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Articles

The MLE of Aigner, Amemiya, and Poirier is not the expectile MLE

Pages 99-114 | Published online: 22 Mar 2021
 

Abstract

This article compares two asymmetric Gaussian likelihood models and their corresponding estimators. Recently, there has been confusion in the literature regarding these models and (1) whether they are the same, or (2) whether both of them can be used to estimate expectiles. After the comparison, it becomes clear that they are not the same and only one of these models is appropriate for that purpose. The similarity between these models is purely superficial. The historical origin of expectiles has also been disputed: some degree of credit can be shared between two papers.

JEL CLASSIFICATION:

Acknowledgment

We would like to thank Minchul Shin, Sebastian Laumer, and two anonymous referees for their helpful comments.

Notes

1 To enrich the physics analogy, the first moment is the point of balance of a lever where the torque from the right and left sides of the fulcrum are equal. The τth expectile is the point of imbalance where the torque from the left of the fulcrum is τ/(1τ) times the torque on the right.

2 For non-degenerate F, Newey and Powell (Citation1987, Theorem 1) prove that the expectile relation μ(τ):(0,1){y|0<F(y)<1} is strictly monotone increasing in τ and injective. Add the convention that μ(0):=inf(Y) and μ(1):=sup(Y), and μ(τ) is a surjective and injective mapping from the closed unit interval to SYR, the convex hull of the support of Y. Thus, every value in the interval [inf(Y),sup(Y)] is an expectile of Y.

3 The exact origin of this asymmetric normal distribution is not obvious. The same distribution was studied previously by Kato et al. (Citation2002) under a different parametrization. It had also appeared elsewhere prior to that date (Liess et al., Citation1997).

4 The estimator in (Equation2.10) has the standard linear form typical of a generalized least squares estimator, βτ̂=(XWX)1XWy. The matrix W is a diagonal matrix of weights [W]ii=wi, with wi=τ if yixiβτ and wi=1τ otherwise.

5 Alternately, we might make the comparison between Equationequation 3.4 and Equationequation 2.2 even clearer by writing ϵi={HN(0,σ21θ)w.p.12HN(0,σ2θ)w.p.12.

6 For θ.5 and arbitrary σ2, a simple calculation will show that n(fθ(y|μ=μo)1/2fθ(y|μ=μo+h/n)1/2)2dy which violates the result in Lemma 12.2.1 in Lehmann and Romano (Citation2006). See also van der Vaart, (Citation2000, p. 64). This demonstrates that the AAP distribution is not differentiable in quadratic mean.

7 Hirano and Porter (Citation2012) also require differentiability in quadratic mean, which is not applicable to our discontinuous density problem.

8 As an aside, it is also worth noting that the difference between the two distributions persists in the “external” case of τ = 0 or τ = 1, where we have limθ0Fθ(y|μ,θ,σ2)=12U(,μ)+12δ(μ)limτ0Fτ(y|μτ,τ,στ2)=U(,μτ). Here, δ is the Dirac delta function.

9 Write τ̂=1θ̂MLE1θ̂MLE+θMLE, and it is trivial to show that μ̂τ̂pμ̂τ.

10 We would quickly point out that this tradeoff will exist for other parametric models with discontinuous density at the location parameter, so long as that parameter is not the minimum or maximum of the distribution. If the model has two moments, (1) the location parameter will be an expectile μτ for some τ(0,1) and (2) the sample expectile μ̂τ will be consistent and asymptotically normal (Holzmann et al., Citation2016).

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