Mathematical Statistics: Exercises and Solutions

1 Mathematical statistics : Exercises and SolutionsJun ShaoMathematicalStatistics: Exercises andSolutionsJun ShaoDepartment of StatisticsUniversity of WisconsinMadison, WI of Congress Control Number: 2005923578 ISBN-10: 0-387-24970-2 Printed on acid-free : 978-0387-24970-4 2005 Springer Science+Business Media, rights reserved. This work may not be translated or copied in whole or in part without thewritten permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street,New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarlyanalysis. Use in connection with any form of information storage and retrieval, electronic adap-tation, computer software, or by similar or dissimilar methodology now known or hereafter de-veloped is use in this publication of trade names, trademarks, service marks, and similar terms, evenif they are not identified as such, is not to be taken as an expression of opinion as to whetheror not they are subject to proprietary in the United States of America.

2 (EB) My ParentsPrefaceSince the publication of my bookMathematical statistics (Shao, 2003), Ihave been asked many times for a solution manual to the Exercises in mybook. Without doubt, Exercises form an important part of a textbookon Mathematical statistics , not only in training students for their researchability in Mathematical statistics but also in presenting many additionalresults as complementary material to the main text. Written solutionsto these Exercises are important for students who initially do not havethe skills in solving these Exercises completely and are very helpful forinstructors of a Mathematical statistics course (whether or not my bookMathematical Statisticsis used as the textbook) in providing answers tostudents as well as finding additional examples to the main text. Moti-vated by this and encouraged by some of my colleagues and Springer-Verlageditor John Kimmel, I have completed this book, Mathematical statistics : Exercises and book consists of Solutions to 400 Exercises , over 95% of which arein my bookMathematical statistics .

3 Many of them are standard exercisesthat also appear in other textbooks listed in the references. It is onlya partial solution manual toMathematical statistics (which contains over900 Exercises ). However, the types of exercise inMathematical Statisticsnotselected in the current book are (1) Exercises that are routine (each exerciseselected in this book has a certain degree of difficulty), (2) Exercises similarto one or several Exercises selected in the current book, and (3) Exercises foradvanced materials that are often not included in a Mathematical statisticscourse for first-year students in statistics ( , Edgeworth expan-sions and second-order accuracy of confidence sets, empirical likelihoods,statistical functionals, generalized linear models, nonparametric tests, andtheory for the bootstrap and jackknife, etc.). On the other hand, this isa stand-alone book, since Exercises and Solutions are comprehensibleindependently of their source for likely readers.

4 To help readers notusing this book together withMathematical statistics , lists of notation,terminology, and some probability distributions are given in the front ofthe notational conventions are the same as or very similar to thoseinMathematical Statisticsand so is the Mathematical level of this are assumed to have a good knowledge in advanced calculus. Acourse in real analysis or measure theory is highly recommended. If thisbook is used with a statistics textbook that does not include probabilitytheory, then knowledge in measure-theoretic probability theory is Exercises are grouped into seven chapters with titles matching thoseinMathematical statistics . A few errors in the Exercises fromMathematicalStatisticswere detected during the preparation of their Solutions and thecorrected versions are given in this book. Although Exercises are numberedindependently of their source, the corresponding number inMathematicalStatisticsis accompanied with each exercise number for convenience ofinstructors and readers who also useMathematical Statisticsas the maintext.

5 For example, exercise 8 (# ) means that exercise 8 in the currentbook is also exercise 19 in Chapter 2 ofMathematical note to students/readers who have a need for Exercises accompaniedby Solutions is that they should not be completely driven by the are encouraged to try each exercise first without readingits solution. If an exercise is solved with the help of a solution, they areencouraged to provide Solutions to similar Exercises as well as to think aboutwhether there is an alternative solution to the one given in this book. Afew Exercises in this book are accompanied by two Solutions and/or notesof brief would like to thank my teaching assistants, Dr. Hansheng Wang, Cheng, and Mr. Fang Fang, who provided valuable help in preparingsome Solutions . Any errors are my own responsibility, and a correction ofthem can be found on my web page , WisconsinJun ShaoApril xvSome 1. Probability 2. Fundamentals of 51 Chapter 3. Unbiased 95 Chapter 4.

6 Estimation in Parametric 141 Chapter 5. Estimation in Nonparametric 209 Chapter 6. Hypothesis 251 Chapter 7. Confidence 353 NotationR: The real :Thek-dimensional Euclidean (c1, .., ck): A vector (element) inRkwithjth componentcj R;cisconsidered as ak 1 matrix (column vector) when matrix algebra : The transpose of a vectorc Rkconsidered as a 1 kmatrix (rowvector) when matrix algebra is involved. c : The Euclidean norm of a vectorc Rk, c 2=c c.|c|: The absolute value ofc : The transpose of a (A)or|A|: The determinant of a (A): The trace of a matrixA. A : The norm of a matrixAdefined as A 2=tr(A A).A 1: The inverse of a : The generalized inverse of a : The square root of a nonnegative definite matrixAdefined byA1/2A1/2= 1/2:TheinverseofA1 (A): The linear space generated by rows of a :Thek kidentity :Thek-dimensional vector of 1 s. : The empty set.(a, b): The open interval fromatob.[a, b]: The closed interval fromatob.(a, b]: The interval fromatobincludingbbut nota.)

7 [a, b): The interval fromatobincludingabut notb.{a, b, c}: The set consisting of the elementsa,b, Ak: The Cartesian product of setsA1, .., Ak,A1 Ak={(a1, .., ak):a1 A1, .., ak Ak}.xixiiNotation (C): The smallest -field that containsC. (X): The smallest -field with respect to whichXis measurable. 1 k: The product measure of 1,.., kon (F1 Fk), where iis a measure onFi,i=1, .., :TheBorel -field :TheBorel -field : The complement of a B: The union of setsAandB. Ai: The union of setsA1,A2, ..A B: The intersection of setsAandB. Ai: The intersection of setsA1,A2, ..IA: The indicator function of a (A): The probability of a setA. fd : The integral of a Borel functionfwith respect to a measure . Afd : The integral offon the setA. f(x)dF(x): The integral offwith respect to the probability measurecorresponding to the cumulative distribution functionF. : The measure is dominated by the measure , , (A)=0always implies (A)= d : The Radon-Nikodym derivative of with respect to.]

8 P: A collection of populations (distributions). : Almost : Almost : A statement holds except on the eventAwithP(A) = 0 for allP P. x: The point mass atx Rkor the distribution degenerated atx Rk.{an}: A sequence of elementsa1,a2, ..an aor limnan=a:{an}converges toaasnincreases to .lim supnan: The largest limit point of{an}, lim supnan=infnsupk infnan: The smallest limit point of{an}, lim infnan=supninfk nak. p: Convergence in probability. d: Convergence in : The derivative of a : The second-order derivative of a (k):Thekth-order derivative of a (x+): The right limit of a functiongatx (x ): The left limit of a functiongatx +(x): The positive part of a functiong,g+(x)=max{g(x),0}.Notationxiii g (x): The negative part of a functiong,g (x)=max{ g(x),0}. g/ x: The partial derivative of a functiongonRk. 2g/ x x : The second-order partial derivative of a {x}: The exponential log(x): The inverse ofex, log(ex)=x. (t): The gamma function defined as (t)= 0xt 1e xdx,t> 1(p): Thepth quantile of a cumulative distribution functionFonR,F 1(t)=inf{x:F(x) t}.

9 E(X)orEX: The expectation of a random variable (vector or matrix) (X): The variance of a random variableXor the covariance matrix ofa random (X, Y): The covariance between random (X|A): The conditional expectation ofXgiven a (X|Y): The conditional expectation (A|A): The conditional probability ofAgiven a (A|Y): The conditional probability (i):Theith order statistic ofX1, .., Xn. Xor X : The sample mean ofX1, .., Xn, X=n 1 ni=1Xi. X j: The average ofXij s over the indexi, X j=n 1 ni= : The sample variance ofX1, .., Xn,S2=(n 1) 1 ni=1(Xi X) : The empirical distribution ofX1, .., Xn,Fn(t)=n 1 ni=1 Xi(t). ( ): The likelihood : The null hypothesis in a testing : The alternative hypothesis in a testing (P, a)orL( , a): The loss function in a decision (P)orRT( ): The risk function of a decision : The Bayes risk of a decision ( , 2): The one-dimensional normal distribution with mean and vari-ance ( , ): Thek-dimensional normal distribution with mean vector andcovariance matrix.

10 (x): The cumulative distribution function ofN(0,1).z :The(1 )th quantile ofN(0,1). 2r: The chi-square distribution with degrees of freedomr. 2r, :The(1 )th quantile of the chi-square distribution 2r. 2r( ): The noncentral chi-square distribution with degrees of freedomrand noncentrality parameter .xivNotationtr: The t-distribution with degrees of , :The(1 )th quantile of the ( ): The noncentral t-distribution with degrees of freedomrand non-centrality parameter .Fa,b: The F-distribution with degrees of ,b, :The(1 )th quantile of the F-distributionFa, ,b( ): The noncentral F-distribution with degrees of freedomaandband noncentrality parameter .: -field: A collectionFof subsets of a set is a -fieldon if(i)theempty set F; (ii) ifA F, then the complementAc F;and(iii) ifAi F,i=1,2, .., then their union Ai F. -finite measure: A measure on a -fieldFon is -finite if there areA1,A2, ..inFsuch that Ai= and (Ai)< for or decision: LetXbe a sample from a populationP.