Introduction to Statistical Inference
Introduction to Statistical Inference STAT 610
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This 6 page Class Notes was uploaded by Mrs. Triston Collier on Thursday September 17, 2015. The Class Notes belongs to STAT 610 at University of Wisconsin - Madison taught by Yi Chai in Fall. Since its upload, it has received 434 views. For similar materials see /class/205071/stat-610-university-of-wisconsin-madison in Statistics at University of Wisconsin - Madison.
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Date Created: 09/17/15
STAT 610 DISCUSSION 6 TA Yi Chai Of ce 1335D MSC Email chaiyistatwiscedu Webpage httpwwwstatwisceduchaiyi Of ce Hours 200 400pm Wednesday or by appointment 1 Summary Maximum Likelihood 7 Likelihood function Lm0 is just the pz0 which is the density function in continuous case or the frequency function in discrete case 7 Maximum likelihood estimate is the estimate of the parameter 2 such that new 111mm 0 0 e o maxLm0 0 e 9 when such a 9 exists 2 Examples Example 1 2221 kiefer Wolfowitz Suppose X1 density 9 m 7 u 1 E4 lt 0 gt EWW I where 4p is the standard normal density and t9 1472 E 9 01472 700 lt u lt 000 lt 02 lt 00 Show that maximum likelihood estimates do not exist but that supapzl02 SupMU2p 1102 if and only if 1 equals one of the numbers m1 zn Assume that z 7 z for i j and that n 2 2 Example 2 2239 Let X denote the number of hits at a certain Web site on day i i 1 n Assume that S 2 X had a Poisson 73n distribution On day 71 1 the Web Master decides to keep track of two types of hitsmoney making and not money making Let and Wj denote the number of hits of type 1 and 2 on day j j n 1 n m Assume that 1 Egg V and 2 Egg W have 73mA1 and 73mA2 distributions where 12 Also assume that S S1 and S2 are independent Find the MLEs of 1 and 2 based on S S1 and S2 Example 3 2240 Let X1 Xn be a sample from the generalized Laplace distribution with density Xn is a sample from population with flt70 exp7z01 z gt 0 expm02 1 70170 01 02 m m lt 0 where 0739 gt 0 j 1 2 a Show that T1 ZXl1Xi gt 0 and T2 Z 7X1Xi lt 0 are sufficient statistics b Find the maximum likelihood estimates of 01 and 02 in terms of T1 and T2 Carefully check the T1 0 or T2 077 case STAT 610 DISCUSSION 1 TA Yi Chai Of ce 1335D MSC Email chaiyistatwiscedu Webpage httpwwwstatwisceduchaiyi Of ce Hours 230 430pm Wednesday or by appointment 1 Summary 0 Parametrization is a map 9 a P9 from the parameter space 9 to P If the parameter space 9 is a nice subset of Euclidean space R then the model P is called parametric otherwise it is called nonparametric A nonparametric model having a parametric component is usually called semi parametric o Identi ability If the map 9 a P9 is oneto one that is if 01 7 02 P91 7 P92 then this parametrizations are called identi able otherwise it is called unidenti able 0 Regular model If P9 is a parametric model and satis es either 1 All of the P9 are continuous with density px 0 2 All of the P9 are discrete with the frequency functions px0 and the set x1 x2 E x px 0 gt 0 is the same set for all 9 E 9 2 Examples 0 Example 1 111 Give a formal statement of the following models identifying the probability laws of the data and the parameter space State whether the model in question is parametric or nonparametric d The number of eggs laid by an insect follows a Poisson distribution with unknown mean Once laid each egg has an unknown chance p of hatching and the hatching of one egg is independent of the hatching of the others An entomologist studies a set of n such insects observing the number of eggs laid and the number of eggs hatching for each nest 0 Example 2 112 Are the following parametrization identi able Prove or disprove b The parametrization of Problem 1d c The parametrization of Problem 1d if the entomologist observes only the number of eggs hatching but not the number of eggs laid in each case 0 Example 3 113 Which of the following parametrizations are identi able Prove or disprove d Xij i 1 p j 1 b are independent with Xij N Naij02 where Mj 1 on 70 041 ap 1 9 1102 and P9 is the distribution of X11 Xpb e Same as d with 041 ap and 1 Ab restricted to the sets where ELI on 0 and 17 271 739 039 0 Example 4 116 Which of the following models are regular Prove or disprove b P9 is the distribution of X when X is uniform on 0 1 2 0 9 1 2 c Suppose X N Na02 Let Y 1ifX 1 and Y X ifX gt1 9 472 and P9 is the distribution of Y STAT 610 DISCUSSION 4 TA Yi Chai Of ce 1335D MSC Email chaiyistatwiscedu Webpage httpwwwstatwisceduchaiyi Of ce Hours 200 400pm Wednesday or by appointment 1 Exponential families o If there exist real valued functions 7707 130 on the parameter space 97 and real functions T and h on R 7 such that the densityfrequency functions 19m7 0 of the P9 may be written 29907 0 WE exp 770T90 309 The family of distributions P9 is said to be an one parameter exponential family 7 TX is sufficient for 0 7 7 T and B are not unique 0 Theorem 162 If the densityor frequency function of X has the canonical form 6190 77 hexp 77T95 140 and 7 is an interior point of the parameter space7 then the MGF of TX is given by M8 exp W8 71 AW for s in some neighborhood of O Moreover7 ETX 1401 VarTX WWI 0 An exponential family is of rank k if and only if the generating statistic T is k dimensional and 17 T1X7 TkX are linearly independent with positive probability 0 Theorem 163 Suppose 73 qzn 7 E 6 is a canonical exponential family generated by T7 h and 6 is open7 then the following are equivalent i 73 is of rank k ii iii 7 is a parameteridenti able VanT is positive de nite iv 7 a is 1 1 on 8 v A is strictly convex on 8 2 Examples 0 Example 1 1418 Assume that the conditional density of Z0 given Y0 yo is Nu 1 33002 Z Z0 MV E Y 5300 a Show that the conditional density of Z given Y y is Ny7 1 b Suppose that Y has the exponential density 7ry AexpPW gt 07y gt 0 Show that the conditional distribution of Y given Z 2 has density 77 71 1 2 7rylz27r 20 exp iglyzikl 7 ygt0 where c ltIgtz 7 x c Find the conditional density 7r0yolzo of Y0 given Z0 20 d Find the best predictor of Y0 given Z0 20 using mean absolute prediction error Ell079Z0 e Show that the best MSPE predictor of Y given Z z is EYlZ z 51 A 7 z 7 7 2 Example 2 1611 Obtain moment generating functions for the sufficient statistics when sam pling from the following distributions b gamma7 F 7 t9 7 p xed Example 3 1618 Suppose Y17 Yn are independent with Y N N l 1 6221702 where 21 2 are known co variate values not all equal Show that the family has rank 3 Give the mean vector and variance matrix of T Example 4 1631 Conjugate Normal Mixture Distributions A Hierarchical Bayesian Normal Model Let WJj 1 lt j lt k be a given collection of pairs with W 6 R777 gt 0 Let p70 a random pair with Paa Mg 7 Aj 0 lt j lt 17 21 j 1 Let 9 be a random variable whose conditional distribution given a7 a W777 is N1jr2 Consider the model X 9 67 where 9 and e are independent and e N N07 737 08 known Not that 9 has the prior density k 7amp0 2 Wm 0 7 M 1 73971 where 17 denote the N07 r2 density Also note that X10 has the Nt97 03 distribution a Find the posterior k 740195 ZPUMU M jllml elelewl 73971 and write it in the form j gt7jm 0 7 MM k 1 u for appropriate jm7 7 7z and lagm This shows that 1 de nes a conjugate prior for the N0ag distribution b Let X 9 q 1lt i lt n where 9 is as previously and 61 767 are iid N0ag Find the posterior 7rt9lz17 7z 7 and show that it belongs to class STAT 610 DISCUSSION 7 TA Yi Chai Of ce 1335D MSC Email chaiyistatwiscedu Webpage httpwwwstatwisceduchaiyi Of ce Hours 200 400pm Wednesday or by appointment 1 The EM Algorithm 2 Bayes Procedures 0 Example 3 329 Suppose we have a sample X1 0 Example 1 241 EM for bivariate data 7 In the bivariate normal Example 246 Complete the E step by nding EZilYi and 7 ln Example 246 verify the M step by showing that E9T M17 M27 0 M 03 Viv30102 mm 0 Example 2 2417 Limitations of the missing value model of Example 246 The assumption underlying Example 246 is that the conditional probability that a component Xj of data vector X is missing given the rest of the data vector is not a function of Xj That is given X7 Xi the process determining whether Xj is missing is independent of Xj This condition is called missing at random For example in Example 246 the probability that Y is missing may depend on Z but not on Y This is given Z the missingness of Y is independent of Y If Y represents the seriousness of a disease this assumption may not be satis ed For instance suppose all subjects with Y 2 2 drop out of the study Then using the E step to impute values for the missing Y s would greatly underpredict the actual Y s because all the Y s in the imputation would have Y 2 ln Example 246 suppose Y is missing iff Y 2 2 If ag 15 01 02 1 and p 05 nd the probability that underpredicts Y r6lm infralm a E A Xn of differences in the effect of generic and namebrand effects for a certain drug where EX 0 A regulatory agency speci es a number 6 gt 0 such that 9 E 76 6 then the generic and name brand drugs are by de nition bioequivalent On the basis of X X1 Xn we want to decide whether or not 9 E 76 6 Assume that given 0 X1 Xn are iid N0 03 where 03 is known and that 9 is random with a Nn0 7393 distribution There are two possible actions a 0 ltgt Bioequivalent a 1 ltgt Not Bioequivalent with losses 00 and 01 Set 1 2 2 w zs071s 1 riexp 72720 0 gt0 C where 0 lt r lt 1 and logr 73126 a Show that the Bayes rule is equivalent to Accept bioequivalence if E0lX z lt 077 and show that it is equivalent to 2 2 Accept bioequivalence if lt T301 62 log lt 2 7 TO 71 where 2 1 71 MM W lt17 man w TamT0 Tan 2 if 0 10 It s proposed that the preceding prior is 77uninforrnative77 if it has no 0 and 73902 large 73902 a 00 Discuss the preceding decision rule for this prior c Discuss the behavior of the preceding decision rule for large n n a Consider the general case a and speci c case
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