Optimal reparametrization and large sample likelihood inference for the location-scale skew-normal model
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OPTIMAL REPARAMETRIZATION AND LARGE SAMPLE LIKELIHOOD INFERENCE FOR THE LOCATION-SCALE SKEW-NORMAL MODEL ´lez-Far´ıas2 Rolando Cavazos-Cadena1 and Graciela M. Gonza 1
Departamento de Estad´ıstica y C´ alculo Universidad Aut´ onoma Agraria Antonio Narro Buenavista, Saltillo COAH, 25315, M´exico E-mail: [email protected] 2
Centro de Invastigaci´ on en Matem´ aticas A. C. Apartado Postal 402, Guanajuato, GTO, 36240, M´exico E-mail: [email protected] (Received July 1, 2010; Accepted October 15, 2010)
Dedicated to the memory of Don Jos´e Mar´ıa Morelos y Pav´ on
[Communicated by Istv´ an Berkes]
Abstract Motivated by results in Rotnitzky et al. (2000), a family of parametrizations of the location-scale skew-normal model is introduced, and it is shown that, under each member of this class, the hypothesis H0 : λ = 0 is invariant, where λ is the asymmetry parameter. Using the trace of the inverse variance matrix associated to a generalized gradient as a selection index, a subclass of optimal parametrizations is identified, and it is proved that a slight variant of Azzalini’s centred parametrization is optimal. Next, via an arbitrary optimal parametrization, a simple derivation of the limit behavior of maximum likelihood estimators is given under H0 , and the asymptotic distribution of the corresponding likelihood ratio statistic for this composite hypothesis is determined.
Mathematics subject classification numbers: 62H12, 62H15. Key words and phrases: singular information matrix, linear dependence restrictions, probability approximations for maximum likelihood estimators, asymptotic independence. This work was supported by the PSF Organization under Grant No. 2007-4, and by CONACYT under Grant 105657 /CB2008. 0031-5303/2012/$20.00 c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
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´ ´IAS R. CAVAZOS-CADENA and G. M. GONZALEZ-FAR
1. Introduction This work concerns asymptotic likelihood estimation and hypothesis testing for the univariate location-scale skew-normal class of probability densities, which is denoted by SN = {ρ(·; θ) | θ ∈ Θ} and is specified as follows: for each θ = (ξ, η, λ) ∈ Θ := R × (0, ∞) × R, 2 x − ξ h x − ξ i ρ(x; θ) := ϕ Φ λ , x ∈ R, η η η
(1.1)
2
where ϕ(x) = (2π)−1/2 e−x /2 is the standard normal density, and Φ(·) stands for the corresponding cumulative distribution function. This family was introduced by Azzalini (1985, 1986) as an extension of the usual location-scale normal model; indeed, when λ = 0, ρ(·; θ) is the normal density with mean ξ and standard deviation η, whereas, if λ 6= 0, then λ controls the asymmetry of ρ(·; θ). The impact of the skew-normal model has been wide and important, and regression models based on this family as well as multivariate extensions have been intensively studied during the last twenty years; see, for instance, Azzalini and Dalla Valle (1996), Azzalini and Capitanio (1999), Genton (2004) and the references therein. Since the seminal works by Azzalini, the problem of testing the symmetry hypothesis H0 : λ = 0 and,
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