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# TPC PSCI 5108

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This 11 page Class Notes was uploaded by Modesto Renner on Thursday October 29, 2015. The Class Notes belongs to PSCI 5108 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 13 views. For similar materials see /class/231929/psci-5108-university-of-colorado-at-boulder in Political Science at University of Colorado at Boulder.

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Date Created: 10/29/15

PSC151087108 Statistical Distributions and Maximum Likelihood Estimation Review of Probability Random Variable rv 7 A variable whose value is determined by a chance experiment 7 A function that associates a point in a sample space With a number 7 Def sample space a set ofall possible outcomes of an experiment Two types of random variables 7 Discrete a random variable X is discrete if the number of values it can assume is countable 7 Continuous a random variable Z is continuous if it takes on any value Within some speci ed interval Convention a random variable is denoted by a capital letter while realizations of the random variable are denoted by lower case letters X7 X2 Probability distribution a function that associates an event the realization ofX denoted X with the probability that X occurs Discrete Probability Function Let Xbe a discrete random variable and assume that the Values it can take are X7 X2 H X De ne fXx PXxfori12 n 0 forx 2 x as the PDF ofX In generale is a probability function if Ark 20 and Emma where the summation is over all points x1 xb xn Important Discrete Tquot quot stcme H F ufnset m equzlly 11er Example sps on cam duws mm a deck of suck wh pxubnbxhty CF17 The pmbabdxty miss functan is a f 1Xp lipquot p q if Bmumulesmbuuun descnbes the numbex Dfsuccesses m a suns ufmdependent mus LEI unis In undomvunbh x scum a bumme summon mmpumm p we when gammy urg mnggmay k u PX k myquot lt lw rm u a hnmnlcueF cmntwhchmxds nchuuie w wm 7 Eat m mimmporp mm mg values in mm m Mg m mum p mm dnsmbmm umh in 39 Poisson Distribution arises in connection With Poisson processes It applies to discrete phenomena those that occur 0 imes during aglven period Oftlme The probability of an occurrence during this period must be constant e Exampl s Number utsanetiuns imposed by euuntry x L Martin Number utuses utturee by npxesidenta Gown Number ufvetus used by pxesident Number utruadlall on npnmculustxetch thighwzy e is KN k 1 k ere ais the base utthe natural lugnmhm e2 71m tel is the taetunal utlt lis npusmve realnumber equaltu the expectednumbex utueeurrenees that ueeur dunngthe given intexvzl Continuous Prob ability Function The probability that a continuous random variable takes on a particular Value is in general zero A function22 where z is a continuous random Vanab e is de d as a continuous random Variable itt fzz20forallz and LfAZMZ Thepmbah39lity that the realization on lies betwem two Values a ampb is 11a ltZltl7J bfzzdz Cumulative Density Function A CDF gives the probability of a random Variable being less than some Value FXO is the CDF of a random Variable X where Mr PX g a For adiscrete random variable Fro z fro For a continuous random variable FAX PX gar Peoo g X no Lfrmdx Properties of the CDF Munro and FXN1 n m then WE Mb 39 PXgtX139FX Pwgtxgtxnynx Example of a CDF Important Continuous Distnbutions m cqu ly 111me Em dssmbunm on 01 m umfmmxs z spmlmc teammate pmbzb mzs of sums Ht 0 1H 7 Wl39lmlampbmtbtgxamthnnnqlltnl mm Mm 111 ch mm dum nmm xsuxful Continuous distnbuhuns on m it w interval 7 Expmmmi distnbuhun dzscnbes time hemquot consecutive uxe undnm events no em uxe undnm events occuxwith 2 Dry 7 Log numzl distnbuhun descnbmg vnnnbles which can be modelled is the 9mm Dfmnny small independent PM vannbles Normal Distribution Norm 31 Distribution X N WM k And hasthePDF 39 1 f I eXp X Jun where roogtxgtoo Notes 39 The standard normal PDF is denoted as 00 39 And the standard normal CDF is D06 Because of symmetry CIgt7x 17 Clgtx The Cl eSquated Dist bution LetZlZZ ane ndependent standard normal wrxables Then 9 6 E a 1 degrees thh freedom Introduction to 1VIainnuIn Likelihood M wumHH n approach to produce esumators thh demble statistical erues The general theory ofmaxxmum hkehhood eehmahon provides a structure thh whxch to denve esumators and standard errors You have need u alread Ls e sts means etc Developed by Sn Ronald Fisher when he was 22 and an undergraduate gt0 8 Example binomial We mnuo eshnme the prubabdxty ngemngnhud on a pnrhculnr mp on com The com o mopenm hmesn1U mdwe get me fulluwmg results HHTHHHTTHH The prubabdxty ufubtmnmgths sequence 5 a funchm onhe unlmuwn parameter n mamapmmmy KHHHHHHHH m embomuenxlen 717lrrl Thus 5 me prubnbxhty ufubtmmg Dursnmple m advance of cuuechngthe dm The Likelihood Function I The parameter TI is fixed but since we do not know what it is it is unobserved we treat it as ifit Varies 7 treating probability of observing the data coin flips as I We can rewrite the probability function as a likelihood function Lparameterdam LIr i HHTHHHTTHH 1 n Key difference between probability and likelihood function 7 The probability function is a function ofthe data wmh the parameter Tr fixed 7 The likelihood function is a function of the parameter Tr wmh the data fixed The whole zdm ofz ezJood zr to nd thepammeter but mmmzer bepmbabzzg oftye alarmed dam The Likelihood Function We can gmph the likelihood function state commands set oh gen 2 allows p to m vary from m e i EPA i T A3 generates the iikeiiheea based on the above formula geepm L P cu em Lls atthe maximumwhm P 7 39 m The Maximum Likelihood Estimator Note that the likelihood ofobtaining the sample results that we have is small regardless ofthe value ofTr max value ofL is 0022236 This is usually the ease as any specific sarnpie has a low probabiiity The likelihood provides useful information about the parameter Tr 7 is unlikely to be close to zero or one and eannotbe eitherzero or one 7 The value otrrthatis rnost supported by the datais the one forwhich the iiireiihood is greatest It is the mamnm 101th 91mm MLE Maximizing Likelihood Functions Prior example used simulation to find the value of1T that maximized the function This is hardly efficient Coin ips follow abinorniai distribution Bernouiii triais v1 for a head and Y0 for a tail LetPP 139 1 Flip coin n times to obtain a sample Y n assumption is that tlips are independent and that the value ofp is fixed aeross Hipstriais From the obtained sample we want to estimate Recall that p is the population mean Eco PY1 1 P0150 0 Heads Tails FYi1 p To estimate the population mean we usually use the sample 1 quot mean Y ZY n iei Note that the proportion ofheads looks similar 1quot p EY si Let L be the llkellhood for observatlonl Xhat ls the llkellhood for the 139 tossgt For each obsematlon we could get a head whlch happens ml probablhty p or we could get a tall whlch occurs wlth probablhty L1 py lp Ifwe have lldsampllng then Iypy lrp 9 Y lrp pl39 lrp 39yquot pz 1 PH We usually take logs because they are much 62516 to deal ml 7 Recall 2quot b lna and mam lna lnb Whlch glves us lnzM ilnL i Inlpz 1 7 mm ZWIKlanHK n rp Key chooslng a value for that maxlmlzes 1n Use calculus to nd the maxlmum ofsome functlon wlth respect to some parameter take the derlvatlve of the functlon wlth respect to that parameter and set the result equal to zero Then solve for the parameter To make sure that the value 15 the maxlmum check to make sure that the second derlvatlve IS negatlve Forexample Conslder the functlon fx x22x dfx72x2 ix Settlng thls equal to zero and solvlng forx glves x1 Checkmg second derlvatlve shows we have a maxlmum deOC 7 dean 7 72 dzx dx Retummg to our 11kel1hood funcnon d1nL 121170710 ti 4 p 17p s1neus1ngthe chem rule ha 7 1 do a 1 1 1 7 171 n Settmgthedenvatwetozem git 917p andsolvmgforp immenemexm 11 M17 ixi rppl wil Liy n1 Properties ofMaXimum Likelihood Estimators ML est1mators are cons1stent ML est1mators are asymptoncally unb1ased though they may be b1ased1n small samples ML est1mators are asymptoncally ef uent 7 no asymptoncally unb1ased est1mator has a smaller asymptonc Vanance

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