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1 Revew for the prevous lecture Cocepts: order statstcs Theorems: Dstrbutos of order statstcs Examples: How to get the dstrbuto of order statstcs Chapter 5 Propertes of a Radom Sample Secto 55 Covergece Cocepts Defto 55: A sequece of radom varables,,,, coverges probablty to a radom varable f, for every ε > 0, P( ε ) = 0, or equvaletly, P( < ε ) = Theorem 55 (Weak Law of Large Numbers): Let,,,be d radom varables wth Var = σ < Defe = Proof: Use Chebychev s Iequalty = The for every ε > 0, P( μ < ε ) = E = μ ad

2 Example 553 (Cosstecy of S ) Let,,,be d radom varables wth E = μ ad Var = σ < ad defe ( S = = ), ca we prove a WLLN for S? Usg Chebychev s Iequalty, we have P E( S σ ) Var( S) ( S σ ε) = ε ε Ad thus, a suffcet codto that S coverges probablty to σ s that Var( S ) 0 whe Theorem 554: Suppose that,, coverges probablty to a radom varable ad that h s cotuous fucto The h ( ), h ( ), coverges probablty to h ( ) Example 555 (Cosstecy of S ) If s a cosstet estmator of σ, the by Theorem 554, the sample S stadard devato 5) S = S s a cosstet estmator of σ Note that E( S ) s a based estmator of σ (Exercse Defto 556: A sequece of radom varables,,,, coverges almost surely to radom varable f, for every ε > 0, P( < ε ) = Example 557 (Almost sure covergece) Let ( ) s = s+ s (0 s ) ad defe P as the uform probablty dstrbuto o the sample space S = [0,], the coverges almost surely to the radom varable () s = s(0 s )

3 Example 558 (Covergece probablty, ot almost surely) Let the sample space S be the closed terval [0,] wth the uform probablty dstrbuto Defe the sequece,, as follows: () s = s+ I[0,] () s, () s = s+ I[0,/ ] (), s 3 () s = s+ I[/, ] () s, () s = s+ I /3]() s, () s = s+ I 3] () s, () s = s+ I ] () s, 4 [0, 5 [/3,/ 6 [/3, Let ( s) = s The t s straghtforward to prove that coverges probablty to but ot coverges almost surely to However, we ca fd a subsequece of that coverges almost surely to Theorem 559 (Strog Law of Large Numbers) Let,,,be d radom varables wth Var = σ < Defe = = The for every ε > 0, P( μ < ε ) = E = μ ad Defto 550: A sequece of radom varables,,,, coverges dstrbuto to a radom varable f F ( x) = F ( x) for all pots x where F ( x) s cotuous Example 55 (Maxmum of Uforms) If,, are d uform (0,) ad ( ) = max, the the radom varable coverges probablty to ad the radom varable ( ) ( ) coverges to a ( ) expoetal () radom varable 3

4 Theorem 55: If the sequece of radom varables,,,, coverges probablty to, the the sequece coverges dstrbuto to Theorem 553: The sequece of radom varable,,,, coverges probablty to a costat μ f ad oly f the sequece also coverges dstrbuto to μ That s, the statemet P( μ > ε ) = 0 for every ε > 0 s equvalet to 0, x < u, P ( x) =, x > u Theorem 554 (Cetral Lmt Theorem) Let,, be a sequece of d radom varables whose mgfs exts eghbor of 0 (that s, ad σ are fte sce the mgf exsts) Defe for ay < x <, M () t exsts for t < h, for some postve h ) Let E = μ ad Var = σ > 0 (Both μ = Let G ) deote the cdf of ( x ( μ)/ σ The, x y G( x) = exp( ) dy π That s, ( μ)/ σ has a tg stadard ormal dstrbuto Proof: Use the theorem 3 Theorem 3: (Covergece of mgfs) Suppose {, =,, } s a sequece of radom varables, each wth mgf 4

5 M () t Furthermore, suppose that M ( t) = M ( ) t for all t a eghborhood of 0, ad M () t s a mgf The there s uque cdf F whose momets are determed by M () t ad for all x where s cotuous, we have F ( x) = F( x) That s, covergece, for t < hof mgfs to a mgf mples covergece of cdfs Theorem 555 (Stroger form of the Cetral Lmt Theorem) Let,, be a sequece of d radom varables wth E = μ ad 0 < Var = σ < Defe = Let G ( ) deote the cdf of x ( μ)/ σ The, for ay < x <, x y G( x) = exp( ) dy π That s, ( μ)/ σ has a tg stadard ormal dstrbuto Theorem 557 (Slutsky s Theorem) If dstrbuto ad Y a, a costat, probablty, the a Y a dstrbuto b + Y + a dstrbuto Example 558 (Normal approxmato wth estmated varace) Suppose that S ( μ) (0,), but the value of σ s ukow We have see example 553 that f Var( S ) 0, the σ σ probablty By exercse 53, we have σ S probablty By theorem 557, we have / 5

6 ( μ) σ ( μ) = (0,) S σ S Notes (relatoshp betwee several covergeces) coverges almost surely coverges probablty coverges dstrbuto coverges probablty exsts a subsequece that coverges almost surely 3 coverges probablty to a costat coverges dstrbuto to a costat 4 Slutsky s Theorem Example 559 (Estmatg the odds) Suppose that,,, are d Beroull( ( p ) radom varables The typcal parameter of terest s p, whch ca be estmated by = = We ca obta the dstrbuto of p, whch s Boormal( p, ) Sometmes we are terested the odds, p, whch may be estmated by The what are propertes of t? For example, how to calculate the varace of t? The exact calculato may be dffcult, but a approxmato ca be obtaed Defto 550: If a fucto gx ( ) has dervatves of order r, that s, costat a, the Tyalor polyomal of order r about a s () r g ( a) Tr ( x) = ( x a) = 0! r ( r d g ) ( x) = g( x ) exsts, the for ay r dx 6

7 Theorem 55 (Taylor) If r ( r d g ) ( a) = g( x) r x= a exsts, the dx gx ( ) Tr ( x) x a = 0 r ( x a) For the statstcal applcato of taylor s Theorem, we are most cocerted wth the frst-order Taylor seres Let be a radom varable wth mea θ ad suppose that g s dfferetable fucto, the gt () g( θ ) + g'( θ)( t θ) The we have E( gt ( )) Eg ( ( θ )) + g'( θ) ET ( θ) = g( θ) ; Ad ( ( )) [ ( ) ( θ)] = [ '( θ)( θ)] = [ '( θ)] ( ) Var g T E g T g E g T g Var T T Example 55 (Cotuato of Example 559) Recall that we are terested the propertes of t gt () =, θ = E( ) = p, the g'( t) =, thus t ( t) p Eg ( ( )) Egp ( ( )) = ; p ad p Var g g p Var p p ( p) ( p) ( p) ( ( )) = [ '( )] ( ) = ( ) = Let 7

8 Example 553 (Approxmate mea ad varace) Suppose s a radom varable wth E( ) = μ 0 If we wat to estmate the mea ad varace of the radom varable g( ), we have E( g( )) g( μ) ad Var( g( )) [ g '( μ)] Var( ) Specfcally, for g( ) = /, we have E(/ ) / μ ad 4 Var(/ ) (/ μ) Var( ) Theorem 554 (Delta Method) Let Y be a sequece of radom varable that satsfes Y ( θ ) (0, σ ) dstrbuto For a gve g ad a specfc value of θ, suppose that g '( θ ) exsts ad s ot 0 The gy g g ' [ ( ) ( θ)] (0, σ [ ( )] ) θ dstrbuto Example 555 (Cotuato of Example 553) Suppose ow that we have the mea of a radom sample 4 the we have ( ) (0,(/ μ) Var ( )) μ, 8

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