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Relative efficiency with equivalence classes of asymptotic covariances
Journal of Econometrics 88 (1999) 79—98
Relative efficiency with equivalence classes of asymptotic covariances David M. Mandy!,*, Carlos Martins-Filho1," ! Department of Economics, 118 Professional Building, University of Missouri, Columbia, MO 65211, USA " Department of Economics, Oregon State University, Ballard Extension Hall 303, Corvallis, OR 97331-3612, USA Received 1 November 1994; received in revised form 1 March 1998
Abstract White’s (1984, Asymptotic Theory for Econometricians. Academic Press (Harcourt Brace Jovanovich), Orlando.) concept of asymptotic variance is shown to allow some ambiguities when used to study asymptotic efficiency. These ambiguities are resolved with some mild conditions on the estimators being studied, because then White’s asymptotic variance is an equivalence class in which efficiency conclusions are invariant across members of the class. Among the extant efficiency definitions, the lim inf-based definition (White, 1994. Estimation, Inference and Specification Analysis. Econometric Society Monograph, vol. 22, Cambridge University Press, Cambridge. p. 136) is most informative even though identical conclusions can be obtained under our conditions with earlier definitions, but there are still some notions of efficiency allowed by White’s asymptotic variance that can only be detected by weaker efficiency definitions. ( 1999 Elsevier Science S.A. All rights reserved. JEL classification: C10; C13; C50 Keywords: Asymptotic efficiency
* Corresponding author. E-mail: [email protected] 1 We benefitted greatly from particularly insightful comments, and much patience, from an associate editor and an anonymous referee. Naturally they bear no responsibility for any remaining errors. 0304-4076/99/$ — see front matter ( 1999 Elsevier Science S.A. All rights reserved. PII S 0 3 0 4 - 4 0 7 6 ( 9 8 ) 0 0 0 2 2 - 0
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1. Introduction Suppose b* and bI are estimators for some parameter vector b and we want to n n determine their relative asymptotic efficiency. A procedure discussed by White (1994), p. 136) is to first find two sequences of matrices, »* and »I , such that n n both »*~1@2Jn(b*!b) and »I ~1@2Jn(bI !b) converge in distribution to stann n n n dard normals. White (1984, p. 66; 1994, p. 91) calls such sequences asymptotic variances, or avars. Next, these sequences are used to show lim inf h@(»I !»*)h*0 ∀h, whence we conclude, for example, that b* is n?= n n n asymptotically efficient relative to bI . This procedure has an advantage over the n traditional ( Fisher, 1925) approach of comparing the covariances of the asymptotic distributions, in that »* and »I need not have limits ( White, 1982). But n n there is a potential disadvantage as well because in all interesting cases there will be other avar sequences, say » , such that »~1@2Jn(b*!b) converges in n n n distribution to a standard normal. Thus, one must inquire whether lim inf h@(»I !»*)h*0 ∀h implies lim inf h@(»I !» )h*0 ∀h for any n?= n n n?= n n such » , that is, whether statements about asymptotic efficiency are invariant to n the avar sequences examined. By Fatou’s lemma, this implication holds if lim (» !»*)"0, and the converse holds as well provided the two lim infs n?= n n are both nonnegative for the same set of »I sequences (that is, for the same rival n estimators bI ). But without lim (» !»*)"0 anything is possible, including n n?= n n a reversal of the original conclusion, in which case the relative efficiency * comparison based on » would appear definitive when it is in fact ambiguous. n Alternatively, »* may yield no conclusion when a conclusion is obtainable using n » , and the same comments apply to all possible substitutes for »* and »I . n n n Thus, at this level of generality, lim (» !»*)"0 for all relevant sequences n?= n n »* and » is sufficient for meaningful efficiency conclusions. The converse is also n n of interest, namely, whether two sequences that satisfy this condition are both avars given that one of them is. More formally, we must investigate whether the relation R defined by M»*N= RM» N= Q lim (» !»*)"0 n n/1 n?= n n n/1 n
(1)
has an equivalence class consisting of all sequences that produce the desired asymptotic normal distribution. In this paper we first show this is the appropriate relation for studying relative asymptotic efficiency, in that efficiency conclusions are unambiguous if and only if all pairs of candidate sequences have the relation R. We then show that an avar collection is not always an equivalence class with respect to R, but some mild conditions on the underlying random vector are sufficient to ensure that an avar collection is indeed an equivalence class with respect to R. Hence, when these conditions hold asymptotic efficiency can be studied without ambiguity using the full generality of the avar concept.
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We demonstrate that the sufficient conditions are equivalent to boundedness of all avar sequences and the corresponding sequences of inverses. When avar sequences, but not their inverses, are bounded the avar collection is a (perhaps proper) subset of an equivalence class with respect to R. This is sufficient for the sign of lim inf h@(»I !»*)h to be invariant across all »* sequences that are n?= n n n avars of Jn(b*!b), and hence for unambiguous efficiency conclusions by this n criterion. However, (White (1984), pp. 78—79) provides another definition of efficiency based on the avar concept that does not always yield the same efficiency conclusions as this lim inf criterion. We show the two definitions are equivalent if avar sequences and their corresponding sequences of inverses are bounded. Hence, for consistency across definitions avar must be an equivalence class with respect to R. This rules out both divergences to infinity and approaches to singularity in the collection. Under these conditions the generalization accomplished by the avar concept over the traditional approach is that avar accommodates nonconvergent but bounded oscillations in the sequences forming the avar class, and their corresponding sequences of inverses. The avar concept does not directly deliver assuredly unambiguous efficiency comparisons when there are divergences to infinity or approaches to singularity in the collection. Bounding these sequences is a mild restriction, however, because the random vector can usually be normalized to eliminate such problems before forming the avar class, as is customary through multiplication by Jn. In general, an element of b* can be normalized by any n nonstochastic sequence in order to obtain bounded avars and inverse avars, and the normalizing sequence can differ across elements of b*. Once all estimators n under consideration are so normalized, our results show that their relative asymptotic efficiencies can be unambiguously compared irrespective of convergence of the avar sequences, the particular sequences examined, or the efficiency definition used. There are two caveats here. First, a multiple correlation between normalized elements of b* might tend to one, in which case some asymptotically redundant n element(s) should be dropped before asymptotic efficiency is studied, since this situation would lead to avar sequences that approach singularity. Second, because relative efficiency conclusions can be affected if different normalizations are used for b* and bI , one should only compare estimators via avars of normalized n n deviations when corresponding elements are normalized with the same sequence. Bounds on avar sequences and their sequences of inverses are usually (White, 1982; 1984, p. 66; 1994, pp. 130—136) although not always (Bates and White, 1993) imposed, but even when they are imposed they are somewhat unsatisfactory as primitive assumptions because they utilize the avar class to restrict itself rather than placing restrictions on the underlying random vector. In contrast, our conditions are imposed directly on the random vector and thus illuminate exactly what is being assumed when bounds are placed on avar sequences. To accomplish this we introduce a new order in probability concept, which we call asymptotic linear independence in probability, or alip.
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Finally, since nonconvergent oscillations are permitted in an avar class it is possible to have ‘one-sided’ relative efficiency that is not addressed by either of White’s definitions, in the form of either a smaller minimal limiting variance (minimin efficiency) or a smaller maximal limiting variance (minimax efficiency). We introduce new definitions to address these possibilities. Naturally, the new definitions are weaker than White’s definitions, so all known avar efficiency conclusions automatically hold for minimin and minimax efficiency. 2. The equivalence relation R for studying relative asymptotic efficiency Let Qq denote the set of all sequences M» N= of real symmetric positiven n/1 definite nonstochastic (q]q) matrices » . For such » , denote by »1@2 the n n n unique real-symmetric positive-definite matrix satisfying »1@2»1@2"» . In esn n n sence, White’s definition of asymptotic covariance is the following. Definition 1 ( ¼hite, 1984, p. 66; 1994, p. 91).2 Let Mx N= be a sequence of n n/1 q-dimensional random vectors (q(R). The asymptotic covariance of Mx N= , n n/1 denoted avar(Mx N= ), is n n/1 $ (0, I )N. avar(Mx N= ),MM» N= 3Qq: »~1@2 x Pz&N n n q n n/1 n n/1 For brevity, we drop the indexes and write » 3avar(x ) to denote that the n n sequence M» N= is an element of avar(Mx N= ). n n/1 n n/1 We say Definition 1 is White’s definition ‘in essence’ because neither White (1984) nor White (1994) acknowledge that avar is a class. Bates and White (1993) explicitly discuss that the avar of their RCASOI class of estimators is a class, but attribute this to the flexibility of RCASOI classes. In fact, in all interesting cases the avar of any single random vector is a class because, if there exists one $ z for any MA N= satisfysequence » satisfying Definition 1, then A »~1@2x P n n n/1 n n n ing lim A "I . Hence, if » 3avar (x ) then A{~1» A~13avar(x ) n n n n n?= n q n n A "I . In other words, a great ∀MA N= satisfying A »~1@23Qq and lim n?= n q n n n n/1 profusion of avar sequences exists whenever a single avar sequence exists. This observation brings to the forefront the issue of whether conclusions about relative asymptotic efficiency are invariant to which element of an avar class is examined. What is needed is an equivalence relation in Qq whose equivalence classes are precisely those collections of sequences for which conclusions about relative asymptotic efficiency are identical. Given this, if avar(x ) n is an equivalence class of the relation, or even a subset of an equivalence class, 2 White (1984) actually only requires positive definiteness of » for large n. Since only the tail of n the sequence is important in the definition, there is no consequential loss of generality from assuming the entire sequence is positive definite.
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then efficiency conclusions are unambiguous. To construct the equivalence relation we first need a formal definition of efficiency. We start with: Definition 2 ( ¼hite, 1994, p. 136).3 Let b* and bI be consistent estimators of n n a nonstochastic q-dimensional vector b. Then b* is asymptotically efficient relative n to bI if there exists »*3avar(Jn(b*!b)) and »I 3avar(Jn(bI !b)) such that n n n n n lim inf h@(»I !»*)h*0 n n n?= for all h3Rq (hO0). An estimator is asymptotically efficient within a class if it is asymptotically efficient relative to every other estimator in the class. Note that one consistent estimator can be asymptotically efficient relative to another only if the avar’s of both normalized estimators are nonempty. According to this definition the relation R in Qq that yields unambiguous efficiency conclusions within an equivalence class is M»*N= RM» N= if and n n/1 n n/1 only if (∀»I ): lim inf h@(»I !»*)h*0 ∀h Q lim inf h@(»I !» )h*0 ∀h. n n n n n n?= n?=
(2)
It is trivial to verify that R is an equivalence relation (satisfying reflexivity, symmetry, and transitivity). More interesting is the fact that this relation is identical to the relation defined in (Eq. (1)). That (Eq. (1)) implies (Eq. (2)) is an application of Fatou’s lemma. For the converse, »I "»* in (Eq. (2)) n n implies lim inf h@(»*!» )h*0 ∀h, while »I "» in (Eq. (2)) implies n?= n n n n lim sup h@(»*!» )h)0 ∀h. We utilize (Eq. (1)) henceforth since it is more n?= n n convenient, and henceforth denote the equivalence class of »* with respect to n R by ER(»*),M» 3Qq: lim (»*!» )"0N. n n n?= n n 3. Sufficient conditions for aar(xn ) to be an equivalence class with respect to R In general, avar(x ) is not an equivalence class with respect to R, so some n restrictions are needed in the form of regularity conditions on the underlying random vector x . These restrictions include the following notion of asymptotic n linear independence in probability. Definition 3. Let Mx N= be a sequence of q-dimensional random vectors n n/1 (q(R) and My N= be a sequence of strictly positive nonstochastic real n n/1 3 Definitions 2 and 6 (below) are rephrased from White’s original statements to reflect the fact that avar(x ) is a class. We use the same terminology to accomplish this that (Bates and White (1993), n Definition 2.5) use for a RCASOI class.
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numbers. Mx N= is asymptotically linearly independent in probability of order n n/1 My N= if for every sequence Mc N= of real nonstochastic q-dimensional vectors n n/1 n n/1 satisfying DDc DD"1 ∀n; there exists a triple (N, e, d), where N is a natural n number, e'0, and d'0; such that
A K KB
c@ x n*N N P e( n n 'd. y n For brevity, this is denoted x "alip(y ).4 n n It is clear that x "alip(y )Nx Oo (y ). The converse fails because n n n 1 n x Oo (y ) still permits subsequences of c@ x /y that converge in probability to n n n n 1 n zero, and also linear combinations of x /y that converge in probability to zero n n as long as individual components do not. Definition 4. A sequence Mx N= of q-dimensional random vectors (q(R) is n n/1 avar-regular if x "O (1) and x "alip(1). n 1 n Avar-regularity places restrictions on the primitive of the problem, the underlying random vector x . Theorem 1 shows this is equivalent to White’s (1984, p. n 66) approach of bounding the avar sequences and the corresponding sequences of inverses. ¹heorem 1. ¸et Mx N= be a sequence of q-dimensional random vectors (q(R) n n/1 for which avar(x ) is nonempty. ¹hen n 1. x "alip(1) if and only if »~1 is bounded ∀» 3avar(x ). n n n n 2. x "O (1) if and only if » is bounded ∀» 3avar (x ). n 1 n n n Hence, x is avar-regular if and only if both » and »~1 are bounded for n n n every » 3avar (x ). n n Proof. All proofs are in the appendix. h The main result of this section is that avar-regularity is sufficient to ensure avar (x ) is an equivalence class with respect to R. Bates and White ((1993), n 4 The alip concept is similar to Mann and Wald’s (1943) notion of u , but these concepts differ in p two ways that are important in the present context. First, alip(y ) is weaker than u (y ) in that u (y ) n p n p n requires the probability that a normalized random variable is nonzero approach one, while alip(y ) n merely requires that this probability not approach zero. Second, alip(y ) is stronger than u (y ) in n p n that alip(y ) places its condition on all bounded nondegenerate linear combinations of a random n vector, while u (y ) merely places its condition on the individual components of a random vector. In p n general, u (1) cannot replace alip(1) in the results below. p
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Theorem (2.3)) investigate whether » 3avar (x )Navar (x )-ER(» ), using the n n n n slightly different relation R in Qq defined by M» N= RM»I N= Q lim »~1@2»I »~1@2"I . q n n n n n/1 n n/1 n?=
(3)
As with R, it is straightforward to verify that R is an equivalence relation. However, R does not have the same equivalence classes as R unless attention is restricted to sequences that are bounded and have bounded inverses. Hence, from (Eq. (2)), R is not always the appropriate relation for studying relative asymptotic efficiency. But Theorem 2 below shows that avar-regularity, which by Theorem 1 bounds the candidate sequences and their inverses, implies avar (x ) is an equivalence class with respect to both R and R. The proof of n Theorem 2 relies on the following preliminary results. ¸emma 1. ¸et Mx N= be a sequence of q-dimensional random vectors (q(R). n/1 $ z&(0, R),n where If x P R is positive definite, then x "alip(1). n n ¸emma 2. ¸et Mx N= be a sequence of q-dimensional random vectors (q(R). n n/1 If » ,»I 3avar (x ) then »I 1@2»~1@2 is bounded. n n n n n ¹heorem 2. ¸et Mx N= be a sequence of q-dimensional random vectors (q(R), n n/1 R be defined by Eq. (1), R be defined by Eq. (3), and » 3avar(x ). ¹hen avar(x )" n n n ER(» ), and n 1. x "alip(1)NER(» ).ER(» ). n n n 2. x "O (1)NER(» )-ER(» ). n 1 n n Hence, avar-regularity implies that avar (x ) is an equivalence class with respect n to both R and R. Theorem 2 shows that, for an avar-regular random vector x and an avar n sequence » 3avar (x ), avar (x ) is precisely the set of sequences M»I N= of real n n n n n/1 symmetric positive-definite matrices such that lim (» !»I )"0 . And from n?= n n q (Eq. (2)), this is precisely the set of sequences that yield unambiguous efficiency conclusions when Definition 2 is used to define relative efficiency. It is worth remarking on the role of normality in obtaining this conclusion. None of the proofs given in the appendix rely on normality of the limiting random vector z, except for the use of the normal characteristic function in establishing avar (x )"ER(» ) in Theorem 2. Moreover, a proof of n n x "alip(1)Navar (x ).ER(» ) that does not rely on normality is available n n n from the authors on request. Hence, most results given here hold even if
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Definition 1 is relaxed to permit convergence to any common random vector z that has zero mean and identity covariance, as in the RCASOI class of estimators discussed by Bates and White (1993). However, for the purpose of obtaining definitive efficiency comparisons within the avar class, avar(x )-ER(» ) is the crucial property. This relies on both normality n n and x "O (1) in our proof, and Example 2 below shows that x " n 1 n O (1) cannot be discarded. Whether normality is necessary for avar(x )1 n ER(» ) or avar(x )-ER(» ) is an open question, as we have no countern n n examples to these when the limit distribution is nonnormal. Bates and White (1993), p. 648) propose a proof of avar(x )-ER(» ) that does not rely on n n normality, but an important and questionable step therein is that lim »~1@2»I 1@2"I is implied by both y and (»~1@2»I 1@2)y converging n n n n q n n?= n in distribution to the same (potentially nonnormal) (0, I ) random vector, q for some sequence of (0, I ) random vectors y . This is not obvious. Indeed, q n our proof resorts to an application of the Mean Value Theorem on the normal characteristic function to make the transformation from convergence in distribution to convergence of sequences of matrix products, and it is not clear how this step might be accomplished for an arbitrary characteristic function. The following example shows that x "alip(1) is needed in Theorems 1 and n 2 and cannot be replaced by x Oo (1) (or by x "u (1)). n 1 n p Example 1. The role of x "alip (1) in making avar (x ) an equivalence class n n with respect to R. Let 1 z 0 2 2 , z" 1 &N , 1 1 z 0 2 4 2 z z ! 2 n#1 1 n , »*~1@2" x" z n n !n z # 2 1 n and
C D AC D C
DB
C D
C
C
n2#1
!n2
!n
D
,
n
D
.
»~1@2" n
!n2
»*~1@2x "
z z !2z ! 2 1 z !2z 1 2 n P 1 2 &N(0, I ), 2 2z 2z 2 2
n2
Then
n
n
C
D C
D
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so »*3avar(x ), while n n z !2nz #z ! 2 2 1 n »~1@2x " n n 2nz 2
C
D
A C
DB
1 1 n2!2n#3# ! 4n2 n , 0 n!n2!1 4
1 n!n2! 4 n2
CD 0
&N
so » N avar(x ). But n n
C
»*" n
D
1 2# 2 n P n#1 2 2 1# n
2
C A B
1 2# n
C
and » " n
A
1 2# n2
so lim (» !»*)"0 , while n 2 n?= n
C
»~1@2»*»~1@2" n n n
D
2 2
D
1 2# 2 n2 P n2#1 2 2 1# n2
2
n2!2n#2
,
B C
D
2 2
D C
n(1!n)
,
n(1!n)
n2
D
R
!R
!R
R
P
.
Note that x Oo (1) here, but x Oalip(1) since c@ "[!2~1@2 2~1@2] ∀n n 1 n 1 0. Hence, nwhen x Oalip(1) elements in Definition 3 yields c@ x "z J2/nP n n n 2 of avar(x ) can have unbounded inverses, and ER(»*) and avar(x ) can both n n n be smaller than ER(»*), even when x Oo (1). Thus alip is strictly stronger n n 1 than o . 1 The next example shows that x "O (1) is needed in Theorems 1 and 2 as n 1 well. Example 2. The role of x "O (1) in making avar(x ) an equivalence n 1 n class with respect to R. Let z&N(0,1), x "nz, »*"n2, and » "(n#1)2. n n n Clearly, »*,» 3avar(x ) and »*R» . But » !»*"(n#1)2!n2"2n n n * n n n n n #1PR, so » and » are not in the same equivalence class with respect n n to R. The problem is x OO (1), in which case elements of avar(x ) n 1 n can be unbounded, and ER(» ) and avar(x ) can both be larger than n n ER (» ). n
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4. Other definitions of asymptotic efficiency for aar equivalence classes White offers another definition of relative asymptotic efficiency based on the avar concept. This is the definition used by Bates and White as well. Definition 6 ( ¼hite, 1984, pp. 78—79). Let b* and bI be consistent estimators of n n a nonstochastic vector b. Then b* is asymptotically efficient relative to bI if there n n exists »*3avar(Jn(b*!b)) and »I 3avar(Jn(bI !b)), and an integer N, such n n n n that »I !»* is positive semidefinite for all n*N. An estimator is asympn n totically efficient within a class if it is asymptotically efficient relative to every other estimator in the class. It is clear that if b* is asymptotically efficient relative to bI according to n n Definition 6 then b* is asymptotically efficient relative to bI according n n to Definition 2 as well. So, those estimators identified as efficient using Definition 6 (for example, those studied by White, 1984, 1994) are unambiguously efficient by Definition 2, provided we impose avar-regularity (or at least O (1)) to 1 resolve ambiguity when using Definition 2. More interesting is that the converse holds as well, as shown by Theorem 3, if the normalized deviation of bI is n avar-regular, but not without avar-regularity, as shown by Examples 3 and 4 below. ¹heorem 3. ¸et b* and bI be consistent estimators of a nonstochastic q-dimenn n sional vector b and assume Jn(bI !b) is avar-regular. If b* is asymptotically n n efficient relative to bI according to Definition 2 then b* is asymptotically efficient n n relative to bI according to Definition 6. n Example 3. The role of x "alip(1) in Theorem 3. Let z&N(0,1), n Jn(b*!b)"2z/n, and Jn(bI !b)"z/n. Since 4/n23avar(Jn(b*!b)), n n n 1/n23avar(Jn(bI !b)), and lim inf (1/n2!4/n2)"0; b* is efficient relative n n n?= to bI according to Definition 2. For arbitrary »*3avar(Jn(b*!b)) we have n n n
A
B
2z 4 $ »*~1@2 &N 0,»*~1 P N(0,1), n n n2 n so »*(n2/4)P1. Similarly, for arbitrary »I 3avar(Jn(bI !b)) we have »I n2P1. n n n n So, there exists N such that
G
2 »*' , n n2 n*NN 3 »I ( , n 2n2
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implying »I !»*(!1/2n2 for n*N. Hence, b* is not efficient relative to n n n bI according to Definition 6, due to the fact that Jn(bI !b)Oalip(1). n n Example 4. The role of x "O (1) in Theorem 3. The condition n 1 Jn(bI !b)"O (1) is only used in the proof of Theorem 3 to establish equiconn 1 tinuity of the sequence of quadratic forms f (h)"h@(»I !»*)h on the unit n n n sphere. Hence, we construct an example in which a sequence of quadratic forms is not equicontinuous on the unit sphere. Let z&N(0, I ), Jn(b*!b)" 2 n »*1@2z, and Jn(bI !b)"»I 1@2z; where n n n
C
»*" n
csc2 / n 0
0 1
D
,
»I "U #»*; n n n
and / and U are chosen to produce the desired quadratic form. Choosing n n / 3(0, n/4) such that / P0 and n n 1 U" n cos4 / !sin4 / n n
C
(cos2 / !sin2 / )(cot2/ !sin2 / ) n n n n sin / cos / (cot2/ !2 sin2 / ) n n n n
sin / cos / (cot2/ !2 sin2 / ) n n n n (cos2 / !sin2 / ) sin2 / n n n
D
makes f (h)"h@U h a saddle (indefinite) rotated / from standard position that n n n collapses around the rotated h axis as nPR but satisfies f (h )"!1 ∀n, 2 n n where h@ "(!sin / cos / ). Note that n n n
C
R UP n R
D
R 0
C
R and »*P n 0
D
0 1
,
so by Theorem 1(2) we have Jn(bI !b)OO (1). It can be shown, however, n 1 that Jn(bI !b)"alip(1) and that »I is positive definite for n large. Because n n the (1,1) element of U is O(csc2 / ) while the off-diagonal elements are O(csc / ), n n n we have
G
R lim inf f (h)" n 0 n?=
if h O0, 1 if h "0, 1
so b* is efficient relative to bI according to Definition 2. To show that b* is not n n n efficient relative to bI according to Definition 6, let ¼* and ¼ I be arbitrary n n n elements of avar(Jn(b*!b)) and avar(Jn(bI !b)), respectively. Then, by the n n
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same arguments used in Example 3 we have »*~1@2¼*»*~1@2 and n n n I »I ~1@2 both converging to I . Now, write »I ~1@2¼ 2 n n n I !¼*)h "h@ »I 1@2[»I ~1@2¼ I »I ~1@2!I ]»I 1@2h #f (h ) h@ (¼ n n 2 n n n n n n n n n n n #h@ »*1@2[I !»*~1@2¼*»*~1@2]»*1@2h . n n 2 n n n n n
(4)
Since h@ »*h "1#cos2 / and h@ »I h "f (h )#h@ »*h "cos2 / , both n n n n n n n n n n n n n h@ »*1@2 and h@ »I 1@2 are bounded. Hence, the first and last terms of (Eq. (4)) n n n n approach zero, leaving only f (h )"!1 for n large, so b* is not efficient relative n n n to bI according to Definition 6. n We have seen that if two estimators are avar-regular then Definition 2 can be used to make an unambiguous efficiency comparison despite the fact that their avar’s are classes. In this case, any elements of the classes are representative and so the researcher can just select a convenient element for each estimator to make the efficiency comparison. This is not true of Definition 6, even though efficiency conclusions by the two definitions are equivalent under avar-regularity. Definition 2, being a limit-based definition, has the advantage that the avar comparisons it calls for always yield the same answer within an equivalence class with respect to R, as noted in (Eq. (2)) above, and Theorem 2 shows that such an equivalence class is exactly the set of sequences under consideration. In contrast, the avar comparisons called for by Definition 6 are not limitbased and therefore do not always yield the same answer within an equivalence class with respect to R. This can happen even when avar sequences have traditional Fisherian limits if the estimators are equally efficient, and does not violate Theorem 3 because Theorem 3 only promises one pair of avar sequences demonstrating the efficiency conclusion that prevails in the limit (i.e., in Definition 2). This ambiguity is noted by Bates and White (1993), p. 639), who define the concept of a canonical avar sequence as a solution. No matter how it is solved, the problem arises only because Definition 6 is not limit-based. Hence, Theorem 3 shows that the problem can also be solved with no change in our concept of asymptotic efficiency by relying on Definition 2 rather than Definition 6, provided we impose avar-regularity. Even though only x "O (1) is n 1 really needed to get unambiguous efficiency conclusions from Definition 2, if we do not impose x "alip(1) as well then the use of Definition 2 in lieu of n Definition 6 comes at the price of a slightly weaker efficiency concept, in that one might conclude an estimator is efficient that would not be found efficient by Definition 6. Put another way, although the two definitions may differ when x Oalip(1), either definition can be used to unambiguously study efficiency n when x Oalip(1), as in the RCANI class of Bates and White, but alip(1) is n relaxed at the expense of either a slightly weaker efficiency concept (if Definition 2 is used) or of having to find canonical avar sequences to resolve ambiguity (if Definition 6 is used). With alip(1) in place these problems do not arise, since use
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of Definition 2 is then unambiguous and equivalent to Definition 6, but of course alip(1) is itself restrictive. Although the fact that avar is a class is not acknowledged in [White (1984, 1994)], and hence the potential ambiguity of the efficiency definition used (Definition 6) is also not mentioned, the efficiency proofs there rely on the use of canonical avar sequences. In some situations Definitions 2 and 6 are both too strong to detect some potentially informative relative efficiencies, because they rely on liminf’s of differences, which are not the same as differences of liminf’s. In these situations the following may prove useful. Definition 7. Let b* and bI be consistent estimators of a nonstochastic qn n dimensional vector b and suppose Jn(b*!b) and Jn(bI !b) are both avarn regular. Further, assume avar (Jn(b*!b)) and avar (Jn(bI !b)) are both n n nonempty. We say b* possesses minimin asymptotic efficiency relative to bI if n n lim inf h@»I h*lim inf h@»*h for all »I 3avar (Jn(bI !b)), n n n n n?= n?= »*3avar (Jn(b*!b)), and h3Rq (hO0). n n We say b* possesses minimax asymptotic efficiency relative to bI if n n lim sup h@»I h*lim sup h@»*h for all »I 3avar (Jn(bI !b)), n n n n n?= n?= »*3avar (Jn(b*!b)), and h3Rq (hO0). n n Minimin efficiency focuses on best asymptotic performance, by which is meant smallest limiting avar’s, while minimax focuses on worst asymptotic performance, by which is meant largest limiting avar’s. Under avar-regularity, efficiency by either Definition 2 or 6 implies both minimin and minimax efficiency. Examples can be constructed using oscillations in which minimin and minimax efficiency of avar-regular sequences hold but Definitions 2 and 6 do not, even though the estimators being compared are conceptually no different from estimators that can be compared with Definitions 2 and 6. Because of Theorem 2, it is equivalent to only require the defining minimin and minimax inequalities to hold for one pair of avars. Note finally that if one avar sequence for each of two avar-regular estimators has a positive-definite limit then the Fisher definition of asymptotic efficiency is applicable to these limiting covariance matrices. In this case all efficiency definitions involve these same limits and are therefore equivalent. That is, if
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$ N(0,»*) there exist positive-definite matrices »* and »I such that Jn(b*!b)P n $ N(0,»I ) then b* is asymptotically efficient relative to bI and Jn(bI !b)P n n n according to any of the definitions discussed herein if and only if »I !»* is positive semidefinite.
Appendix A. The proof of Theorem 1 uses the following fact. Fact. ¸et x be a random variable with E(x)"k and »(x)"p2(R. ¹hen P(p)Dx!kD)'0. Proof. If P(p'Dx!kD)"1 then p'0, in which case p2": (x!k)2 @x~k@:p dF(x)(: p2 dF(x)"p2, a contradiction. h @x~k@:p Proof of ¹heorem 1. Fix » 3avar (x ), let (e 2 e ) be an orthonormal linearly n n n1 nq independent set of eigenvectors for » , and (j 2 j ) be the corresponding n n1 nq real strictly positive eigenvalues (a full set of real strictly positive eigenvalues and orthonormal linearly independent eigenvectors exists since » is symmetric and n positive definite). (1) First, assume x "alip(1) and suppose the elements of »~1 are not n n bounded. Then there is an unbounded inverse eigenvalue sequence j~1, in which ni case there is a subsequence Mj N= such that lim j "0. Since (j1@2, e ) is kni n/1 n?= kni ni ni »1@2e " an (eigenvalue, eigenvector) pair for »1@2, this implies lim n n?= kn kni lim j1@2e "0 (using orthonormality of the eigenvectors to bound e ). That n?= kni kni kni is, e@ »1@2"o(1). But then e@ x "e@ »1@2»~1@2x "o(1)O (1)"o (1), which kni kn kni kn kni kn kn kn 1 1 contradicts x "alip(1). For the converse, assume »~1 is bounded and suppose n kn x Oalip(1). Since (j~1 2 j~1) are the eigenvalues for »~1, j~1 is bounded for ni n nq n1 n i"1,2,q. That is, there exists M3(0,R) such that j~1(M, implying ni j 'M~1 for i"1,2, q; ∀n. ni Also, since x Oalip(1) there exists a sequence Mc N= on the unit ball in Rq with n n n/1 the following property: To each n there corresponds k *n such that P(M~1@2/2)Dc@ x D))1/n. kn kn n That is, lim P n?=
A
B
M~1@2 )Dc@ x D "0. kn kn 2
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Since (e 2 e ) is an orthonormal basis for Rq we may write c as n1 nq n q c " + (c@ e )e , n ni ni n i/1 so +q (c@ e )2"c@ c "1 ∀n, since c is on the unit ball. Thus, by definition of n n n i/1 n ni e and j , and orthonormality of (e 2 e ), ni ni n1 nq
A
B
1@2 q DDc@ »1@2DD"(c@ » c )1@2" + (c@ e )2j n n n n n n ni ni i/1
A
B
1@2 q "M~1@2 ∀n. ' M~1 + (c@ e )2 n ni i/1 Now, write c@ x "DDc@ »1@2DDa@ »~1@2x , kn kn kn kn kn kn kn
c@ »1@2 where a@ , kn kn , n k DDc@ »1@2DD kn kn
so that DDa DD"1 ∀n. Then a has a convergent subsequence al nPa , where k 0 kn $ kna@ z&(0, a@ a )"(0, 1).5 By the Fact, P(1)D DDa DD"1, so a@l n»~1@2 xl nP l n k k k 0 0 0 0 a@ zD)'0. Moreover, since all distribution functions are continuous off a count0 able set, there exists e3[1, 1] at which the distribution function of Da@ zD is 2 0 continuous. Then, by convergence in distribution there exists N such that
A
n*NNP
B A
M~1@2 M~1@2 )Dc@l nxl nD " P )DDc@l n»1@2 xl nD l DDDa@l »~1@2 l k k k kn kn kn k 2 2 * P(1)Da@l n»~1@2 xl nD) l k kn k 2
B
xl D) * P(e)Da@l »~1@2 l kn
kn
kn
' P(e)Da@0zD) 2 P(1)Da@ zD) 0 '0, * 2 a contradiction.
5 In Definition 1, z is a standard normal random variable. Theorem 1 actually holds for any limiting random vector z with 0 mean and identity covariance, irrespective of normality. To demonstrate this, we avoid use of normality in the present proof.
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(2) First, assume x "O (1) and suppose the elements of » are not bounded. n 1 n Then, as in (1), there is an eigenvalue subsequence Mj N= such that kni n/1 lim »~1@2e "lim j~1@2e "0. That is, e@ »~1@2"o(1). Since e is on n?= kn kni kn n?= kni kni kni kni the unit ball in Rq there is a convergent subsequence el Pe , where DDe DD"1, so kn i 0 0 $ e@ z&(0, e@ e )"(0, 1), implying e@l »~1@2 e@l n »~1@2 xl nP xl nOo (1) by the l l ki kn k kni kn k 0 0 0 1 x Fact. But since x "O (1) we have e@l n »~1@2 l l "o(1)O (1)"o (1), a contraki kn kn 1 1 n 1 diction. For the converse, just note that »1@2 is bounded whenever » is n n bounded, so x "»1@2»~1@2x "O(1)O (1)"O (1). h n n 1 1 n n Proof of ¸emma 1. Since c@Rc is a continuous function of c, there exists cN 3B (0) 1 (the boundary of the unit ball in Rq) such that 0(cN @RcN )c@Rc ∀c3B (0). 1 Now, suppose x Oalip(1). Then as in Theorem 1(1) there exists a sequence n Mc N= on the unit ball with the following property: n n/1 To each n there corresponds k *n such that P(cN @RcN /2)Dc@ x D))1/n. kn kn n That is, lim P(cN @RcN /2)Dc@ x D)"0. Since c is on the unit ball there exists kn kn kn n?= subsequence c Pc 3B (0). Hence, c@ x P $ c@ z&(0, c@ Rc ). a convergent l l l n k kn k n 0 0 0 0 1 By the Fact, P(c@ Rc )Dc@ zD)'0. Moreover, since cN @RcN /2(cN @RcN )c@ Rc , and 0 0 0 0 0 since all distribution functions are continuous off a countable set, there exists e3[cN @RcN /2, c@ Rc ] at which the distribution function of Dc@ zD is continuous. Thus, 0 0 0 by convergence in distribution there exists N such that n*NNP
A
B
cN @RcN P(e)Dc@ zD) 0 )Dc@l nxl nD *P(e)Dc@l nxl nD)' k k k k 2 2
P(c@ Rc )Dc@ zD) 0 '0, * 0 0 2 a contradiction. h
Proof of ¸emma 2. Suppose not. Then (»I 1@2»~1@2)@(»I 1@2»~1@2) is an unn n n n bounded, symmetric, positive-definite matrix. Hence, it can be represented as E K E@ , where E is orthogonal. As in Theorem 1, by unboundedness there exists n n n n an eigenvalue subsequence j~1P0. Denote by f the unit vector in direction i, kni i and let f@E@ »1@2»I ~1@2 n c@ , i n n , n DD f@E@ »1@2»I ~1@2DD n i n n
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so that DDc DD"1 ∀n. Note that n DD f @E@ »1@2»I ~1@2DD " ( f {E@ (»1@2»I ~1»1@2)E f )1@2 n n i n i n n n i n n " ( f {E@ E K~1E@ E f )1@2 i n n n n ni " ( f {K~1f )1@2"j~1@2. i n i ni Hence, c@ »I ~1@2x " c@ »I ~1@2»1@2»~1@2x n n n n n n n n " j1@2[ f {E@ »1@2»I ~1@2»I ~1@2»1@2]»~1@2x n n n n n ni i n n " j1@2[ f {E@ E K~1E@ ]»~1@2x ni i n n n n n n " j1@2[ f {K~1E@ ]»~1@2x n ni i n n n " j~1@2e@ »~1@2x , n ni n ni where e is column i of E . Since e is on the unit ball in Rq, there is ni n kni $ e@ z a convergent subsequence el n Pe 3LB (0), so e@l n »~1@2 xl nP l ki ki kn k 0 0 1 1 &(0, e@ e )"(0, 1).6 Hence, j~1@2 xl nP 0. But, by Lemma 1, e@l n »~1@2 l l kni ki kn k 0 0 xl "alip(1), a contradiction. h »I ~1@2 l kn
kn
Proof of ¹heorem 2. Suppose first that »I 3ER(» ). Then n n lim »1@2»I ~1»1@2"I , so »1@2»I ~1@2"O(1). Denote the characteristic funcn n n q n n?= n tion of a random vector y by f . By the continuity theorem ( Lukacs, 1970, p. 48), y f ~1@2 (h)Pf (h) pointwise. Hence, by (White (1984), p. 66) Lemma 4.23, Vn xn z f I ~1@2 (h)!f (»1@2»I ~1@2h)"f I ~1@2 1@2 ~1@2 (h) n (Vn Vn )Vn xn Vn xn z n !f (»1@2»I ~1@2h)P0 pointwise in h. z n n Recalling that f (h)"exp(!h@h/2), z
A
B
A B
h@»I ~1@2» »I ~1@2h h@h n n f (»1@2»I ~1@2h)"exp ! n Pexp ! , n z n 2 2 so f I ~1@2 (h)Pf (h). Applying the continuity theorem again shows that Vn xn z »I 3avar (x ). n n
6 As in Theorem 1, normality of z is not needed here.
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Now, consider the converse by supposing »I 3avar (x ). By Lemma 2, M*1 n n can denote a common bound for all elements of »I 1@2»~1@2. Fix e'0 and set n n r"q2M. Recalling that f ~1@2 (h)"f (»~1@2h), by the second continuity theVn xn xn n orem ( Lukacs, 1970, p. 53) there exists N such that e h@h e r2 f (»~1@2h)!exp ! ( exp ! ∀h3B (0), xn n r 2 4 2 n*N N e t@t e r2 f (»I ~1@2t)!exp ! ( exp ! ∀t3B (0), xn n r 2 4 2
G
K K
G
K K
A BK A BK
A B A B
where B (0) is the closed ball of radius r about 0 in Rq. Since »I 1@2»~1@2h3B (0) n r n r for every n and h3LB (0), setting t"»I 1@2»~1@2h yields n n 1 h@h e r2 f (»~1@2h)!exp ! ( exp ! , xn n 2 2 4 n*N N e h@»~1@2»I »~1@2h e r2 n n f (»~1@2h)!exp ! n ( exp ! , n x n 2 2 4 for every h3LB (0). Hence, 1
K A A B
A BK A
BK
B
A BK
A B A B
h@»~1@2»I »~1@2h h@h n n n*N N exp ! n !exp ! e 2 2 e r2 ( exp ! 2 2
∀h3LB (0). 1
By the Mean Value theorem,
K A K A BK
B
A BK
h@»~1@2»I »~1@2h h@h n n exp ! n !exp ! 2 2
1 c " ! exp ! Dh@»~1@2»I »~1@2h!h@hD n n n 2 2 for some c between h@»~1@2»I »~1@2h and h@h. Thus n n n
A B
n*N NDh@»~1@2»I »~1@2h!h@hD(e exp n n n e
c!r2 2
for every h3LB (0) and the corresponding c (which also depends on n). Since 1 h@»~1@2»I »~1@2h)r2 for every n and h3LB (0), and h@h"1)r2 for such h, we n n n 1 have c)r2. Thus, exp((c!r2)/2))1 for every n and h3LB (0). That is, 1 n*N NDh@»~1@2»I »~1@2h!h@hD(e ∀h3LB (0). 1 n n n e
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Since h can be any vector on the unit ball, this implies lim »~1@2 n?= n »I »~1@2"I , or »I 3ER(» ). q n n n n Next, consider (1). Since »~1 is bounded by Theorem 1(1) and »~1@2»I n n n [»~1@2»I »~1@2!I ]"0 when »~1@2!I "»~1@2[»I !» ]»~1@2, lim q n n n?= n n n n n n q lim (»I !» )"0. n?= n n Finally, consider (2). Since » is bounded by Theorem 1(2) and »I !» " n n n lim (»I !» )"0 when lim »1@2[»~1@2»I »~1@2!I ]»1@2, n?= n n n?= q n n n n n [»~1@2»I »~1@2!I ]"0. h q n n n Proof of ¹heorem 3. Select »*3avar (Jn(b*!b)) and »I 3avar (Jn(bI !b)) n n n n such that f (h),lim inf f (h)"lim g (h)*0 ∀h3Rq (hO0), where n?= n n?= n f (h),h@(»I !»*)h and g (h),infM f (h): m*nN. We first establish unin n n n m form equicontinuity of the sequence f (h) on the boundary of the unit ball n in Rq. By Theorem 1(2), »I is bounded. It is straightforward to use this n along with f (h)*0 ∀h to conclude that »* is bounded as well, so denote n by M a common bound for all elements of »I and »*. For any h, h 3B (0) n n 0 1 we have D f (h)!f (h )D)DDh!h DD[DD(»I !»*)hDD#DDh@ (»I !»*)DD] 0 n n n 0 0 n n n )DDh!h DD4Mq. 0 Hence, for every e'0 there exists d"e/4Mq such that DDh!h DD(dND f (h)!f (h )D(e ∀n, or f (h) is uniformly equicontinuous on 0 n n 0 n LB (0). Now, by definition of g (h) there exists a subsequence f (h) such that 1 n in g (h)#1/n'f (h)*g (h). Thus f (h) converges pointwise to f (h), so by equiconn in n in tinuity this convergence is actually uniform on LB (0). This implies 1 lim f (h)!1/n"f (h) uniformly on LB (0). Hence, since f (h)*g (h) by defn?= in 1 n n inition of g (h), for every natural number k there exists a natural number N such n k that 1 1 n*N Nf (h)*g (h)'f (h)! 'f (h)! ∀h3LB (0). k n n in 1 n k We may summarize this by saying that the convergence to the limit inferior is uniform on LB (0) in the sense that for every k there exists N such 1 k that n*N Nf (h)#1/k'0 ∀h3LB (0), and without loss of generality k n 1 we may assume 1(N (N (2. Let k be the largest natural number 1 2 n k satisfying N )n for n*N and note that n*N ∀n*N and k 1 kn 1 lim 1/k "0. Thus »I #1/k I 3avar (Jn(bI !b)) by Theorem 2(1), so that n?= n n n q n b* is asymptotically efficient relative to bI according to Definition 6 if n n (»I #(1/k )I )!»* is positive semidefinite for n large. For any n*N and n n q n 1
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hO0 we have
A BCA h@ DDhDD
B DA B A B A B
A B
h h@ h h@h " [»I !»*] # n DDhDD n DDhDD DDhDD k DDhDD2 n h 1 "f # '0, n DDhDD k n * so h@[(»I #1 I )!» ]h'0 ∀n*N and ∀hO0. That is, (»I #(1/k )I )!»* n 1 n n q n n kn q is positive-definite for n*N . h 1 1 »I # I !»* n n k q n
References Bates, C.E., White, H., 1993. Determination of estimators with minimum asymptotic covariance matrices. Econometric Theory 9, 633—648. Fisher, R.A., 1925. Theory of statistical estimation. Proceedings of the Cambridge Philosophical Society 22, 700—725. Lukacs, E., 1970. Characteristic Functions. Hafner, New York. Mann, H.B., Wald, A., 1943. On stochastic limit and order relationships. Annals of Mathematical Statistics 14, 217—226. White, H., 1982. Instrumental variables regression with independent observations. Econometrica 50, 483—499. White, H., 1984. Asymptotic Theory for Econometricians. Academic Press (Harcourt Brace Jovanovich), Orlando. White, H., 1994. Estimation, Inference and Specification Analysis. Econometric Society Monograph, vol. 22. Cambridge University Press, Cambridge.
Relative efficiency with equivalence classes of asymptotic covariances David M. Mandy!,*, Carlos Martins-Filho1," ! Department of Economics, 118 Professional Building, University of Missouri, Columbia, MO 65211, USA " Department of Economics, Oregon State University, Ballard Extension Hall 303, Corvallis, OR 97331-3612, USA Received 1 November 1994; received in revised form 1 March 1998
Abstract White’s (1984, Asymptotic Theory for Econometricians. Academic Press (Harcourt Brace Jovanovich), Orlando.) concept of asymptotic variance is shown to allow some ambiguities when used to study asymptotic efficiency. These ambiguities are resolved with some mild conditions on the estimators being studied, because then White’s asymptotic variance is an equivalence class in which efficiency conclusions are invariant across members of the class. Among the extant efficiency definitions, the lim inf-based definition (White, 1994. Estimation, Inference and Specification Analysis. Econometric Society Monograph, vol. 22, Cambridge University Press, Cambridge. p. 136) is most informative even though identical conclusions can be obtained under our conditions with earlier definitions, but there are still some notions of efficiency allowed by White’s asymptotic variance that can only be detected by weaker efficiency definitions. ( 1999 Elsevier Science S.A. All rights reserved. JEL classification: C10; C13; C50 Keywords: Asymptotic efficiency
* Corresponding author. E-mail: [email protected] 1 We benefitted greatly from particularly insightful comments, and much patience, from an associate editor and an anonymous referee. Naturally they bear no responsibility for any remaining errors. 0304-4076/99/$ — see front matter ( 1999 Elsevier Science S.A. All rights reserved. PII S 0 3 0 4 - 4 0 7 6 ( 9 8 ) 0 0 0 2 2 - 0
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1. Introduction Suppose b* and bI are estimators for some parameter vector b and we want to n n determine their relative asymptotic efficiency. A procedure discussed by White (1994), p. 136) is to first find two sequences of matrices, »* and »I , such that n n both »*~1@2Jn(b*!b) and »I ~1@2Jn(bI !b) converge in distribution to stann n n n dard normals. White (1984, p. 66; 1994, p. 91) calls such sequences asymptotic variances, or avars. Next, these sequences are used to show lim inf h@(»I !»*)h*0 ∀h, whence we conclude, for example, that b* is n?= n n n asymptotically efficient relative to bI . This procedure has an advantage over the n traditional ( Fisher, 1925) approach of comparing the covariances of the asymptotic distributions, in that »* and »I need not have limits ( White, 1982). But n n there is a potential disadvantage as well because in all interesting cases there will be other avar sequences, say » , such that »~1@2Jn(b*!b) converges in n n n distribution to a standard normal. Thus, one must inquire whether lim inf h@(»I !»*)h*0 ∀h implies lim inf h@(»I !» )h*0 ∀h for any n?= n n n?= n n such » , that is, whether statements about asymptotic efficiency are invariant to n the avar sequences examined. By Fatou’s lemma, this implication holds if lim (» !»*)"0, and the converse holds as well provided the two lim infs n?= n n are both nonnegative for the same set of »I sequences (that is, for the same rival n estimators bI ). But without lim (» !»*)"0 anything is possible, including n n?= n n a reversal of the original conclusion, in which case the relative efficiency * comparison based on » would appear definitive when it is in fact ambiguous. n Alternatively, »* may yield no conclusion when a conclusion is obtainable using n » , and the same comments apply to all possible substitutes for »* and »I . n n n Thus, at this level of generality, lim (» !»*)"0 for all relevant sequences n?= n n »* and » is sufficient for meaningful efficiency conclusions. The converse is also n n of interest, namely, whether two sequences that satisfy this condition are both avars given that one of them is. More formally, we must investigate whether the relation R defined by M»*N= RM» N= Q lim (» !»*)"0 n n/1 n?= n n n/1 n
(1)
has an equivalence class consisting of all sequences that produce the desired asymptotic normal distribution. In this paper we first show this is the appropriate relation for studying relative asymptotic efficiency, in that efficiency conclusions are unambiguous if and only if all pairs of candidate sequences have the relation R. We then show that an avar collection is not always an equivalence class with respect to R, but some mild conditions on the underlying random vector are sufficient to ensure that an avar collection is indeed an equivalence class with respect to R. Hence, when these conditions hold asymptotic efficiency can be studied without ambiguity using the full generality of the avar concept.
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We demonstrate that the sufficient conditions are equivalent to boundedness of all avar sequences and the corresponding sequences of inverses. When avar sequences, but not their inverses, are bounded the avar collection is a (perhaps proper) subset of an equivalence class with respect to R. This is sufficient for the sign of lim inf h@(»I !»*)h to be invariant across all »* sequences that are n?= n n n avars of Jn(b*!b), and hence for unambiguous efficiency conclusions by this n criterion. However, (White (1984), pp. 78—79) provides another definition of efficiency based on the avar concept that does not always yield the same efficiency conclusions as this lim inf criterion. We show the two definitions are equivalent if avar sequences and their corresponding sequences of inverses are bounded. Hence, for consistency across definitions avar must be an equivalence class with respect to R. This rules out both divergences to infinity and approaches to singularity in the collection. Under these conditions the generalization accomplished by the avar concept over the traditional approach is that avar accommodates nonconvergent but bounded oscillations in the sequences forming the avar class, and their corresponding sequences of inverses. The avar concept does not directly deliver assuredly unambiguous efficiency comparisons when there are divergences to infinity or approaches to singularity in the collection. Bounding these sequences is a mild restriction, however, because the random vector can usually be normalized to eliminate such problems before forming the avar class, as is customary through multiplication by Jn. In general, an element of b* can be normalized by any n nonstochastic sequence in order to obtain bounded avars and inverse avars, and the normalizing sequence can differ across elements of b*. Once all estimators n under consideration are so normalized, our results show that their relative asymptotic efficiencies can be unambiguously compared irrespective of convergence of the avar sequences, the particular sequences examined, or the efficiency definition used. There are two caveats here. First, a multiple correlation between normalized elements of b* might tend to one, in which case some asymptotically redundant n element(s) should be dropped before asymptotic efficiency is studied, since this situation would lead to avar sequences that approach singularity. Second, because relative efficiency conclusions can be affected if different normalizations are used for b* and bI , one should only compare estimators via avars of normalized n n deviations when corresponding elements are normalized with the same sequence. Bounds on avar sequences and their sequences of inverses are usually (White, 1982; 1984, p. 66; 1994, pp. 130—136) although not always (Bates and White, 1993) imposed, but even when they are imposed they are somewhat unsatisfactory as primitive assumptions because they utilize the avar class to restrict itself rather than placing restrictions on the underlying random vector. In contrast, our conditions are imposed directly on the random vector and thus illuminate exactly what is being assumed when bounds are placed on avar sequences. To accomplish this we introduce a new order in probability concept, which we call asymptotic linear independence in probability, or alip.
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Finally, since nonconvergent oscillations are permitted in an avar class it is possible to have ‘one-sided’ relative efficiency that is not addressed by either of White’s definitions, in the form of either a smaller minimal limiting variance (minimin efficiency) or a smaller maximal limiting variance (minimax efficiency). We introduce new definitions to address these possibilities. Naturally, the new definitions are weaker than White’s definitions, so all known avar efficiency conclusions automatically hold for minimin and minimax efficiency. 2. The equivalence relation R for studying relative asymptotic efficiency Let Qq denote the set of all sequences M» N= of real symmetric positiven n/1 definite nonstochastic (q]q) matrices » . For such » , denote by »1@2 the n n n unique real-symmetric positive-definite matrix satisfying »1@2»1@2"» . In esn n n sence, White’s definition of asymptotic covariance is the following. Definition 1 ( ¼hite, 1984, p. 66; 1994, p. 91).2 Let Mx N= be a sequence of n n/1 q-dimensional random vectors (q(R). The asymptotic covariance of Mx N= , n n/1 denoted avar(Mx N= ), is n n/1 $ (0, I )N. avar(Mx N= ),MM» N= 3Qq: »~1@2 x Pz&N n n q n n/1 n n/1 For brevity, we drop the indexes and write » 3avar(x ) to denote that the n n sequence M» N= is an element of avar(Mx N= ). n n/1 n n/1 We say Definition 1 is White’s definition ‘in essence’ because neither White (1984) nor White (1994) acknowledge that avar is a class. Bates and White (1993) explicitly discuss that the avar of their RCASOI class of estimators is a class, but attribute this to the flexibility of RCASOI classes. In fact, in all interesting cases the avar of any single random vector is a class because, if there exists one $ z for any MA N= satisfysequence » satisfying Definition 1, then A »~1@2x P n n n/1 n n n ing lim A "I . Hence, if » 3avar (x ) then A{~1» A~13avar(x ) n n n n n?= n q n n A "I . In other words, a great ∀MA N= satisfying A »~1@23Qq and lim n?= n q n n n n/1 profusion of avar sequences exists whenever a single avar sequence exists. This observation brings to the forefront the issue of whether conclusions about relative asymptotic efficiency are invariant to which element of an avar class is examined. What is needed is an equivalence relation in Qq whose equivalence classes are precisely those collections of sequences for which conclusions about relative asymptotic efficiency are identical. Given this, if avar(x ) n is an equivalence class of the relation, or even a subset of an equivalence class, 2 White (1984) actually only requires positive definiteness of » for large n. Since only the tail of n the sequence is important in the definition, there is no consequential loss of generality from assuming the entire sequence is positive definite.
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then efficiency conclusions are unambiguous. To construct the equivalence relation we first need a formal definition of efficiency. We start with: Definition 2 ( ¼hite, 1994, p. 136).3 Let b* and bI be consistent estimators of n n a nonstochastic q-dimensional vector b. Then b* is asymptotically efficient relative n to bI if there exists »*3avar(Jn(b*!b)) and »I 3avar(Jn(bI !b)) such that n n n n n lim inf h@(»I !»*)h*0 n n n?= for all h3Rq (hO0). An estimator is asymptotically efficient within a class if it is asymptotically efficient relative to every other estimator in the class. Note that one consistent estimator can be asymptotically efficient relative to another only if the avar’s of both normalized estimators are nonempty. According to this definition the relation R in Qq that yields unambiguous efficiency conclusions within an equivalence class is M»*N= RM» N= if and n n/1 n n/1 only if (∀»I ): lim inf h@(»I !»*)h*0 ∀h Q lim inf h@(»I !» )h*0 ∀h. n n n n n n?= n?=
(2)
It is trivial to verify that R is an equivalence relation (satisfying reflexivity, symmetry, and transitivity). More interesting is the fact that this relation is identical to the relation defined in (Eq. (1)). That (Eq. (1)) implies (Eq. (2)) is an application of Fatou’s lemma. For the converse, »I "»* in (Eq. (2)) n n implies lim inf h@(»*!» )h*0 ∀h, while »I "» in (Eq. (2)) implies n?= n n n n lim sup h@(»*!» )h)0 ∀h. We utilize (Eq. (1)) henceforth since it is more n?= n n convenient, and henceforth denote the equivalence class of »* with respect to n R by ER(»*),M» 3Qq: lim (»*!» )"0N. n n n?= n n 3. Sufficient conditions for aar(xn ) to be an equivalence class with respect to R In general, avar(x ) is not an equivalence class with respect to R, so some n restrictions are needed in the form of regularity conditions on the underlying random vector x . These restrictions include the following notion of asymptotic n linear independence in probability. Definition 3. Let Mx N= be a sequence of q-dimensional random vectors n n/1 (q(R) and My N= be a sequence of strictly positive nonstochastic real n n/1 3 Definitions 2 and 6 (below) are rephrased from White’s original statements to reflect the fact that avar(x ) is a class. We use the same terminology to accomplish this that (Bates and White (1993), n Definition 2.5) use for a RCASOI class.
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numbers. Mx N= is asymptotically linearly independent in probability of order n n/1 My N= if for every sequence Mc N= of real nonstochastic q-dimensional vectors n n/1 n n/1 satisfying DDc DD"1 ∀n; there exists a triple (N, e, d), where N is a natural n number, e'0, and d'0; such that
A K KB
c@ x n*N N P e( n n 'd. y n For brevity, this is denoted x "alip(y ).4 n n It is clear that x "alip(y )Nx Oo (y ). The converse fails because n n n 1 n x Oo (y ) still permits subsequences of c@ x /y that converge in probability to n n n n 1 n zero, and also linear combinations of x /y that converge in probability to zero n n as long as individual components do not. Definition 4. A sequence Mx N= of q-dimensional random vectors (q(R) is n n/1 avar-regular if x "O (1) and x "alip(1). n 1 n Avar-regularity places restrictions on the primitive of the problem, the underlying random vector x . Theorem 1 shows this is equivalent to White’s (1984, p. n 66) approach of bounding the avar sequences and the corresponding sequences of inverses. ¹heorem 1. ¸et Mx N= be a sequence of q-dimensional random vectors (q(R) n n/1 for which avar(x ) is nonempty. ¹hen n 1. x "alip(1) if and only if »~1 is bounded ∀» 3avar(x ). n n n n 2. x "O (1) if and only if » is bounded ∀» 3avar (x ). n 1 n n n Hence, x is avar-regular if and only if both » and »~1 are bounded for n n n every » 3avar (x ). n n Proof. All proofs are in the appendix. h The main result of this section is that avar-regularity is sufficient to ensure avar (x ) is an equivalence class with respect to R. Bates and White ((1993), n 4 The alip concept is similar to Mann and Wald’s (1943) notion of u , but these concepts differ in p two ways that are important in the present context. First, alip(y ) is weaker than u (y ) in that u (y ) n p n p n requires the probability that a normalized random variable is nonzero approach one, while alip(y ) n merely requires that this probability not approach zero. Second, alip(y ) is stronger than u (y ) in n p n that alip(y ) places its condition on all bounded nondegenerate linear combinations of a random n vector, while u (y ) merely places its condition on the individual components of a random vector. In p n general, u (1) cannot replace alip(1) in the results below. p
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Theorem (2.3)) investigate whether » 3avar (x )Navar (x )-ER(» ), using the n n n n slightly different relation R in Qq defined by M» N= RM»I N= Q lim »~1@2»I »~1@2"I . q n n n n n/1 n n/1 n?=
(3)
As with R, it is straightforward to verify that R is an equivalence relation. However, R does not have the same equivalence classes as R unless attention is restricted to sequences that are bounded and have bounded inverses. Hence, from (Eq. (2)), R is not always the appropriate relation for studying relative asymptotic efficiency. But Theorem 2 below shows that avar-regularity, which by Theorem 1 bounds the candidate sequences and their inverses, implies avar (x ) is an equivalence class with respect to both R and R. The proof of n Theorem 2 relies on the following preliminary results. ¸emma 1. ¸et Mx N= be a sequence of q-dimensional random vectors (q(R). n/1 $ z&(0, R),n where If x P R is positive definite, then x "alip(1). n n ¸emma 2. ¸et Mx N= be a sequence of q-dimensional random vectors (q(R). n n/1 If » ,»I 3avar (x ) then »I 1@2»~1@2 is bounded. n n n n n ¹heorem 2. ¸et Mx N= be a sequence of q-dimensional random vectors (q(R), n n/1 R be defined by Eq. (1), R be defined by Eq. (3), and » 3avar(x ). ¹hen avar(x )" n n n ER(» ), and n 1. x "alip(1)NER(» ).ER(» ). n n n 2. x "O (1)NER(» )-ER(» ). n 1 n n Hence, avar-regularity implies that avar (x ) is an equivalence class with respect n to both R and R. Theorem 2 shows that, for an avar-regular random vector x and an avar n sequence » 3avar (x ), avar (x ) is precisely the set of sequences M»I N= of real n n n n n/1 symmetric positive-definite matrices such that lim (» !»I )"0 . And from n?= n n q (Eq. (2)), this is precisely the set of sequences that yield unambiguous efficiency conclusions when Definition 2 is used to define relative efficiency. It is worth remarking on the role of normality in obtaining this conclusion. None of the proofs given in the appendix rely on normality of the limiting random vector z, except for the use of the normal characteristic function in establishing avar (x )"ER(» ) in Theorem 2. Moreover, a proof of n n x "alip(1)Navar (x ).ER(» ) that does not rely on normality is available n n n from the authors on request. Hence, most results given here hold even if
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Definition 1 is relaxed to permit convergence to any common random vector z that has zero mean and identity covariance, as in the RCASOI class of estimators discussed by Bates and White (1993). However, for the purpose of obtaining definitive efficiency comparisons within the avar class, avar(x )-ER(» ) is the crucial property. This relies on both normality n n and x "O (1) in our proof, and Example 2 below shows that x " n 1 n O (1) cannot be discarded. Whether normality is necessary for avar(x )1 n ER(» ) or avar(x )-ER(» ) is an open question, as we have no countern n n examples to these when the limit distribution is nonnormal. Bates and White (1993), p. 648) propose a proof of avar(x )-ER(» ) that does not rely on n n normality, but an important and questionable step therein is that lim »~1@2»I 1@2"I is implied by both y and (»~1@2»I 1@2)y converging n n n n q n n?= n in distribution to the same (potentially nonnormal) (0, I ) random vector, q for some sequence of (0, I ) random vectors y . This is not obvious. Indeed, q n our proof resorts to an application of the Mean Value Theorem on the normal characteristic function to make the transformation from convergence in distribution to convergence of sequences of matrix products, and it is not clear how this step might be accomplished for an arbitrary characteristic function. The following example shows that x "alip(1) is needed in Theorems 1 and n 2 and cannot be replaced by x Oo (1) (or by x "u (1)). n 1 n p Example 1. The role of x "alip (1) in making avar (x ) an equivalence class n n with respect to R. Let 1 z 0 2 2 , z" 1 &N , 1 1 z 0 2 4 2 z z ! 2 n#1 1 n , »*~1@2" x" z n n !n z # 2 1 n and
C D AC D C
DB
C D
C
C
n2#1
!n2
!n
D
,
n
D
.
»~1@2" n
!n2
»*~1@2x "
z z !2z ! 2 1 z !2z 1 2 n P 1 2 &N(0, I ), 2 2z 2z 2 2
n2
Then
n
n
C
D C
D
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so »*3avar(x ), while n n z !2nz #z ! 2 2 1 n »~1@2x " n n 2nz 2
C
D
A C
DB
1 1 n2!2n#3# ! 4n2 n , 0 n!n2!1 4
1 n!n2! 4 n2
CD 0
&N
so » N avar(x ). But n n
C
»*" n
D
1 2# 2 n P n#1 2 2 1# n
2
C A B
1 2# n
C
and » " n
A
1 2# n2
so lim (» !»*)"0 , while n 2 n?= n
C
»~1@2»*»~1@2" n n n
D
2 2
D
1 2# 2 n2 P n2#1 2 2 1# n2
2
n2!2n#2
,
B C
D
2 2
D C
n(1!n)
,
n(1!n)
n2
D
R
!R
!R
R
P
.
Note that x Oo (1) here, but x Oalip(1) since c@ "[!2~1@2 2~1@2] ∀n n 1 n 1 0. Hence, nwhen x Oalip(1) elements in Definition 3 yields c@ x "z J2/nP n n n 2 of avar(x ) can have unbounded inverses, and ER(»*) and avar(x ) can both n n n be smaller than ER(»*), even when x Oo (1). Thus alip is strictly stronger n n 1 than o . 1 The next example shows that x "O (1) is needed in Theorems 1 and 2 as n 1 well. Example 2. The role of x "O (1) in making avar(x ) an equivalence n 1 n class with respect to R. Let z&N(0,1), x "nz, »*"n2, and » "(n#1)2. n n n Clearly, »*,» 3avar(x ) and »*R» . But » !»*"(n#1)2!n2"2n n n * n n n n n #1PR, so » and » are not in the same equivalence class with respect n n to R. The problem is x OO (1), in which case elements of avar(x ) n 1 n can be unbounded, and ER(» ) and avar(x ) can both be larger than n n ER (» ). n
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4. Other definitions of asymptotic efficiency for aar equivalence classes White offers another definition of relative asymptotic efficiency based on the avar concept. This is the definition used by Bates and White as well. Definition 6 ( ¼hite, 1984, pp. 78—79). Let b* and bI be consistent estimators of n n a nonstochastic vector b. Then b* is asymptotically efficient relative to bI if there n n exists »*3avar(Jn(b*!b)) and »I 3avar(Jn(bI !b)), and an integer N, such n n n n that »I !»* is positive semidefinite for all n*N. An estimator is asympn n totically efficient within a class if it is asymptotically efficient relative to every other estimator in the class. It is clear that if b* is asymptotically efficient relative to bI according to n n Definition 6 then b* is asymptotically efficient relative to bI according n n to Definition 2 as well. So, those estimators identified as efficient using Definition 6 (for example, those studied by White, 1984, 1994) are unambiguously efficient by Definition 2, provided we impose avar-regularity (or at least O (1)) to 1 resolve ambiguity when using Definition 2. More interesting is that the converse holds as well, as shown by Theorem 3, if the normalized deviation of bI is n avar-regular, but not without avar-regularity, as shown by Examples 3 and 4 below. ¹heorem 3. ¸et b* and bI be consistent estimators of a nonstochastic q-dimenn n sional vector b and assume Jn(bI !b) is avar-regular. If b* is asymptotically n n efficient relative to bI according to Definition 2 then b* is asymptotically efficient n n relative to bI according to Definition 6. n Example 3. The role of x "alip(1) in Theorem 3. Let z&N(0,1), n Jn(b*!b)"2z/n, and Jn(bI !b)"z/n. Since 4/n23avar(Jn(b*!b)), n n n 1/n23avar(Jn(bI !b)), and lim inf (1/n2!4/n2)"0; b* is efficient relative n n n?= to bI according to Definition 2. For arbitrary »*3avar(Jn(b*!b)) we have n n n
A
B
2z 4 $ »*~1@2 &N 0,»*~1 P N(0,1), n n n2 n so »*(n2/4)P1. Similarly, for arbitrary »I 3avar(Jn(bI !b)) we have »I n2P1. n n n n So, there exists N such that
G
2 »*' , n n2 n*NN 3 »I ( , n 2n2
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implying »I !»*(!1/2n2 for n*N. Hence, b* is not efficient relative to n n n bI according to Definition 6, due to the fact that Jn(bI !b)Oalip(1). n n Example 4. The role of x "O (1) in Theorem 3. The condition n 1 Jn(bI !b)"O (1) is only used in the proof of Theorem 3 to establish equiconn 1 tinuity of the sequence of quadratic forms f (h)"h@(»I !»*)h on the unit n n n sphere. Hence, we construct an example in which a sequence of quadratic forms is not equicontinuous on the unit sphere. Let z&N(0, I ), Jn(b*!b)" 2 n »*1@2z, and Jn(bI !b)"»I 1@2z; where n n n
C
»*" n
csc2 / n 0
0 1
D
,
»I "U #»*; n n n
and / and U are chosen to produce the desired quadratic form. Choosing n n / 3(0, n/4) such that / P0 and n n 1 U" n cos4 / !sin4 / n n
C
(cos2 / !sin2 / )(cot2/ !sin2 / ) n n n n sin / cos / (cot2/ !2 sin2 / ) n n n n
sin / cos / (cot2/ !2 sin2 / ) n n n n (cos2 / !sin2 / ) sin2 / n n n
D
makes f (h)"h@U h a saddle (indefinite) rotated / from standard position that n n n collapses around the rotated h axis as nPR but satisfies f (h )"!1 ∀n, 2 n n where h@ "(!sin / cos / ). Note that n n n
C
R UP n R
D
R 0
C
R and »*P n 0
D
0 1
,
so by Theorem 1(2) we have Jn(bI !b)OO (1). It can be shown, however, n 1 that Jn(bI !b)"alip(1) and that »I is positive definite for n large. Because n n the (1,1) element of U is O(csc2 / ) while the off-diagonal elements are O(csc / ), n n n we have
G
R lim inf f (h)" n 0 n?=
if h O0, 1 if h "0, 1
so b* is efficient relative to bI according to Definition 2. To show that b* is not n n n efficient relative to bI according to Definition 6, let ¼* and ¼ I be arbitrary n n n elements of avar(Jn(b*!b)) and avar(Jn(bI !b)), respectively. Then, by the n n
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same arguments used in Example 3 we have »*~1@2¼*»*~1@2 and n n n I »I ~1@2 both converging to I . Now, write »I ~1@2¼ 2 n n n I !¼*)h "h@ »I 1@2[»I ~1@2¼ I »I ~1@2!I ]»I 1@2h #f (h ) h@ (¼ n n 2 n n n n n n n n n n n #h@ »*1@2[I !»*~1@2¼*»*~1@2]»*1@2h . n n 2 n n n n n
(4)
Since h@ »*h "1#cos2 / and h@ »I h "f (h )#h@ »*h "cos2 / , both n n n n n n n n n n n n n h@ »*1@2 and h@ »I 1@2 are bounded. Hence, the first and last terms of (Eq. (4)) n n n n approach zero, leaving only f (h )"!1 for n large, so b* is not efficient relative n n n to bI according to Definition 6. n We have seen that if two estimators are avar-regular then Definition 2 can be used to make an unambiguous efficiency comparison despite the fact that their avar’s are classes. In this case, any elements of the classes are representative and so the researcher can just select a convenient element for each estimator to make the efficiency comparison. This is not true of Definition 6, even though efficiency conclusions by the two definitions are equivalent under avar-regularity. Definition 2, being a limit-based definition, has the advantage that the avar comparisons it calls for always yield the same answer within an equivalence class with respect to R, as noted in (Eq. (2)) above, and Theorem 2 shows that such an equivalence class is exactly the set of sequences under consideration. In contrast, the avar comparisons called for by Definition 6 are not limitbased and therefore do not always yield the same answer within an equivalence class with respect to R. This can happen even when avar sequences have traditional Fisherian limits if the estimators are equally efficient, and does not violate Theorem 3 because Theorem 3 only promises one pair of avar sequences demonstrating the efficiency conclusion that prevails in the limit (i.e., in Definition 2). This ambiguity is noted by Bates and White (1993), p. 639), who define the concept of a canonical avar sequence as a solution. No matter how it is solved, the problem arises only because Definition 6 is not limit-based. Hence, Theorem 3 shows that the problem can also be solved with no change in our concept of asymptotic efficiency by relying on Definition 2 rather than Definition 6, provided we impose avar-regularity. Even though only x "O (1) is n 1 really needed to get unambiguous efficiency conclusions from Definition 2, if we do not impose x "alip(1) as well then the use of Definition 2 in lieu of n Definition 6 comes at the price of a slightly weaker efficiency concept, in that one might conclude an estimator is efficient that would not be found efficient by Definition 6. Put another way, although the two definitions may differ when x Oalip(1), either definition can be used to unambiguously study efficiency n when x Oalip(1), as in the RCANI class of Bates and White, but alip(1) is n relaxed at the expense of either a slightly weaker efficiency concept (if Definition 2 is used) or of having to find canonical avar sequences to resolve ambiguity (if Definition 6 is used). With alip(1) in place these problems do not arise, since use
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of Definition 2 is then unambiguous and equivalent to Definition 6, but of course alip(1) is itself restrictive. Although the fact that avar is a class is not acknowledged in [White (1984, 1994)], and hence the potential ambiguity of the efficiency definition used (Definition 6) is also not mentioned, the efficiency proofs there rely on the use of canonical avar sequences. In some situations Definitions 2 and 6 are both too strong to detect some potentially informative relative efficiencies, because they rely on liminf’s of differences, which are not the same as differences of liminf’s. In these situations the following may prove useful. Definition 7. Let b* and bI be consistent estimators of a nonstochastic qn n dimensional vector b and suppose Jn(b*!b) and Jn(bI !b) are both avarn regular. Further, assume avar (Jn(b*!b)) and avar (Jn(bI !b)) are both n n nonempty. We say b* possesses minimin asymptotic efficiency relative to bI if n n lim inf h@»I h*lim inf h@»*h for all »I 3avar (Jn(bI !b)), n n n n n?= n?= »*3avar (Jn(b*!b)), and h3Rq (hO0). n n We say b* possesses minimax asymptotic efficiency relative to bI if n n lim sup h@»I h*lim sup h@»*h for all »I 3avar (Jn(bI !b)), n n n n n?= n?= »*3avar (Jn(b*!b)), and h3Rq (hO0). n n Minimin efficiency focuses on best asymptotic performance, by which is meant smallest limiting avar’s, while minimax focuses on worst asymptotic performance, by which is meant largest limiting avar’s. Under avar-regularity, efficiency by either Definition 2 or 6 implies both minimin and minimax efficiency. Examples can be constructed using oscillations in which minimin and minimax efficiency of avar-regular sequences hold but Definitions 2 and 6 do not, even though the estimators being compared are conceptually no different from estimators that can be compared with Definitions 2 and 6. Because of Theorem 2, it is equivalent to only require the defining minimin and minimax inequalities to hold for one pair of avars. Note finally that if one avar sequence for each of two avar-regular estimators has a positive-definite limit then the Fisher definition of asymptotic efficiency is applicable to these limiting covariance matrices. In this case all efficiency definitions involve these same limits and are therefore equivalent. That is, if
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$ N(0,»*) there exist positive-definite matrices »* and »I such that Jn(b*!b)P n $ N(0,»I ) then b* is asymptotically efficient relative to bI and Jn(bI !b)P n n n according to any of the definitions discussed herein if and only if »I !»* is positive semidefinite.
Appendix A. The proof of Theorem 1 uses the following fact. Fact. ¸et x be a random variable with E(x)"k and »(x)"p2(R. ¹hen P(p)Dx!kD)'0. Proof. If P(p'Dx!kD)"1 then p'0, in which case p2": (x!k)2 @x~k@:p dF(x)(: p2 dF(x)"p2, a contradiction. h @x~k@:p Proof of ¹heorem 1. Fix » 3avar (x ), let (e 2 e ) be an orthonormal linearly n n n1 nq independent set of eigenvectors for » , and (j 2 j ) be the corresponding n n1 nq real strictly positive eigenvalues (a full set of real strictly positive eigenvalues and orthonormal linearly independent eigenvectors exists since » is symmetric and n positive definite). (1) First, assume x "alip(1) and suppose the elements of »~1 are not n n bounded. Then there is an unbounded inverse eigenvalue sequence j~1, in which ni case there is a subsequence Mj N= such that lim j "0. Since (j1@2, e ) is kni n/1 n?= kni ni ni »1@2e " an (eigenvalue, eigenvector) pair for »1@2, this implies lim n n?= kn kni lim j1@2e "0 (using orthonormality of the eigenvectors to bound e ). That n?= kni kni kni is, e@ »1@2"o(1). But then e@ x "e@ »1@2»~1@2x "o(1)O (1)"o (1), which kni kn kni kn kni kn kn kn 1 1 contradicts x "alip(1). For the converse, assume »~1 is bounded and suppose n kn x Oalip(1). Since (j~1 2 j~1) are the eigenvalues for »~1, j~1 is bounded for ni n nq n1 n i"1,2,q. That is, there exists M3(0,R) such that j~1(M, implying ni j 'M~1 for i"1,2, q; ∀n. ni Also, since x Oalip(1) there exists a sequence Mc N= on the unit ball in Rq with n n n/1 the following property: To each n there corresponds k *n such that P(M~1@2/2)Dc@ x D))1/n. kn kn n That is, lim P n?=
A
B
M~1@2 )Dc@ x D "0. kn kn 2
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Since (e 2 e ) is an orthonormal basis for Rq we may write c as n1 nq n q c " + (c@ e )e , n ni ni n i/1 so +q (c@ e )2"c@ c "1 ∀n, since c is on the unit ball. Thus, by definition of n n n i/1 n ni e and j , and orthonormality of (e 2 e ), ni ni n1 nq
A
B
1@2 q DDc@ »1@2DD"(c@ » c )1@2" + (c@ e )2j n n n n n n ni ni i/1
A
B
1@2 q "M~1@2 ∀n. ' M~1 + (c@ e )2 n ni i/1 Now, write c@ x "DDc@ »1@2DDa@ »~1@2x , kn kn kn kn kn kn kn
c@ »1@2 where a@ , kn kn , n k DDc@ »1@2DD kn kn
so that DDa DD"1 ∀n. Then a has a convergent subsequence al nPa , where k 0 kn $ kna@ z&(0, a@ a )"(0, 1).5 By the Fact, P(1)D DDa DD"1, so a@l n»~1@2 xl nP l n k k k 0 0 0 0 a@ zD)'0. Moreover, since all distribution functions are continuous off a count0 able set, there exists e3[1, 1] at which the distribution function of Da@ zD is 2 0 continuous. Then, by convergence in distribution there exists N such that
A
n*NNP
B A
M~1@2 M~1@2 )Dc@l nxl nD " P )DDc@l n»1@2 xl nD l DDDa@l »~1@2 l k k k kn kn kn k 2 2 * P(1)Da@l n»~1@2 xl nD) l k kn k 2
B
xl D) * P(e)Da@l »~1@2 l kn
kn
kn
' P(e)Da@0zD) 2 P(1)Da@ zD) 0 '0, * 2 a contradiction.
5 In Definition 1, z is a standard normal random variable. Theorem 1 actually holds for any limiting random vector z with 0 mean and identity covariance, irrespective of normality. To demonstrate this, we avoid use of normality in the present proof.
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(2) First, assume x "O (1) and suppose the elements of » are not bounded. n 1 n Then, as in (1), there is an eigenvalue subsequence Mj N= such that kni n/1 lim »~1@2e "lim j~1@2e "0. That is, e@ »~1@2"o(1). Since e is on n?= kn kni kn n?= kni kni kni kni the unit ball in Rq there is a convergent subsequence el Pe , where DDe DD"1, so kn i 0 0 $ e@ z&(0, e@ e )"(0, 1), implying e@l »~1@2 e@l n »~1@2 xl nP xl nOo (1) by the l l ki kn k kni kn k 0 0 0 1 x Fact. But since x "O (1) we have e@l n »~1@2 l l "o(1)O (1)"o (1), a contraki kn kn 1 1 n 1 diction. For the converse, just note that »1@2 is bounded whenever » is n n bounded, so x "»1@2»~1@2x "O(1)O (1)"O (1). h n n 1 1 n n Proof of ¸emma 1. Since c@Rc is a continuous function of c, there exists cN 3B (0) 1 (the boundary of the unit ball in Rq) such that 0(cN @RcN )c@Rc ∀c3B (0). 1 Now, suppose x Oalip(1). Then as in Theorem 1(1) there exists a sequence n Mc N= on the unit ball with the following property: n n/1 To each n there corresponds k *n such that P(cN @RcN /2)Dc@ x D))1/n. kn kn n That is, lim P(cN @RcN /2)Dc@ x D)"0. Since c is on the unit ball there exists kn kn kn n?= subsequence c Pc 3B (0). Hence, c@ x P $ c@ z&(0, c@ Rc ). a convergent l l l n k kn k n 0 0 0 0 1 By the Fact, P(c@ Rc )Dc@ zD)'0. Moreover, since cN @RcN /2(cN @RcN )c@ Rc , and 0 0 0 0 0 since all distribution functions are continuous off a countable set, there exists e3[cN @RcN /2, c@ Rc ] at which the distribution function of Dc@ zD is continuous. Thus, 0 0 0 by convergence in distribution there exists N such that n*NNP
A
B
cN @RcN P(e)Dc@ zD) 0 )Dc@l nxl nD *P(e)Dc@l nxl nD)' k k k k 2 2
P(c@ Rc )Dc@ zD) 0 '0, * 0 0 2 a contradiction. h
Proof of ¸emma 2. Suppose not. Then (»I 1@2»~1@2)@(»I 1@2»~1@2) is an unn n n n bounded, symmetric, positive-definite matrix. Hence, it can be represented as E K E@ , where E is orthogonal. As in Theorem 1, by unboundedness there exists n n n n an eigenvalue subsequence j~1P0. Denote by f the unit vector in direction i, kni i and let f@E@ »1@2»I ~1@2 n c@ , i n n , n DD f@E@ »1@2»I ~1@2DD n i n n
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so that DDc DD"1 ∀n. Note that n DD f @E@ »1@2»I ~1@2DD " ( f {E@ (»1@2»I ~1»1@2)E f )1@2 n n i n i n n n i n n " ( f {E@ E K~1E@ E f )1@2 i n n n n ni " ( f {K~1f )1@2"j~1@2. i n i ni Hence, c@ »I ~1@2x " c@ »I ~1@2»1@2»~1@2x n n n n n n n n " j1@2[ f {E@ »1@2»I ~1@2»I ~1@2»1@2]»~1@2x n n n n n ni i n n " j1@2[ f {E@ E K~1E@ ]»~1@2x ni i n n n n n n " j1@2[ f {K~1E@ ]»~1@2x n ni i n n n " j~1@2e@ »~1@2x , n ni n ni where e is column i of E . Since e is on the unit ball in Rq, there is ni n kni $ e@ z a convergent subsequence el n Pe 3LB (0), so e@l n »~1@2 xl nP l ki ki kn k 0 0 1 1 &(0, e@ e )"(0, 1).6 Hence, j~1@2 xl nP 0. But, by Lemma 1, e@l n »~1@2 l l kni ki kn k 0 0 xl "alip(1), a contradiction. h »I ~1@2 l kn
kn
Proof of ¹heorem 2. Suppose first that »I 3ER(» ). Then n n lim »1@2»I ~1»1@2"I , so »1@2»I ~1@2"O(1). Denote the characteristic funcn n n q n n?= n tion of a random vector y by f . By the continuity theorem ( Lukacs, 1970, p. 48), y f ~1@2 (h)Pf (h) pointwise. Hence, by (White (1984), p. 66) Lemma 4.23, Vn xn z f I ~1@2 (h)!f (»1@2»I ~1@2h)"f I ~1@2 1@2 ~1@2 (h) n (Vn Vn )Vn xn Vn xn z n !f (»1@2»I ~1@2h)P0 pointwise in h. z n n Recalling that f (h)"exp(!h@h/2), z
A
B
A B
h@»I ~1@2» »I ~1@2h h@h n n f (»1@2»I ~1@2h)"exp ! n Pexp ! , n z n 2 2 so f I ~1@2 (h)Pf (h). Applying the continuity theorem again shows that Vn xn z »I 3avar (x ). n n
6 As in Theorem 1, normality of z is not needed here.
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Now, consider the converse by supposing »I 3avar (x ). By Lemma 2, M*1 n n can denote a common bound for all elements of »I 1@2»~1@2. Fix e'0 and set n n r"q2M. Recalling that f ~1@2 (h)"f (»~1@2h), by the second continuity theVn xn xn n orem ( Lukacs, 1970, p. 53) there exists N such that e h@h e r2 f (»~1@2h)!exp ! ( exp ! ∀h3B (0), xn n r 2 4 2 n*N N e t@t e r2 f (»I ~1@2t)!exp ! ( exp ! ∀t3B (0), xn n r 2 4 2
G
K K
G
K K
A BK A BK
A B A B
where B (0) is the closed ball of radius r about 0 in Rq. Since »I 1@2»~1@2h3B (0) n r n r for every n and h3LB (0), setting t"»I 1@2»~1@2h yields n n 1 h@h e r2 f (»~1@2h)!exp ! ( exp ! , xn n 2 2 4 n*N N e h@»~1@2»I »~1@2h e r2 n n f (»~1@2h)!exp ! n ( exp ! , n x n 2 2 4 for every h3LB (0). Hence, 1
K A A B
A BK A
BK
B
A BK
A B A B
h@»~1@2»I »~1@2h h@h n n n*N N exp ! n !exp ! e 2 2 e r2 ( exp ! 2 2
∀h3LB (0). 1
By the Mean Value theorem,
K A K A BK
B
A BK
h@»~1@2»I »~1@2h h@h n n exp ! n !exp ! 2 2
1 c " ! exp ! Dh@»~1@2»I »~1@2h!h@hD n n n 2 2 for some c between h@»~1@2»I »~1@2h and h@h. Thus n n n
A B
n*N NDh@»~1@2»I »~1@2h!h@hD(e exp n n n e
c!r2 2
for every h3LB (0) and the corresponding c (which also depends on n). Since 1 h@»~1@2»I »~1@2h)r2 for every n and h3LB (0), and h@h"1)r2 for such h, we n n n 1 have c)r2. Thus, exp((c!r2)/2))1 for every n and h3LB (0). That is, 1 n*N NDh@»~1@2»I »~1@2h!h@hD(e ∀h3LB (0). 1 n n n e
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Since h can be any vector on the unit ball, this implies lim »~1@2 n?= n »I »~1@2"I , or »I 3ER(» ). q n n n n Next, consider (1). Since »~1 is bounded by Theorem 1(1) and »~1@2»I n n n [»~1@2»I »~1@2!I ]"0 when »~1@2!I "»~1@2[»I !» ]»~1@2, lim q n n n?= n n n n n n q lim (»I !» )"0. n?= n n Finally, consider (2). Since » is bounded by Theorem 1(2) and »I !» " n n n lim (»I !» )"0 when lim »1@2[»~1@2»I »~1@2!I ]»1@2, n?= n n n?= q n n n n n [»~1@2»I »~1@2!I ]"0. h q n n n Proof of ¹heorem 3. Select »*3avar (Jn(b*!b)) and »I 3avar (Jn(bI !b)) n n n n such that f (h),lim inf f (h)"lim g (h)*0 ∀h3Rq (hO0), where n?= n n?= n f (h),h@(»I !»*)h and g (h),infM f (h): m*nN. We first establish unin n n n m form equicontinuity of the sequence f (h) on the boundary of the unit ball n in Rq. By Theorem 1(2), »I is bounded. It is straightforward to use this n along with f (h)*0 ∀h to conclude that »* is bounded as well, so denote n by M a common bound for all elements of »I and »*. For any h, h 3B (0) n n 0 1 we have D f (h)!f (h )D)DDh!h DD[DD(»I !»*)hDD#DDh@ (»I !»*)DD] 0 n n n 0 0 n n n )DDh!h DD4Mq. 0 Hence, for every e'0 there exists d"e/4Mq such that DDh!h DD(dND f (h)!f (h )D(e ∀n, or f (h) is uniformly equicontinuous on 0 n n 0 n LB (0). Now, by definition of g (h) there exists a subsequence f (h) such that 1 n in g (h)#1/n'f (h)*g (h). Thus f (h) converges pointwise to f (h), so by equiconn in n in tinuity this convergence is actually uniform on LB (0). This implies 1 lim f (h)!1/n"f (h) uniformly on LB (0). Hence, since f (h)*g (h) by defn?= in 1 n n inition of g (h), for every natural number k there exists a natural number N such n k that 1 1 n*N Nf (h)*g (h)'f (h)! 'f (h)! ∀h3LB (0). k n n in 1 n k We may summarize this by saying that the convergence to the limit inferior is uniform on LB (0) in the sense that for every k there exists N such 1 k that n*N Nf (h)#1/k'0 ∀h3LB (0), and without loss of generality k n 1 we may assume 1(N (N (2. Let k be the largest natural number 1 2 n k satisfying N )n for n*N and note that n*N ∀n*N and k 1 kn 1 lim 1/k "0. Thus »I #1/k I 3avar (Jn(bI !b)) by Theorem 2(1), so that n?= n n n q n b* is asymptotically efficient relative to bI according to Definition 6 if n n (»I #(1/k )I )!»* is positive semidefinite for n large. For any n*N and n n q n 1
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hO0 we have
A BCA h@ DDhDD
B DA B A B A B
A B
h h@ h h@h " [»I !»*] # n DDhDD n DDhDD DDhDD k DDhDD2 n h 1 "f # '0, n DDhDD k n * so h@[(»I #1 I )!» ]h'0 ∀n*N and ∀hO0. That is, (»I #(1/k )I )!»* n 1 n n q n n kn q is positive-definite for n*N . h 1 1 »I # I !»* n n k q n
References Bates, C.E., White, H., 1993. Determination of estimators with minimum asymptotic covariance matrices. Econometric Theory 9, 633—648. Fisher, R.A., 1925. Theory of statistical estimation. Proceedings of the Cambridge Philosophical Society 22, 700—725. Lukacs, E., 1970. Characteristic Functions. Hafner, New York. Mann, H.B., Wald, A., 1943. On stochastic limit and order relationships. Annals of Mathematical Statistics 14, 217—226. White, H., 1982. Instrumental variables regression with independent observations. Econometrica 50, 483—499. White, H., 1984. Asymptotic Theory for Econometricians. Academic Press (Harcourt Brace Jovanovich), Orlando. White, H., 1994. Estimation, Inference and Specification Analysis. Econometric Society Monograph, vol. 22. Cambridge University Press, Cambridge.