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Robert V.Hogg, Allen T.Craig - Introduction to Mathematical Statistics (4th Edition).pdf
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Robert V. Hogg
Allen
T.
Craig
THE
UNIVERSITY
OF
IOWA
Introduction
to
Mathematical
Statistics
Fourth Edition
Macmillan
Publishing
Co.,
Inc.
NEW
YORK
Collier
Macmillan
Publishers
LONDON
Preface
Copyright
© 1978,
Macmillan
Publishing
Co.,
Inc.
Printed
in
the
United
States
of
America
Earlier
editions
© 1958
and
1959
and
copyright
© 1965
and
1970
by
Macmillan
Publishing
Co.,
Inc.
Macmillan
Publishing
Co.,
Inc.
866
Third
Avenue, New
York,
New
York
10022
Collier
Macmillan
Canada,
Ltd.
Library
of Congress
Cataloging
in
Publication
Data
Hogg,
Robert
V
Introduction
to
mathematical
statistics.
We are
much
indebted
to
our
colleagues
throughout
the
country
who
have so generously provided us
with
suggestions on
both
the
order of
presentation
and
the
kind
of
material
to
be included
in
this
edition of
Introduction to Mathematical Statistics. We believe
that
you
will find
the
book
much
more
adaptable
for classroom use
than
the
previous
edition. Again, essentially all
the
distribution
theory
that
is needed is
found in
the
first five chapters.
Estimation
and
tests
of
statistical
hypotheses, including nonparameteric methods, follow in Chapters 6, 7,
8,
and
9, respectively. However, sufficient
statistics
can
be
introduced
earlier
by
considering
Chapter
10 immediately
after
Chapter
6 on
estimation. Many of
the
topics of
Chapter
11 are such
that
they
may
also be
introduced
sooner:
the
Rae-Cramer
inequality
(11.1)
and
robust
estimation (11.7)
after
measures of
the
quality
of estimators
(6.2), sequential analysis (11.2)
after
best
tests
(7.2), multiple com-
parisons (11.3)
after
the
analysis of variance (8.5),
and
classification
(11.4)
after
material
on
the
sample correlation coefficient (8.7).
With
this
flexibility
the
first eight
chapters
can
easily be covered in courses of
either six semester hours or eight
quarter
hours, supplementing
with
the
various topics from Chapters 9
through
11 as
the
teacher
chooses
and
as
the
time
permits.
In
a longer course, we hope
many
teachers
and
students
will be
interested
in
the
topics of stochastic independence
(11.5), robustness (11.6
and
11.7),
multivariate
normal distributions
(12.1),
and
quadratic
forms (12.2
and
12.3).
We are obligated to Catherine M.
Thompson
and
Maxine Merrington
and
to Professor E. S. Pearson for permission to include Tables
II
and
V, which are
abridgments
and
adaptations
of tables published in
Biometrika. We wish to
thank
Oliver &
Boyd
Ltd.,
Edinburgh,
for
permission to include
Table
IV, which is an
abridgment
and
adaptation
v
56789
131415
YEAR
PRINTING
Bibliography:
p.
Includes
index.
1.
Mathematical
statistics.
I. Craig, Allen
Thornton,
(date)
joint
author.
II.
Title.
QA276.H59 1978 519 77-2884
ISBN
0-02-355710-9
(Hardbound)
ISBN
0-02-978990-7
(International Edition)
All
rights
reserved. No
part
of
this
book
may
be
reproduced
or
transmitted
in
any
form
or by
any
means,
electronic or
mechanical,
including
photocopying,
recording, or
any
information
storage
and
retrieval
system,
without
permission
in
writing
from
the
Publisher.
T~RN
vi
Preface
of
Table
III
from
the
book
Statistical Tables for Biological, Agricultural,
and Medical Research
by
the
late
Professor
Sir
Ronald
A.
Fisher,
Cambridge,
and
Dr.
Frank
Yates,
Rothamsted.
Finally,
we
wish
to
thank
Mrs.
Karen
Horner
for
her
first-class
help
in
the
preparation
of
the
manuscript.
R. V. H.
A. T. C.
Contents
Chapter 1
Distributions
of
Random
Variables
1
1.1 Introduction 1
1.2 Algebra of Sets 4
1.3 Set Functions 8
1.4 The Probability Set Function 12
1.5
Random
Variables 16
1.6
The
Probability Density
Function
23
1.7 The Distribution
Function
31
1.8 Certain Probability Models 38
1.9 Mathematical
Expectation
44
1.10 Some Special Mathematical Expectations 48
1.11 Chebyshev's
Inequality
58
Chapter 2
Conditional
Probability
and
Stochastic
Independence
61
2.1 Conditional Probability 61
2.2 Marginal
and
Conditional Distributions 65
2.3 The Correlation Coefficient 73
2.4 Stochastic Independence 80
Chapter 3
Some
Special
Distributions
3.1 The Binomial, Trinomial,
and
Multinomial Distributions 90
3.2 The Poisson Distribution 99
3.3 The
Gamma
and
Chi-Square Distributions 103
3.4 The Normal Distribution 109
3.5 The Bivariate Normal Distribution 117
vii
90
viii
Contents
Contents
ix
Chapter 4
Distributions
of
Functions
of
Random
'Variables
4.1 Sampling Theory 122
4.2 Transformations of Variables of the Discrete
Type
128
4.3 Transformations of Variables of
the
Continuous Type 132
4.4 The
t
and
F Distributions 143
4.5 Extensions of
the
Change-of-Variable Technique 147
4.6 Distributions of Order Statistics 154
4.7 The Moment-Generating-Function Technique 164
4.8 The Distributions of
X
and
nS
2
ja
2
172
4.9
Expectations
of Functions of
Random
Variables 176
Chapter 5
Limiting
Distributions
5.1 Limiting Distributions 181
5.2 Stochastic Convergence 186
5.3 Limiting Moment-Generating Functions 188
5.4 The Central
Limit
Theorem 192
5.5 Some Theorems on Limiting Distributions 196
122
181
Chapter 8
Other
Statistical
Tests
8.1 Chi-Square Tests 269
8.2 The Distributions of Certain Quadratic Forms 278
8.3 A
Test
of
the
Equality
of Several Means 283
8.4 Noncentral X
2
and
Noncentral F 288
8.5 The Analysis of Variance 291
8.6 A Regression Problem 296
8.7 A
Test
of Stochastic Independence 300
Chapter 9
Nonparametric
Methods
9.1 Confidence
Intervals
for Distribution Quantiles 304
9.2 Tolerance Limits for Distributions 307
9.3 The Sign
Test
312
9.4 A
Test
of Wilcoxon 314
9.5 The
Equality
of Two Distributions 320
9.6 The Mann-Whitney-Wilcoxon
Test
326
9.7 Distributions
Under
Alternative Hypotheses 331
9.8 Linear
Rank
Statistics 334
269
304
Chapter 6
Estimation
6.1
Point
Estimation
200
6.2 Measures of Quality of
Estimators
207
6.3 Confidence
Intervals
for Means 212
6.4 Confidence Intervals for Differences of Means 219
6.5 Confidence
Intervals
for Variances 222
6.6 Bayesian
Estimates
227
200
Chapter 10
Sufficient
Statistics
1v.1 A Sufficient
Statistic
for a
Parameter
341
10.2 The Rao-Blackwell Theorem 349
10.3 Completeness
and
Uniqueness 353
10.4 The Exponential Class of Probability Density Functions 357
10.5 Functions of a
Parameter
361
10.6 The Case of Several
Parameters
364
341
Chapter 7
Statistical
Hypotheses
7.1 Some Examples
and
Definitions 235
7.2 Certain
Best
Tests 242
7.3 Uniformly Most Powerful Tests 251
7.4 Likelihood
Ratio
Tests 257
235
Chapter 11
Further
Topics
in
Statistical
Inference
11.1 The
Rae-Cramer
Inequality
370
11.2 The Sequential Probability
Ratio
Test
374
11.3 Multiple Comparisons
380
11.4 Classification 385
370
x
Contents
11.5 Sufficiency, Completeness,
and
Stochastic Independence 389
11.6
Robust
Nonparametric Methods 396
11.7
Robust
Estimation
400
Chapter 12
Further
Normal
Distribution
Theory
12.1 The Multivariate Normal Distribution 405
12.2
The
Distributions of Certain Quadratic
Forms
12.3 The Independence of Certain Quadratic
Forms
Appendix
A
References
Appendix
B
Tables
Appendix
C
Answers
to
Selected
Exercises
Index
410
414
405
421
423
429
435
Chapter I
Distributions
of
Random
Variables
1.1
Introduction
Many kinds of investigations
may
be characterized in
part
by
the
fact
that
repeated
experimentation,
under
essentially
the
same con-
ditions, is more or less
standard
procedure.
For
instance, in medical
research,
interest
may
center on
the
effect of a
drug
that
is to be
administered; or an economist
may
be concerned
with
the
prices of
three specified commodities
at
various
time
intervals; or
the
agronomist
may
wish to
study
the
effect
that
a chemical fertilizer
has
on
the
yield
of a cereal grain.
The
only
way
in which
an
investigator
can
elicit
information
about
any
such phenomenon is to perform his experiment.
Each
experiment
terminates
with
an outcome.
But
it
is characteristic of
these experiments
that
the
outcome
cannot
be predicted
with
certainty
prior to
the
performance of
the
experiment.
Suppose
that
we
have
such an experiment,
the
outcome of which
cannot
be predicted
with
certainty,
but
the
experiment is of such a
nature
that
the
collection of every possible outcome
can
be described
prior to
its
performance.
If
this
kind
of experiment
can
be
repeated
under
the
same conditions, it is called a random experiment,
and
the
collection of every possible outcome is called
the
experimental space or
the
sample space.
Example 1.
In
the
toss of a coin, let
the
outcome tails be denoted by
T
and
let
the
outcome heads be denoted by H.
If
we assume
that
the
coin
may
be repeatedly tossed
under
the
same conditions,
then
the
toss of this
coin is an example of a random experiment in which
the
outcome is one of
1
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