' r4 M* \2 u; c(4) Sample surveys (variance estimation, imputation for nonrespondents); c; R* N8 `# q- O1 w+ r
(5) Longitudinal data analysis with missing data/covariates! P e- q" B, h& M- j' C
(6) Medical statistics (bioequivalence, shelf-life estimation, clinical trials) h/ W3 t _# j4 E4 z
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Mathematical Statistics 2nd ed - J. Shao (Springer Texts in Statistics, 2003) .pdf
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' F' o% N( B W% _7 m4 s& c9 `目录Preface to the First Edition) k" H; w: R2 T6 p2 y* q$ V
Preface to the Second Edition+ ~8 k% n! F2 K: N
Chapter 1.Probability Theory }9 n) z& L6 m7 ]" Z
1.1 Probability Spaces and Random Elements
2 `, a" F: m* V" _' W w( d; R1.1.1σ－fields and measures* t0 @2 A4 O% z& ]" e
1.1.2 Measurable functions and distributions# E, s$ v+ @/ j6 C1 y8 Y
1.2 Integration and Differentiation
" g) g* [$ F# }4 v* d* n1.2.1 Integration
2 A* G8 [1 `3 G1.2.2 Radon.Nikodym derivative
6 i! m1 U2 W7 I1 I! o1.3 Distributions and Their Characteristics
+ b J: ^& A. S0 n( D! F1.3.1 Distributions and probability densities
( J* [/ o0 v4 G2 c/ [2 N Q; q1.3.2 Moments and moment inequalities5 V- ]( s6 q. @. k' t
1.3.3 Moment generating and characteristic functions
4 e7 y6 q5 J! n0 H1 h( b1.4 Conditional Expectations4 {. J) z% r s9 K& {2 {- F; T' p \8 e
1.4.1 Conditional expectations
) l) |2 g5 v2 [$ @1.4.2 Independence
5 M3 g R3 \6 o9 ~; R1.4.3 Conditional distributions
) b; h5 x0 ^' g4 R1 h1 S7 X1.4.4 Markov chains and martingales
% v" R6 g4 o4 Z/ g# a1.5 Asymptotic Theory) v% D2 K' }& }" K. h, @
1.5.1 Convergence modes and stochastic orders
$ ]$ f% N" n; v, ], e1.5.2 Weak convergence
$ q( J2 o8 P" p5 z; W: V9 X5 j1.5.3 Convergence of transformations
$ ~* r. A5 f: H; t% E7 J, j( I1.5.4 The law of large numbers
/ V% J7 y- J# o2 T: A1.5.5 The central limit theorem. C, R7 x* I | k4 j
1.5.6 Edgeworth and Cornish-Fisher expansions
. r7 P& {% b! r* i* H1.6 Exercises) z0 i2 s+ c( _ C6 x4 {* H
! [/ k* |. X7 e( XChapter 2. Fundamentals of Statistics
2 Q0 `2 C$ [" |6 Y2.1 Populations，Samples，and Models
5 T* t# |- {0 d- y2.1.1 Populations and samples4 q1 x0 K( A! B8 O! G
2.1.2 Parametric and nonparametric models1 Z f9 ~& W0 U) e
2.1.3 Exponential and location.scale families( X5 k: I* j: [( ?5 z0 g
2.2 Statistics.Sufficiency,and Completeness0 P; q8 Y+ ~# g% S" i4 n
2.2.1 Statistics and their distributions: p1 V8 _7 l/ L! F$ A
2.2.2 Sufficiency and minimal sufficiency
8 X+ q6 l, @3 }, E+ S2.2.3 Complete statistics
% J8 J5 ?/ r/ k) {6 `; C2.3 Statistical Decision Theory
4 R* Y4 M1 n: A4 }2.3.1 Decision rules，lOSS functions，and risks; c3 \1 P7 m C/ V" \7 e
2.3.2 Admissibility and optimality
( y% w! ], J$ E2.4 Statistical Inference
/ F2 _+ F j7 o2.4.1 P0il）t estimators
, e/ C. F. T6 Q$ i* ?2.4.2 Hypothesis tests
/ h% q# E. s, W }; E5 m9 G+ [4 E2.4.3 Confidence sets2 k2 h$ z& T" C) m( |
2.5 Asymptotic Criteria and Inference! D) S, Z/ F9 C
2.5.1 Consistency
$ v) i- `3 j8 Y- p" b$ n2.5.2 Asymptotic bias，variance，and mse
0 @" T$ h* L c& u2.5.3 Asymptotic inference, ? p; k# D$ A3 q
2.6 Exercises
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Chapter 3.Unbiased Estimation
5 A4 v" B3 Y+ C8 t& s. ]7 \' o( t4 G3.1 The UMVUE
7 _- D/ R6 P9 K) K3.1.1 Sufficient and complete statistics( @$ W# S9 |0 A6 |
3.1.2 A necessary and.sufficient condition5 D( }/ Q9 R( Y# J" N3 ]
3.1.3 Information inequality
% Z- }2 ]% m; ?6 W9 V+ C3.1.4 Asymptotic properties of UMVUE's
3 w3 \6 p. o! n V3.2 U-Statistics, \% Q6 h8 T0 M; ?
3.2.1 Some examples
. s1 Z7 J/ f+ j, l: ?/ y" X* _4 z3.2.2 Variances of U-statistics3 `& i* Z9 a$ Y7 w
3.2.3 The projection method
) }. r* l+ k. _, }' A) ?3.3 The LSE in Linear Models1 E5 k% r M# l6 S* P# ?
3.3.1 The LSE and estimability5 I" N# y. J4 h5 F
3.3.2 The UMVUE and BLUE
2 Q8 n4 ~1 |; D/ y0 ~$ c3.3.3 R0bustness of LSE's! \ |' Q. C! u1 v, \ O
3.3.4 Asymptotic properties of LSE's
4 Z* ?* ?& I/ ?' a" c% f3.4 Unbiased Estimators in Survey Problems
6 C5 R6 ]: P# T9 d3.4.1 UMVUE's of population totals
) f, U6 l7 b% a8 g' w4 Z7 O. L3.4.2 Horvitz-Thompson estimators
& _ E0 _# e. x* B3.5 Asymptotically Unbiased Estimators% [9 E2 ?( i# H* X
3.5.1 Functions of unbiased estimators D$ t8 \6 g G6 ]/ i% R
3.5.2 The method ofmoments
8 c" ~, m2 Q' a; V, y" v D; {3.5.3 V-statistics, d; r! y* m& H% D. J& h& Q
3.5.4 The weighted LSE# G' m3 a" D8 i% h; c$ |% r# Q
3.6 Exercises
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Chapter 4.Estimation in Parametric Models% }. M, G1 p" h, O8 g) s) g: |) i
4.1 Bayes Decisions and Estimators
1 d% D9 L1 R/ Z E+ N4.1.1 Bayes actions5 I! L/ n, S3 A% z' y; d. L
4.1.2 Empirical and hierarchical Bayes methods0 s! w; c2 h6 _) B k5 e
4.1.3 Bayes rules and estimators: k' A1 @& d$ a! f& M
4.1.4 Markov chain Mollte Carlo3 ?+ Y, [% V6 X5 j; i) r
4.2 Invariance......
4 V. ^0 P- C( _8 C, O4 `4.2.1 One-parameter location families) p4 Z: U. I! R. C0 z. E1 {3 r
4.2.2 One-parameter seale families
& Q4 W8 i0 `8 z; w& G( e4.2.3 General location-scale families
% d& ^% p( I, y. P6 Y3 t4.3 Minimaxity and Admissibility
7 Q* M% G. W3 N: R6 {4.3.1 Estimators with constant risks
( M3 `6 a, T+ B0 ]6 W9 |4.3.2 Results in one-parameter exponential families5 T2 M5 d; P! n5 v; f' L% O2 p
4.3.3 Simultaneous estimation and shrinkage estimators
1 H% w9 r+ M3 X/ f4.4 The Method of Maximum Likelihood
* c+ h) a9 v1 S4 a w9 h, K, H4.4.1 The likelihood function and MLE's
" X* y9 j9 I% Z4 h+ s& ~8 [4.4.2 MLE's in generalized linear models" z) x \% g) {; z/ ? @9 F
4.4.3 Quasi-likelihoods and conditional likelihoods
' e h! J) y( |. M# w4.5 Asymptotically Efficient Estimation: M0 T7 V" G% ^+ }7 y
4.5.1 Asymptotic optimality' I8 w- E5 z- G5 f' M: _# c
4.5.2 Asymptotic efficiency of MLE's and RLE's
: |2 z4 |# n5 ]; q) J' E" x4.5.3 Other asymptotically efficient estimators q3 ]+ u5 Q$ V) c- Z
4.6 Exercises
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Chapter 5.Estimation in Nonparametric Models
; V' U0 p& h" I$ L5.1 Distribution Estimators" F, P) ~% A0 {7 c; X
5.1.1 Empirical C.d.f.'s in i.i.d.cases5 \( K% r- D: X3 q4 ^9 _
5.1.2 Empirical likelihoods
: q, n, f' Z" q, F! L1 G5 w. l5.1.3 Density estimation
/ W9 C7 F6 V) `+ E: }0 e5.1.4 Semi-parametric methods
7 h. |( M: U2 c6 |$ `9 `5.2 Statistical Functionals
# B* ]5 A, Q& _$ C" i5.2.1 Differentiability and asymptotic normality
4 g0 n- `% C, b3 G6 J. ] Q5.2.2 L-.M-.and R-estimators and rank statistics
" j+ w- x K& u7 L7 |8 X. O" O7 q5.3 Linear Functions of Order Statistics* a1 U, T3 f7 Z8 V% P
5.3.1 Sample quantiles) b6 L9 Q$ ]4 \& y& C
5.3.2 R0bustness and efficiency+ g, M( N) t3 _6 P+ e
5.3.3 L-estimators in linear models
+ q! p" W V6 w$ V5.4 Generalized Estimating Equations
. S1 e4 r! v& m# H* H7 @5.4.1 The GEE method and its relationship with others: p8 ?& t3 @. i ]- } |# ~
5.4.2 Consistency of GEE estimators
, U& I! K2 L5 B. }9 h5.4.3 Asymptotic normality of GEE estimators9 Z( E0 a f: F8 q
5.5 Variance Estimation2 @, F \/ x0 r7 R- T
5.5.1 The substitution.method
2 r. @+ `8 e9 E0 f5.5.2 The jackknife
, n6 }+ R' ?& l3 a7 P7 S A- O2 m3 [% \5.5.3 The bootstrap$ d3 y' x5 ~& v- p l
5.6 Exercises. z6 Y9 ?( I! D4 n
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Chapter 6.Hypothesis Tests
" o" |7 D& o# l* `& l6 r+ B6.1 UMP Tests2 k7 d' s' z Q \4 ]" ^
6.1.1 The Neyman-Pearson lemma9 Z' ?) ]) \1 g
6.1.2 Monotone likelihood ratio
1 U3 W& W. s0 z) @. \5 s6.1.3 UMP tests for two-sided hypotheses
* O R# X {% [) O6.2 UMP Unbiased Tests/ S2 }6 n# k8 g7 h/ j
6.2.1 Unbiasedness，similarity，and Neyman structure) k }5 q2 [9 ]/ P( c5 f F' j
6.2.2 UMPU tests in exponential families
' j8 a8 X% S0 R* o. U6.2.3 UMPU tests in normal families
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Chapter 7 Confidence Sets: `1 k+ K* X, ]; J' L5 x% [
References n8 c3 F$ q5 \% P* \4 {
List of Notation8 \8 {# T& ?' X0 s9 W
List of Abbreviations7 n O7 S# B9 e! z3 J
Index of Definitions,Main Results,and Examples
$ W. Y. p( [9 J! n, ? ~6 uAuthor Index
* t' j/ h8 @& d- M) eSubject Index / s" \8 N" b# p5 F
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Mathematical Statistics -- Exercises and Solutions (Shao Jun).pdf
) C- O! {! s: g6 ^《数理统计:问题与解答》内容简介：this book consists of solutions to 400 exercises, over 95% of which arein my book Mathematical Statistics. Many of them are standard exercisesthat also appear in other textbooks listed in the references. It is onlya partial solution manual to Mathematical Statistics （which contains over900 exercises）. However, the types of exercise in Mathematical Statistics notselected in the current book are （1） exercises that are routine （each exerciseselected in this book has a certain degree of difficulty）, （2） exercises similarto one or several exercises selected in the current book, and （3） exercises foradvanced materials that are often not included in a mathematical statisticscourse for first-year Ph.D. students in statistics （e.g., Edgeworth expan-sions and second-order accuracy of confidence sets, empirical likelihoods,statistical functionals, generalized linear models, nonparametric tests, andtheory for the bootstrap and jackknife, etc.）. On the other hand, this isa stand-alone book, since exercises and solutions are comprehensibleindependently of their source for likely readers. To help readers notusing this book together with Mathematical Statistics, lists of notation,terminology, and some probability distributions are given in the front ofthe book.
" i3 `5 P/ b x3 j/ [4 t# tPreface( o! r) p$ J, r9 h$ z U% H
Notation
+ F( B2 P& y) _3 _Terminology$ B% ] w# i7 |3 o
Some Distributions! o. @* Z/ J5 s5 m& ]/ T0 M
Chapter 1. Probability Theory6 X1 _ V6 E7 }, z2 V" \
Chapter 2. Fundamentals of Statistics
. u8 R/ F' S. bChapter 3. Unbiased Estimation
0 I& Q4 M# k8 U% @2 S9 t) uChapter 4. Estimation in Parametric Models
+ Z9 {6 M+ D; L" JChapter 5. Estimation in Nonparametric Models
+ r- u8 r( |9 t9 j' D" i* Z3 i0 ~; F5 KChapter 6. Hypothesis Tests7 C9 y3 D; A. U2 Z4 o: N
Chapter 7. Confidence Sets
/ @& s/ H* p) F* OReferences$ R$ b& N- o0 e7 ?8 W E5 @: _
Index & |* [! E) V: i4 {2 o" O- }
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