Efficiency combined with simplicity: new testing procedures for Generalized Inverse Gaussian models

Abstract

The standard efficient testing procedures in the Generalized Inverse Gaussian (GIG) family (also known as Halphen Type A family) are likelihood ratio tests, and hence rely on Maximum Likelihood (ML) estimation of the three parameters of the GIG. The particular form of GIG densities, involving modified Bessel functions, prevents in general form a closed-form expression for ML estimators, which are obtained at the expense of complex numerical approximation methods. On the contrary, Method of Moments (MM) estimators allow for concise expressions, but tests based on these estimators suffer from a lack of efficiency as compared to likelihood ratio tests. This is why, in recent years, trade-offs between ML and MM estimators have been proposed, resulting in simpler yet not completely efficient estimators and tests. In the present paper, we do not propose such a trade-off but rather an optimal combination of both methods, our tests inheriting efficiency from an ML-like construction and simplicity from the MM estimators of the nuisance parameters. This goal shall be reached by attacking the problem from a new angle, namely via the Le Cam methodology. Besides providing simple efficient testing methods, the theoretical background of this methodology further allows us to write out explicitly power expressions for our tests. A Monte Carlo simulation study shows that, also at small sample sizes, our simpler procedures do at least as good as the complex likelihood ratio tests. We conclude the paper by applying our findings on two real-data sets.

A Appendix

Proof of Theorem 2.1 Establishing the ULAN property of $\mathrm{GIG}(p,a,b)$ with respect to all three parameters $p,a,b$ is quite straightforward since we are not working within a semiparametric family of distributions (hence we do not have to deal with an infinite-dimensional parameter); the problem considered involves a parametric family of distributions with densities meeting the most classical regularity conditions. In particular, one readily obtains that (i) $(p,a,b)\mapsto \sqrt{c(p,a,b) x^{p-1}\mathrm{e}^{-(ax+b/x)/2}}$ is continuously differentiable for every $x>0$ and (ii) the associated Fisher information matrix is well defined and continuous in $p,a$ and $b$. Thus, by Lemma 7.6 of van der Vaart (1998), $(p,a,b)\mapsto \sqrt{c(p,a,b) x^{p-1}\mathrm{e}^{-(ax+b/x)/2}}$ is differentiable in quadratic mean, and the ULAN property follows from Theorem 7.2 of van der Vaart (1998). This completes the proof. $\square$

Proof of Proposition 3.1  It follows from the asymptotic linearity property of the central sequence given in (3) that, under $\mathrm{P}_{{\pmb \vartheta }}^{(n)}$ and for $n\rightarrow \infty$,

$$\begin{aligned} {{\pmb \varDelta }}^{(n)\mathrm{eff}}_p\left( {\pmb \vartheta }+n^{-1/2} (0,\tau _2^{(n)},\tau _3^{(n)})'\right) ={{\pmb \varDelta }}^{(n)\mathrm{eff}}_p({\pmb \vartheta })-{\pmb P}({\pmb \vartheta }){\pmb \varGamma }({\pmb \vartheta }) \left( \begin{array}{cll} 0\\ \tau _2^{(n)}\\ \tau _3^{(n)} \end{array} \right) +o_\mathrm{{P}}(1), \end{aligned}$$

where

$$\begin{aligned} {\pmb P}({\pmb \vartheta })&= \left( 1, -(\varGamma _{p,a}({\pmb \vartheta }),\varGamma _{p,b}({\pmb \vartheta }))\left( \begin{array}{ll} \varGamma _{b,b}({\pmb \vartheta })&{}\quad -\varGamma _{a,b}({\pmb \vartheta })\\ -\varGamma _{a,b}({\pmb \vartheta })&{}\quad \varGamma _{a,a}({\pmb \vartheta }) \end{array}\right) \right. \\&\left. \quad /(\varGamma _{b,b}({\pmb \vartheta })\varGamma _{a,a}({\pmb \vartheta })-(\varGamma _{a,b}({\pmb \vartheta }))^2)\right) \end{aligned}$$

is the projection matrix onto the subspace orthogonal to the central sequences for $a$ and $b$. One easily checks that the product ${\pmb P}({\pmb \vartheta }){\pmb \varGamma }({\pmb \vartheta })$ is of the form $(\cdot ,0,0)$, leading to the announced asymptotic linearity (4). The other asymptotic equality in probability follows by continuity. $\square$

Proof of Theorem 3.1 The statement in Part (i) is easily proved thanks to the asymptotic linearity in (6) under the null, since $\varDelta ^{(n)\mathrm{eff}}_p(p_0,a,b)$ is asymptotically $\mathcal {N}(0, \varGamma _{p,p}^{\mathrm{eff}}(p_0,a,b))$ under $\bigcup _{a\in \mathbb R_0^+} \bigcup _{b\in \mathbb R_0^+}\mathrm{P}^{(n)}_{p_0,a,b}$ by the central limit theorem.

In order to prove the more delicate Part (ii), observe that, under $\mathrm{P}^{(n)}_{p_0,a,b}$ and for any bounded sequence ${{\pmb \tau }}^{(n)}=(\tau _1^{(n)}, {\tau }_2^{(n)}, \tau _3^{(n)}) '\in \mathbb R^3$, we see that, as $n\rightarrow \infty$,

$$\begin{aligned}&\begin{pmatrix} \varDelta _p^{(n)\mathrm{eff}}(p_0,a,b) \\ \Lambda ^{(n)}_{(p_0,a,b)'+n^{-1/2}{\pmb \tau }^{(n)}/(p_0,a,b)'} \end{pmatrix}\\&\quad \mathop {\longrightarrow }\limits ^{\mathcal {L}}\mathcal {N}_2 \left( \begin{pmatrix} 0 \\ -\frac{1}{2} {\pmb \tau }'{\pmb \varGamma }({\pmb \vartheta }){\pmb \tau }\end{pmatrix}, \begin{pmatrix} \varGamma _{p,p}^{\mathrm{eff}}(p_0,a,b) &{}\quad \tau _1\varGamma _{p,p}^{\mathrm{eff}}(p_0,a,b)\\ \tau _1 \varGamma _{p,p}^{\mathrm{eff}}(p_0,a,b) &{}\quad {\pmb \tau }'{\pmb \varGamma }({\pmb \vartheta })\tau \end{pmatrix}\right) , \end{aligned}$$

where $\Lambda ^{(n)}_{(p_0,a,b)'+n^{-1/2}{\pmb \tau }^{(n)}/(p_0,a,b)'}$ is the log-likelihood ratio and ${\pmb \tau }=\lim _{n \rightarrow \infty } {\pmb \tau }^{(n)}$. We can then apply Le Cam’s third lemma which implies that $\varDelta _{p}^{(n)\mathrm{eff}}(p_0,a,b)$ is asymptotically $\mathcal {N}\left( \tau _1\varGamma _{p,p}^{\mathrm{eff}}(p_0,a, b), \varGamma _{p,p}^{\mathrm{eff}}(p_0,a,b)\right)$ under $\mathrm{P}^{(n)}_{p_0+n^{-1/2}\tau _1^{(n)},a,b}$. Since the asymptotic linearity (6) holds as well under $\mathrm{P}^{(n)}_{p_0+n^{-1/2}\tau _1^{(n)},a,b}$ by contiguity, Part (ii) of the theorem readily follows.

As regards Part (iii), the fact that $\phi ^{(n)}_{p_0}$ has asymptotic level $\alpha$ follows directly from the asymptotic null distribution given in Part (i), while local asymptotic maximinity is a consequence of the convergence of the local experiments to the Gaussian shift experiment (see Le Cam and Yang 2000). $\square$