A Central Limit Theorem for Belief Functions

Transcription

1 A Cetral Limit Theorem for Belief Fuctios Larry G. Estei Kyougwo Seo November 7, 2. CLT for Belief Fuctios The urose of this Note is to rove a form of CLT (Theorem.4) that is used i Estei ad Seo (2). More geeral cetral limit results ad other alicatios will follow i later drafts. Let S fb; Ng ad K (S) ffbg; fng; fb; Ngg the set of oemty subsets of S. Deote by s (s ; s 2 ; :::) the geeric elemet of S ad by (s ) the emirical frequecy of the outcome B i the rst exerimets i samle s. Let be a belief fuctio o S, that is, there exists m 2 (K (S)) such that, for every A S, (A) m (fk 2 K (S) : K Ag). Its cojugate is give by (A) (SA), ad the roduct is the belief fuctio o S satisfyig, for every A S, (A) m f K e K K 2 ::: 2 K (S ) : K e Ag. (.) Here m is the ordiary i.i.d. roduct of the measure m. Bosto Uiversity, lestei@bu.edu ad Deartmet of Maagerial Ecoomics ad Decisio Scieces, Northwester Uiversity, k-seo@kellogg.orthwester.edu. We gratefully ackowledge the acial suort of the Natioal Sciece Foudatio (awards SES-9774 ad SES ) ad discussios with egjig Che. This aer is relimiary; its mai urose is to rovide a CLT that is used i our aer Bayesia iferece ad o-bayesia redictio ad choice: foudatios ad a alicatio to etry games with multile equilibria.

2 The LLN asserts certaity that asymtotic emirical frequecies will lie i the iterval [ (B) ; (N)], that is, fs : [lim if (s ) ; lim su (s )] [ (B) ; (N)]g. The CLT describes (u to aroximatio) beliefs about ite samle frequecies. A simle CLT is rovided rst because it rovides ersective o later results. Let N () be the cdf of a stadard ormal distributio. Theorem.. ( lim! lim! ( s : (s ) (fng) ( s : (s ) (fbg) ( )! (B) (fng)) )! (fbg) > (fbg)) N () N (). The roof follows readily from the ext lemma showig that for the evets idicated, the miimizig measures are i.i.d. As a result classical limit theorems alied to these measures deliver corresodig limit theorems for the i.i.d. roduct. Lemma.2. Let P ad P be the measures o S with P (B) (B) ad P (B) (B) resectively. The their i.i.d. roducts, deoted P ad P, both lie i core ( ); ad, for ay t, (fs : t (s )g) P (fs : t (s )g) (.2) ad (fs : (s ) tg) P (fs : (s ) tg). (.3) See the aedix for a roof. A CLT for two-sided itervals is less trivial because miimizig measures are ot easily ideti ed. Thus the followig theorem uses a di eret roof strategy ad alies a versio of the multidimesioal Berry-Essee Theorem (Dasguta (28,. 45-6)): If the d-dimesioal radom variables X ; X 2 ; ::: are i.i.d., E (X ), V ar (X ) is the idetity matrix, ad if E (kx k) is ite, the there exists a costat K such that, for all, su C2C Pr X + ::: + X 2 C 2 Pr ( 2 C) K.

3 Here C is the collectio of all covex subsets of d, ad is stadard ormal ad d -valued. Let N 2 (; ; ) be the cdf for the bivariate ormal with zero meas, uit variaces ad correlatio coe ciet, that is, N 2 ( ; 2 ; ) Pr ( ; 2 2 ) where ( ; 2 ) is bivariate ormal with the idicated momets. Theorem.3. There is a costat K that does ot deed o, 2 or, such that 8 >< (B) + 9 ( (B)) (B) (s B ) >: (B) + C 2 A ( (B)) (B) >; K : N 2 ; 2 ; (B)(N) (B)( (B)) (B)( (B)) Moreover, the same holds if ad 2 deed o. This theorem is a secial case of the ext oe, but it also serves as a lemma i the roof of the more geeral result. emark. Whe is additive, the idicated correlatio coe ciet equals ad the iequality becomes j < (s )(B) (B) 2 Pr ( < 2 ) j K, ( (B))(B) where is stadard ormal. Proof. De e radom variables X i I (K i 2 K (S) : K i fbg) ad Y i I (K i 2 K (S) : K i fng). (.4) Note that I (K i fbg)) + I (K i fng)) I (K i fb; Ng)) 6. Thus the rst two idicators are ot erfectly egatively correlated. 3

4 Observe rst that X i Y i. Comute, usig m, that E (X i ) (B), E (Y i ) (N), V ar (X i ) (B) ( (B)), V ar (Y i ) ( (N)) (N) ad cov (X i ; Y i ) E (X i Y i ) E (X i ) E (Y i ) E (X i ) E (X i ) E (Y i ) (B) (B) ( (N)) (B) (N) : Here, the secod equality follows because X i imlies Y i. The 2 3 E 4@ 2 V ar 4@ X i (B) ( (B))(B) Y i ( (N)) ( (N))(N) X i (B) ( (B))(B) Y i ( (N)) ( (N))(N) A5 ad 3 A5, where corr (X i ; Y i ) (B) (N) (B) ( (B)) ( (N)) (N) : Note that < K K 2 ::: mi s 2K K 2 ::: i i s : < P P I (s i B) i I (s i B) 2 max s 2K K 2 ::: i () P I (s i B) 2 () P < mi I (s P s i B) max I (s 2K K 2 ::: s i B) 2 () 2K K 2 ::: i P < P I (K i fbg) [ I (K i fng)] 2 () i Coclude from (.) that i P < P X i Y i 2 i i 4

5 Cosequetly, s : < P i I (s i B) 2 m K K 2 ::: 2 (K (S)) : < P i X i P i Y i 2 : < (s ) (B), (s ) ( (N)) 2 ( (B))(B) ( (N))(N) ( (B)) (B) + (B) < (s ) ( (N)) (N) 2 + ( (N)) P m ( (B)) (B) + (B) < i X i P i Y i ( (N)) (N) 2 + ( (N)) P P m i < X i (B) i, Y i ( (N)) 2 : ( (B))(B) ( (N))(N) This ermits traslatio of the assertio to be rove ito oe about i.i.d. robability measures ad thus classical results ca be alied. Use the Cholesky decomositio of the variace-covariace matrix to obtai V such that (V ad [ < X i (B) ( (B))(B) Y i ( (N)) ( (N))(N) P i X i (B) ad ( (B)) (B) P (V ) A is stadard ormal (with correlatio ), P i Y i ( (N)) ( (N)) (N) 2 ] () i X i (B) P ( (B))(B) i Y i ( (N)) ( (N))(N) C A 2 C, for some covex C 2. Therefore, by the multidimesioal Berry-Essee Theorem, P i X i (B) C m ) B ( P C C A 2 CA Pr i Y i ( (N)) 2 K ( (N))(N) 5

6 for some costat K that does ot deed o C or, where ormal. De e ~ ~ V. The Pr 2 C Pr <, ~ ~ 2 Pr ~ <, ~ 2 N 2 ( ; 2 ; ) : is stadard Therefore, < (s ) (B), (s ) ( (N)) 2 ( (B))(B) ( (N))(N) (B)(N) N 2 ; 2 ; (B)( (B))( (N))(N) K : Fially, the same roof works whe ad 2 are relaced by ; ad 2;. The ext theorem geeralizes the recedig ad is the mai objective of this Note. Theorem.4. Suose that G :! is bouded, quasi-cocave ad uersemicotiuous. The G ( (s )) d (s ) E [mi fg (X) ; G (X2)g] + O, where (X; X2) is ormally distributed with mea ( (B) ; (B)) ad variace (B) ( (B)) (B) (N) : (B) (N) ( (N)) (N) That is, lim su G ( (s ) (B)) d (s )! for some costat K. su ;a jg (a)j <. E [mi fg (X) ; G (X2)g] K Moreover, the same holds whe G deeds o ad 6

7 The recedig theorem is the secial case where G () [a;b ] () ad a (B)+ 2 ( (B)) (B), b (B)+ 2 2 ( (B)) (B). Proof. Without loss of geerality, suose that if ;a G (a). De e M su ;a G (a). Sice G is quasicocave, there are two (iverse) fuctios G L ad G such that fa : G (a) t g G L (t) ; G (t) for all t 2 [; M]. (The iverses may ot be de ed at t, but this is of o cosequece below.) By the de itio of Choquet itegratio, M G ( (s )) d (s ) (G ( (s )) t) dt Note that Thus, by Theorem.3, M G L (t) (s ) G (t) dt: G L (t) (s ) G (t) G L (t) (B) B (s ) (B), ( (B))(B) ( (s ) ( (N)) G ( (N))(N) (t) ( (N)) ( (N))(N) C A : G L (t) (s ) G (t)! G L (t) (B) N 2 ; G (t) ( (N)) ; ( (B)) (B) ( (N)) (N) with t, M M (B)(N). Because the term O (B)( (B))( (N))(N) G L (t) (s ) G (t) dt! G L (t) (B) N 2 ; G (t) ( (N)) ; ( (B)) (B) ( (N)) (N) 7 + O does ot deed o dt + O :

8 Let ad 2 be joitly ormally distributed with mea (; ) ad variace, ad let X ad X2 be ormally distributed as i the theorem statemet. The! M G L (t) (B) N 2 ; G (t) ( (N)) ; dt ( (B)) (B) ( (N)) (N) M M M * M M G L (t) (B) + ( (B))(B) ; A ( (N))(N) dt ( (N)) + 2 G (t) Pr G L (t) X, X2 G (t) dt Pr G L (t) X G (t), G Pr (G (X ) t, G (X 2) t) dt + O L (t) X 2 G Pr (mi fg (X) ; G (X2)g t) dt + O E [mi fg (X ) ; G (X 2)g] + O : (t) dt + O To comlete the roof, we eed oly rove the equality marked with a asterisk. De e radom variables (X i ; Y i ) i as i (.4). The they are i.i.d. uder m ad, because X i Y i, P P i Pr a X i i, Y i b P P i Pr a X i i Y i b P P Pr a b, a b : i X i i Y i 8

9 By the multidimesioal Berry-Essee Theorem, P P i j Pr a X i i, Y i b Pr(a X, X2 b) j j Pr (a E [X ]) P P i (X i E[X ]) i, (Y i E[Y ]) (b E [Y ]) Pr (a E [X ]) (X E [X ]), (X2 E [Y ]) (b E [Y ]) P P i su j Pr (X i E[X ]) i ; (Y i E[Y ]) 2 C Pr(( ; 2 ) 2 C) j K ; C2 2 covex ad similarly j Pr a P i X i b, a P i Y i b Pr (a X b, a X 2 b) j K. It follows that, for all a b ad, j Pr(a X, X 2 b) Pr (a X b, a X 2 b) j 2K. This roves the marked equatio. Fially, the above roof works also if G is relaced by G such that su ;a jg (a)j. A. Aedix: Proof of Lemma.2 Prove (.2). Ste : K K 2 ::: fs : ii (s i B) tg i ii (K i fbg) t. Here is a roof: K K 2 ::: s : () mi P i s 2K K 2 ::: i I (s i B) t P I (s i B) t P () mi I (s i B) t s i 2K i i P () I (K i fbg) t. i 9

10 Ste 2: Let m 2 (K (S )) be the measure for. The P s : I (s i B) t i P m K K 2 ::: 2 K (S ) : I (K i fbg) t : Argue as follows: P s : I (s i B) t i m K 2 K (S ) : K s : P i i I (s i B) t s : m K K 2 ::: 2 K (S ) : K K 2 ::: Next aly Ste. Ste 3: Comlete the roof. By Ste 2, (fs : (s ) tg) P s : I (s i B) t i m K K 2 ::: 2 K (S ) : Pr iy i t, P i P i I (K i fbg) t I (s i B) t : where Y i or, Pr (Y i ) (B) ad the Y i s are i.i.d. Therefore, the recedig equals P (fs : (s ) tg) : To rove (.3), reverse the roles of B ad N i the recedig argumet. efereces [] A. Dasguta, Asymtotic Theory of Statistics ad Probability, Sriger, 28. [2] L.G. Estei ad K. Seo, Bayesia iferece ad o-bayesia redictio ad choice: foudatios ad a alicatio to etry games with multile equilibria, 2.