RELATIVE EFFICIENCY WITH EQUIVALENCE

CLASSES OF ASYMPTOTIC COVARIANCES

David M. Mandy and Carlos Martins-Filho^{∗}

University of Missouri and Oregon State University, respectively

February 1998

Abstract. White’s [1984] 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 liminf-based definition [White 1994, 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.

JEL Classifications C10, C13, C50.

David M. Mandy Department of Economics 118 Professional Building University of Missouri Columbia, MO 65211 USA Voice 573-882-1763 FAX 573-882-2697 Internet Mandy @ econ.missouri.edu

Carlos Martins-Filho Department of Economics Oregon State University Ballard Extension Hall 303 Corvallis, OR 97331-3612 USA Voice 541-737-1476 FAX 541-737-5917 Internet Carlos.Martins @ orst.edu

Key words and phrases. Asymptotic Efficiency.

∗ 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.

Typeset by AMS-TEX RELATIVE EFFICIENCY WITH EQUIVALENCE CLASSES OF ASYMPTOTIC COVARIANCES 1

1. Introduction

Suppose β_{n}^{∗} and β̃_{n} are estimators for some parameter vector β and we want to determine their relative asymptotic efficiency. A procedure discussed by White [1994, p. 136] is to first find two sequences of matrices, V_{n}^{∗} and Ṽ_{n}, such that both V_{n}^{∗−1/2}\sqrt{n}(β_{n}^{∗} − β) and Ṽ_{n}^{−1/2}\sqrt{n}(β̃_{n} − β) converge in distribution to standard normals. White [1984, p. 66; 1994, p. 91] calls such sequences asymptotic variances, or avars. Next these sequences are used to show lim inf_{n→∞} θ′(Ṽ_{n} − V_{n}^{∗})θ ≥ 0 ∀ θ, whence we conclude, for example, that β_{n}^{∗} is asymptotically efficient relative to β̃_{n}. This procedure has an advantage over the traditional (Fisher [1925]) approach of comparing the covariances of the asymptotic distributions, in that V_{n}^{∗} and Ṽ_{n} need not have limits (White [1982]). But there is a potential disadvantage as well because in all interesting cases there will be other avar sequences, say V_{n}, such that V_{n}^{−1/2}\sqrt{n}(β_{n}^{∗} − β) converges in distribution to a standard normal. Thus, one must inquire whether lim inf_{n→∞} θ′(Ṽ_{n} − V_{n}^{∗})θ ≥ 0 ∀ θ implies lim inf_{n→∞} θ′(Ṽ_{n} − V_{n})θ ≥ 0 ∀ θ for any such V_{n}, that is, whether statements about asymptotic efficiency are invariant to the avar sequences examined. By Fatou’s Lemma, this implication holds if lim_{n→∞}(V_{n} − V_{n}^{∗}) = 0, and the converse holds as well provided the two liminfs are both nonnegative for the same set of Ṽ_{n} sequences (that is, for the same rival estimators β̃_{n}). But without lim_{n→∞}(V_{n} − V_{n}^{∗}) = 0 anything is possible, including a reversal of the original conclusion, in which case the relative efficiency comparison based on V_{n}^{∗} would appear definitive when it is in fact ambiguous. Alternatively, V_{n}^{∗} may yield no conclusion when a conclusion is obtainable using V_{n}, and the same comments apply to all possible substitutes for V_{n}^{∗} and Ṽ_{n}. Thus, at this level of generality, lim_{n→∞}(V_{n} − V_{n}^{∗}) = 0 for all relevant sequences V_{n}^{∗} and V_{n} is sufficient for meaningful efficiency conclusions. The converse is also 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

{V_{n}^{∗}}_{n=1}^{∞}R{V_{n}}_{n=1}^{∞} ⇔ lim_{n→∞}(V_{n} − V_{n}^{∗}) = 0  (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

2 DAVID M. MANDY AND CARLOS MARTINS-FILHO

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.

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_{n→∞} θ′(Ṽ_{n} − V_{n}^{∗})θ to be invariant across all V_{n}^{∗} sequences that are avars of \sqrt{n}(β_{n}^{∗} − β), and hence for unambiguous efficiency conclusions by this criterion. However, White [1984, pp. 78-9] provides another definition of efficiency based on the avar concept that does not always yield the same efficiency conclusions as this liminf 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 \sqrt{n}. In general, an element of β_{n}^{∗} can be normalized by any nonstochastic sequence in order to obtain bounded avars and inverse avars, and the normalizing sequence can differ across elements of β_{n}^{∗}. Once all estimators 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 β_{n}^{∗} might tend to one, in which case some asymptotically redundant 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 normaliza-

RELATIVE EFFICIENCY WITH EQUIVALENCE CLASSES OF ASYMPTOTIC COVARIANCES 3

tions are used for β_{n}^{∗} and β̃_{n}, one should only compare estimators via avars of normalized 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.

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 Q^{q} denote the set of all sequences {V_{n}}_{n=1}^{∞} of real symmetric positive definite nonstochastic (q × q)

matrices V_{n}. For such V_{n}, denote by V_{n}^{1/2} the unique real symmetric positive definite matrix satisfying V_{n}^{1/2}V_{n}^{1/2} = V_{n}. In essence, White’s definition of asymptotic covariance is the following.

Definition 1 [White 1984, p. 66; White 1994, p. 91]^{1}. Let {x_{n}}_{n=1}^{∞} be a sequence of q-dimensional random vectors (q < ∞). The asymptotic covariance of {x_{n}}_{n=1}^{∞}, denoted avar ({x_{n}}_{n=1}^{∞}), is

avar ({x_{n}}_{n=1}^{∞}) ≡ {{V_{n}}_{n=1}^{∞} ∈ Q^{q}: V_{n}^{−1/2}x_{n} →^{d} z ∼ N(0, I_{q})} .

For brevity, we drop the indexes and write V_{n} ∈ avar (x_{n}) to denote that the sequence {V_{n}}_{n=1}^{∞} is an element of avar ({x_{n}}_{n=1}^{∞}).

1White [1984] actually only requires positive definiteness of V_{n} for large n. Since only the tail of the sequence is important in the definition, there is no consequential loss of generality from assuming the entire sequence is positive definite.

4 DAVID M. MANDY AND CARLOS MARTINS-FILHO

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 sequence V_{n} satisfying Definition 1, then A_{n}V_{n}^{−1/2}x_{n} →^{d} z for any {A_{n}}_{n=1}^{∞} satisfying lim_{n→∞} A_{n} = I_{q}. Hence, if V_{n} ∈ avar (x_{n}) then A_{n}^{′−1}V_{n}A_{n}^{−1} ∈ avar (x_{n}) ∀ {A_{n}}_{n=1}^{∞} satisfying A_{n}V_{n}^{−1/2} ∈ Q^{q} and lim_{n→∞} A_{n} = I_{q}. In other words, a great 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 Q^{q} 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, then efficiency conclusions are unambiguous. To construct the equivalence relation we first need a formal definition of efficiency. We start with:

Definition 2 [White 1994, p. 136]^{2}. Let β_{n}^{∗} and β̃_{n} be consistent estimators of a nonstochastic q-dimensional vector β. Then β_{n}^{∗} is asymptotically efficient relative to β̃_{n} if there exists V_{n}^{∗} ∈ avar (\sqrt{n}(β_{n}^{∗} − β)) and Ṽ_{n} ∈ avar (\sqrt{n}(β̃_{n} − β)) such that

lim inf_{n→∞} θ′(Ṽ_{n} − V_{n}^{∗})θ ≥ 0

for all θ ∈ ℜ^{q} (θ ≠ 0). An estimator is asymptotically efficient within a class if it is asymptotically efficient relative to every other estimator in the class.

Notice 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 Q^{q} that yields unambiguous efficiency conclusions within an equivalence class is {V_{n}^{∗}}_{n=1}^{∞}R{V_{n}}_{n=1}^{∞} if and only if

(∀ Ṽ_{n}): lim inf_{n→∞} θ′(Ṽ_{n} − V_{n}^{∗})θ ≥ 0 ∀ θ ⇔ lim inf_{n→∞} θ′(Ṽ_{n} − V_{n})θ ≥ 0 ∀ θ.  (2)

2Definitions 2 and 6 (below) are rephrased from White’s original statements to reflect the fact that avar (x_{n}) is a class. We use the same terminology to accomplish this that Bates and White [1993, Definition 2.5] use for a RCASOI class.

RELATIVE EFFICIENCY WITH EQUIVALENCE CLASSES OF ASYMPTOTIC COVARIANCES 5

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 equation (1). That (1) implies (2) is an application of Fatou’s Lemma. For the converse, Ṽ_{n} = V_{n}^{∗} in (2) implies lim inf_{n→∞} θ′(V_{n}^{∗}−V_{n})θ ≥ 0 ∀ θ, while Ṽ_{n} = V_{n} in (2) implies lim sup_{n→∞} θ′(V_{n}^{∗} − V_{n})θ ≤ 0 ∀ θ. We utilize (1) henceforth since it is more convenient, and henceforth denote the equivalence class of V_{n}^{∗} with respect to R by E_{R}(V_{n}^{∗}) ≡ {V_{n} ∈ Q^{q}: lim_{n→∞}(V_{n}^{∗} − V_{n}) = 0}.

3. Sufficient Conditions for avar (x_{n}) to be an Equivalence Class with Respect to R In general avar (x_{n}) is not an equivalence class with respect to R, so some restrictions are needed in the

form of regularity conditions on the underlying random vector x_{n}. These restrictions include the following notion of asymptotic linear independence in probability.

Definition 3. Let {x_{n}}_{n=1}^{∞} be a sequence of q-dimensional random vectors (q < ∞) and {y_{n}}_{n=1}^{∞} be a sequence of strictly positive nonstochastic real numbers. {x_{n}}_{n=1}^{∞} is asymptotically linearly independent in probability of order {y_{n}}_{n=1}^{∞} if for every sequence {c_{n}}_{n=1}^{∞} of real nonstochastic q-dimensional vectors satisfying ∥c_{n}∥ = 1 ∀ n; there exists a triple (N, ϵ, δ), where N is a natural number, ϵ > 0, and δ > 0; such that

n ≥ N ⇒ P (ϵ < |\frac{c′_{n}x_{n}}{y_{n}} |) > δ.

For brevity, this is denoted x_{n} = alip(y_{n}).^{3}

It is clear that x_{n} = alip(y_{n}) ⇒ x_{n} ≠ o_{p}(y_{n}). The converse fails because x_{n} ≠ o_{p}(y_{n}) still permits subsequences of \frac{c′_{n}x_{n}}{y_{n}} that converge in probability to zero, and also linear combinations of \frac{x_{n}}{y_{n}} that converge in probability to zero as long as individual components do not.

Definition 4. A sequence {x_{n}}_{n=1}^{∞} of q-dimensional random vectors (q < ∞) is avar-regular if x_{n} = O_{p}(1) and x_{n} = alip(1).

3The alip concept is similar to Mann and Wald’s [1943] notion of ω_{p}, but these concepts differ in two ways that are important in the present context. First, alip(y_{n}) is weaker than ω_{p}(y_{n}) in that ω_{p}(y_{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_{n}) is stronger than ω_{p}(y_{n}) in that alip(y_{n}) places its condition on all bounded nondegenerate linear combinations of a random vector, while ω_{p}(y_{n}) merely places its condition on the individual components of a random vector. In general, ω_{p}(1) cannot replace alip(1) in the results below.

6 DAVID M. MANDY AND CARLOS MARTINS-FILHO

Avar-regularity places restrictions on the primitive of the problem, the underlying random vector x_{n}. Theorem 1 shows this is equivalent to White’s [1984, p. 66] approach of bounding the avar sequences and the corresponding sequences of inverses. Theorem 1. Let {x_{n}}_{n=1}^{∞} be a sequence of q-dimensional random vectors (q < ∞) for which avar (x_{n}) is nonempty. Then

(i) x_{n} = alip(1) if and only if V_{n}^{−1} is bounded ∀ V_{n} ∈ avar (x_{n}). (ii) x_{n} = O_{p}(1) if and only if V_{n} is bounded ∀ V_{n} ∈ avar (x_{n}). Hence x_{n} is avar-regular if and only if both V_{n} and V_{n}^{−1} are bounded for every V_{n} ∈ avar (x_{n}). Proof. All proofs are in the Appendix.

The main result of this section is that avar-regularity is sufficient to ensure avar (x_{n}) is an equivalence class with respect to R. Bates and White [1993, Theorem 2.3] investigate whether V_{n} ∈ avar (x_{n}) ⇒ avar (x_{n}) ⊆ E_{R}(V_{n}), using the slightly different relation R in Q^{q} defined by

{V_{n}}_{n=1}^{∞}R{Ṽ_{n}}_{n=1}^{∞} ⇔ lim_{n→∞} V_{n}^{−1/2}Ṽ_{n}V_{n}^{−1/2} = I_{q}.  (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 equation (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_{n}) is an equivalence class with respect to both R and R. The proof of Theorem 2 relies on the following preliminary results. Lemma 1. Let {x_{n}}_{n=1}^{∞} be a sequence of q-dimensional random vectors (q < ∞). If x_{n} →^{d} z ∼ (0, Σ), where Σ is positive definite, then x_{n} = alip(1). Lemma 2. Let {x_{n}}_{n=1}^{∞} be a sequence of q-dimensional random vectors (q < ∞). If V_{n}, Ṽ_{n} ∈ avar (x_{n}) then Ṽ_{n}^{1/2}V_{n}^{−1/2} is bounded. Theorem 2. Let {x_{n}}_{n=1}^{∞} be a sequence of q-dimensional random vectors (q < ∞), R be defined by equation (1), R be defined by equation (3), and V_{n} ∈ avar (x_{n}). Then avar (x_{n}) = E_{R}(V_{n}), and

RELATIVE EFFICIENCY WITH EQUIVALENCE CLASSES OF ASYMPTOTIC COVARIANCES 7

(i) x_{n} = alip(1) ⇒ E_{R}(V_{n}) ⊇ E_{R}(V_{n}). (ii) x_{n} = O_{p}(1) ⇒ E_{R}(V_{n}) ⊆ E_{R}(V_{n}). Hence avar-regularity implies that avar (x_{n}) is an equivalence class with respect to both R and R.

Theorem 2 shows that, for an avar-regular random vector x_{n} and an avar sequence V_{n} ∈ avar (x_{n}), avar (x_{n}) is precisely the set of sequences {Ṽ_{n}}_{n=1}^{∞} of real symmetric positive definite matrices such that lim_{n→∞}(V_{n} − Ṽ_{n}) = 0_{q}. And from equation (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_{n}) = E_{R}(V_{n}) in Theorem 2. Moreover, a proof of x_{n} = alip(1) ⇒ avar (x_{n}) ⊇ E_{R}(V_{n}) that does not rely on normality is available from the authors on request. Hence most results given here hold even if 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_{n}) ⊆ E_{R}(V_{n}) is the crucial property. This relies on both normality and x_{n} = O_{p}(1) in our proof, and Example 2 below shows that x_{n} = O_{p}(1) cannot be discarded. Whether normality is necessary for avar (x_{n}) ⊆ E_{R}(V_{n}) or avar (x_{n}) ⊆ E_{R}(V_{n}) is an open question, as we have no counter-examples to these when the limit distribution is non-normal. Bates and White [1993, p. 648] propose a proof of avar (x_{n}) ⊆ E_{R}(V_{n}) that does not rely on normality, but an important and questionable step therein is that lim_{n→∞} V_{n}^{−1/2}Ṽ_{n}^{1/2} = I_{q} is implied by both y_{n} and (V_{n}^{−1/2}Ṽ_{n}^{1/2})y_{n} converging in distribution to the same (potentially non-normal) (0, I_{q}) random vector, for some sequence of (0, I_{q}) random vectors y_{n}. This is not obvious. Indeed, 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_{n} = alip(1) is needed in Theorems 1 and 2 and cannot be replaced by x_{n} ≠ o_{p}(1) (or by x_{n} = ω_{p}(1)).

Example 1: The role of x_{n} = alip(1) in making avar (x_{n}) an equivalence class with respect to R. Let

z = [ \begin{matrix}z_{1} \\ z_{2} \end{matrix} ] ∼ N ([ \begin{matrix}0 \\ 0 \end{matrix} ] , [ 2 \frac{1}{2}_{\frac{1}{2} \frac{1}{4}} ]) , x_{n} = [ \begin{matrix}z_{1} − \frac{z_{2}}{n} \\ z_{1} + \frac{z_{2}}{n} \end{matrix} ] , V_{n}^{∗−1/2} = [ \begin{matrix}n + 1 & −n \\ −n & n \end{matrix} ] , and

V_{n}^{−1/2} = [ \begin{matrix}n^{2} + 1 & −n^{2} \\ −n^{2} & n^{2} \end{matrix} ] .

Then

V_{n}^{∗−1/2}x_{n} = [ \begin{matrix}z_{1} − 2z_{2} − \frac{z_{2}}{n} \\ 2z_{2} \end{matrix} ] →^{p} [ \begin{matrix}z_{1} − 2z_{2} \\ 2z_{2} \end{matrix} ] ∼ N(0, I_{2}),

so V_{n}^{∗} ∈ avar (x_{n}), while

V_{n}^{−1/2}x_{n} = [ \begin{matrix}−2nz_{2} + z_{1} − \frac{z_{2}}{n} \\ 2nz_{2} \end{matrix} ] ∼ N ([ \begin{matrix}0 \\ 0 \end{matrix} ] , [ \begin{matrix}n^{2} − 2n + 3 + \frac{1}{4n^{2}} − \frac{1}{n} & n − n^{2} − \frac{1}{4} \\ n − n^{2} − \frac{1}{4} & n^{2} \end{matrix} ]) ,

so V_{n} ∉ avar (x_{n}). But

V_{n}^{∗} = [ \begin{matrix}2 & 2 + \frac{1}{n} \\ 2 + \frac{1}{n} & 1 + (\frac{n+1}{n})^{2} \end{matrix} ] → [ \begin{matrix}2 & 2 \\ 2 & 2 \end{matrix} ] and V_{n} = [ \begin{matrix}2 & 2 + \frac{1}{n^{2}} \\ 2 + \frac{1}{n^{2}} & 1 + (\frac{n^{2}+1}{n^{2}})^{2} \end{matrix} ] → [ \begin{matrix}2 & 2 \\ 2 & 2 \end{matrix} ] ,

so lim_{n→∞}(V_{n} − V_{n}^{∗}) = 0_{2}, while

V_{n}^{−1/2}V_{n}^{∗}V_{n}^{−1/2} = [ \begin{matrix}n^{2} − 2n + 2 & n(1 − n) \\ n(1 − n) & n^{2} \end{matrix} ] → [ \begin{matrix}∞ & −∞ \\ −∞ & ∞ \end{matrix} ] .

Note that x_{n} ≠ o_{p}(1) here, but x_{n} ≠ alip(1) since c′_{n} = [ −2^{−1/2} 2^{−1/2} ] ∀ n in Definition 3 yields c′_{n}x_{n} = \frac{z_{2}\sqrt{2}}{n} →^{p} 0. Hence when x_{n} ≠ alip(1) elements of avar (x_{n}) can have unbounded inverses, and E_{R}(V_{n}^{∗}) and avar (x_{n}) can both be smaller than E_{R}(V_{n}^{∗}), even when x_{n} ≠ o_{p}(1). Thus alip is strictly stronger than o_{p}. □

The next example shows that x_{n} = O_{p}(1) is needed in Theorems 1 and 2 as well.

Example 2: The role of x_{n} = O_{p}(1) in making avar (x_{n}) an equivalence class with respect to R. Let z ∼ N(0, 1), x_{n} = nz, V_{n}^{∗} = n^{2}, and V_{n} = (n + 1)^{2}. Clearly V_{n}^{∗}, V_{n} ∈ avar (x_{n}) and V_{n}^{∗}RV_{n}. But V_{n} − V_{n}^{∗} = (n + 1)^{2} − n^{2} = 2n + 1 → ∞, so V_{n}^{∗} and V_{n} are not in the same equivalence class with respect to R. The problem is x_{n} ≠ O_{p}(1), in which case elements of avar (x_{n}) can be unbounded, and E_{R}(V_{n}) and avar (x_{n}) can both be larger than E_{R}(V_{n}). □

RELATIVE EFFICIENCY WITH EQUIVALENCE CLASSES OF ASYMPTOTIC COVARIANCES 9

4. Other Definitions of Asymptotic Efficiency for avar 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 [White 1984, pp. 78-79]. Let β_{n}^{∗} and β̃_{n} be consistent estimators of a nonstochastic vector β. Then β_{n}^{∗} is asymptotically efficient relative to β̃_{n} if there exists V_{n}^{∗} ∈ avar (\sqrt{n}(β_{n}^{∗} − β)) and Ṽ_{n} ∈ avar (\sqrt{n}(β̃_{n} − β)), and an integer N, such that Ṽ_{n} − V_{n}^{∗} is positive semidefinite for all n ≥ N. An estimator is asymptotically efficient within a class if it is asymptotically efficient relative to every other estimator in the class.

It is clear that if β_{n}^{∗} is asymptotically efficient relative to β̃_{n} according to Definition 6 then β_{n}^{∗} is asymptotically efficient relative to β̃_{n} according to Definition 2 as well. So, those estimators identified as efficient using Definition 6 (for example, those studied by White [1984 and 1994]) are unambiguously efficient by Definition 2, provided we impose avar-regularity (or at least O_{p}(1)) to 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 β̃_{n} is avar-regular, but not without avar-regularity, as shown by Examples 3 and 4 below. Theorem 3. Let β_{n}^{∗} and β̃_{n} be consistent estimators of a nonstochastic q-dimensional vector β and assume \sqrt{n}(β̃_{n} − β) is avar-regular. If β_{n}^{∗} is asymptotically efficient relative to β̃_{n} according to Definition 2 then β_{n}^{∗} is asymptotically efficient relative to β̃_{n} according to Definition 6. Example 3: The role of x_{n} = alip(1) in Theorem 3. Let z ∼ N(0, 1), \sqrt{n}(β_{n}^{∗}−β) = \frac{2z}{n}, and \sqrt{n}(β̃_{n}−β) = \frac{z}{n}. Since \frac{4}{n^{2}} ∈ avar (\sqrt{n}(β_{n}^{∗} − β)), \frac{1}{n^{2}} ∈ avar (\sqrt{n}(β̃_{n} − β)), and lim inf_{n→∞}(\frac{1}{n^{2}} − \frac{4}{n^{2}} ) = 0; β_{n}^{∗} is efficient relative to β̃_{n} according to Definition 2. For arbitrary V_{n}^{∗} ∈ avar (\sqrt{n}(β_{n}^{∗} − β)) we have

V_{n}^{∗−1/2} \frac{2z}{n} ∼ N (0, V_{n}^{∗−1} \frac{4}{n^{2}} ) →^{d} N(0, 1),

so V_{n}^{∗} \frac{n^{2}}{4} → 1. Similarly, for arbitrary Ṽ_{n} ∈ avar (\sqrt{n}(β̃_{n} − β)) we have Ṽ_{n}n^{2} → 1. So, there exists N such that

n ≥ N ⇒ \{ \begin{matrix}V_{n}^{∗} > \frac{2}{n^{2}} \\ Ṽ_{n} < \frac{3}{2n^{2}} , \end{matrix}

implying Ṽ_{n} − V_{n}^{∗} < −\frac{1}{2n^{2}} for n ≥ N. Hence β_{n}^{∗} is not efficient relative to β̃_{n} according to Definition 6, due to the fact that \sqrt{n}(β̃_{n} − β) ≠ alip(1). □

10 DAVID M. MANDY AND CARLOS MARTINS-FILHO

Example 4: The role of x_{n} = O_{p}(1) in Theorem 3. The condition \sqrt{n}(β̃_{n} − β) = O_{p}(1) is only used in the proof of Theorem 3 to establish equicontinuity of the sequence of quadratic forms f_{n}(θ) = θ′(Ṽ_{n}−V_{n}^{∗})θ on the unit 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_{2}), \sqrt{n}(β_{n}^{∗} − β) = V_{n}^{∗1/2}z, and \sqrt{n}(β̃_{n} − β) = Ṽ_{n}^{1/2}z; where

V_{n}^{∗} = [ \begin{matrix}csc^{2} ϕ_{n} & 0 \\ 0 & 1 \end{matrix} ] , Ṽ_{n} = Φ_{n} + V_{n}^{∗};

and ϕ_{n} and Φ_{n} are chosen to produce the desired quadratic form. Choosing ϕ_{n} ∈ (0, \frac{π}{4} ) such that ϕ_{n} → 0 and

Φ_{n} = \frac{1}{cos^{4} ϕ_{n} − sin^{4} ϕ_{n}} [ \begin{matrix}(cos^{2} ϕ_{n} − sin^{2} ϕ_{n})(cot^{2} ϕ_{n} − sin^{2} ϕ_{n}) & sin ϕ_{n} cos ϕ_{n}(cot^{2} ϕ_{n} − 2 sin^{2} ϕ_{n}) \\ sin ϕ_{n} cos ϕ_{n}(cot^{2} ϕ_{n} − 2 sin^{2} ϕ_{n}) & (cos^{2} ϕ_{n} − sin^{2} ϕ_{n}) sin^{2} ϕ_{n} \end{matrix} ]

makes f_{n}(θ) = θ′Φ_{n}θ a saddle (indefinite) rotated ϕ_{n} from standard position that collapses around the rotated θ_{2} axis as n → ∞ but satisfies f_{n}(θ_{n}) = −1 ∀ n, where θ′_{n} = ( − sin ϕ_{n} cos ϕ_{n}). Note that

Φ_{n} → [ \begin{matrix}∞ & ∞ \\ ∞ & 0 \end{matrix} ] and V_{n}^{∗} → [ \begin{matrix}∞ & 0 \\ 0 & 1 \end{matrix} ] ,

so by Theorem 1(ii) we have \sqrt{n}(β̃_{n} − β) ≠ O_{p}(1). It can be shown, however, that \sqrt{n}(β̃_{n} − β) = alip(1) and that Ṽ_{n} is positive definite for n large. Because the (1, 1) element of Φ_{n} is O(csc^{2} ϕ_{n}) while the off-diagonal elements are O(csc ϕ_{n}), we have

lim inf_{n→∞} f_{n}(θ) = \{ \begin{matrix}∞ & if θ_{1} ≠ 0 \\ 0 & if θ_{1} = 0, \end{matrix}

so β_{n}^{∗} is efficient relative to β̃_{n} according to Definition 2. To show that β_{n}^{∗} is not efficient relative to β̃_{n} according to Definition 6, let W_{n}^{∗} and W̃_{n} be arbitrary elements of avar (\sqrt{n}(β_{n}^{∗} −β)) and avar (\sqrt{n}(β̃_{n} −β)), respectively. Then, by the same arguments used in Example 3 we have V_{n}^{∗−1/2}W_{n}^{∗}V_{n}^{∗−1/2} and Ṽ_{n}^{−1/2}W̃_{n}Ṽ_{n}^{−1/2} both converging to I_{2}. Now write

θ′_{n}(W̃_{n} − W_{n}^{∗})θ_{n} = θ′_{n}Ṽ_{n}^{1/2}[Ṽ_{n}^{−1/2}W̃_{n}Ṽ_{n}^{−1/2} − I_{2}]Ṽ_{n}^{1/2}θ_{n} + f_{n}(θ_{n})
+ θ′_{n}V_{n}^{∗1/2}[I_{2} − V_{n}^{∗−1/2}W_{n}^{∗}V_{n}^{∗−1/2}]V_{n}^{∗1/2}θ_{n}.  (4)

Since θ′_{n}V_{n}^{∗}θ_{n} = 1+cos^{2} ϕ_{n} and θ′_{n}Ṽ_{n}θ_{n} = f_{n}(θ_{n})+θ′_{n}V_{n}^{∗}θ_{n} = cos^{2} ϕ_{n}, both θ′_{n}V_{n}^{∗1/2} and θ′_{n}Ṽ_{n}^{1/2} are bounded. Hence the first and last terms of (4) approach zero, leaving only f_{n}(θ_{n}) = −1 for n large, so β_{n}^{∗} is not efficient relative to β̃_{n} according to Definition 6. □

RELATIVE EFFICIENCY WITH EQUIVALENCE CLASSES OF ASYMPTOTIC COVARIANCES 11

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 equation (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 limit-based 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_{n} = O_{p}(1) is really needed to get unambiguous efficiency conclusions from Definition 2, if we do not impose x_{n} = alip(1) as well then the use of Definition 2 in lieu of 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_{n} ≠ alip(1), either definition can be used to unambiguously study efficiency when x_{n} ≠ alip(1), as in the RCANI class of Bates and White, but alip(1) is 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 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 and 1994], and hence the potential ambiguity of the efficiency definition used (Definition 6) is also not mentioned, the

12 DAVID M. MANDY AND CARLOS MARTINS-FILHO

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 β_{n}^{∗} and β̃_{n} be consistent estimators of a nonstochastic q-dimensional vector β and suppose \sqrt{n}(β_{n}^{∗} − β) and \sqrt{n}(β̃− β) are both avar-regular. Further assume avar (\sqrt{n}(β_{n}^{∗} − β)) and avar (\sqrt{n}(β̃_{n} − β)) are both nonempty. We say β_{n}^{∗} possesses minimin asymptotic efficiency relative to β̃_{n} if

lim inf_{n→∞} θ′Ṽ_{n}θ ≥ lim inf_{n→∞} θ′V_{n}^{∗}θ for all Ṽ_{n} ∈ avar (\sqrt{n}(β̃_{n} − β)), V_{n}^{∗} ∈ avar (\sqrt{n}(β_{n}^{∗} − β)), and θ ∈ ℜ^{q} (θ ≠ 0).

We say β_{n}^{∗} possesses minimax asymptotic efficiency relative to β̃_{n} if

lim sup_{n→∞} θ′Ṽ_{n}θ ≥ lim sup_{n→∞} θ′V_{n}^{∗}θ for all Ṽ_{n} ∈ avar (\sqrt{n}(β̃_{n} − β)), V_{n}^{∗} ∈ avar (\sqrt{n}(β_{n}^{∗} − β)), and θ ∈ ℜ^{q} (θ ≠ 0).

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 avarregularity, 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 there exist positive definite matrices V^{∗} and Ṽ such that \sqrt{n}(β_{n}^{∗} − β) →^{d} N(0, V^{∗}) and \sqrt{n}(β̃_{n} − β) →^{d} N(0, Ṽ) then β_{n}^{∗} is asymptotically efficient relative to β̃_{n} according to any of the definitions discussed herein if and only if Ṽ − V^{∗} is positive semidefinite.

Appendix

The proof of Theorem 1 uses the following fact.

RELATIVE EFFICIENCY WITH EQUIVALENCE CLASSES OF ASYMPTOTIC COVARIANCES 13

Fact. Let x be a random variable with E(x) = μ and V (x) = σ^{2} < ∞. Then P (σ ≤ |x − μ|) > 0. Proof. If P (σ > |x−μ|) = 1 then σ > 0, in which case σ^{2} = ∫_{|x−μ|<σ}(x−μ)^{2}dF (x) < ∫_{|x−μ|<σ} σ^{2}dF (x) = σ^{2}, a contradiction. □ Proof of Theorem 1. Fix V_{n} ∈ avar (x_{n}), let ( e_{n1} … e_{nq}) be an orthonormal linearly independent set of eigenvectors for V_{n}, and ( λ_{n1} … λ_{nq}) be the corresponding real strictly positive eigenvalues (a full set of real strictly positive eigenvalues and orthonormal linearly independent eigenvectors exists since V_{n} is symmetric and positive definite).

(i) First assume x_{n} = alip(1) and suppose the elements of V_{n}^{−1} are not bounded. Then there is an unbounded inverse eigenvalue sequence λ_{ni}^{−1}, in which case there is a subsequence {λ_{k_{n}i}}_{n=1}^{∞} such that lim_{n→∞} λ_{k_{n}i} = 0. Since (λ_{ni}^{1/2}, e_{ni}) is an (eigenvalue,eigenvector) pair for V_{n}^{1/2}, this implies lim_{n→∞} V_{k_{n}}^{1/2}e_{k_{n}i} = lim_{n→∞} λ_{k_{n}i}^{1/2}e_{k_{n}i} = 0 (using orthonormality of the eigenvectors to bound e_{k_{n}i}). That is, e′_{k_{n}i}V_{k_{n}}^{1/2} = o(1). But then e′_{k_{n}i}x_{k_{n}} = e′_{k_{n}i}V_{k_{n}}^{1/2}V_{k_{n}}^{−1/2}x_{k_{n}} = o(1)O_{p}(1) = o_{p}(1), which contradicts x_{k_{n}} = alip(1). For the converse, assume V_{n}^{−1} is bounded and suppose x_{n} ≠ alip(1). Since ( λ_{n1}^{−1} … λ_{nq}^{−1} ) are the eigenvalues for V_{n}^{−1}, λ_{ni}^{−1} is bounded for i = 1, …, q. That is, there exists M ∈ (0, ∞) such that λ_{ni}^{−1} < M, implying

λ_{ni} > M^{−1}, for i = 1, …, q; ∀ n.

Also, since x_{n} ≠ alip(1) there exists a sequence {c_{n}}_{n=1}^{∞} on the unit ball in ℜ^{q} with the following property:

To each n there corresponds k_{n} ≥ n such that P (\frac{M^{−1/2}}{2} ≤ |c′_{k_{n}}x_{k_{n}}|) ≤ \frac{1}{n}.

That is,

lim_{n→∞} P (\frac{M^{−1/2}}{2} ≤ |c′_{k_{n}}x_{k_{n}}|) = 0.

Since ( e_{n1} … e_{nq}) is an orthonormal basis for ℜ^{q} we may write c_{n} as

c_{n} = ∑_{i=1}^{q}(c′_{n}e_{ni})e_{ni},

so ∑_{i=1}^{q}(c′_{n}e_{ni})^{2} = c′_{n}c_{n} = 1 ∀ n, since c_{n} is on the unit ball. Thus, by definition of e_{ni} and λ_{ni}, and orthonormality of ( e_{n1} … e_{nq}),

∥c′_{n}V_{n}^{1/2}∥ = (c′_{n}V_{n}c_{n})^{1/2} = (∑_{i=1}^{q}(c′_{n}e_{ni})^{2}λ_{ni})^{1/2} > (M^{−1} ∑_{i=1}^{q}(c′_{n}e_{ni})^{2})^{1/2} = M^{−1/2} ∀ n.

14 DAVID M. MANDY AND CARLOS MARTINS-FILHO

Now write

c′_{k_{n}}x_{k_{n}} = ∥c′_{k_{n}}V_{k_{n}}^{1/2}∥a′_{k_{n}}V_{k_{n}}^{−1/2}x_{k_{n}}, where a′_{k_{n}} ≡ \frac{c′_{k_{n}}V_{k_{n}}^{1/2}}{∥c′_{k_{n}}V_{k_{n}}^{1/2}∥},

so that ∥a_{k_{n}}∥ = 1 ∀ n. Then a_{k_{n}} has a convergent subsequence a_{ℓ_{kn}} → a_{0}, where ∥a_{0}∥ = 1, so a′_{ℓ_{kn}} V_{ℓ_{kn}}^{−1/2}x_{ℓ_{kn}} →^{d} a′_{0}z ∼ (0, a′_{0}a_{0}) = (0, 1).^{4} By the Fact, P (1 ≤ |a′_{0}z|) > 0. Moreover, since all distribution functions are continuous off a countable set, there exists ϵ ∈ [\frac{1}{2}, 1] at which the distribution function of |a′_{0}z| is continuous. Then, by convergence in distribution there exists N such that

n ≥ N ⇒ P (\frac{M^{−1/2}}{2} ≤ |c′_{ℓ_{kn}} x_{ℓ_{kn}} |) = P (\frac{M^{−1/2}}{2} ≤ ∥c′_{ℓ_{kn}} V_{ℓ_{kn}}^{1/2}∥|a′_{ℓ_{kn}} V_{ℓ_{kn}}^{−1/2}x_{ℓ_{kn}} |)

≥ P (\frac{1}{2} ≤ |a′_{ℓ_{kn}} V_{ℓ_{kn}}^{−1/2}x_{ℓ_{kn}} |)

≥ P (ϵ ≤ |a′_{ℓ_{kn}} V_{ℓ_{kn}}^{−1/2}x_{ℓ_{kn}} |)

> \frac{P (ϵ ≤ |a′_{0}z|)}{2}

≥ \frac{P (1 ≤ |a′_{0}z|)}{2} > 0,

a contradiction. (ii) First assume x_{n} = O_{p}(1) and suppose the elements of V_{n} are not bounded. Then, as in (i), there is an eigenvalue subsequence {λ_{k_{n}i}}_{n=1}^{∞} such that lim_{n→∞} V_{k_{n}}^{−1/2}e_{k_{n}i} = lim_{n→∞} λ_{k_{n}i}^{−1/2}e_{k_{n}i} = 0. That is, e′_{k_{n}i}V_{k_{n}}^{−1/2} = o(1). Since e_{k_{n}i} is on the unit ball in ℜ^{q} there is a convergent subsequence e_{ℓ_{kn} i} → e_{0}, where

∥e_{0}∥ = 1, so e′_{ℓ_{kn} i}V_{ℓ_{kn}}^{−1/2}x_{ℓ_{kn}} →^{d} e′_{0}z ∼ (0, e′_{0}e_{0}) = (0, 1), implying e′_{ℓ_{kn} i}V_{ℓ_{kn}}^{−1/2}x_{ℓ_{kn}} ≠ o_{p}(1) by the Fact. But

since x_{n} = O_{p}(1) we have e′_{ℓ_{kn} i}V_{ℓ_{kn}}^{−1/2}x_{ℓ_{kn}} = o(1)O_{p}(1) = o_{p}(1), a contradiction. For the converse, just note that V_{n}^{1/2} is bounded whenever V_{n} is bounded, so x_{n} = V_{n}^{1/2}V_{n}^{−1/2}x_{n} = O(1)O_{p}(1) = O_{p}(1). □ Proof of Lemma 1. Since c′Σc is a continuous function of c, there exists c̄ ∈ ∂B_{1}(0) (the boundary of the unit ball in ℜ^{q}) such that

0 < c̄′Σc̄ ≤ c′Σc ∀ c ∈ ∂B_{1}(0).

Now suppose x_{n} ≠ alip(1). Then as in Theorem 1(i) there exists a sequence {c_{n}}_{n=1}^{∞} on the unit ball with the following property:

To each n there corresponds k_{n} ≥ n such that P (\frac{c̄′Σc̄}{2} ≤ |c′_{k_{n}}x_{k_{n}}|) ≤ \frac{1}{n}.

4In 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.

RELATIVE EFFICIENCY WITH EQUIVALENCE CLASSES OF ASYMPTOTIC COVARIANCES 15

That is, lim_{n→∞} P (\frac{c̄′Σc̄}{2} ≤ |c′_{k_{n}}x_{k_{n}}|) = 0. Since c_{k_{n}} is on the unit ball there exists a convergent subsequence

c_{ℓ_{kn}} → c_{0} ∈ ∂B_{1}(0). Hence c′_{ℓ_{kn}} x_{ℓ_{kn}} →^{d} c′_{0}z ∼ (0, c′_{0}Σc_{0}). By the Fact, P (c′_{0}Σc_{0} ≤ |c′_{0}z|) > 0. Moreover,

since \frac{c̄′Σc̄}{2} < c̄′Σc̄ ≤ c′_{0}Σc_{0}, and since all distribution functions are continuous off a countable set, there exists ϵ ∈ [c̄′Σc̄/2, c′_{0}Σc_{0}] at which the distribution function of |c′_{0}z| is continuous. Thus, by convergence in distribution there exists N such that

n ≥ N ⇒ P (\frac{c̄′Σc̄}{2} ≤ |c′_{ℓ_{kn}} x_{ℓ_{kn}} |) ≥ P (ϵ ≤ |c′_{ℓ_{kn}} x_{ℓ_{kn}} |) > \frac{P (ϵ ≤ |c′_{0}z|)}{2} ≥ \frac{P (c′_{0}Σc_{0} ≤ |c′_{0}z|)}{2} > 0,

a contradiction. □ Proof of Lemma 2. Suppose not. Then (Ṽ_{n}^{1/2}V_{n}^{−1/2})′(Ṽ_{n}^{1/2}V_{n}^{−1/2}) is an unbounded, symmetric, positive definite matrix. Hence it can be represented as E_{n}Λ_{n}E′_{n}, where E_{n} is orthogonal. As in Theorem 1, by unboundedness there exists an eigenvalue subsequence λ_{k_{n}i}^{−1} → 0. Denote by f_{i} the unit vector in direction i, and let

c′_{n} ≡ \frac{f′_{i}E′_{n}V_{n}^{1/2}Ṽ_{n}^{−1/2}}{∥f′_{i}E′_{n}V_{n}^{1/2}Ṽ_{n}^{−1/2}∥},

so that ∥c_{n}∥ = 1 ∀ n. Note that

∥f′_{i}E′_{n}V_{n}^{1/2}Ṽ_{n}^{−1/2}∥ = (f′_{i}E′_{n}(V_{n}^{1/2}Ṽ_{n}^{−1}V_{n}^{1/2})E_{n}f_{i})^{1/2}
= (f′_{i}E′_{n}E_{n}Λ_{n}^{−1}E′_{n}E_{n}f_{i})^{1/2}
= (f′_{i}Λ_{n}^{−1}f_{i})^{1/2} = λ_{ni}^{−1/2}.

Hence

c′_{n}Ṽ_{n}^{−1/2}x_{n} = c′_{n}Ṽ_{n}^{−1/2}V_{n}^{1/2}V_{n}^{−1/2}x_{n}
= λ_{ni}^{1/2}[f′_{i}E′_{n}V_{n}^{1/2}Ṽ_{n}^{−1/2}Ṽ_{n}^{−1/2}V_{n}^{1/2}]V_{n}^{−1/2}x_{n}
= λ_{ni}^{1/2}[f′_{i}E′_{n}E_{n}Λ_{n}^{−1}E′_{n}]V_{n}^{−1/2}x_{n}
= λ_{ni}^{1/2}[f′_{i}Λ_{n}^{−1}E′_{n}]V_{n}^{−1/2}x_{n}
= λ_{ni}^{−1/2}e′_{ni}V_{n}^{−1/2}x_{n},

where e_{ni} is column i of E_{n}. Since e_{k_{n}i} is on the unit ball in ℜ^{q}, there is a convergent subsequence

e_{ℓ_{kn} i} → e_{0} ∈ ∂B_{1}(0), so e′_{ℓ_{kn} i}V_{ℓ_{kn}}^{−1/2}x_{ℓ_{kn}} →^{d} e′_{0}z ∼ (0, e′_{0}e_{0}) = (0, 1).^{5} Hence λ_{ℓ_{kn} i}^{−1/2}e′_{ℓ_{kn} i}V_{ℓ_{kn}}^{−1/2}x_{ℓ_{kn}} →^{p} 0. But,

by Lemma 1, Ṽ_{ℓ_{kn}}^{−1/2}x_{ℓ_{kn}} = alip(1), a contradiction. □

5As in Theorem 1, normality of z is not needed here.

16 DAVID M. MANDY AND CARLOS MARTINS-FILHO

Proof of Theorem 2. Suppose first that Ṽ_{n} ∈ E_{R}(V_{n}). Then lim_{n→∞} V_{n}^{1/2}Ṽ_{n}^{−1}V_{n}^{1/2} = I_{q}, so V_{n}^{1/2}Ṽ_{n}^{−1/2} = O(1). Denote the characteristic function of a random vector y by f_{y}. By the continuity theorem [Lukacs 1970, p. 48], f_{V_{n}^{−1/2}x_{n}}(θ) → f_{z}(θ) pointwise. Hence, by White [1984, p. 66] Lemma 4.23,

f_{Ṽ_{n}^{−1/2}x_{n}}(θ) − f_{z}(V_{n}^{1/2}Ṽ_{n}^{−1/2}θ) = f_{(Ṽ_{n}^{−1/2}V_{n}^{1/2})V_{n}^{−1/2}x_{n}}(θ) − f_{z}(V_{n}^{1/2}Ṽ_{n}^{−1/2}θ) → 0 pointwise in θ.

Recalling that f_{z}(θ) = exp (−\frac{θ′θ}{2} ),

f_{z}(V_{n}^{1/2}Ṽ_{n}^{−1/2}θ) = exp (−\frac{θ′Ṽ_{n}^{−1/2}V_{n}Ṽ_{n}^{−1/2}θ}{2} ) → exp (−\frac{θ′θ}{2} ) ,

so f_{Ṽ_{n}^{−1/2}x_{n}}(θ) → f_{z}(θ). Applying the continuity theorem again shows that Ṽ_{n} ∈ avar (x_{n}). Now consider the converse by supposing Ṽ_{n} ∈ avar (x_{n}). By Lemma 2, M ≥ 1 can denote a common bound for all elements of Ṽ_{n}^{1/2}V_{n}^{−1/2}. Fix ϵ > 0 and set r = q^{2}M. Recalling that f_{V_{n}^{−1/2}x_{n}}(θ) = f_{x_{n}}(V_{n}^{−1/2}θ), by the second continuity theorem [Lukacs 1970, p. 53] there exists N_{ϵ} such that

n ≥ N_{ϵ} ⇒ \{ \begin{matrix}|f_{x_{n}}(V_{n}^{− \frac{1}{2}} θ) − exp (−\frac{θ′θ}{2} )| & < \frac{ϵ}{4}exp (−\frac{r^{2}}{2} ) ∀ θ ∈ \overline{B_{r}(0)} \\ |f_{x_{n}}(Ṽ_{n}^{− \frac{1}{2}} t) − exp (−\frac{t′t}{2})| & < \frac{ϵ}{4}exp (−\frac{r^{2}}{2} ) ∀ t ∈ \overline{B_{r}(0)}, \end{matrix}

where \overline{B_{r}(0)} is the closed ball of radius r about 0 in ℜ^{q}. Since Ṽn^{\frac{1}{2}} V_{n}^{− \frac{1}{2}} θ ∈ \overline{B_{r}(0)} for every n and θ ∈ ∂B_{1}(0), setting t = Ṽn^{\frac{1}{2}} V_{n}^{− \frac{1}{2}} θ yields

n ≥ N_{ϵ} ⇒ \{ \begin{matrix}|f_{x_{n}}(V_{n}^{− \frac{1}{2}} θ) − exp (−\frac{θ′θ}{2} )| & < \frac{ϵ}{4}exp (−\frac{r^{2}}{2} ) \\ |f_{x_{n}}(V_{n}^{− \frac{1}{2}} θ) − exp (−^{\frac{θ′V_{n}^{− \frac{1}{2}} Ṽ_{n}V_{n}^{− \frac{1}{2}} θ}{2}} )| & < \frac{ϵ}{4}exp (−\frac{r^{2}}{2} ) \end{matrix}

for every θ ∈ ∂B_{1}(0). Hence

n ≥ N_{ϵ} ⇒ |exp (−\frac{θ′V_{n}^{− \frac{1}{2}} Ṽ_{n}V_{n}^{− \frac{1}{2}} θ}{2} ) − exp (−\frac{θ′θ}{2} )| < \frac{ϵ}{2}exp (−\frac{r^{2}}{2} ) ∀ θ ∈ ∂B_{1}(0).

By the Mean Value Theorem,

|exp (−\frac{θ′V_{n}^{− \frac{1}{2}} Ṽ_{n}V_{n}^{− \frac{1}{2}} θ}{2} ) − exp (−\frac{θ′θ}{2} )| = |−\frac{1}{2}exp (−\frac{c}{2})| |θ′V_{n}^{− \frac{1}{2}} Ṽ_{n}V_{n}^{− \frac{1}{2}} θ − θ′θ|

for some c between θ′V_{n}^{− \frac{1}{2}} Ṽ_{n}V_{n}^{− \frac{1}{2}} θ and θ′θ. Thus

n ≥ N_{ϵ} ⇒ |θ′V_{n}^{− \frac{1}{2}} Ṽ_{n}V_{n}^{− \frac{1}{2}} θ − θ′θ| < ϵexp (\frac{c − r^{2}}{2} )

RELATIVE EFFICIENCY WITH EQUIVALENCE CLASSES OF ASYMPTOTIC COVARIANCES 17

for every θ ∈ ∂B_{1}(0) and the corresponding c (which also depends on n). Since θ′V_{n}^{− \frac{1}{2}} Ṽ_{n}V_{n}^{− \frac{1}{2}} θ ≤ r^{2} for every n and θ ∈ ∂B_{1}(0), and θ′θ = 1 ≤ r^{2} for such θ, we have c ≤ r^{2}. Thus exp (\frac{c−r^{2}}{2} ) ≤ 1 for every n and θ ∈ ∂B_{1}(0). That is,

n ≥ N_{ϵ} ⇒ |θ′V_{n}^{− \frac{1}{2}} Ṽ_{n}V_{n}^{− \frac{1}{2}} θ − θ′θ| < ϵ ∀ θ ∈ ∂B_{1}(0).

Since θ can be any vector on the unit ball, this implies lim_{n→∞} V_{n}^{−1/2}Ṽ_{n}V_{n}^{−1/2} = I_{q}, or Ṽ_{n} ∈ E_{R}(V_{n}). Next consider (i). Since V_{n}^{−1} is bounded by Theorem 1(i) and V_{n}^{−1/2}Ṽ_{n}V_{n}^{−1/2} − I_{q} = V_{n}^{−1/2}[Ṽ_{n} − V_{n}]V_{n}^{−1/2}, lim_{n→∞}[V_{n}^{−1/2}Ṽ_{n}V_{n}^{−1/2} − I_{q}] = 0 when lim_{n→∞}(Ṽ_{n} − V_{n}) = 0. Finally, consider (ii). Since V_{n} is bounded by Theorem 1(ii) and Ṽ_{n} − V_{n} = V_{n}^{1/2}[V_{n}^{−1/2}Ṽ_{n}V_{n}^{−1/2} − I_{q}]V_{n}^{1/2}, lim_{n→∞}(Ṽ_{n} − V_{n}) = 0 when lim_{n→∞}[V_{n}^{−1/2}Ṽ_{n}V_{n}^{−1/2} − I_{q}] = 0. □ Proof of Theorem 3. Select V_{n}^{∗} ∈ avar (\sqrt{n}(β_{n}^{∗} − β)) and Ṽ_{n} ∈ avar (\sqrt{n}(β̃_{n} − β)) such that f(θ) ≡ lim inf_{n→∞} f_{n}(θ) = lim_{n→∞} g_{n}(θ) ≥ 0 ∀ θ ∈ ℜ^{q} (θ ≠ 0), where f_{n}(θ) ≡ θ′(Ṽ_{n} − V_{n}^{∗})θ and g_{n}(θ) ≡ inf{f_{m}(θ): m ≥ n}. We first establish uniform equicontinuity of the sequence f_{n}(θ) on the boundary of the unit ball in ℜ^{q}. By Theorem 1(ii), Ṽ_{n} is bounded. It is straightforward to use this along with f(θ) ≥ 0 ∀ θ to conclude that V_{n}^{∗} is bounded as well, so denote by M a common bound for all elements of Ṽ_{n} and V_{n}^{∗}. For any θ, θ_{0} ∈ ∂B_{1}(0) we have

|f_{n}(θ) − f_{n}(θ_{0})| ≤ ∥θ − θ_{0}∥ [∥(Ṽ_{n} − V_{n}^{∗})θ∥ + ∥θ′_{0}(Ṽ_{n} − V_{n}^{∗})∥] ≤ ∥θ − θ_{0}∥4Mq.

Hence for every ϵ > 0 there exists δ = \frac{ϵ}{4Mq} such that ∥θ − θ_{0}∥ < δ ⇒ |f_{n}(θ) − f_{n}(θ_{0})| < ϵ ∀ n, or f_{n}(θ) is uniformly equicontinuous on ∂B_{1}(0). Now, by definition of g_{n}(θ) there exists a subsequence f_{i_{n}}(θ) such that g_{n}(θ) + \frac{1}{n} > f_{i_{n}}(θ) ≥ g_{n}(θ). Thus f_{i_{n}}(θ) converges pointwise to f(θ), so by equicontinuity this convergence is actually uniform on ∂B_{1}(0). This implies lim_{n→∞} f_{i_{n}}(θ) − \frac{1}{n} = f(θ) uniformly on ∂B_{1}(0). Hence, since f_{n}(θ) ≥ g_{n}(θ) by definition of g_{n}(θ), for every natural number k there exists a natural number N_{k} such that

n ≥ N_{k} ⇒ f_{n}(θ) ≥ g_{n}(θ) > f_{i_{n}}(θ) − \frac{1}{n} > f(θ) − \frac{1}{k} ∀ θ ∈ ∂B_{1}(0).

We may summarize this by saying that the convergence to the limit inferior is uniform on ∂B_{1}(0) in the sense that for every k there exists N_{k} such that n ≥ N_{k} ⇒ f_{n}(θ)+ \frac{1}{k} > 0 ∀ θ ∈ ∂B_{1}(0), and without loss of generality we may assume 1 < N_{1} < N_{2} < · · ·. Let k_{n} be the largest natural number k satisfying N_{k} ≤ n for n ≥ N_{1} and note that n ≥ N_{k_{n}} ∀ n ≥ N_{1} and lim_{n→∞} \frac{1}{k_{n}} = 0. Thus Ṽ_{n} + \frac{1}{k_{n}} I_{q} ∈ avar (\sqrt{n}(β̃_{n} − β)) by Theorem

2(i), so that β_{n}^{∗} is asymptotically efficient relative to β̃_{n} according to Definition 6 if (Ṽ_{n} + \frac{1}{k_{n}} I_{q}) − V_{n}^{∗} is

positive semidefinite for n large. For any n ≥ N_{1} and θ ≠ 0 we have

(\frac{θ′}{∥θ∥}) [(Ṽ_{n} + \frac{1}{k_{n}} I_{q}) − V_{n}^{∗}] (\frac{θ}{∥θ∥}) = (\frac{θ′}{∥θ∥}) [Ṽ_{n} − V_{n}^{∗}] (\frac{θ}{∥θ∥}) + \frac{θ′θ}{k_{n}∥θ∥^{2}}

= f_{n} (\frac{θ}{∥θ∥}) + \frac{1}{k_{n}} > 0,

so θ′ [(Ṽ_{n} + \frac{1}{k_{n}} I_{q}) − V_{n}^{∗}] θ > 0 ∀ n ≥ N_{1} and ∀ θ ≠ 0. That is, (Ṽ_{n} + \frac{1}{k_{n}} I_{q}) − V_{n}^{∗} is positive definite for

n ≥ N_{1}. □

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