LECTURE 14 NOTES. A sequence of α-level tests {ϕ n (x)} is consistent if

Transcription

1 LECTURE 14 NOTES 1. Asymptotic power of tests. Defiitio 1.1. A sequece of -level tests {ϕ x)} is cosistet if β θ) := E θ [ ϕ x) ] 1 as, for ay θ Θ 1. Just like cosistecy of a sequece of estimators, Defiitio 1.1 is a basic otio of correctess. I fact, most tests are cosistet. I the rest of the sectio, we refrai from presetig mathematically rigorous results because the level of the subject is such that it is difficult to state the eve the assumptios without itroducig additioal techical cocepts. Cosider testig H 0 : θ = θ 0 versus H 1 : θ θ 0 by a Wald test. That is, cosider the test ϕx) = 1 χ p,, ) w ), where w is the Wald test statistic: 1.1) w := V 1 ) ˆθ θ 0 ˆθ is a asymptotically ormal estimator of θ, ad V is a cosistet estimator of its asymptotic variace). The power is βθ) = w χ ) p, At ay θ 1 θ 0, the Wald statistic diverges. Ideed, V 1 ˆθ θ 0 ) = V 1 ) ˆθ θ 1 + V 1 ) θ1 θ 0. We recogize the first term is O P 1) ˆθ is asymptotically ormal), but the secod term diverges. Thus the power teds to oe. It is possible to show that the LR ad score tests are cosistet by similar argumets. Cosistecy esures the power of a test grows to oe as the sample size grows. However, the rate of covergece is uclear. Whe we ecoutered a similar problem whe evaluatig poit estimators, we blew up the error by ad studied the limitig distributio of ˆθ θ ). 1

2 STAT 01B The aalogous trick here is to study the limitig distributio of the Wald statistic uder a sequece of local alteratives: 1.) θ := θ 0 + h. Formally, cosider a triagular array x 1,1 x,1 x,..... x,1 x,... x,....., where x,i i.i.d. F θ. We remark that observatios i differet rows of the array are ot idetically distributed. Let ˆθ be a asymptotically ormal estimator of θ based o observatios {x,i } i []. The Wald statistic is V 1 ) ˆθ θ 0 = V 1 ) ˆθ θ + V 1 h. Ituitively, the first term coverges i distributio to a N 0, I p ) radom variable, ad the secod term coverges to Avar ˆθ ) 1 h. Thus 1.3) V 1 ) d ) 1 ) ˆθ θ 0 N Avar ˆθ h, I p. ad the power fuctio coverges to z ) 1 βθ ) P + Avar ˆθ h ) χ p,, where z N 0, I p ). The precedig limit of the power fuctio is called the asymptotic power of the Wald test. Evidetly, the larger V 1 h is, the higher is the asymptotic power. Thus Wald tests based o efficiet estimators are more powerful. There is a similar story for the LR ad score tests. We remark that z + µ is distributed as a o-cetral χ radom variable: µ is the o-cetrality parameter. Example 1.. test. We kow Let x i i.i.d. Berp). We wish to test H 0 : p = p 0 by a Wald 1. the MLE of p is ˆp = x,. the asymptotic variace of ˆp is p1 p).

3 LECTURE 14 NOTES 3 If p p 0, the power of the Wald test is approximately ) ) βp) = P ˆp p 0 ) p χ p1 p) 1, ) ) = P ˆp p) p + p p0 ) χ p1 p) p1 p) 1, ) P z + p p0 ) χ p1 p) 1, ), where z is a stadard ormal radom variable. The o-cetrality parameter is p p 0) p1 p). To illustrate use of the precedig approximate power fuctio i a cocrete settig, cosider the desig questio: how may samples are required to achieve 0.9 power agaist the alterative H 1 : p 1 = p ? By the properties of the o-cetral χ 1 distributio, to esure P z + µ) χ 1,) 0.90, the o-cetrality parameter µ must be at least Recall the o-cetrality parameter is p 1 p 0 ) p 1 1 p 1 ). We solve for to deduce > 10.51p 11 p 1 ) p 1 p 0 ) = It is possible to rigorously justify 1.1) uder suitable coditios by appeaig to the theory of local asymptotic ormality, much of which was developed by Lucie Le Cam at Berkeley.. Iterval estimatio. I the first part of the course, we cosidered the task of poit estimatio, where the goal is to provide a sigle poit that is a guess for the value of the ukow parameter. The goal of iterval estimatio is to provide a set that cotais the ukow parameter with some prescribed probability. Defiitio.1. Let Cx) Θ be a set-valued radom variable. It is a 1 -cofidece set for a parameter θ if θ Cx)) 1. If Cx) is a iterval o R, we call Cx) a cofidece iterval. We emphasize that the set Cx) ot the parameter θ is the radom quatity i Defiitio.1. Observig [ lx), ux) ] = [l, u] should ot be iterpreted as θ [l, u] with probability at least 1 : θ is a determiistic

4 4 STAT 01B quatity, so it is osese to cosider the probability of θ [l, u]. A correct statemet is θ [lx), ux)] with probability at least 1. As we shall see, formig iterval estimators essetially boil dow to ivertig hypothesis tests. Lemma.. Let Aθ 0 ) X be the acceptace regio of a -level test of H 0 : θ = θ 0. The set A 1 x), where A 1 x) := {θ Θ : x Aθ)} is a 1 cofidece set for θ 0. That is, the set of parameters θ 0 at which a -level test of H 0 : θ = θ 0 accepts is a 1 -cofidece set for θ 0. Proof. By defiitio of A 1 x), the evet {x Aθ 0 )} is equivalet to {θ 0 A 1 x)}. Uder H 0, a -level test accepts with probability at least 1 : 0 x Aθ 0 )). Thus A 1 x) is a 1 cofidece set: 0 θ 0 A 1 x)) = 0 x Aθ 0 )) 1. Lemma. is essetially a tautology: A 1 x) is a 1 -cofidece set because the evet { θ A 1 x) } is equivalet to { x Aθ) }. Sice Aθ) is the acceptace regio of a -level test, 0 x Aθ 0 )) 1. Example.3. Let x N µ, 1). Cosider testig H 0 : µ = µ 0 versus H 1 : µ µ 0. We showed that the -level LRT of H 0 : µ = µ 0 rejects if x µ 0 > z. The acceptace regio Aµ 0 ) is { } x R : x µ 0 z. Thus A 1 x) = { } µ 0 R : x µ 0 z is a 1 -cofidece iterval for µ 0. = [ x z, x + z i.i.d. Example.4. Let x i N µ, σ ), where σ is ukow. Cosider testig H 0 : µ = µ 0 versus H 1 : µ µ 0. We showed that the -level LRT of H 0 : µ = µ 0 rejects if φx) > t, where φx) = x µ0 ) ŝ. ]

5 LECTURE 14 NOTES 5 is the t-statistic. The acceptace regio is { } x R : φx) t. Thus A 1 x) = { } µ 0 R : φx) t = [ x ŝ t, x + ŝ ] t is a 1 -cofidece iterval for µ 0. We remark that Lemma. has a coverse: it is possible to obtai a -level test from a 1 -cofidece set. Lemma.5. Let Cx) is a 1 -cofidece iterval for θ. The test ϕx) = 1 1 Cx) θ 0 ), H 0 : θ = θ 0 versus H 1 : θ θ 0. That is, the test that rejects whe Cx) does ot cotai θ 0 is a -level test. Proof. Sice Cx) is a 1 cofidece iterval for θ 0, P 0 θ0 Cx) ) 1. Thus E 0 [ 1 ϕx) ] = 1 P0 θ0 Cx) ) 1 1 ). A coveiet formalism that highlights the coectio betwee hypothesis testig ad iterval is that of pivot or pivotal quatity. Defiitio.6. A fuctio φx, θ) is a pivot for a parametric model if the its distributio uder x F θ does ot deped o θ. That is, ) φx, θ) C does ot deped o θ. We remark that a pivot is techically ot a statistic because the fuctio depeds o the ukow parameter θ. The caoical example of a pivot is the z-statistic x µ σ. It is a pivot for the ormal locatio-scale model. Give a pivot, it is possible to 1. test the hypothesis H 0 : θ = θ 0. Uder H 0, we kow the distributio of the pivot φx, θ 0 ). Thus comparig the observatio φx, θ 0 ) to the kow distributio is the basis of a test.

6 6 STAT 01B. form a cofidece iterval for θ. As we shall see, ivertig the pivot i its secod argumet leads to a cofidece iterval for θ. Let C φx, θ) be a set of 1 mass uder the distributio of the pivot: φx, θ) C ). If we pi the secod argumet of the pivot at θ 0 lettig φ θ0 x) = φx, θ 0 )) ad ivert the pivot i its first argumet, we obtai φ 1 θ 0 C) := { x X : φx, θ 0 ) C }, which is the acceptace regio of a -level test of H 0 : θ = θ 0. Ideed, 0 x φ 1 θ 0 C) ) = 0 φx, θ0 ) C ) = 1, If we pi the first argumet of the pivot at x lettig φ x θ) = φx, θ 0 )) ad ivert the pivot i its secod argumet, we obtai which is a 1 -cofidece set for θ. φ 1 x C) := { θ Θ : φx, θ) C }, Example.7 Example.3 cotiued). We kow x µ is a pivot for the Gaussia locatio model: if x N µ, 1), φx, µ) := x µ is a pivot. If we pi the secod argumet of the pivot at µ 0, the pre-image of [ z, z ] uder the pivot is φ 1 [ µ 0 z, z ]) { [ = x R : x µ z, z ]}, which is the acceptace regio of the -level LRT of H 0 : µ = µ 0 versus H 1 : µ µ 0. The pre-image of, z ] is φ 1 µ 0, z ]) = { x R : x µ, z ]}, which is the acceptace regio of the -level UMPU test of H 0 : µ = µ 0 versus H 1 : µ > µ 0. Fially, if we pi the first argumet of the pivot at µ 0 ad ivert the pivot i its secod argumet, we obtai φ 1 [ x z, z ]) { [ = µ R : x µ z, z ]}, which is a 1 -cofidece iterval for µ.

7 LECTURE 14 NOTES 7 I practice, it is usually ot possible to derive a exact pivot. However, asymptotic pivots are easier to obtai. Formally, a asymptotic pivot for a parametric model is a fuctio whose asymptotic distributio does ot deped o the parameter. The caoical example of a asymptotic pivot is: V 1 ˆθ θ ) d ) N 0, Ip. Usurprisigly, ivertig asymptotic pivots leads to asymptotically -level tests ad 1 -cofidece itervals..1. Most accurate iterval estimators. Defiitio.8. A 1 cofidece set Cx) is most accurate at θ if θ Cx)) θ C x)) for ay θ θ ad ay other 1 cofidece set C x). It is uifomly most accurate UMA) o Θ if it is most accurate at ay θ Θ. Ituitively, a most accurate 1 cofidece set is least likely amog 1 cofidece sets) to iclude icorrect parameters. A straightforward calculatio shows that a most accurate 1 cofidece set has the smallest expected Lesbegue) volume amog 1 cofidece sets. Ideed, ) [ [ ]] E θ vol Cx) = 1 Cx) θ )dθ f θ x)dx X Θ ) = 1 Cx) θ )f θ x)dx dθ Θ X = θ Cx) ) dθ, which is miimized whe the itegrad is miimized o Θ. Θ Theorem.9. A 1 -UMA cofidece set o Θ is the iverse of - level UMP tests of H 0 : θ = θ versus H 1 : θ Θ \ {θ }. Proof. Let Aθ ) be the acceptace area of a UMP test of H 0 : θ = θ versus H 1 : θ Θ \ {θ }. Sice the test is UMP, x X \ Aθ ) ) x X \ A θ ) ) for ay θ Θ \ {θ } ad ay other the acceptace area of ay other -level test A θ ). Equivaletly, x Aθ ) ) x A θ ) ).

8 8 STAT 01B We ivert the acceptace area to obtai θ A 1 x) ) θ A 1 x) ). We observe that to obtai a most accurate cofidece set at a poit, it is ecessary to ivert a family of most powerful tests. Ideed, by Theorem.9, the most accurate cofidece set at θ is the iverse of -level most powerful tests of H 0 : θ = θ versus H 1 : θ = θ. I hypothesis testig, there is ofte o UMP tests for a prescribed pair of ull ad alterative hypotheses. Similarly, i iterval estimatio, there is usually o UMA cofidece set. To make the otio of most powerful test tractable, we restricted our attetio to ubiased tests ad studied UMPU tests. There is a similar otio of ubiasedess for cofidece sets ad ivertig ubiased tests lead to ubiased cofidece sets. Defiitio.10. A 1 cofidece set is ubiased if θ Cx) ) 1 for ay θ θ. It is a uiformly most accurate ubiased UMAU) cofidece set o Θ if it is uiformly most accurate o Θ amog ubiased cofidece sets. Lemma.11. A 1 -ubiased cofidece set o Θ is the iverse of -level ubiased tests of H 0 : θ = θ versus H 1 : θ Θ \ {θ }. Proof. The lemma is almost a tautology. Let Aθ ) be the acceptace regio of a ubiased test of H 0 : θ = θ versus H 1 : θ Θ \ {θ }: x A 1 θ ) ) 1. We ivert the acceptace regio to obtai the stated coclusio. We combie Lemma.11 ad Theorem.9 to obtai the usurprisig) result that the iverse of the acceptace regio of a UMPU test is a UMAU cofidece set. The proof is very similar to the proof of Theorem.9, ad we skip the details here. Theorem.1. A 1 -UMAU cofidece set o Θ is the iverse of -level UMPU tests of H 0 : θ = θ versus H 1 : θ Θ \ {θ }. Yuekai Su Berkeley, Califoria December, 015