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G.IVCHENKf YU.MEDVEDEV A.CHISTYAKOV + PROBLEMS IN MATHEMA TICAL I - --+-- - - MIR PUBLISHERS MOSCOW ~ PR01JLEMSIN MATHEMATICAL STATISTICS r. lil. HB~BHko, 10. iii. MBABBABB, A. B. LfHCTfiKOB C60pHHK ·3alla ll no MaTeMaTHlleCKOH CTaTHCTHKe "SblCUl3R WKona~ MOCKB3 G. I.Ivchenko, Yu. I.Medvedev, A. V. Chislyakov P/(OBI.EMSIN MATHEMATICAL STATISTICS Mir Publishers Moscow l'ranllatal from the Rus$ian by Elena Trcshneva FitSot published 1991 Revised from Ihe /989 Ruuian edirlon Ha IIH'?flUU(WOM If:Jb'~ Printcd in 'hI!! Union 0/Soviet Socialist R~publiN ISBN s-oJ-otl5JW e r. 11.11"'lCHJ(O, 10.11. Me,D.DC,l(CIl. ISBN 5-OlS-OOOO4,.... r\.B.'fHcrllJ(OB. 1989 @ Enjlish rranslarton, E.Troshneva. 1?91 Contents Preface IS Theory and Problems 1 Chllpler I PriDeIp~ ofStatbtbl Description. S&mpIlna~ lies 'illS Their Dl~tributloal 7 ClIapler1 Esumatioo of D!Slrihutlon Paramttt~ J7 Estimators llnd :rtl.eir General ProPCrlKs 45 Optimum Estimators 56 Mall/mum Uke!ihood estimates ci6 Confidence Estimation 71 Chapter 3 TeSlSof Stalll1tltal Hypotheses 79 Goodnen of Fit nsts 87 A Choice Bc:ty..cen 1\vo Simple Hypotheses 96 Compq§ill: Hypotheses 99 lestsof HypOtheses and Confidence EMinuuion lOt Likellhood ~tio lest 102 Various Problems 104 Chapkr 4 UDell, RegreSsion and Ihe Last Squlrfl Melbod 107 Chapin S Decision'Funetlons 119 Chapter 6 Slatbitip; of Stationary Sequences 131 Answers aDd Solutions 137 Appendix 263 L~l of Dlslrib<llkltls 277 BlblloCn.phy 279 PrefiJce This problem book coversall thetraditionaltopicsinmodernsaatisti caltheoryandisdesigned forstudents-attechnicalcollegesand univer sities who have mathematical statistics as an obligatory course. The problems arc mostly analytical The student is asked to prove the validity or an assertion or carry-nut an investlgauon. This will help him grasp the main aspectsor mamemaucatstatistics. Some of the problemsare more difficult and can be used M individualassign menu for course papers. Wehaveincluded problemson computersimulationofrandomvari ables in order to obtain the data for statistical interpretation. Any "theoretical" problem which contains a starisrica!algorithm for data analysiscanbe used(with theappropriate(practicallyinnnite) choice of the model parameters) to formulate a "practical" problem. At the first stagetheoriginaldatashould besimulated usingeitherpublished tables of random numbers or special computer programs. Then, by Interpreting these "experimental" results according to the algorithm in question, the student can compare the theoretical hypothesis with the originalparameters which are known as they were used when the sample was simulated. All the problems differ in complexity. More diffiCUlt problems are marked with an asterisk and may require a significant effort on the part of the reader. Problems that cannot be reduced to standard al goruhms are answered in detail or hints are given. Each chapter contains the basic nouons, assertions, and formulas fromtherespectivetheoreticalsection.Thestatisticaltablesat theend of the book will help the reader Obtain numerical results. The list of distributions will help him choose problems on different aspects of the same model. The Authors THEORY AND PROBLEMS CHAPTER 1 Principles of Statistical Description. Sampling Characteristics and Their Distributions 1.1.'Problems in mathematical statisticsare based on seausricaldata obtained by observations on a finite set of random variables X _ (Xl, ..", X~) which describe theoutcome of an experiment. We sa)' that theexperiment tonsis~ of n trials. where the ilh trial results in a randomvariableXi. J- I, ...,n.Asetofobservablerandomvaria bles X _ (X" ..., X,.) i~ ~Iled asample, the values XI, I = I, ..., fl. are called the elements (units) ofa sample. and the number n is called the samplesize. A set~r- Ix = {Xl, ...•x...)I of all possible x: "" realizationsofthesample (Xl• ..., X..)iscalledasamplespace. When the True distribut.ion of X (the distribution function FlI.(Xl, •..,x,,)- P(X, '" XI, .._, X" 0;;; x,,»is unknown (completely or partially) and only the class (family) of admissible distributions :;F=: IF{xl• ...•X~)I which contains thedjstribution F" of the sam ple Xisspecified. thenwehavea stattsttcatmoaet(;-Y.;.9"'(or simply model,~). Mathematical statistics reveals (within a given model.7 ) the properties of the true distribution Fx using the results of obser various on the sample X. Someexpenmemsconsistofrepealed independent observationson a random variable ~ (with the distribution L(~». Then the sample X _ (X,•...,X~)isIIsetofindependentsimilarlydistributed random variables. where f (X1) ,., ..({a. I'" I. .... n. 10 be concise, we say that.X '" (Xl, ...•Xn) is a samplefrom the distribution / {O. The statisticalmodel for repeated independent observations is written as .'7= IFf~ i.e., we ~n1y indicate:the class of admissible disteibution functions of the original random variable ~. If,:/-= IF(X;(/). {JESJ, t.e.. the admissible dlstrtburlcn functions aredefined up toaparameter'8,rbenthemodelissaid tobeporometrle. e and the set of the possible values of 6 is called a parametricset. Wewillonlyconsiderabsolutelycontinuous ordiscretemodelsand use!t(x}'"f {.n U(;c,6)forparametricmodels)to denote thedistribu- • THeory and Problems crr nondensityoftherandom variable thedistribulionFtisabsolutely continuous, and the probability P(f '" xl if it is discrete. In the case of a parametric model the distribution of probabitines onasamplespace.:Y"-which corresponds10 the parameter0isdenoted P~. Similarly, E,T(X), D.T(X) are used 10 denote the moments of a given function T(X) ofthe sample X when F"(x; 8) is the distribution (unction of the sample. . 1.2. Many problems in mathemattcaf statistics concern sequences ofrandom variables ('1~Iwhichconverge to a limitT/(a random varia ble or a constant) as n .... 00. We will use two forms of convergence, i.e., convergence in probability ('l~ .!. 1J ... PO".. - "I> &) ..... 0 ~e > 0) and conver&5J1ce in distribution, Of weak convergence (.L'(",,) - 1 (.,)or "..- ., ~ F••(x) - F,(x) VXEC(F~), where C(F) is the set of points of continuity of the function F(x». Note: that the: P.convcrgence implies the .zeconvergence, The inference on the p. convergenceofvarioussampling characteristicsoften follows from the generalass~rtjo.n on~econ\lergenc~offunctions ofrandom vUiabl~s 17. P. 27J. I.e.. if""I... c/ '" const, 1= 1,• ..., r. and ~(xl•...• x,) IS an arbitrary function continuous in,fhe neighbourhoodofthe point (c" ...• c,), then ""<'II"" ..., '1",) ... ~(c,• ..., c,). 1.3. If X :. (X" .._,X..)isasamplefrom a distributionL (E).then F~(x) = F(x) is called a theoretical disrribution function. and • F,,(x) = J_j,,n(x_) 0:: -nI ~".".',."1.(X; ~ x) (1.1) isan emptrtcatdistributionfunction (here ,I(IO(x) is the number orete mems in a sample, which satisfy the condition X; " x, and I(A) is the indicator of the event A). Bythe Bernoulli theorem, F,,(x) !.F(~)vxasn -- co. i.e., for large n the value of F..(x) can be an estimate for F(x). The Glivenko and Kcbnogprcv theorems on the asymptotic properties ofF..(x) for large n [7, p. 22] prove that the empirical distribution function can be an estimator for the theoretical distribution function. If a random variable ~ is discrete and assumes the values QI. 01• ...• then thedislributlon law for ~maybe convenientlyrepresent ed byIhefrequenciesh,ln,where,11,isthe.numberofunitsinasample, which areequal to a,.Thenh,ln ... PeE _ a.), r "'" 1.2•._..as n ... co. If the values of Ehave the density J((x) ""f(x), we may investigate the frequencies hkln of the events [~e.4..1, where f.6kI is a system e.. of nonintersecting intervals into which the region of the possible