The Mathematical Appendix

Transcription

1 The Mathematcal Appedx Defto A: If ( Λ, Ω, where ( λ λ λ whch the probablty dstrbutos,,..., Defto A. uppose that ( Λ,,..., s a expermet type, the σ-algebra o λ λ λ are defed s deoted by ( (,,...,, σ Ω. λ λ λ Ω s a expermet type, ad that D (, M,, s a decso problem. We deote by ( cossts of those subsets C of for whch o σ ( Ω, we use the symbol σ the σ-algebra o whch C belogs to σ ( Ω. Wheever m s a measure m to deote the measure o ( formula m ( C m( C for each set C belogg to ( σ. σ whch s defed by the Theorem. All decso problems have lelhood vectors. Proof of Theorem. uppose that (, M,, (,,...,, the form ( defed by D s a decso problem, so that s of λ λ λ Ω for some measures λ, λ,..., λ o a set Ω. If the measure : s for all sets C ( ( C ( C µ λ σ, each of the measures,,..., λ λ λ s absolutely cotuous wth respect to :. Now the Rado-Nodym Theorem (see e.g. Rud, 986, -3 mples that there are fuctos L, L,..., L whch satsfy the codto for all sets C whch belogs to ( vector of the decso problem D.! ( C L ( x d ( x λ C µ σ. Hece, the (+-tuple( µ, L, L,..., L a lelhood

2 (,,...,, Theorem. uppose that ( (, M,, λ λ λ Ω s a expermet type, ad that D s a decso problem. If the measures λ ad have the lelhood fuctos L ad λ (where {,..., } L relatve to the measure µ, ad the lelhood fuctos L ad L relatve to the measure µ, t must be the case that ( / ( ( / ( L x L x L x L x wth the possble excepto of a subset of whch has zero measure wth respect to both ad λ. (Here the value of a rato ab couts as + f a ad b, ad as udefed f a b. Proof of Theorem. Together wth the defto of a lelhood fucto (Defto 6, the assumptos of ths theorem mply that, whe or, C C (A ( ( ( for all sets C ( σ. L x dµ ( x λ C L x dµ ( x The measure µ ca be dvded to two parts: a part µ whch s absolutely cotuous wth respect to µ, ad aother part µ whch s sgular wth respect to µ. Equato (A mples that ( µ ( for all sets C σ ( C L x d x C C L x d ( x C L x d ( x (A ( µ λ ( ( µ for all sets C ( σ ad for both ad. ce, so that we ca coclude that µ s absolutely cotuous wth respect to µ, Rado-Nodym Theorem (see e.g. Rud, 986, -3 mples that there s a real-valued fucto r o whch s such that µ C r x dµ ( x C (A3 ( ( for all sets C σ (. But together (A ad (A3 mply that λ

3 ( ( ( ( ( L x dµ ( x λ C L x dµ ( x L x r x dµ ( x C C C ce ths result s vald for all sets C σ ( everywhere L ( x L ( x r( x, t ca be cocluded that µ -almost. Ths result s vald both whe ad, a fact whch further mples that µ -almost everywhere (A4 L ( x / L ( x L ( x r( x / L ( x r( x L ( x / L ( x ce equato (A mples that the measures λ ad λ are absolutely cotuous wth respect to µ, t ca be cocluded that the subset of whch ths formula s ot vald must have zero measure also wth respect to the measures λ ad λ.! (,,...,, Defto A3. uppose that ( D (, M,, λ λ λ Ω s a expermet type, ad that s a decso problem. A fucto f from to M s sad to satsfy the DF-codtos f t satsfes the followg codtos: Wheever {,,..., }, the fucto h whch maps each x oto ( ( ( h x pay f x λ s measurable wth respect to the σ-algebra σ ( measure λ., ad tegrable wth respect to the Remar. The DF-codtos whch are defed by the above deftos are regularty codtos whch we are metoed our defto of a decso fucto, Defto 7. It s clear that the requremet of measurablty whch s cluded these codtos s a reasoable mmum requremet whch actual rules for choosg a elemet of M o the bass of a observed elemet of ormally trvally satsfy. Also the addto whch the other codto of Defto A3 maes to ths requremet s a rather small oe. The other codto meas that the tegral 3

4 exsts for each {,,..., } ( ( λ λ ( pay f x d x. If the frst regularty codto s vald, ths tegral wll be { pay m m M } always exst whe or M s fte, ad also whe e.g. ether sup ( f { pay ( m m M } λ < or λ > (that s, whe the values of the payoff fucto have ether a fte upper lmt or a fte lower lmt. ( Theorem 3. If (,,...,, Ω s a expermet type, (, M,, λ λ λ D s a decso problem, ad ( µ L L L s a lelhood vector of D, the quatty ( the formula where,,,..., ( ( µ ( pay f pay x d x, f, ( λ ( pay ( x p pay f ( x L x f, Proof of Theorem 3. ( By defto, pay( f λ s gve by ( λ ( ( λ ( µ ( pay f pay f x L x d x, pay f s gve by ad whe ths formula s combed wth the defto of pay ( f, t mples that ( ( ( ( ( pay f p pay f p pay f x L x d x pay x d x ( ( λ ( λ µ, µ! f (,,...,, Theorem 4 (Ma Theorem. uppose that ( D (, M,, s a decso problem, ad that ( L L L λ λ λ Ω s a expermet type, that µ a lelhood vector of the,,,..., decso problem D. If the decso fucto f F (, M ;, each x a measure f ( x M for whch the quatty s such that t chooses for 4

5 ( ( λ ( Q ppay f x L x s largest amog the measures of M, the decso fucto f s optmal for. Proof of Theorem 4. Ths follows mmedately from Theorem 3, because the tegral whch s metoed ths theorem obtas ts largest value whe f (x s at each pot x of chose so that ( ( λ ( ppay f x L x receves ts largest possble value at that pot.! (,,...,, Corollary. uppose that ( (, M,, paysmple λ λ λ Ω s a expermet type, that D s a deal decso problem wth smple payoffs, ad that ( L L L µ,,,..., a lelhood vector of D. If a decso fucto f F (, M ;, paysmple s such that, for each x, f ( x s a measure λ { λ, λ,..., λ } for whch pl( ts largest value, the f s optmal for. x receves Proof of the Corollary. Whe the assumptos of ths corollary are vald, f ( x s for each x a measure x ( λ, where ( x {,..., }. Further, ths case the value of the quatty whch s maxmzed by a optmal decso fuctos accordg to Theorem 4 s ( ( λ ( ( ( ( p pay f x L x p L x x x Hece, Theorem 4 mples that whe the assumptos of ths corollary are vald, a decso fucto wll be optmal f t chooses for each x a measure λ for whch the value of the pl x s largest.! quatty ( (,, Theorem 5. uppose that ( λ λ dstrbutos, that (, M,, Ω s a expermet type wth two probablty D s a deal decso problem, ad that pay s a strctly epstemc payoff fucto. If (, p s a esemble of toe expermets whch p (,, ad f the decso fucto f * F (, M ;, test of λ agast λ. s optmal wth, f* s a best 5

6 Proof. ce the decso problem (, M,, M { m, m }, where m λ D s deal, M must be the set ad m λ. The esemble of toe expermets must, of course, be of the form (, p, where p ( p, p for some p [,] ths case p. Let us ow assume that f F (, M ;, to Defto the expected payoff of f wth equals ( ( ( ( pay f p pay f m + p pay f m., ad by assumpto s a arbtrary decso fucto Accordg Whe we defe, wth Defto md, the umbers sze( f ad power ( f as ( ( ( ad power ( f m f ( m sze f m f m respectvely, ad the umbers ad ( pay( m m (, as pay( m m pay( m m ad pay m m, respectvely, Defto 8 mples that pay( f m pay( f ( x m dm ( x m ( f ( m pay( mm + m ( f ( m pay( mm ( sze( f pay( m m + sze( f pay( m m ( sze( f + ( sze( f + sze( f pay( m m pay( m m + ( sze( f ad smlarly that pay( f m pay( f ( x m dm ( x m ( f ( m pay( mm + m ( f ( m pay( mm ( power ( f pay( mm + power ( f pay( mm ( power ( f + power ( f pay( m m + power ( f pay( m m + power ( f Hece, 6

7 pay ( f p pay( f m + ( p pay( f m ( ( ( ( ( ( p pay m m + sze f + p pay m m + power f ( ( ( C p sze f + p power f where ( ( ( C p pay m m + p pay m m + p s costat whose value does ot deped o the choce of f. Assume ow that f * F (, M ;, (, ;, s optmal wth, ad that g F M pay s a arbtrary test of m agast m whch s of the same sze wth the test f* of m agast m. I ths case t has to be the case that sze( f * sze( g f* s optmal, t must also be the case that pay ( f * pay ( g of pay ( f whch we ust deduced these two facts mply that ( p power ( f * ( p power ( g ad, sce. Together wth the formula However, sce pay s strctly epstemc, t must be the case that s postve. ce we have also assumed that so that p, we ca ow coclude that the multpler ( ( * ( power f power g. p s postve Accordg to the defto of the fucto power ths codto meas that, as a test of m agast m the decso fucto f* has a power whch s at least as large as the power of the test g m agast m. ce the test g was arbtrary, t ca be cocluded that f* s a best test of m agast m.! (,, Theorem 6. uppose that ( λ λ dstrbutos, that (, M,, Ω s a expermet type wth two probablty D s a deal decso problem, ad that pay s a strctly epstemc payoff fucto. uppose further that (, p s a esemble of toe expermets whch ( p, p p, p, ad defe the umber K by 7

8 p K p, where pay( λ λ pay( λ λ, ad pay( λ λ pay( λ λ decso fucto f * F (, M ;, wth the crtcal rato K.. Now a s optmal wth f t s a lelhood rato test Proof of Theorem 6. ce the decso problem (, M,, be the set M { m, m }, where m λ ad m λ lelhood vector of D. Whe the fuctos Q ( the formula D s deal, M must. Let (, L, L µ be a arbtrary x, where or, are defed wth ( ( ( + ( ( ( Q x p pay m m L x p pay m m L x for each x, our Ma Theorem shows that the decso fucto f * F (, M ;, s optmal wth f t chooses the measure m f Q ( x Q ( x ( ( Q x > Q x. However, f oe puts t s easy to see that > ad the measure m f ( ( ( ( ( ( Q x p pay m m L x + p pay m m L x, ( ( + ( + ( ( ( ( + ( Q x p pay mm L x p pay mm L x Q x p L x ad that smlarly ( ( + ( ( Q x Q x p L x Hece, Q ( x > Q ( x f ad oly f p L ( x ( p L ( x f > or, equvaletly, f ad oly ( / ( ( /( L x L x > p p K. 8

9 mlarly, Q ( x > Q ( x f ad oly f ( / ( L x L x < K. Together wth Defto 4, ths shows that the decso fucto f * F (, M ;, lelhood rato test wth the lmtg rato K.! s optmal wth f t s a (,, The Neyma-Pearso Theorem. uppose that ( λ λ two probablty dstrbutos, ad that (, M,, (, ;, Ω s a expermet type wth D s a deal decso problem. If f F M pay s a lelhood rato test, t s a best test of λ agast λ. Proof of the Neyma-Pearso Theorem. Let f F (, M ;, be a arbtrary lelhood rato test, ad let us deote the crtcal rato of the lelhood rato test f of λ agast, λ by K. ce the decso problem D s deal, M must be the set M { m, m } where m λ ad m λ. Let us ow cosder the modfed decso problem (, M,, paysmple D, whch pay smple s the smple payoff fucto for whch smple ( smple ( ad paysmple ( m m paysmple ( m m pay m m pay m m D s a deal decso problem wth a strctly epstemc payoff fucto.. Clearly, Whe the esemble of toe expermets s defed to be (, p where ( ( K, K ( K p + +, our Theorem 6 mples that whe f s vewed as a decso fucto whch s assocated wth the decso problem D, t s optmal wth. Together wth Theorem 5 ths mples that, whe f s vewed as a decso fucto whch s assocated wth the decso problem D, t s a best test of λ agast λ. Together wth the defto of a best test ths mmedately mples that f must be a best test also whe t s vewed as a decso fucto whch s assocated wth the orgal decso problem D.! Referece Rud, W. [986]: Real ad Complex Aalyss. Thrd Edto. New Yor: McGraw-Hll. 9