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# Mathematical Statistics STAT 709

UW

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This 3 page Class Notes was uploaded by Mrs. Triston Collier on Thursday September 17, 2015. The Class Notes belongs to STAT 709 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 33 views. For similar materials see /class/205087/stat-709-university-of-wisconsin-madison in Statistics at University of Wisconsin - Madison.

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Date Created: 09/17/15

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11101111121 011 quotquotZX l X 301 1 3909391 9111111123131 1100111 0113 1111 011 0p M011 3011 11 108120 112101108 0113 111 X 1030011 11101111121 0113 S11 11001211101 81 4 11011111 11111211 11138 31101110118112 81101A0111 0113 81 394 1 X 111011 SMO1101 05 4 Xf 0011011 139211 XHM S 9 21Wquot XWIM p1112 1 gt no 101111 1121 11106 01011061111 0011011 3931 gt 10A011011m 9 gt 05 15 312113 110118 0 lt 31 12 81 010113 0 lt 9 K1112 101 1 10 5311111131100 0113 111013 3931112381100 4 X 10 08120 112100118 0113 1011181100 011 11 AAqu 01 81101111131100 p1112 110111111011 81 f 011 001118 Xf 1g 1 Xf 1g 11 81101111131100 p1112 110111111011 K1112 101 16391 1110100111 111011 SMO1101 111 8111110112 111 3111801 12 8111811 110118111112380 011 11120 1 gooxd 1Xquot lt XVJ S611111111 X lt X 111 XXquot lt XVJ S611111111 X lt X 11 1Xquot XVJ S611111111 X X 1 110111 39Xd 81101111131100 81 1 312113 0801111115 15 ya 03 1g 111011 11013011111 0111121118120111 12 011 1 111112 0012118 531111112110111 12 110 110111101 SX3JGA j 11101111121 011 ZX 1 X X 301 011 1119110911 811101110111 111112111 111 1101380111 81113 03 XGAASUR 1112 80p1A0111 1110100113 81111111121111 81101111131100 3111801 8111111101101 0111 1081108 0111128 0113 111 Xf 03 80310AI11D Xf 81 081108 011108 111 X 03 80310AI11D X 11 80138131238 111 1003 31112310111111 1112 81 11013121111018111211 p01119mg p111 11191109111 S XEISQHIS suonemxogsuen 10 93119319111103 91 9111113911 Theorem 111 Slutsky s theorem Let XX1X2 171172 be random variables on a probability space Suppose that X gtd X and YR gt c where c is a constant Then i X Y gtd X i ii 1an gtd OX iii Xnyg gtd Xc if c 0 Proof We prove only The proofs of ii and iii are left exercises Let t 6 R and 6 gt 0 be fixed constants Then Fx t 1 an YR S f PXn Y S t D Y c lt 6 PYn cf Z 6 PXn 3 t fl PYn fl Z 6 and similarly Pixn5 Z PXn 3 t cf PUTn cf Z 6 If t c t c 6 and t c 6 are continuity points of F X then it follows from the previous two inequalities and the hypotheses of the theorem that FX t c e g lirrigianXnHn 3 lim sup Fxnyn 3 FX t c 6 R Since 6 can be arbitrary why39 Fxvt Fxft 6 The result follows from F X m F X t C An application of Theorem 111 is given in the proof of the following important result Theorem 112 Let X1 X2 and Y be random k vectors satisfying WAX c gtd Y 1 where c 6 Rk and an is a sequence of positive numbers with limHoo an Let g be a function from Rk to R i If g is differentiable at i then anMXn 9M gtd VWWYZ 2 where Vgx denotes the k vector of partial derivatives of g at 3 ii Suppose that g has continuous partial derivatives of order m gt 1 in a neighborhood of f with all the partial derivatives of order j 1 g j g m 1 vanishing at i but with the mth order partial derivatives not all vanishing at c Then 1 k k 8mg IZ MXR 9M gtd w i 2 32m 3 16 I I I Elfin 1 l imzl where is the jth component of Y Proof We prove only The proof of ii is similar Let Z rzngXn gci rznVgciTXn c If we can show that Z 0 1 then by 1 Theorem 19iii and Theorem 111i result 2 holds The differentiability of g at c implies that for any 6 gt 0 there is a 55 gt 0 such that gx gci Vgclx c 3 c 4 whenever c lt 55 Let 7 gt 0 be fixed By 1 0an Z 7 1 S 1 0an ill 2 5e Pl lnlan Z7 1 Since an gt 30 1 and Theorem 111ii imply X7 gt C By Theorem 110iii 1 implies anHXn c gtd Without loss of generality we can assume that 776 is a continuity point of FEDH Then limnsup PZn Z n PXn c 2 55 PrznXn c 2 776 PlllYH 2 776 The proof is complete since 6 can be arbitrary In statistics we often need a nondegenerated limiting distribution of r1ngXn gc so that probabilities involving rzngXn gci can be approximated by the cdf of VgcTl if 2 holds Hence result is not useful for this purpose if Vgc 0 and in such cases result 3 may be applied A useful method in statistics called the delta method is based on the following corollary of Theorem 112 Corollary 11 Assume the conditions of Theorem 112 If Y has the Nk0 2 distribution then rzngXn gci gtd N 0 VgcT2Vgc Example 131 Let Xn be a sequence of random variables satisfying FAX c gtd NO 1 Consider the function 332 If c 7 0 then an application of Corollary 11 gives that c2 gtd N04c2 If a 0 the first order derivative of g at 0 is 0 but the second order derivative of g E 2 Hence an application of result 3 gives that nXg gtd N0 12 which has the chi square distribution x Example 114 The last result can also be obtained by applying Theorem 110iii

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