In our paper published in vol. 22, no. 1, of Econometric Theory (2006, pp. 69–97) the proof of (22) in Lemma 3.1 is incorrect as given. The problem in the proof occurs after the first display on p. 93 where we attempted to deduce (22) from the already established relation (21). The argument given there is incorrect, because an existence-quantor was mistaken for an all-quantor. We show how the problem can be repaired. In the course of this, we also note a useful variant of another result in Lemma 3.1.

There are two ways of fixing the problem. First, if the contiguity assumption in Lemma 3.1 is strengthened to the assumption that any two sequences of probability measures P_{n,βn⁽¹⁾} and P_{n,βn⁽²⁾} with β_n⁽¹⁾ ∈ B_n and β_n⁽²⁾ ∈ B_n are mutually contiguous, then (22) is correct as it stands. To see that this claim is true, assume that (22) is false. Then there exists a δ₀, 0 ≤ δ₀ < δ*/2, such that

Consequently, there exist a subsequence n_i with n_i → ∞ as i → ∞ and a sequence of estimators

such that

Furthermore, by definition of δ* there exist β_n⁽¹⁾ and β_n⁽²⁾ in B_n such that

which implies

for n large enough, and hence d(φ_ni(β_ni⁽¹⁾),φ_ni(β_ni⁽²⁾)) > 2δ₀ for large enough i. For large enough i we thus obtain

The first term on the far right-hand side of the preceding display now converges to zero for i → ∞ in view of (1). In view of the same reference and mutual contiguity (which carries over to subsequences) we also see that the second term converges to zero as i → ∞. This leads to a contradiction and establishes the claim. (We note that this derivation does not refer to the element α appearing in the formulation of Lemma 3.1 and, consequently, also does not rely on the assumption that the sets B_n have nonempty intersection.)

Relation (22) in Lemma 3.1 has been used to derive results in Leeb and Pötscher (2006b, 2008) and Pötscher (2006). All these results are unaffected, because the statistical models considered in these references satisfy the mutual contiguity assumption made here.

Second, without strengthening the contiguity assumption in Lemma 3.1 (i.e., leaving the assumptions of that lemma unchanged), the relation (22) can be established if the range of δ is restricted from δ < δ*/2 to δ < δ*/4. To see this, assume that (22) with this restriction on the range of δ is false. Then there exists a δ₀, 0 ≤ δ₀ < δ*/4 such that the same argument as given before shows the existence of a subsequence n_i with n_i → ∞ as i → ∞ and a sequence of estimators

such that (1) holds. Also, exactly as in the preceding proof, there exist β_n⁽¹⁾ and β_n⁽²⁾ in B_n such that

holds for large enough n. It follows that for large enough n

or

holds. Set β_n = β_n⁽¹⁾ if (2) holds and set β_n = β_n⁽²⁾ otherwise. Then

holds for all n large enough. Consequently, for i large enough we arrive at

The first term on the far right-hand side of the preceding display now converges to zero for i → ∞ in view of (1). In view of the same reference and because α ∈ B_n for all n, we obtain

Because P_n,βn is contiguous with respect to P_n,α by assumption, we also see that the second term on the right-hand side in (4) converges to zero as i → ∞. This leads to a contradiction and establishes the claim.

We also take the opportunity to point out that a variant of the proof just given allows one to derive the following variant of (21) in Lemma 3.1: Let B_n, α, and the contiguity assumption be as in Lemma 3.1 and assume that δ* > 0. Fix δ satisfying 0 ≤ δ < δ*/4. Then any estimator

satisfying

also satisfies

This can be proved as follows: As in the preceding proof, construct a sequence β_n satisfying (3) with δ in place of δ₀. It follows that

holds for large enough n. By (5) and contiguity the second term on the right-hand side of the preceding inequality converges to zero for n → ∞, which establishes (6). (We note that the incorrect reasoning in Leeb and Pötscher (2006a) attempting to derive (22) from (21) actually delivers (22) with δ restricted to δ < δ*/4 when this reasoning is applied to (6) here instead of to (21) in Leeb and Pötscher (2006a).)

We finally note that Remarks 3.1(vi) and B.1 in Leeb and Pötscher (2006a) also apply to the results established here.