NOTESFORQUANTITATIVEMETHODS

1 NOTES FOR QUANTITATIVE METHODS:SOME ADVANCED MATHEMATICS FROM AN ELEMENTARY POINT OF VIEWMAXWELL B. STINCHCOMBEC ontents0. Organizational Stu 41. Some Basics About Numbers and Lengths and Why we want Valuing sequences of Convex Self-guided tour to di erentiability and concavity112. Some Basic Results in Metric Probability distributions as cdf' Compactness and the existence of The Theorem of the The Separating Hyperplane Problems203. Dynamic Programming, Deterministic and Compactness and continuity in spaces of Deterministic Dynamic Stochastic Dynamic Problems254.

2 An Overview of the Statistics Part of this Other properties of Classical statistics29 Date: November 25, Semester, 2003. Unique # B. An Information Mis-Speci Problems325. Basic Probability, Transformations, and Basic Probability and Transformations and Problems346. Some Continuous Uniform distributions,U[ 1; 2] The normal or Gaussian family of distributions,N( ; 2) A useful The gamma family, ( ; ) Special cases of ( ; ) Cauchy random Exponential Some (in) Problems427. Random Vectors, Conditional Expectations, Dependence, conditional probabilities and Causality and conditional Independence, sums of independent rv' Covariance and Bivariate A pair of discrete, portfolio management The matrix Problems518.

3 Sampling Distributions and Normal Approximations529. Su cient Statistics as Data Su cient Problems5610. Finding and Evaluating The basic Gaussian Some examples of nding Problems6011. Evaluating di erent estimators61 QUANTITATIVE Mean Squared Error (MSE) Desirable properties for The Cram er-Rao lower Problems6212. Hypothesis The perfect power function and types of Some generalities about the probabilities of the di erent types of errors The Likelihood Ratio Con dence intervals,p-values, and hypothesis Problems684 MAXWELL B.

4 StuffMeetings: Mondays and Wednesdays, 2-3:30 and Wednesdays 8:30-9:30,in BRB : Myo cehoursMondays andWednesdays 10-12. Youarevery lucky tohaveLoriStuntz for this course. Her o ce is , e-mail address cehours : Forthestatisticalpartofthecourse, we'll useGeorgeCasella andRogerBerger'sStatistical Inference, 2'nd ed. (Duxbury 2002), following it fairly closely. For the optimiza-tionandanalysispartsofthecourse ,we' 'sApplied ProbabilityModelswith Optimization Applications(Dover Publications, 1992) and A. N. Kolmogorov and 'sIntroductory Real Analysis(DoverPublications, 1970).

5 Throughout, youwillberefering to themicroeconomics textbook,Microeconomic Theory,byMas-Colell, Whinston,and : Completeness properties ofRandR`, summability and valuation of streams ofutilities, convex analysisandduality; furtherpropertiesofR`andrelated spaces, (includingcompactness, continuity and measurability of functions onR`, summability of sequences,existence of optima, xed point theorems, cdf's, other metrics, other metric spaces, theTheorem of the Maximum); Probabilities and expectations (including domains, modes ofconvergence, convergence theorems, orders of stochastic dominance, conditional expecta-tionsandprobabilities);Dynamicpr ogramming(includingpropertiesofsequences pacesandprobabilities on them, Bellman and Euler equations, the role of the Theorem of the Max-imum, growth models).

6 Statistics (including speci c distributions [uniform, gamma, beta,Gaussian,t,F, 2, Poisson, negative exponential, Weibull, logistic], estimators and theirproperties[consistency, Glivenko-Cantelli, di erent kinds of\best"estimators, Bayesian es-timators, MLE estimators, information inequalities, su ciency, Blackwell-Rao], propertiesof hypothesis tests [types of errors and their associated distributions, the Neyman-PearsonLemma]).QUANTITATIVE Basics About Numbers and QuantitiesReadings: Marinacci's \An Axiomatic Approach to Complete Patience and Time Invari-ance,"Journal of Economic Theory83, 105-144 (1998).

7 Mas-Colell, Whinston, and Greenon support functions and the supporting hyperplane below is for you to readand work on, either by yourself or in a study and elementary school. As models of mea-surements of quantities, we're we want of easily described lengths, clt and inQ, convergence implies settling down, but not the reverse. sequences andRas the completion of completeness: decreasing and increasing bounded sequences have limits,equivalently, every bounded set hasa sup and an inf. The idea ofcompletion also shows upin the major limit theorem in statistics ( the CLT).

8 Sequences of section is based on classic analyses as well asthe more recent Marinacci's \An Axiomatic Approach to Complete Patience and Time In-variance,"JournalofEconomicTheory83,1 05-144(1998). Patienceabout nitesequences,(r1;r2;:::;rt),ofrewardsse emstobeaboutbeingindi erentbetweenalltimepermutationsof the sequence. In the dynamic programming models used in game theory and macro, oneoften achieves in nite sequences of rewards. These may not beentirely believable, but theydo a pretty good job of capturing the idea of an inde nite limsuptrt, equality for , 's 23,R`, :R!

9 R,u(x):=(u(xt)t2N)2RN,ubounded. From here on, we'll just usexfor theelements inRNand try to value them, thinking that they are bounded streams of (x):=liminftxt(thatis,liminftu(xt))which alwaysexists(bycompleteness). Thein nite extension of a simple of idea of patience is here | any permutation of the integersfails to changeVliminf(x).Thinking about a nite sequence of rewards, a useful de nition of patience is that anypermutation of the reward sequence is indi erent, having the good stu early is just asdesirable as having it late.

10 If :N!Nis 1-1 and onto, thenx denotes the sequence(x (t))t2N. is any permutation ,Vliminf(x)=Vliminf(x ).6 MAXWELL B. STINCHCOMBEO ther ideas of patience include taking the limit of time averages and discounting with adiscount factorclose (x):=limT1 TPTt=1xtexists i V2(x):=lim "1(1 )P1t=1 txtexists, in which caseV1(x)=V2(x) (FroebeniusandLittlewood).The sequence ofxt0101010101010101 hasVliminf(x)=0<V1(x)=V2(x)= 's extension of the classic sequence ofxt11|{z}210000|{z}2211111111|{z}23 is of the formx for a peculiar kind of.