Advanced calculus with applications in statistics:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Hoboken, NJ
Wiley-Interscience
2003
|
| Ausgabe: | 2. ed., rev. and expanded |
| Schriftenreihe: | Wiley series in probability and statistics
|
| Schlagworte: | |
| Online-Zugang: | Contributor biographical information Publisher description Table of contents Inhaltsverzeichnis Klappentext |
| Beschreibung: | XIX, 673 S. |
| ISBN: | 0471391042 |
Internformat
MARC
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| 100 | 1 | |a Khuri, André I. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Advanced calculus with applications in statistics |c André I. Khuri |
| 250 | |a 2. ed., rev. and expanded | ||
| 264 | 1 | |a Hoboken, NJ |b Wiley-Interscience |c 2003 | |
| 300 | |a XIX, 673 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Wiley series in probability and statistics | |
| 650 | 4 | |a Calcul infinitésimal | |
| 650 | 4 | |a Cálculo | |
| 650 | 7 | |a Cálculo diferencial e integral |2 larpcal | |
| 650 | 4 | |a Estadística matemática | |
| 650 | 4 | |a Statistique mathématique | |
| 650 | 4 | |a Calculus | |
| 650 | 4 | |a Mathematical statistics | |
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| 856 | 4 | |u http://www.loc.gov/catdir/toc/wiley031/2002068986.html |3 Table of contents | |
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Datensatz im Suchindex
| _version_ | 1804133155935879168 |
|---|---|
| adam_text | Contents
Preface
xv
Preface
to the First Edition
xvii
1.
An Introduction to Set Theory
1
1.1.
The Concept of a Set,
1
1.2.
Set Operations,
2
1.3.
Relations and Functions,
4
1.4.
Finite, Countable, and Uncountable Sets,
6
1.5.
Bounded Sets,
9
1.6.
Some Basic Topological Concepts,
10
1.7.
Examples in Probability and Statistics,
13
Further Reading and Annotated Bibliography,
15
Exercises,
17
2.
Basic Concepts in Linear Algebra
21
2.1.
Vector Spaces and Subspaces,
21
2.2.
Linear Transformations,
25
2.3.
Matrices and Determinants,
27
2.3.1.
Basic Operations on Matrices,
28
2.3.2.
The Rank of a Matrix,
33
2.3.3.
The Inverse of a Matrix,
34
2.3.4.
Generalized Inverse of a Matrix,
36
2.3.5.
Eigenvalues and Eigenvectors of a Matrix,
36
2.3.6.
Some Special Matrices,
38
2.3.7.
The Diagonalization of a Matrix,
38
2.3.8.
Quadratic Forms,
39
vii
viii CONTENTS
2.3.9.
The Simultaneous Diagonalization
of Matrices,
40
2.3.10.
Bounds on Eigenvalues,
41
2.4.
Applications of Matrices in Statistics,
43
2.4.1.
The Analysis of the Balanced Mixed Model,
43
2.4.2.
The Singular-Value Decomposition,
45
2.4.3.
Extrema
of Quadratic Forms,
48
2.4.4.
The Parameterization of Orthogonal
Matrices,
49
Further Reading and Annotated Bibliography,
50
Exercises,
53
3.
Limits and Continuity of Functions
57
3.1.
Limits of a Function,
57
3.2.
Some Properties Associated with Limits of Functions,
63
3.3.
The
о, О
Notation,
65
3.4.
Continuous Functions,
66
3.4.1.
Some Properties of Continuous Functions,
71
3.4.2.
Lipschitz Continuous Functions,
75
3.5.
Inverse Functions,
76
3.6.
Convex Functions,
79
3.7.
Continuous and Convex Functions in Statistics,
82
Further Reading and Annotated Bibliography,
87
Exercises,
88
4.
Differentiation
93
4.1.
The Derivative of a Function,
93
4.2.
The Mean Value Theorem,
99
4.3.
Taylor s Theorem,
108
4.4.
Maxima and Minima of a Function,
112
4.4.1.
A Sufficient Condition for a Local Optimum,
114
4.5.
Applications in Statistics,
115
4.5.1
Functions of Random Variables,
116
4.5.2.
Approximating Response Functions,
121
4.5.3.
The
Poisson
Process,
122
4.5.4.
Minimizing the Sum of Absolute Deviations,
124
Further Reading and Annotated Bibliography,
125
Exercises,
127
CONTENTS
¡χ
5.
Infinite Sequences and Series
132
5.1.
Infinite Sequences,
132
5.1.1.
The Cauchy Criterion,
137
5.2.
Infinite Series,
140
5.2.1.
Tests of Convergence for Series
of Positive Terms,
144
5.2.2.
Series of Positive and Negative Terms,
158
5.2.3.
Rearrangement of Series,
159
5.2.4.
Multiplication of Series,
162
5.3.
Sequences and Series of Functions,
165
5.3.1.
Properties of Uniformly Convergent Sequences
and Series,
169
5.4.
Power Series,
174
5.5.
Sequences and Series of Matrices,
178
5.6.
Applications in Statistics,
182
5.6.1.
Moments of a Discrete Distribution,
182
5.6.2.
Moment and Probability Generating
Functions,
186
5.6.3.
Some Limit Theorems,
191
5.6.3.1.
The Weak Law of Large Numbers
(Khinchine s Theorem),
192
5.6.3.2.
The Strong Law of Large Numbers
(Kolmogorov s Theorem),
192
5.6.3.3.
The Continuity Theorem for Probability
Generating Functions,
192
5.6.4.
Power Series and Logarithmic Series
Distributions,
193
5.6.5.
Poisson
Approximation to Power Series
Distributions,
194
5.6.6.
A Ridge Regression Application,
195
Further Reading and Annotated Bibliography,
197
Exercises,
199
6.
Integration
205
6.1.
Some Basic Definitions,
205
6.2.
The Existence of the Riemann Integral,
206
6.3.
Some Classes of Functions That Are Riemann
Integrable,
210
6.3.1.
Functions of Bounded Variation,
212
:
CONTENTS
6.4.
Properties of the Riemann Integral,
215
6.4.1.
Change of Variables in Riemann Integration,
219
6.5.
Improper Riemann Integrals,
220
6.5.1.
Improper Riemann Integrals of the Second
Kind,
225
6.6.
Convergence of a Sequence of Riemann Integrals,
227
6.7.
Some Fundamental Inequalities,
229
6.7.1.
The Cauchy-Schwarz Inequality,
229
6.7.2.
Holder s Inequality,
230
6.7.3.
Minkowski s Inequality,
232
6.7.4.
Jensen s Inequality,
233
6.8.
Riemann-Stieltjes Integral,
234
6.9.
Applications in Statistics,
239
6.9.1.
The Existence of the First Negative Moment of a
Continuous Distribution,
242
6.9.2.
Transformation of Continuous Random
Variables,
246
6.9.3.
The Riemann-Stieltjes Representation of the
Expected Value,
249
6.9.4.
Chebyshev s Inequality,
251
Further Reading and Annotated Bibliography,
252
Exercises,
253
7.
Multidimensional Calculus
261
7.1.
Some Basic Definitions,
261
7.2.
Limits of
a
Multivariable
Function,
262
7.3.
Continuity of
a
Multivariable
Function,
264
7.4.
Derivatives of
a
Multivariable
Function,
267
7.4.1.
The Total Derivative,
270
7.4.2.
Directional Derivatives,
273
7.4.3.
Differentiation of Composite Functions,
276
7.5.
Taylor s Theorem for
a
Multivariable
Function,
277
7.6.
Inverse and Implicit Function Theorems,
280
7.7.
Optima of
a
Multivariable
Function,
283
7.8.
The Method of
Lagrange
Multipliers,
288
7.9.
The Riemann Integral of
a
Multivariable
Function,
293
7.9.1.
The Riemann Integral on Cells,
294
7.9.2.
Iterated Riemann Integrals on Cells,
295
7.9.3.
Integration over General Sets,
297
7.9.4.
Change of Variables in
η
-Tuple Riemann
Integrals,
299
CONTENTS Xi
7.10. Differentiation
under the
Integral
Sign,
301
7.11. Applications in
Statistics,
304
7.11.1.
Transformations of Random Vectors,
305
7.11.2.
Maximum Likelihood Estimation,
308
7.11.3.
Comparison of Two Unbiased
Estimators,
310
7.11.4.
Best Linear Unbiased Estimation,
311
7.11.5.
Optimal Choice of Sample Sizes in Stratified
Sampling,
313
Further Reading and Annotated Bibliography,
315
Exercises,
316
8.
Optimization in Statistics
327
8.1.
The Gradient Methods,
329
8.1.1.
The Method of Steepest Descent,
329
8.1.2.
The Newton-Raphson Method,
331
8.1.3.
The Davidon-Fletcher-Powell Method,
331
8.2.
The Direct Search Methods,
332
8.2.1.
The Nelder-Mead Simplex Method,
332
8.2.2.
Price s Controlled Random Search
Procedure,
336
8.2.3.
The Generalized Simulated Annealing
Method,
338
8.3.
Optimization Techniques in Response Surface
Methodology,
339
8.3.1.
The Method of Steepest Ascent,
340
8.3.2.
The Method of Ridge Analysis,
343
8.3.3.
Modified Ridge Analysis,
350
8.4.
Response Surface Designs,
355
8.4.1.
First-Order Designs,
356
8.4.2.
Second-Order Designs,
358
8.4.3.
Variance and Bias Design Criteria,
359
8.5.
Alphabetic Optimality of Designs,
362
8.6.
Designs for Nonlinear Models,
367
8.7.
Multiresponse Optimization,
370
8.8.
Maximum Likelihood Estimation and the
EM Algorithm,
372
8.8.1.
The EM Algorithm,
375
8.9.
Minimum Norm Quadratic Unbiased Estimation of
Variance Components,
378
x¡¡
CONTENTS
8.10.
Scheffé s
Confidence Intervals,
382
8.10.1.
The Relation of
Scheffé s
Confidence Intervals
to the F-Test,
385
Further Reading and Annotated Bibliography,
391
Exercises,
395
9.
Approximation of Functions
403
9.1.
Weierstrass
Approximation,
403
9.2.
Approximation by Polynomial Interpolation,
410
9.2.1.
The Accuracy of
Lagrange
Interpolation,
413
9.2.2.
A Combination of Interpolation and
Approximation,
417
9.3
Approximation by Spline Functions,
418
9.3.1.
Properties of Spline Functions,
418
9.3.2.
Error Bounds for Spline Approximation,
421
9.4.
Applications in Statistics,
422
9.4.1.
Approximate Linearization of Nonlinear Models
by
Lagrange
Interpolation,
422
9.4.2.
Splines in Statistics,
428
9.4.2.1.
The Use of Cubic Splines in
Regression,
428
9.4.2.2.
Designs for Fitting Spline Models,
430
9.4.2.3.
Other Applications of Splines in
Statistics,
431
Further Reading and Annotated Bibliography,
432
Exercises,
434
10.
Orthogonal Polynomials
437
10.1.
Introduction,
437
10.2.
Legendre Polynomials,
440
10.2.1.
Expansion of a Function Using Legendre
Polynomials,
442
10.3.
Jacobi Polynomials,
443
10.4.
Chebyshev Polynomials,
444
10.4.1.
Chebyshev Polynomials of the First Kind,
444
10.4.2.
Chebyshev Polynomials of the Second Kind,
445
10.5.
Hermite Polynomials,
447
10.6.
Laguerre Polynomials,
451
10.7.
Least-Squares Approximation with Orthogonal
Polynomials,
453
CONTENTS Xiii
10.8. Orthogonal
Polynomials Defined on a Finite Set,
455
10.9.
Applications in Statistics,
456
10.9.1.
Applications of Hermite Polynomials,
456
10.9.1.1.
Approximation of Density Functions
and Quantiles of Distributions,
456
10.9.1.2.
Approximation of a Normal
Integral,
460
10.9.1.3.
Estimation of Unknown
Densities,
461
10.9.2.
Applications of Jacobi and Laguerre
Polynomials,
462
10.9.3.
Calculation of Hypergeometric Probabilities
Using Discrete Chebyshev Polynomials,
462
Further Reading and Annotated Bibliography,
464
Exercises,
466
11.
Fourier Series
471
11.1.
Introduction,
471
11.2.
Convergence of Fourier Series,
475
11.3.
Differentiation and Integration of Fourier Series,
483
11.4.
The Fourier Integral,
488
11.5.
Approximation of Functions by Trigonometric
Polynomials,
495
11.5.1.
Parseval s Theorem,
496
11.6.
The Fourier Transform,
497
11.6.1.
Fourier Transform of a Convolution,
499
11.7.
Applications in Statistics,
500
11.7.1
Applications in Time Series,
500
11.7.2.
Representation of Probability Distributions,
501
11.7.3.
Regression Modeling,
504
11.7.4.
The Characteristic Function,
505
11.7.4.1.
Some Properties of Characteristic
Functions,
510
Further Reading and Annotated Bibliography,
510
Exercises,
512
12.
Approximation of Integrals
517
12.1.
The Trapezoidal Method,
517
12.1.1.
Accuracy of the Approximation,
518
12.2.
Simpson s Method,
521
12.3.
Newton-Cotes Methods,
523
XIV
CONTENTS
12.4.
Gaussian Quadrature,
524
12.5.
Approximation over an Infinite Interval,
528
12.6.
The Method of Laplace,
531
12.7.
Multiple Integrals,
533
12.8.
The Monte Carlo Method,
535
12.8.1.
Variation Reduction,
537
12.8.2.
Integrals in Higher Dimensions,
540
12.9.
Applications in Statistics,
541
12.9.1.
The Gauss-Hermite Quadrature,
542
12.9.2.
Minimum Mean Squared Error
Quadrature,
543
12.9.3.
Moments of a Ratio of Quadratic Forms,
546
12.9
A. Laplace s Approximation in Bayesian
Statistics,
548
12.9.5.
Other Methods of Approximating Integrals
in Statistics,
549
Further Reading and Annotated Bibliography,
550
Exercises,
552
Appendix. Solutions to Selected Exercises
557
Chapter
1, 557
Chapter
2, 560
Chapter
3, 565
Chapter
4, 570
Chapter
5, 577
Chapter
6, 590
Chapter
7, 600
Chapter
8, 613
Chapter
9, 622
Chapter
10, 627
Chapter
11, 635
Chapter
12, 644
General Bibliography
652
Index
665
Knowledge
of advanced calculus has become imperative to the understanding of the
recent advances in statistical methodology. The First Edition of Advanced Calculus with
Applications in Statistics has served as a reliable resource for both practicing statisticians
and students alike. In light of the tremendous growth of the field of statistics since the
book s publication,
André Khuri
has reexamined his popular work and substantially
expanded it to provide the most up-to-date and comprehensive coverage of the subject.
Retaining the original s much-appreciated application-oriented approach, Advanced
Calculus with Applications in Statistics, Second Edition supplies a rigorous introduction to
the central themes of advanced calculus suitable for both statisticians and mathemati¬
cians alike. The Second Edition adds significant new material on:
•
Basic topological concepts
•
Orthogonal polynomials
•
Fourier series
•
Approximation of integrals
•
Solutions to selected exercises
The volume s user-friendly text is notable for its end-of-chapter applications, designed
to be flexible enough for both statisticians and mathematicians. Its well thought-out
solutions to exercises encourage independent study and reinforce mastery of the
content. Any statistician, mathematician, or student wishing to master advanced cal¬
culus and its applications in statistics will find this new edition a welcome resource.
ANDRE
L
KHÎJRI,.
ΡκΡ«
is a Professor in the Department of Statistics at the
Unwersity of Florida, Gainesville.
|
| any_adam_object | 1 |
| author | Khuri, André I. |
| author_facet | Khuri, André I. |
| author_role | aut |
| author_sort | Khuri, André I. |
| author_variant | a i k ai aik |
| building | Verbundindex |
| bvnumber | BV019704722 |
| callnumber-first | Q - Science |
| callnumber-label | QA303 |
| callnumber-raw | QA303.2.K48 2003 |
| callnumber-search | QA303.2.K48 2003 |
| callnumber-sort | QA 3303.2 K48 42003 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | QH 150 SK 920 |
| ctrlnum | (OCoLC)50123248 (DE-599)BVBBV019704722 |
| dewey-full | 515 51521 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515 515 21 |
| dewey-search | 515 515 21 |
| dewey-sort | 3515 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | 2. ed., rev. and expanded |
| format | Book |
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| id | DE-604.BV019704722 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T20:04:14Z |
| institution | BVB |
| isbn | 0471391042 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-013032220 |
| oclc_num | 50123248 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-29T DE-355 DE-BY-UBR DE-521 DE-578 |
| owner_facet | DE-19 DE-BY-UBM DE-29T DE-355 DE-BY-UBR DE-521 DE-578 |
| physical | XIX, 673 S. |
| publishDate | 2003 |
| publishDateSearch | 2003 |
| publishDateSort | 2003 |
| publisher | Wiley-Interscience |
| record_format | marc |
| series2 | Wiley series in probability and statistics |
| spelling | Khuri, André I. Verfasser aut Advanced calculus with applications in statistics André I. Khuri 2. ed., rev. and expanded Hoboken, NJ Wiley-Interscience 2003 XIX, 673 S. txt rdacontent n rdamedia nc rdacarrier Wiley series in probability and statistics Calcul infinitésimal Cálculo Cálculo diferencial e integral larpcal Estadística matemática Statistique mathématique Calculus Mathematical statistics Differentialrechnung (DE-588)4012252-9 gnd rswk-swf Statistik (DE-588)4056995-0 gnd rswk-swf Analysis (DE-588)4001865-9 gnd rswk-swf Differentialrechnung (DE-588)4012252-9 s Statistik (DE-588)4056995-0 s DE-604 Analysis (DE-588)4001865-9 s 1\p DE-604 http://www.loc.gov/catdir/bios/wiley042/2002068986.html Contributor biographical information http://www.loc.gov/catdir/description/wiley035/2002068986.html Publisher description http://www.loc.gov/catdir/toc/wiley031/2002068986.html Table of contents Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013032220&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013032220&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Khuri, André I. Advanced calculus with applications in statistics Calcul infinitésimal Cálculo Cálculo diferencial e integral larpcal Estadística matemática Statistique mathématique Calculus Mathematical statistics Differentialrechnung (DE-588)4012252-9 gnd Statistik (DE-588)4056995-0 gnd Analysis (DE-588)4001865-9 gnd |
| subject_GND | (DE-588)4012252-9 (DE-588)4056995-0 (DE-588)4001865-9 |
| title | Advanced calculus with applications in statistics |
| title_auth | Advanced calculus with applications in statistics |
| title_exact_search | Advanced calculus with applications in statistics |
| title_full | Advanced calculus with applications in statistics André I. Khuri |
| title_fullStr | Advanced calculus with applications in statistics André I. Khuri |
| title_full_unstemmed | Advanced calculus with applications in statistics André I. Khuri |
| title_short | Advanced calculus with applications in statistics |
| title_sort | advanced calculus with applications in statistics |
| topic | Calcul infinitésimal Cálculo Cálculo diferencial e integral larpcal Estadística matemática Statistique mathématique Calculus Mathematical statistics Differentialrechnung (DE-588)4012252-9 gnd Statistik (DE-588)4056995-0 gnd Analysis (DE-588)4001865-9 gnd |
| topic_facet | Calcul infinitésimal Cálculo Cálculo diferencial e integral Estadística matemática Statistique mathématique Calculus Mathematical statistics Differentialrechnung Statistik Analysis |
| url | http://www.loc.gov/catdir/bios/wiley042/2002068986.html http://www.loc.gov/catdir/description/wiley035/2002068986.html http://www.loc.gov/catdir/toc/wiley031/2002068986.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013032220&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013032220&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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