Mathematical methods of statistics:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Princeton
Princeton University Press
1971
|
| Ausgabe: | 12. printing |
| Schriftenreihe: | Princeton mathematical series
9 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVI, 575 Seiten Illustrationen |
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Datensatz im Suchindex
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| adam_text | Table op Contents.
First Part.
MATHEMATICAL INTRODUCTION.
Chapters 1—3. Sets op Points.
Page
Chapter 1. General properties of sets 3
1. Sets. — 2. Subsets, space. — 3. Operations on sets. — 4. Sequences of
sets. — 5. Monotone sequences. — 6. Additive classes of sets.
Chapter 2. Linear point sets 10
1. Intervals. — 2. Various properties of sets in H,. — 3. Borel sets.
Chapter 3. Point sets in n dimensions 15
1. Intervals. — 2. Various properties of sets in Rn. — 3. Borel sets. — 4.
Linear sets. — 5. Suhspace, product space.
References to chapters 1—3 18
Chapters 4—7. Theory of Measure and Integration
in JRj.
Chapter 4. The Lebesgue measure of a linear point set 19
1. Length of an interval. — 2. Generalization. — 3. The measure of a sum
of intervals. — 4. Outer and inner measure of a bounded set. — 5. Measurable
sets and Lebesgue measure. — 6. The class of measurable sets. — 7. Mea¬
surable sets and Borel sets.
Chapter 5. The Lebesgue integral for functions of one variable. 33
1. The integral of a bounded function over a set of finite measure. — 2. im¬
measurable functions. — 3. Properties of the integral. — 4. The integral of
an unbounded function over a set of finite measure. — 5. The integral over
a set of infinite measure. — 6. The Lebesgue integral as an additive set
function.
Chapter 6. Non negative additive set functions in J?j 48
1. Generalization of the Lebesgue measure and the Lebesgue integral. — 2.
Set functions and point functions. — 3. Construction of a set function. —
4. P iaeasure. — 6. Bounded set functions. — 6. Distributions. — 7. Sequen¬
ces ot distributions. — 8. A convergence theorem.
XI
P»ge
Chapter 7. The Lebesgue Stieltjes integral for functions of one
variable 62
1. The integral of a bounded fnnction over a set of finite P measure. — 2.
Unbounded functions and sets of infinite P measure. — 3. Lebesgne Stieltjes
integrals with a parameter. — 4. Lebesgne Stieltjes integrals with respect
to a distribution. — 5. The Riemann Stieltjes integral.
References to chapters 4—7 75
Chapters 8—9. Theory of Measure and Integration
INJRn.
Chapter 8. Lebesgue measure and other additive set functions
in JRn 76
1. Lebesgue measure in Rn. — 2. Non negative additive set functions in Rn.
—• 3. Bounded set functions. — 4. Distributions. — 5. Sequences of distri¬
butions. — 6. Distributions in a product space.
Chapter 9. The Lebesgue Stieltjes integral for functions of n
variables 85
1. The Lebesgne Stieltjes integral. — 2. Lebesgue Stieltjes integrals with
respect to a distribution. — 8. A theorem on repeated integrals. — 4. The
Riemann Stieltjes integral. — 6. The Schwarz inequality.
Chapters 10—12. Various Questions.
Chapter 10. Fourier integrals 89
1. The characteristic function of a distribution in Rt. — 2. Some auxiliary
functions. — 3. Uniqueness theorem for characteristic functions in Rt. —
i. Continuity theorem for characteristic functions in H,. — 6. Some particular
integrals. — 6. The characteristic function of a distribution in Rn. — 7.
Continuity theorem for characteristic functions in Rn.
Chapter 11. Matrices, determinants and quadratic forms 103
1. Matrices. — 2. Vectors. — 3. Matrix notation for i: ¦ : ¦ ¦ :
tions. — 4. Matrix notation for bilinear and quadratic .or^. b. Deter¬
minants. — 6. Bank. — 7. Adjugate and reciprocal matrices. •— 8. Linear
equations. — 9. Orthogonal matrices. Characteristic numbers. — 10. Non
negative quadratic forms. — 11. Decomposition of S aj|. ¦— 12. Some inte¬
gral formulae.
Chapter 12. Miscellaneous complements 122
1. The symbols 0, o and co. — 2. The Euler MacLanrin sum formula. —
3. The Gamma function. — 4. The Beta fnnction. — 5. Stirling s formula.
— 6. Orthogonal polynomials.
XII
Second Part.
RANDOM VARIABLES AND PROBABILITY DISTRIBU¬
TIONS.
Chapters 13—14. Foundations.
Page
Chapter 13. Statistics and probability 137
1. Random experiments. — 2. Examples. — 3. Statistical regularity. — 4.
Object of a mathematical tbeory. — 6. Mathematical probability.
Chapter 14. Fundamental definitions and axioms 151
1. Eandom variables. (Axioms 1—2.) — 2. Combined variables. (Axiom 3.)
— 3. Conditional distributions. — 4. Independent variables. •— 6. Functions
of random variables. — 6. Conclusion.
Chapters 15—20. Variables and Distributions in JJx.
Chapter 15. General properties 166
1. Distribution function and frequency function. — 2. Two simple types of
distributions. —• 3. Mean values. — 4. Moments. — 5. Measures of location.
— 6. Measures of dispersion. — 7. Tchebyeheff s theorem. — 8. Measures
of skewness and excess. — 9. Characteristic functions. — 10. Semi invari¬
ants. — 11. Independent variables. — 12. Addition of independent variables.
Chapter 16. Various discrete distributions 192
1. The function 6 (a;). — 2. The binomial distribution. — 3. Bernoulli s
theorem. — 4. De Moivre s theorem. — 5. The Poisson distribution. — 6.
The generalized binomial distribution of Poisson.
Chapter 17. The normal distribution 208
1. The normal functions. — 2. The normal distribution. — 3. Addition * f
independent normal variables. — 4. The central limit theorem. — 6. Comple¬
mentary remarks to the central limit theorem. — 6. Orthogonal expansion
derived from the normal distribution. — 7. Asymptotic expansion derived
from the normal distribution. — 8. The role of the normal distribution in
statistics.
Chapter 18. Various distributions related to the normal...... 233
1. The x* distribution. — 2. Student s distribution. — 3. Fisher s z distribu
tion. •—¦ 4. The Beta distribution.
Chapter 19. Further continuous distributions 244
1. The rectangular distribution. — 2. Canchy s and Laplace s distributions.
¦— 8. Truncated distributions. — 4. The Pearson system.
XIII
P»ge
Chapter 20. Some convergence theorems 250
1. Convergence of distributions and variables. — 2. Convergence of certain
distributions to the normal. — 3. Convergence in probability. — 4. Tche
bycheff s theorem. — 6. Khintchine s theorem. — 6. A convergence theorem.
Exercises to chapters 15—20 255
Chapters 21—24. Variables and Distributions in Rn.
Chapter 21. The two dimensional case 260
1. Two simple types of distributions. —• 2. Mean values, moments. — 3.
Characteristic functions. — 4. Conditional distributions. — 6. Regression, I.
— 6. Regression, II. — 7. The correlation coefficient. — 8. Linear trans¬
formation of variables. — 9. The correlation ratio and the mean square
contingency. — 10. The ellipse of concentration. — 11. Addition of inde¬
pendent variables. — 12. The normal distribution.
Chapter 22. General properties of distributions in R^ 291
1. Two simple types of distributions. Conditional distributions. — 2.
Change of variables in a continuous distribution. — 3. Mean values, mo¬
ments. — 4. Characteristic functions. — 5. Eank of a distribution. — 6.
Linear transformation of variables. — 7. The ellipsoid of concentration.
Chapter 23. Regression and correlation in n variables 301
1. Regression surfaces. — 2. Linear mean square regression. — 3. Residuals.
4. Partial correlation. — 6. The multiple correlation coefficient. — 6. Or¬
thogonal mean square regression.
Chapter 24. The normal distribution 310
1. The characteristic function. — 2. The non singular normal distribution.
— 3. The singular normal distribution. — 4. Linear transformation of nor¬
mally distributed variables. — 6. Distribution of a sum of squares. — 6.
Conditional distributions. — 7. Addition of independent variables. The cen¬
tral limit theorem.
Exercises to chapters 21—24 317
Third Part.
STATISTICAL INFERENCE.
Chapters 25—26. Generalities.
Chapter 25. Preliminary notions on sampling 323
1. Introductory remarks. — 2. Simple random sampling. — 3. The distribu¬
tion of the sample. — 4. The sample values as random variables. Sampling
XIV
Page
distributions. — 5. Statistical image of a distribution. — 6. Biased sampling.
Random sampling numbers. — 7. Sampling without replacement. The
representative method.
Chapter 26. Statistical inference 332
1. Introductory remaris. — 2. Agreement between theory and facts. Tests
of significance. — 3. Description. — 4. Analysis. — 5. Prediction.
Chapters 27—29. Sampling Distributions.
Chapter 27. Characteristics of sampling distributions 341
1. Notations. — 2. The sample mean x. — 3. The moments ar. — 4. The
variance w,. — 5. Higher central moments and semi invariants. — 6. Un¬
biased estimates. — 7. Functions of moments. — 8. Characteristics of multi¬
dimensional distributions. — 9. Corrections for grouping.
Chapter 28. Asymptotic properties of sampling distributions .. 363
1. Introductory remarks. — 2. The moments. — 3. The central moments.
— 4. Functions of moments. — 6. The qnantiles. — 6. The extreme values
and the range.
Chapter 29. Exact sampling distributions 378
1. The problem. — 2. Fisher s lemma. Degrees of freedom. — 3. The joint
distribution of x and 8* in samples from a normal distribution. — 4. Stu¬
dent s ratio. — 5. A lemma. — 6. Sampling from a two dimensional normal
distribution. — 7. The correlation coefficient. — 8. The regression coeffici¬
ents. — 9. Sampling from a ^ dimensional normal distribution. — 10. The
generalized variance. — 11. The generalized Student ratio. — 12. Regression
coefficients. — 13. Partial and multiple correlation coefficients.
Chapters 30—31. Tests of Significance, I.
Chapter 30. Tests of goodness of fit and allied tests 416
1. The x* t^ *n *ne case °f a completely specified hypothetical distribu¬
tion. — 2. Examples. — 3. The jj* test when certain parameters are estimated
from the sample. — 4. Examples. — 5. Contingency tables. — 6. jj* as a test
of homogeneity. — 7. Criterion of differential death rates. — 8. Further
tests of goodness of fit.
Chapter 31. Tests of significance for parameters 452
1. Tests based on standard errors. — 2. Tests based on exact distributions.
— 3. Examples.
Chapters 32—34. Theory of Estimation.
Chapter 32. Classification of estimates 473
1. The problem. ¦— 2. Two lemmas. — 3. Minimum variance of an estimate.
XT
P»ge
Efficient estimates. — 4. Sufficient estimates. — 6. Asymptotically efficient
estimates. — 6. The case of two unknown parameters. — 1. Several unknown
parameters. — 8. Generalisation.
Chapter 33. Methods of estimation 497
1. The method of moments. — 2. The method of maximum likelihood. —
3. Asymptotic properties of maximum likelihood estimates. — 4. The y*
minimum method.
Chapter 34. Confidence regions 507
1. Introductory remarks. — 2. A single unknown parameter. — 8. The
general case. — 4. Examples.
Chapters 35—37. Tests of Significance, II.
Chapter 35. General theory of testing statistical hypotheses... 525
1. The choice of a test of significance. — 2. Simple and composite hy¬
potheses. — 3. Tests of simple hypotheses. Most powerful tests. — 4. Un¬
biased tests. — 6. Tests of composite hypotheses.
Chapter 36. Analysis of variance 536
1. Variability of mean values. — 2. Simple grouping of variables. — 3.
Generalization. — 4. .Randomized blocks. — 6. Latin squares.
Chapter 37. Some regression problems 548
1. Problems involving non random variables. — 2. Simple regression. — 3.
Multiple regression. — i. Further regression problems.
Tables 1—2. The Normal Distribution 557
Table 3. The X^Distribution 559
Table 4. The ^ Distribution 560
List of References 561
Index 571
XVI
|
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| dewey-full | 519.5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.5 |
| dewey-search | 519.5 |
| dewey-sort | 3519.5 |
| dewey-tens | 510 - Mathematics |
| discipline | Psychologie Mathematik Wirtschaftswissenschaften |
| edition | 12. printing |
| format | Book |
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| language | English |
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| spelling | Cramér, Harald 1893-1985 (DE-588)119133415 aut Mathematical methods of statistics by Harald Cramér 12. printing Princeton Princeton University Press 1971 XVI, 575 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Princeton mathematical series 9 Mathematical statistics Mathematische Methode (DE-588)4155620-3 gnd rswk-swf Statistik (DE-588)4056995-0 gnd rswk-swf Methode (DE-588)4038971-6 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Statistik (DE-588)4056995-0 s Mathematische Methode (DE-588)4155620-3 s 1\p DE-604 Methode (DE-588)4038971-6 s 2\p DE-604 Princeton mathematical series 9 (DE-604)BV000019035 9 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001266694&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Cramér, Harald 1893-1985 Mathematical methods of statistics Princeton mathematical series Mathematical statistics Mathematische Methode (DE-588)4155620-3 gnd Statistik (DE-588)4056995-0 gnd Methode (DE-588)4038971-6 gnd |
| subject_GND | (DE-588)4155620-3 (DE-588)4056995-0 (DE-588)4038971-6 (DE-588)4151278-9 |
| title | Mathematical methods of statistics |
| title_auth | Mathematical methods of statistics |
| title_exact_search | Mathematical methods of statistics |
| title_full | Mathematical methods of statistics by Harald Cramér |
| title_fullStr | Mathematical methods of statistics by Harald Cramér |
| title_full_unstemmed | Mathematical methods of statistics by Harald Cramér |
| title_short | Mathematical methods of statistics |
| title_sort | mathematical methods of statistics |
| topic | Mathematical statistics Mathematische Methode (DE-588)4155620-3 gnd Statistik (DE-588)4056995-0 gnd Methode (DE-588)4038971-6 gnd |
| topic_facet | Mathematical statistics Mathematische Methode Statistik Methode Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001266694&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000019035 |
| work_keys_str_mv | AT cramerharald mathematicalmethodsofstatistics |