Small-sample one-sided testing in extreme value regression models

Abstract

We derive adjusted signed likelihood ratio statistics for a general class of extreme value regression models. The adjustments reduce the error in the standard normal approximation to the distribution of the signed likelihood ratio statistic. We use Monte Carlo simulations to compare the finite-sample performance of the different tests. Our simulations suggest that the signed likelihood ratio test tends to be liberal when the sample size is not large and that the adjustments are effective in shrinking the size distortion. Two real data applications are presented and discussed.

Appendix

Let $v$ be a vector, and $v_{(r)}$ be the vector $v$ without its $r$th component. Analogously, let $\mathcal {V}$ be a matrix and $\mathcal {V}_{(r)}$ be the matrix $\mathcal {V}$ without the $r$th column. Assume that $\nu =\beta _r$ is the parameter of interest. The Fisher information matrix (see Ferrari and Pinheiro 2012) can be written as

$$\begin{aligned} I(\theta )=I(\beta _r,\theta _{(r)})=I(\beta _r,\beta _{(r)},\gamma )= \begin{pmatrix} I_{\beta _r\beta _r} &{}\quad I_{\beta _r\beta _{(r)}} &{}\quad I_{\beta _r\gamma } \\ I_{\beta _{(r)}\beta _r} &{}\quad I_{\beta _{(r)}\beta _{(r)}} &{}\quad I_{\beta _{(r)}\gamma } \\ I_{\gamma \beta _r} &{}\quad I_{\gamma \beta _{(r)}} &{}\quad I_{\gamma \gamma } \\ \end{pmatrix}, \end{aligned}$$

where

$$\begin{aligned} \begin{array}{ll} I_{\beta _r\beta _r}={\mathbf {x}}_{\cdot r}^{\!\top }\Phi ^{-1}T^2\Phi ^{-1}{\mathbf {x}}_{\cdot r},&{}\quad I_{\beta _r\beta _{(r)}}\!=\!I_{\beta _{(r)}\beta _r}^{\!\top }={\mathbf {x}}_{\cdot r}^{\!\top }\Phi ^{-1}T^2\Phi ^{-1}X_{(r)},\\ I_{\beta _r\gamma }=I_{\gamma \beta _r}^{\!\top }=({\mathcal {E}}-1){\mathbf {x}}_r^{\!\top }\Phi ^{-1}TH\Phi ^{-1}Z, &{}\quad I_{\beta _{(r)}\beta _{(r)}}=X_{(r)}^{\!\top }\Phi ^{-1}T^2\Phi ^{-1}X_{(r)},\\ I_{\beta _{(r)}\gamma }=I_{\gamma \beta _{(r)}}^{\!\top }=({\mathcal {E}}-1)X_{(r)}^{\!\top }\Phi ^{-1}TH\Phi ^{-1}Z, &{}\quad I_{\gamma \gamma }=\bigl (1 + \Gamma ^{(2)}(2)\bigr )Z^{\!\top }\Phi ^{-1}H^2\Phi ^{-1}Z, \end{array} \end{aligned}$$

${\mathbf {x}}_{\cdot r}=(x_{1r},\ldots ,x_{nr})^{\!\top }$ is the $r$th column of $X$ and $\Phi$, $T$, $H$, $X$ and $Z$ as defined above.

Consider a parameterization $\vartheta =(\beta _r,\vartheta _{(r)})$, $\vartheta _{(r)}=(\kappa ,\tau )$, where $\vartheta$, $\vartheta _{(r)}$, $\kappa$ and $\tau$ have dimensions $(k+m)$, $(k+m-1)$, $(k-1)$ and $m$, respectively, in such a way that $\beta _r$ is orthogonal to $\vartheta _{(r)}$. Let

$$\begin{aligned} I_{\beta _r\theta _{(r)}}= \left( \begin{array}{ll} I_{\beta _r\beta _{(r)}}&I_{\beta _r\gamma } \end{array} \right) , \quad I_{\theta _{(r)}\theta _{(r)}}= \left( \begin{array}{ll} I_{\beta _{(r)}\beta _{(r)}} &{} I_{\beta _{(r)}\gamma }\\ I_{\gamma \beta _{(r)}} &{} I_{\gamma \gamma } \end{array}\right) , \quad I_{\theta _{(r)}\theta _{(r)}}^{-1}= \left( \begin{array}{ll} I^{\beta _{(r)}\beta _{(r)}} &{} I^{\beta _{(r)}\gamma }\\ I^{\gamma \beta _{(r)}} &{} I^{\gamma \gamma } \end{array} \right) . \end{aligned}$$

Define $A \equiv I_{\beta _r\beta _{(r)}}I^{\beta _{(r)}\beta _{(r)}}+I_{\beta _r\gamma }I^{\gamma \beta _{(r)}},$ $B \equiv I_{\beta _r\beta _{(r)}}I^{\beta _{(r)}\gamma }+I_{\beta _r\gamma }I^{\gamma \gamma },$ $\beta _{(r)}=\kappa -\beta _rA^{\!\top }$ and $\gamma =\tau -\beta _rB^{\!\top }$. From Cox and Reid (1987), it can be shown that $\beta _r$ is orthogonal to $(\kappa , \tau )$.

In order to obtain $R_0^*$, one should first re-write (4)–(6) in the orthogonal parameterization $\vartheta =(\beta _r,\kappa , \tau )$, and define the systematic components $\eta ^*(\vartheta )$ and $\delta ^*(\vartheta )$. The derivative of the log-likelihood function with respect to the parameter of interest $\beta _r$, $\ell _{\beta _r}^*(\vartheta )$, the observed information matrix, $J^*(\vartheta )$ and the element of the expected information matrix that corresponds to $\beta _r$, $I_{\beta _r\beta _r}^*(\vartheta )$, should then be computed in this parameterization as in Ferrari and Pinheiro (2012, p. 584), and inserted into (9).

Consider, for instance, the maximum extreme value model regression model (4) with location and dispersion sub-models $g(\mu _t) = \eta _t = \eta (x_{t},\beta )=x_{t}^\top \beta$ and $h(\sigma _t) = \delta _t = \delta (z_{t},\gamma )=z_{t}^\top \gamma .$ We have

$$\begin{aligned} g(\mu _t)= x_{tr}\beta _r+x_{t(r)}^\top \beta _{(r)}= x_{tr}\beta _r+x_{t(r)}^\top (\kappa -\beta _r A^{\!\top })= (x_{tr}-x_{t(r)}^\top A^{\!\top })\beta _r+x_{t(r)}^\top \kappa \equiv \eta ^*_t(\vartheta ) \end{aligned}$$

(13)

and

$$\begin{aligned} h(\sigma _t)= z_{t}^\top \tau -\beta _r z_{t}^\top B^{\!\top }\equiv \delta ^*_t(\vartheta ). \end{aligned}$$

(14)

For the reparameterized model (4) with (13)–(14), we have

$$\begin{aligned} \ell ^*_{\beta _r}(\vartheta )= & {} \frac{\partial \ell ^*(\vartheta )}{\partial \beta _r}= \iota ^{\!\top }\Bigl ( \Phi ^{-1}T\left( {\mathcal {I}} - \breve{\mathcal {Z}} \right) {\mathbf {v}}_1 + \Phi ^{-1}H\Bigl (-{\mathcal {I}}+{\mathcal {Z}}-{\mathcal {Z}}\breve{\mathcal {Z}} \Bigr ){\mathbf {v}}_2 \Bigr ),\\ J^*_{\beta _r\beta _r}= & {} {\mathbf {v}}_1^{\!\top } V_{\beta \beta } {\mathbf {v}}_1 +2{\mathbf {v}}_1^{\!\top }V_{\beta \gamma } \; {\mathbf {v}}_2 +{\mathbf {v}}_2^{\!\top }V_{\gamma \gamma } \; {\mathbf {v}}_2, \quad J^*_{\kappa \kappa }=X_{(r)}^{\!\top } \; V_{\beta \beta } \; X_{(r)}, \\ J^*_{\beta _r\kappa }= & {} J_{\kappa \beta _r}^{\!\top }= {\mathbf {v}}_1^{\!\top } V_{\beta \beta } \; X_{(r)}+{\mathbf {v}}_2^{\!\top } \; V_{\beta \gamma } \; X_{(r)}, \quad J^*_{\kappa \tau }=J_{\tau \kappa }^{\!\top }=X_{(r)}^{\!\top }V_{\beta \gamma } \; Z, \\ J^*_{\beta _r\tau }= & {} J_{\tau \beta _r}^{\!\top }= {\mathbf {v}}_1^{\!\top }V_{\beta \gamma } \; Z +{\mathbf {v}}_2^{\!\top }V_{\gamma \gamma } \; Z, \quad J^*_{\tau \tau }= Z^{\!\top }V_{\gamma \gamma } \; Z \end{aligned}$$

and

$$\begin{aligned} I^*_{\beta _r\beta _r}= & {} {\mathbf {v}}_1^{\!\top } \Phi ^{-1}T^2\Phi ^{-1} {\mathbf {v}}_1 +2({\mathcal {E}}-1){\mathbf {v}}_1^{\!\top } \Phi ^{-1}T H \Phi ^{-1} {\mathbf {v}}_2\\&+\bigl (1+\Gamma ^{(2)}(2)\bigr ){\mathbf {v}}_2^{\!\top }\Phi ^{-1}H^2\Phi ^{-1}{\mathbf {v}}_2, \end{aligned}$$

where ${\mathbf {v}}_1=\left( {\mathbf {x}}_{\cdot r}-X_{(r)}A^{\!\top } \right)$, ${\mathbf {v}}_2=-(ZB^{\!\top })$, $V_{\beta \beta }=\Phi ^{-1}T\Bigl (\breve{\mathcal {Z}}\Phi ^{-1} + ({\mathcal {I}}-\breve{\mathcal {Z}})ST\Bigr )T$, $V_{\beta \gamma }=\Phi ^{-1}T({\mathcal {I}}-\breve{\mathcal {Z}}+{\mathcal {Z}}\breve{\mathcal {Z}})H\Phi ^{-1}$ and $V_{\gamma \gamma }=\Phi ^{-1}H\Bigl ((-{\mathcal {I}}+2{\mathcal {Z}}-2{\mathcal {Z}}\breve{\mathcal {Z}}+{\mathcal {Z}}^2\breve{\mathcal {Z}})\Phi ^{-1}+(-{\mathcal {I}}+{\mathcal {Z}}-{\mathcal {Z}}\breve{\mathcal {Z}})QH\Bigr )H$. DiCiccio & Martin’s adjusted signed likelihood ratio statistic can now be obtained by replacing the formulas above in (9).