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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, Andr 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 mathematicians 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 calculus and its applications in statistics will find this new edition a welcome resource.

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Editorial Reviews

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"This is an exceptional book, which I would recommend for anyone beginning a career in statistical research." (Journal of the American Statistical Association, September 2004)
An application-oriented and highly rigorous introduction to the central themes of advanced calculus for statistics students, with enough theoretical explanation to be suitable for mathematics students. Topics include an introduction to set theory, linear algebra, limits and continuity, differentiation, infinite sequences and series, integration, multidimensional calculus, optimization in statistics, and approximation of functions. Exercises and an annotated bibliography are included. Annotation c. Book News, Inc., Portland, OR (
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Product Details

Meet the Author

ANDRE I. KHURI, PhD, is a Professor in the Department of Statistics at the University of Florida, Gainesville.

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Advanced Calculus with Applications in Statistics

By André I. Khuri

John Wiley & Sons

ISBN: 0-471-39104-2

Chapter One

An Introduction to Set Theory

The origin of the modern theory of sets can be traced back to the Russian-born German mathematician Georg Cantor (1845-1918). This chapter introduces the basic elements of this theory.


A set is any collection of well-defined and distinguishable objects. These objects are called the elements, or members, of the set and are denoted by lowercase letters. Thus a set can be perceived as a collection of elements united into a single entity. Georg Cantor stressed this in the following words: "A set is a multitude conceived of by us as a one."

If x is an element of a set A, then this fact is denoted by writing x [element of] A. If, however, x is not an element of A, then we write x [??] A. Curly brackets are usually used to describe the contents of a set. For example, if a set A consists of the elements [x.sub.1], [x.sub.2],..., [x.sub.n], then it can be represented as A = {[x.sub.1], [x.sub.2],..., [x.sub.n]}. In the event membership in a set is determined by the the satisfaction of a certain property or a relationship, then the description of the same can be given within the curly brackets. For example, if A consists of all real numbers x such that [chi square] > 1, then it can be expressed as A = {x|[chi square] > 1}, where the bar | is used simply to mean "such that." The definition of sets in this manner is based on the axiom of abstraction, which states that given any property, there exists a set whose elements are just those entities having that property.

Definition 1.1.1. The set that contains no elements is called the empty set and is denoted by [empty set].

Definition 1.1.2. A set A is a subset of another set B, written symbolically as A [subset] B, if every element of A is an element of B. If B contains at least one element that is not in A, then A is said to be a proper subset of B.

Definition 1.1.3. A set A and a set B are equal if A [subset] B and B [subset] A. Thus, every element of A is an element of B and vice versa.

Definition 1.1.4. The set that contains all sets under consideration in a certain study is called the universal set and is denoted by [Omega].


There are two basic operations for sets that produce new sets from existing ones. They are the operations of union and intersection.

Definition 1.2.1. The union of two sets A and B, denoted by A [union] B, is the set of elements that belong to either A or B, that is,

A [union] B = {x|x [element of] A or x [element of] B}.

This definition can be extended to more than two sets. For example, if [A.sub.1], [A.sub.2],..., [A.sub.n] are n given sets, then their union, denoted by [[union].sup.n.sub.i=1][A.sub.i], is a set such that x is an element of it if and only if x belongs to at least one of the [A.sub.i] (i = 1, 2,..., n).

Definition 1.2.2. The intersection of two sets A and B, denoted by A [intersection] B, is the set of elements that belong to both A and B. Thus

A [intersection] B = {x|x [element of] A and x [element of] B}.

This definition can also be extended to more than two sets. As before, if [A.sub.1], [A.sub.2],..., [A.sub.n] are n given sets, then their intersection, denoted by [[intersection].sup.n.sub.i]=1 [A.sub.i], is the set consisting of all elements that belong to all the [A.sub.i] (i = 1, 2,..., n).

Definition 1.2.3. Two sets A and B are disjoint if their intersection is the empty set, that is, A [intersection] B = [empty set].

Definition 1.2.4. The complement of a set A, denoted by [bar.A], is the set consisting of all elements in the universal set that do not belong to A. In other words, x [element of] [bar.A] if and only if x [??] A.

The complement of A with respect to a set B is the set B - A which consists of the elements of B that do not belong to A. This complement is called the relative complement of A with respect to B.

From Definitions 1.1.1-1.1.4 and 1.2.1-1.2.4, the following results can be concluded:

Result 1.2.1. The empty set [empty set] is a subset of every set. To show this, suppose that A is any set. If it is false that [empty set] [subset] A, then there must be an element in [empty set] which is not in A. But this is not possible, since [empty set] is empty. It is therefore true that [empty set] [subset] A.

Result 1.2.2. The empty set [empty set] is unique. To prove this, suppose that [[empty set].sub.1] and [[empty set].sub.2] are two empty sets. Then, by the previous result, [[empty set].sub.1] [subset] [[empty set].sub.2] and [[empty set].sub.2] [greater than or equal to] [[empty set].sub.2]. Hence, [[empty set].sub.1] = [[empty set].sub.2].

Result 1.2.3. The complement of [empty set] is [Omega]. Vice versa, the complement of [Omega] is [empty set].

Result 1.2.4. The complement of [bar.A] is A.

Result 1.2.5. For any set A, A [union] [bar.A] = [Omega] and A [intersection] [bar.A] = [empty set].

Result 1.2.6. A - B = A - A [intersection] B.

Result 1.2.7. A [union] (B [union] ITLITL) = (A [union] B) [union] ITLITL.

Result 1.2.8. A [intersection] (B [intersection] ITLITL) = (A [intersection] B) [intersection] ITLITL).

Result 1.2.9. A [union] (B [intersection] ITLITL) = (A [union] B) [intersection] (A [union] ITLITL).

Result 1.2.10. A [intersection] (B [union] ITLITL) = (A [intersection] B) [union] (A [intersection] ITLITL).



Definition 1.2.5. Let A and B be two sets. Their Cartesian product, denoted by A x B, is the set of all ordered pairs (a, b) such that a [element of] A and b [element of] B, that is,

A x B = {(a, b)|a [element of] A and b [element of] B}.

The word "ordered" means that if a and c are elements in A and b and d are elements in B, then (a, b) = (c, d) if and only if a = c and b = d.

The preceding definition can be extended to more than two sets. For example, if [A.sub.1], [A.sub.2],..., [A.sub.n] are n given sets, then their Cartesian product is denoted by [x.sup.n.sub.i=1][A.sub.i] and defined by


Here, ([a.sub.1], [a.sub.2],..., [a.sub.n]), called an ordered n-tuple, represents a generalization of the ordered pair. In particular, if the [A.sub.i] are equal to A for i = 1, 2,..., n, then one writes [A.sup.n] for [x.sup.n.sub.i=1]A.

The following results can be easily verified:

Result 1.2.13. A x B = [empty set] if and only if A = [empty set] or B = [empty set].

Result 1.2.14. (A [union] B]) x ITLITL = (A x ITLITL) [union] (B x ITLITL).

Result 1.2.15. (A [intersection] B) x ITLITL = (A x ITLITL) [intersection] x (B x ITLITL).

Result 1.2.16. (A x B) [intersection] (ITLITL x D) = (A [intersection] ITLITL) x (B [intersection] D).


Let A x B be the Cartesian product of two sets, A and B.

Definition 1.3.1. A relations [rho] from A to B is a subset of A x B, that is, [rho] consists of ordered pairs (a, b) such that a [element of] A and b [element of] B. In particular, if A = B, then [rho] is said to be a relation in A.

For example, if A = {7, 8, 9} and B = {7, 8, 9, 10}, then [rho] {(a, b)|a < b, a [element of] A, b [element of] B} is a relation from A to B that consists of the six ordered pairs (7, 8), (7, 9), (7, 10), (8, 9), (8, 10), and (9, 10).

Whenever [rho] is a relation and (x, y) [element of] [rho], then x and y are said to be [rho]-related. This is denoted by writing x p y.

Definition 1.3.2. A relation [rho] in a set A is an equivalence relation if the following properties are satisfied:

1. [rho] is reflexive, that is, a [rho] a for any a in A.

2. [rho] is symmetric, that is, if a [rho] b, then b [rho] a for any a, b in A.

3. [rho] is transitive, that is, if a [rho] b and b [rho] c, then a [rho] c for any a, b, c in A.

If [rho] is an equivalence relation in a set A, then for a given [a.sub.0] in A, the set

ITLITL([a.sub.0]) = {a [element of] A|[a.sub.0] [rho] a},

which consists of all elements of A that are [rho]-related to [a.sub.0], is called an equivalence class of [a.sub.0].

Result 1.3.1. a [element of] ITLITL(a) for any a in A. Thus each element of A is an element of an equivalence class.

Result 1.3.2. If ITLITL([a.sub.1]) and ITLITL([a.sub.2]) are two equivalence classes, then either ITLITL([a.sub.1]) = ITLITL([a.sub.2]), or ITLITL([a.sub.1) and ITLITL([a.sub.2]) are disjoint subsets.

It follows from Results 1.3.1 and 1.3.2 that if A is a nonempty set, the collection of distinct [rho]-equivalence classes of A forms a partition of A.

As an example of an equivalence relation, consider that a [rho] b if and only if a and b are integers such that a - b is divisible by a nonzero integer n. This is the relation of congruence modulo n in the set of integers and is written symbolically as a [equivalent to] b (mod n). Clearly, a [equivalent to] a (mod n), since a - a = 0 is divisible by n. Also, if a [equivalent to] b (mod n), then b [equivalent to] a (mod n), since if a - b is divisible by n, then so is b - a. Furthermore, if a [equivalent to] _ b (mod n) and b [equivalent to] c (mod n), then a [equivalent to] c (mod n). This is true because if a - b and b - c are both divisible by n, then so is (a - b) + (b - c) = a - c. Now, if [a.sub.0] is a given integer, then a [rho]-equivalence class of [a.sub.0] consists of all integers that can be written as a = [a.sub.0] + kn, where k is an integer. This in this example ITLITL([a.sub.0]) is the set {[a.sub.0] + kn|k [element of] J}, where J denotes the set of all integers.

Definition 1.3.3. Let [rho] be a relation from A to B. Suppose that [rho] has the property that for all x in A, if x [rho] y and x [rho] z, where y and z are elements in B, then y = z. Such a relation is called a function.

Thus a function is a relation [rho] such that any two elements in B that are [rho]-related to the same x in A must be identical. In other words, to each element x in A, there corresponds only one element y in B. We call y the value of the function at x and denote it by writing y = f(x). The set A is called the domain of the function f, and the set of all values of f(x) for x in A is called the range of f, or the image of A under f, and is denoted by f(A). In this case, we say that f is a function, or a mapping, from A into B. We express this fact by writing f: A [right arrow] B. Note that f(A) is a subset of B. In particular, if B = f(A), then f is said to be a function from A onto B. In this case, every element b in B has a corresponding element a in A such that b = f(a).

Definition 1.3.4. A function f defined on a set A is said to be a one-to-one function if whenever f([x.sub.1]) = f([x.sub.2]) for [x.sub.1], [x.sub.2] in A, one has [x.sub.1] = [x.sub.2]. Equivalently, f is a one-to-one function if whenever [x.sub.1] [not equal to] [x.sub.2], one has f([x.sub.1])[not equal to]f([x.sub.2]).


Excerpted from Advanced Calculus with Applications in Statistics by André I. Khuri Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents


Preface to the First Edition.

1. An Introduction to Set Theory.

1.1. The Concept of a Set.

1.2. Set Operations.

1.3. Relations and Functions.

1.4. Finite, Countable, and Uncountable Sets.

1.5. Bounded Sets.

1.6. Some Basic Topological Concepts.

1.7. Examples in Probability and Statistics.

2. Basic Concepts in Linear Algebra.

2.1. Vector Spaces and Subspaces.

2.2. Linear Transformations.

2.3. Matrices and Determinants.

2.4. Applications of Matrices in Statistics.

3. Limits and Continuity of Functions.

3.1. Limits of a Function.

3.2. Some Properties Associated with Limits of Functions.

3.3. The o, O Notation.

3.4. Continuous Functions.

3.5. Inverse Functions.

3.6. Convex Functions.

3.7. Continuous and Convex Functions in Statistics.

4. Differentiation.

4.1. The Derivative of a Function.

4.2. The Mean Value Theorem.

4.3. Taylor's Theorem.

4.4. Maxima and Minima of a Function.

4.5. Applications in Statistics.

5. Infinite Sequences and Series.

5.1. Infinite Sequences.

5.2. Infinite Series.

5.3. Sequences and Series of Functions.

5.4. Power Series.

5.5. Sequences and Series of Matrices.

5.6. Applications in Statistics.

6. Integration.

6.1. Some Basic Definitions.

6.2. The Existence of the Riemann Integral.

6.3. Some Classes of Functions That Are Riemann Integrable.

6.4. Properties of the Riemann Integral.

6.5. Improper Riemann Integrals.

6.6. Convergence of a Sequence of Riemann Integrals.

6.7. Some Fundamental Inequalities.

6.8. RiemannStieltjes Integral.

6.9. Applications in Statistics.

7. Multidimensional Calculus.

7.1. Some Basic Definitions.

7.2. Limits of a Multivariable Function.

7.3. Continuity of a Multivariable Function.

7.4. Derivatives of a Multivariable Function.

7.5. Taylor's Theorem for a Multivariable Function.

7.6. Inverse and Implicit Function Theorems.

7.7. Optima of a Multivariable Function.

7.8. The Method of Lagrange Multipliers.

7.9. The Riemann Integral of a Multivariable Function.

7.10. Differentiation under the Integral Sign.

7.11. Applications in Statistics.

8. Optimization in Statistics.

8.1. The Gradient Methods.

8.2. The Direct Search Methods.

8.3. Optimization Techniques in Response Surface Methodology.

8.4. Response Surface Designs.

8.5. Alphabetic Optimality of Designs.

8.6. Designs for Nonlinear Models.

8.7. Multiresponse Optimization.

8.8. Maximum Likelihood Estimation and the EM Algorithm.

8.9. Minimum Norm Quadratic Unbiased Estimation of Variance Components.

8.10. Scheffé's Confidence Intervals.

9. Approximation of Functions.

9.1. Weierstrass Approximation.

9.2. Approximation by Polynomial Interpolation.

9.3. Approximation by Spline Functions.

9.4. Applications in Statistics.

10. Orthogonal Polynomials.

10.1. Introduction.

10.2. Legendre Polynomials.

10.3. Jacobi Polynomials.

10.4. Chebyshev Polynomials.

10.5. Hermite Polynomials.

10.6. Laguerre Polynomials.

10.7. Least-Squares Approximation with Orthogonal Polynomials.

10.8. Orthogonal Polynomials Defined on a Finite Set.

10.9. Applications in Statistics.

11. Fourier Series.

11.1. Introduction.

11.2. Convergence of Fourier Series.

11.3. Differentiation and Integration of Fourier Series.

11.4. The Fourier Integral.

11.5. Approximation of Functions by Trigonometric Polynomials.

11.6. The Fourier Transform.

11.7. Applications in Statistics.

12. Approximation of Integrals.

12.1. The Trapezoidal Method.

12.2. Simpson’s Method.

12.3. NewtonCotes Methods.

12.4. Gaussian Quadrature.

12.5. Approximation over an Infinite Interval.

12.6. The Method of Laplace.

12.7. Multiple Integrals.

12.8. The Monte Carlo Method.

12.9. Applications in Statistics.

Appendix. Solutions to Selected Exercises.

General Bibliography.


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