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Introduction to Mathematical Statistics PDF

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I n t r o d u c t i o n t o M a t h e m a t i c a l S t a t i s t i c s H o g g e t a l Introduction to Mathematical Statistics . Robert V. Hogg Joeseph McKean 7 Allen T. Craig e ISBN 978-1-29202-499-8 Seventh Edition 9 781292 024998 Introduction to Mathematical Statistics Robert V. Hogg Joeseph McKean Allen T. Craig Seventh Edition ISBN 10: 1-292-02499-2 ISBN 13: 978-1-292-02499-8 Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk © Pearson Education Limited 2014 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affi liation with or endorsement of this book by such owners. ISBN 10: 1-292-02499-2 ISBN 13: 978-1-292-02499-8 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Printed in the United States of America 122334456667409283831231175739577115 P E A R S O N C U S T O M L I B R AR Y Table of Contents 1. Probability and Distributions Robert V. Hogg/Joeseph McKean/Allen T. Craig 1 2. Multivariate Distributions Robert V. Hogg/Joeseph McKean/Allen T. Craig 75 3. Some Special Distributions Robert V. Hogg/Joeseph McKean/Allen T. Craig 141 4. Some Elementary Statistical Inferences Robert V. Hogg/Joeseph McKean/Allen T. Craig 207 5. Consistency and Limiting Distributions Robert V. Hogg/Joeseph McKean/Allen T. Craig 295 6. Maximum Likelihood Methods Robert V. Hogg/Joeseph McKean/Allen T. Craig 327 7. Sufficiency Robert V. Hogg/Joeseph McKean/Allen T. Craig 383 8. Optimal Tests of Hypotheses Robert V. Hogg/Joeseph McKean/Allen T. Craig 439 9. Inferences About Normal Models Robert V. Hogg/Joeseph McKean/Allen T. Craig 485 10. Nonparametric and Robust Statistics Robert V. Hogg/Joeseph McKean/Allen T. Craig 537 11. Appendix: Mathematical Comments Robert V. Hogg/Joeseph McKean/Allen T. Craig 617 12. Appendix: R Functions Robert V. Hogg/Joeseph McKean/Allen T. Craig 621 13. Appendix: Tables of Distributions Robert V. Hogg/Joeseph McKean/Allen T. Craig 631 I 664415 14. Appendix: Lists of Common Distributions Robert V. Hogg/Joeseph McKean/Allen T. Craig 641 Index 645 II Probability and Distributions 1 Introduction Manykindsofinvestigationsmaybecharacterizedinpartbythefactthatrepeated experimentation, under essentially the same conditions, 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 the 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. Supposethatwehavesuchanexperiment,buttheexperimentisofsuchanature thatacollectionofeverypossibleoutcomecanbedescribedpriortoitsperformance. 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 1.1. In the toss of a coin, let the outcome tails be denoted by T and let theoutcomeheadsbedenotedbyH. Ifweassumethatthecoinmayberepeatedly 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 the two symbols T and H; that is, the sample space is the collection of these two symbols. Example 1.2. In the cast of one red die and one white die, let the outcome be the ordered pair (number of spots up on the red die, number of spots up on the whitedie). Ifweassumethatthesetwodicemayberepeatedlycastunderthesame conditions, then the cast of this pair of dice is a random experiment. The sample spaceconsistsofthe36orderedpairs: (1,1),...,(1,6),(2,1),...,(2,6),...,(6,6). LetC denoteasamplespace, letcdenoteanelementofC, andletC representa collectionofelementsofC. If,upontheperformanceoftheexperiment,theoutcome FromChapter1ofIntroductiontoMathematicalStatistics,SeventhEdition.RobertV.Hogg, JosephW.McKean,AllenT.Craig. Copyright(cid:2)c 2013byPearsonEducation,Inc. Allrightsreserved. 1 Probability and Distributions is in C, we shall say that the event C has occurred. Now conceive of our having made N repeated performances of the random experiment. Then we can count the number f of times (the frequency) that theevent C actually occurredthroughout the N performances. The ratio f/N is called the relative frequency of the event C in these N experiments. A relative frequency is usually quite erratic for small values of N, as you can discover by tossing a coin. But as N increases, experience indicates that we associate with the event C a number, say p, that is equal or approximately equal to that number about which the relative frequency seems to stabilize. Ifwedothis,thenthenumberpcanbeinterpretedasthatnumberwhich, infutureperformancesoftheexperiment, therelativefrequencyoftheeventC will either equal or approximate. Thus, although we cannot predict the outcome of a random experiment, we can, for a large value of N, predict approximately the relative frequency with which the outcome will be in C. The number p associated with the event C is given various names. Sometimes it is called the probability that theoutcomeoftherandomexperimentisinC;sometimesitiscalledtheprobability oftheeventC;andsometimesitiscalledtheprobability measure ofC. Thecontext usually suggests an appropriate choice of terminology. Example 1.3. Let C denote the sample space of Example 1.2 and let C be the collection of every ordered pair of C for which the sum of the pair is equal to seven. ThusC isthecollection(1,6),(2,5),(3,4),(4,3),(5,2), and(6,1). Supposethatthe dice are cast N =400 times and let f, the frequency of a sum of seven, be f =60. ThentherelativefrequencywithwhichtheoutcomewasinC isf/N = 60 =0.15. 400 Thus we might associate with C a number p that is close to 0.15, and p would be called the probability of the event C. Remark 1.1. The preceding interpretation of probability is sometimes referred to as the relative frequency approach, and it obviously depends upon the fact that an experiment can be repeated under essentially identical conditions. However, many persons extend probability to other situations by treating it as a rational measure ofbelief. Forexample,thestatementp= 2 wouldmeantothemthattheirpersonal 5 orsubjective probabilityoftheeventC isequalto 2. Hence,iftheyarenotopposed 5 to gambling, this could be interpreted as a willingness on their part to bet on the outcome of C so that the two possible payoffs are in the ratio p/(1−p)= 2/3 = 2. 5 5 3 Moreover,iftheytrulybelievethatp= 2 iscorrect,theywouldbewillingtoaccept 5 either side of the bet: (a) win 3 units if C occurs and lose 2 if it does not occur, or (b) win 2 units if C does not occur and lose 3 if it does. However, since the mathematical properties of probability given in Section 3 are consistent with either oftheseinterpretations,thesubsequentmathematicaldevelopmentdoesnotdepend upon which approach is used. The primary purpose of having a mathematical theory of statistics is to provide mathematical models for random experiments. Once a model for such an experi- ment has been provided and the theory worked out in detail, the statistician may, within this framework, make inferences (that is, draw conclusions) about the ran- domexperiment. Theconstructionofsuchamodelrequiresatheoryofprobability. One of the more logically satisfying theories of probability is that based on the concepts of sets and functions of sets. These concepts are introduced in Section 2. 2 Probability and Distributions 2 Set Theory The concept of a set or a collection of objects is usually left undefined. However, a particular set can be described so that there is no misunderstanding as to what collection of objects is under consideration. For example, the set of the first 10 positiveintegersissufficientlywelldescribedtomakeclearthatthenumbers 3 and 4 14 are not in the set, while the number 3 is in the set. If an object belongs to a set, it is said to be an element of the set. For example, if C denotes the set of real numbers x for which 0 ≤ x ≤ 1, then 3 is an element of the set C. The fact that 4 3 is an element of the set C is indicated by writing 3 ∈ C. More generally, c ∈ C 4 4 means that c is an element of the set C. The sets that concern us are frequently sets of numbers. However, the language of sets of points proves somewhat more convenient than that of sets of numbers. Accordingly, we briefly indicate how we use this terminology. In analytic geometry considerable emphasis is placed on the fact that to each point on a line (on which an origin and a unit point have been selected) there corresponds one and only one number,sayx;andthattoeachnumberxtherecorrespondsoneandonlyonepoint on the line. This one-to-one correspondence between the numbers and points on a line enables us to speak, without misunderstanding, of the “point x” instead of the “number x.” Furthermore, with a plane rectangular coordinate system and with x andynumbers,toeachsymbol(x,y)therecorrespondsoneandonlyonepointinthe plane; and to each point in the plane there corresponds but one such symbol. Here again,wemayspeakofthe“point(x,y),”meaningthe“orderednumberpairxand y.” This convenient language can be used when we have a rectangular coordinate system in a space of three or more dimensions. Thus the “point (x ,x ,...,x )” 1 2 n meansthenumbersx ,x ,...,x intheorderstated. Accordingly,indescribingour 1 2 n sets, we frequently speak of a set of points (a set whose elements are points), being careful, of course, to describe the set so as to avoid any ambiguity. The notation C = {x : 0 ≤ x ≤ 1} is read “C is the one-dimensional set of points x for which 0 ≤ x ≤ 1.” Similarly, C = {(x,y) : 0 ≤ x ≤ 1,0 ≤ y ≤ 1} can be read “C is the two-dimensional set of points (x,y) that are interior to, or on the boundary of, a square with opposite vertices at (0,0) and (1,1).” We say a set C is countable if C is finite or has as many elements as there are positiveintegers. Forexample,thesetsC ={1,2,...,100}andC ={1,3,5,7,...} 1 2 are countable sets. The interval of real numbers (0,1], though, is not countable. We now give some definitions (together with illustrative examples) that lead to an elementary algebra of sets adequate for our purposes. Definition 2.1. If each element of a set C is also an element of set C , the set C 1 2 1 is called a subset of the set C . This is indicated by writing C ⊂C . If C ⊂C 2 1 2 1 2 and also C ⊂ C , the two sets have the same elements, and this is indicated by 2 1 writing C =C . 1 2 Example 2.1. Let C = {x : 0 ≤ x ≤ 1} and C = {x : −1 ≤ x ≤ 2}. Here the 1 2 one-dimensional set C is seen to be a subset of the one-dimensional set C ; that 1 2 is, C ⊂ C . Subsequently, when the dimensionality of the set is clear, we do not 1 2 make specific reference to it. 3 Probability and Distributions Example 2.2. DefinethetwosetsC ={(x,y):0≤x=y ≤1}andC ={(x,y): 1 2 0 ≤ x ≤ 1,0 ≤ y ≤ 1}. Because the elements of C are the points on one diagonal 1 of the square, then C ⊂C . 1 2 Definition 2.2. If a set C has no elements, C is called the null set. This is indicated by writing C =φ. Definition 2.3. The set of all elements that belong to at least one of the sets C 1 and C is called the union of C and C . The union of C and C is indicated by 2 1 2 1 2 writing C ∪C . The union of several sets C ,C ,C ,... is the set of all elements 1 2 1 2 3 thatbelong to at least one of the several sets, denoted by C ∪C ∪C ∪···=∪∞ C 1 2 3 j=1 j or by C ∪C ∪···∪C =∪k C if a finite number k of sets is involved. 1 2 k j=1 j We refer to a union of the form ∪∞ C as a countable union. j=1 j Example 2.3. Define the sets C = {x : x = 8,9,10,11, or 11 < x ≤ 12} and 1 C ={x:x=0,1,...,10}. Then 2 C ∪C = {x:x=0,1,...,8,9,10,11, or 11<x≤12} 1 2 = {x:x=0,1,...,8,9,10 or 11≤x≤12}. Example 2.4. Define C and C as in Example 2.1. Then C ∪C =C . 1 2 1 2 2 Example 2.5. Let C =φ. Then C ∪C =C , for every set C . 2 1 2 1 1 Example 2.6. For every set C, C∪C =C. Example 2.7. Let (cid:2) (cid:3) C = x: 1 ≤x≤1 , k =1,2,3,... . k k+1 Then∪∞ C ={x:0<x≤1}. Notethatthenumberzeroisnotinthisset,since k=1 k it is not in one of the sets C ,C ,C ,.... 1 2 3 Definition 2.4. The set of all elements that belong to each of the sets C and C 1 2 is called the intersection of C and C . The intersection of C and C is indicated 1 2 1 2 by writing C ∩C . The intersection of several sets C ,C ,C ,... is the set of all 1 2 1 2 3 elements that belong to each of the sets C ,C ,C ,.... This intersection is denoted 1 2 3 by C ∩C ∩C ∩···=∩∞ C or by C ∩C ∩···∩C =∩k C if a finite number 1 2 3 j=1 j 1 2 k j=1 j k of sets is involved. We refer to an intersection of the form ∩∞ C as a countable intersection. j=1 j Example 2.8. Let C ={(0,0),(0,1),(1,1)} and C ={(1,1),(1,2),(2,1)}. Then 1 2 C ∩C ={(1,1)}. 1 2 Example 2.9. Let C = {(x,y) : 0 ≤ x+y ≤ 1} and C = {(x,y) : 1 < x+y}. 1 2 Then C and C have no points in common and C ∩C =φ. 1 2 1 2 Example 2.10. For every set C, C∩C =C and C∩φ=φ. 4 Probability and Distributions C C C C 1 2 1 2 (a) (b) Figure 2.1: (a) C ∪C and (b) C ∩C . 1 2 1 2 Example 2.11. Let (cid:4) (cid:5) C = x:0<x< 1 , k =1,2,3,... k k Then ∩∞ C = φ, because there is no point that belongs to each of the sets k=1 k C ,C ,C ,.... 1 2 3 Example 2.12. Let C and C represent the sets of points enclosed, respectively, 1 2 by two intersecting ellipses. Then the sets C ∪C and C ∩C are represented, 1 2 1 2 respectively, by the shaded regions in the Venn diagrams in Figure 2.1. Definition2.5. Incertaindiscussionsorconsiderations,thetotalityofallelements that pertain to the discussion can be described. This set of all elements under consideration is given a special name. It is called the space. We often denote spaces by letters such as C and D. Example 2.13. Let the number of heads, in tossing a coin four times, be denoted by x. Of necessity, the number of heads is of the numbers 0,1,2,3,4. Here, then, the space is the set C ={0,1,2,3,4}. Example 2.14. Consider all nondegenerate rectangles of base x and height y. To be meaningful, both x and y must be positive. Then the space is given by the set C ={(x,y):x>0,y >0}. Definition2.6. LetC denoteaspaceandletC beasubsetofthesetC. Thesetthat consists of all elements of C that are not elements of C is called the complement of C (actually, with respect to C). The complement of C is denoted by Cc. In particular, Cc =φ. 5

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