Improved heteroskedasticity likelihood ratio tests in symmetric nonlinear regression models

Abstract

In this paper we address the issue of testing inference of the dispersion parameter in heteroscedastic symmetric nonlinear regression models considering small samples. We derive Bartlett corrections to improve the likelihood ratio as well modified profile likelihood ratio tests. Our results extend some of those obtained in Cordeiro (J Stat Comput Simul 74:609–620, 2004) and Ferrari et al. (J Stat Plan Inference 124:423–437, 2004), who consider a symmetric nonlinear regression model and normal linear regression model, respectively. We also present the bootstrap and bootstrap Bartlett corrected likelihood ratio tests. Monte Carlo simulations are carried out to compare the finite sample performances of the three corrected tests and their uncorrected versions. The numerical evidence shows that the corrected modified profile likelihood ratio test, the bootstrap and bootstrap Bartlett corrected likelihood ratio test perform better than the other ones. We also present an empirical application.

Appendices

Appendix A

In this appendix we obtain the matrix $j^*({\varvec{\delta }}; \varvec{\hat{\beta }_\delta }, \hat{\gamma }_\delta )$ expressed in (3) which denotes the observed information matrix for the nuisance parameters $({\varvec{\beta }}^\top , \gamma )^\top .$ We can express $j^*({\varvec{\delta }}; \varvec{\hat{\beta }_\delta }, \hat{\gamma }_\delta )$ by

$$\begin{aligned} j^{*}({\varvec{\delta }}; \varvec{\hat{\beta }_\delta },\hat{\gamma }_{{\varvec{\delta }}}) = - \left( \begin{array}{cc} j^{*}_{\beta \beta } &{} \varvec{0} \\ \varvec{0} &{} j^{*}_{\gamma \gamma } \end{array} \right) , \end{aligned}$$

where $j^{*}_{\beta \beta }$ is a square matrix of order p whose entries are given by

$$\begin{aligned} j^{*}_{\beta \beta } = -\frac{\partial ^2 l^*(\varvec{y};{\varvec{\theta }}^*)}{\partial \beta _j \partial \beta _l} = -\frac{1}{\gamma } \sum _{\ell =1}^{n} t(z_\ell )^{(2)} q_\ell (j,l)_\ell + \frac{1}{\gamma ^{1/2}}\sum _{\ell =1}^n t(z_\ell )^{(1)}q_\ell ^{1/2}(jl)_\ell \end{aligned}$$

and

$$\begin{aligned} j^{*}_{\gamma \gamma } = -\frac{\partial ^2 l^*(\varvec{y};{\varvec{\theta }}^*)}{\partial \gamma ^2} = - \frac{n}{2 \gamma ^2} - \frac{3}{4 \gamma ^2} \sum _{\ell =1}^n t(z_\ell )^{(1)}z_\ell - \frac{1}{4 \gamma ^2} \sum _{\ell =1}^n t(z_\ell )^{(2)}z_\ell ^2 \end{aligned}$$

is a scalar, with $\varvec{\theta ^*}=({\varvec{\beta }}^\top , {\varvec{\delta }}^\top ,\gamma )^\top \,\,q_\ell =\frac{(\prod _{s=1}^n m_s)^{1/n}}{m_\ell }$, $(j,l)_\ell = (\partial \mu _\ell /\partial \beta _j)\,\,(\partial \mu _\ell / \partial \beta _l)$ and $(jl)_\ell =\partial ^2 \mu _\ell / \partial \beta _j \partial \beta _l, \ j,l=1,\ldots ,p.$

Appendix B

In this appendix we present the required derivatives of the log-likelihood function in (2) up to the fourth order and their respective moments, considering $m_\ell =\exp (\varvec{\omega }_\ell ^\top {\varvec{\delta }}),$ to obtain the constants c and $c_m$ from the Bartlett correction factors. For this, we introduce the following notation: $\lambda _{rs}=E(\partial ^2 l/\partial {\varvec{\theta }}_r \partial {\varvec{\theta }}_s)$, $\lambda _{rst}=E(\partial ^3 l/\partial {\varvec{\theta }}_r \partial {\varvec{\theta }}_s \partial {\varvec{\theta }}_t),$ etc, $a, b, c,\ldots$ index parameters of interest ${\varvec{\delta }}$, i, j, $l,\ldots$ index nuisance parameters ${\varvec{\beta }}$ and $r,s,t,\ldots$ index all $p+q+1$ parameters of the model. Thus, we have

$$\begin{aligned} \lambda _{ab \gamma }= & {} \frac{\varDelta _1}{\gamma }\sum _{\ell =1}^n (\varvec{\omega }_\ell - \bar{\varvec{\omega }})_{ab},\ \ \lambda _{acd} = \varDelta _1 \sum _{\ell =1}^n (\varvec{\omega }_\ell - \bar{\varvec{\omega }})_{acd},\\ \lambda _{abcd}= & {} \varDelta _2 \sum _{\ell =1}^n (\varvec{\omega }_\ell - \bar{\varvec{\omega }})_{abcd},\\ \lambda _{\gamma \gamma }= & {} (\varDelta _4 \gamma ^2)^{-1}, \ \ \lambda _{ija}= \frac{1}{2\gamma } (\delta _{(0,0,1,0,1)}+2\delta _{(0,1,0,0,0)})\sum _{\ell =1}^n \varvec{q}_{\ell ab},\\ \lambda _{abij}= & {} \frac{1}{\gamma }\varDelta _6\sum _{\ell =1}^n (\varvec{\omega }_\ell - \bar{\varvec{\omega }})_{ab}q_\ell (i,j)_\ell + \frac{1}{\gamma }\varDelta _7\sum _{\ell =1}^n (\varvec{\omega }_\ell - \bar{\varvec{\omega }})_{ab}(i,j)_\ell ,\\ \lambda _{ab\gamma \gamma }= & {} \frac{\varDelta _8}{\gamma ^2}\sum _{\ell =1}^n(\varvec{\omega }_\ell - \bar{\varvec{\omega }})_{ab} \quad \ \text{ and }\\ \lambda _{ai}= & {} \lambda _{a\gamma }=\lambda _{i\gamma }= \lambda _{ai\gamma }=\lambda _{abi}=\lambda _{a\gamma \gamma }=0, \end{aligned}$$

and the derived cumulants required to calculate c and $c_m$ are given by:

$$\begin{aligned} (\lambda _{\gamma \gamma })_\gamma= & {} \frac{\varDelta _3}{\gamma ^3}, \ \ (\lambda _{ab \gamma })_\gamma = - \frac{\varDelta _1}{\gamma ^2}\sum _{\ell =1}^n (\varvec{\omega }_\ell - \bar{\varvec{\omega }})_{ab},\\ (\lambda _{ija})_b= & {} \frac{1}{2\gamma }(\delta _{(0,0,1,0,1)}+2\delta _{(0,1,0,0,0)})\sum _{\ell =1}^n (\varvec{\omega }_\ell - \bar{\varvec{\omega }})_{ab}q_l(i,j)_\ell \ \text{ and }\\ (\lambda _{ac})_d= & {} (\lambda _{ac})_{bd} = (\lambda _{acd})_b = \lambda _{iab}^{(c)} = (\lambda _{iab})_j=\lambda _{ai}^{(bj)}=\lambda _{a\gamma \gamma }^{(b)}=\lambda _{a\gamma \gamma }^{(\gamma )}=0. \end{aligned}$$

Appendix C

Here we obtain in detail the expression c for the Bartlett correction factor for the LR statistic. As defined in Sect. 3.1, the Bartlett correction factor is expressed by $1+c/k,$ where

$$\begin{aligned} c=\epsilon _{p,k}-\epsilon _p. \end{aligned}$$

From Lawley’s expansion (1956), we have that $E(LR)=k+\epsilon _{p,k}-\epsilon _p+O(n^{-2}),$ where

$$\begin{aligned} \epsilon _{p,k}=\sum _{{\varvec{\theta }}^*} (l_{rstu}-l_{rstuvw}), \end{aligned}$$

(6)

being

$$\begin{aligned} l_{rstu}= & {} \lambda ^{rs}\lambda ^{tu} \left\{ \frac{\lambda _{rstu}}{4} - \lambda _{rst}^{(u)}+\lambda _{rt}^{(su)} \right\} \ \text{ and }\\ l_{rstuvw}= & {} \lambda ^{rs}\lambda ^{tu}\lambda ^{vw} \left\{ \lambda _{rtv} \left( \frac{\lambda _{suw}}{6}-\lambda _{sw}^{(u)} \right) \right. \nonumber \\&+\left. \lambda _{rtu}\left( \frac{\lambda _{svw}}{4} - \lambda _{sw}^{(v)} +\lambda _{rt}^{(v)}\lambda _{sw}^{(u)} +\lambda _{rt}^{(u)}\lambda _{sw}^{(v)} \right) \right\} , \end{aligned}$$

where $-\lambda ^{rs}$ is the (r, s) element from the inverse of Fisher’s information matrix. The sum in (6) varies in all components of ${\varvec{\theta }}^*$. The expression $\epsilon _p$ is obtained from (6) considering that the sum varies across all nuisance parameters. Thus, we can write c as

$$\begin{aligned} c=\epsilon _k ({\varvec{\delta }}) +\epsilon _{p,k}({\varvec{\beta }},{\varvec{\delta }})+\epsilon _{p,k}({\varvec{\delta }},\gamma )+\epsilon _{p,q}({\varvec{\beta }},{\varvec{\delta }},\gamma ), \end{aligned}$$

where $\epsilon _k ({\varvec{\delta }})$ indicates that the sum (6) was performed on all components of ${\varvec{\delta }}$, analogously to other cases, considering the respective parameters. Replacing $\lambda '$s obtained in Appendix B in the $\epsilon '$s and after of extensive algebra, we have the constant c in (4).

Appendix D

Finally, here we obtain the constant $c_m$ for the Bartlett correction factor for the $LR_m$ statistic, considering Equation (5) of Ferrari et al. (2004). Thus, we have that $c_m$ can be expressed by

$$\begin{aligned} c_m= & {} \frac{1}{4} \lambda ^{ab}\lambda ^{cd}\lambda _{abcd} - \lambda ^{ab}\lambda ^{cd}(\lambda _{acd})_b + \lambda ^{ab}\lambda ^{cd}(\lambda _{ac})_{db} -\lambda ^{ij}\lambda ^{ab}(\lambda _{iab})_j \nonumber \\&-\, \lambda ^{\gamma \gamma }\lambda ^{ab}(\lambda _{ab\gamma })_\gamma - \left( \frac{1}{4} \lambda ^{ab}\lambda ^{cd}\lambda ^{ef} + \frac{1}{2} \lambda ^{ab}\lambda ^{cf}\lambda ^{de} -\frac{1}{3} \lambda ^{ab}\lambda ^{cf}\lambda ^{de} \right) \lambda _{acd}\lambda _{bef} \nonumber \\&+\,(\lambda ^{ab}\lambda ^{cd}\lambda ^{ef}+\lambda ^{ab}\lambda ^{cf}\lambda ^{de})\lambda _{acd}(\lambda _{be})_f - (\lambda ^{ab}\lambda ^{cd}\lambda ^{ef} +\lambda ^{ab}\lambda ^{cf}\lambda ^{de})(\lambda _{ac})_d(\lambda ^{be})_f \nonumber \\&-\, \left( \frac{1}{4} \lambda ^{ij}\lambda ^{ab}\lambda ^{cd} + \frac{1}{2} \lambda ^{ij}\lambda ^{ad}\lambda ^{bc} \right) \lambda _{iab}\lambda _{jcd} + (\lambda ^{ij}\lambda ^{ab}\lambda ^{kl})\lambda _{iab}(\lambda _{jk})_l \nonumber \\&- \left( \frac{1}{4}\lambda ^{\gamma \gamma }\lambda ^{ab}\lambda ^{cd} + \frac{1}{2} \lambda ^{\gamma \gamma }\lambda ^{ad}\lambda ^{bc} \right) \lambda _{ab \gamma }\lambda _{cd \gamma } + (\lambda ^{\gamma \gamma }\lambda ^{ab}\lambda ^{\gamma \gamma })\lambda _{ab\gamma }(\lambda _{\gamma \gamma })_{\gamma }. \end{aligned}$$

(7)

Replacing the cumulant $\lambda '$s obtained in Appendix B in Equation (7) and after intense algebra, we have

$$\begin{aligned} c_m= & {} \frac{1}{4} \varDelta _2 \sum _{\ell =1}^n (\varvec{\omega }_\ell - \bar{\varvec{\omega }})_a \lambda ^{ab}(\varvec{\omega }_\ell - \bar{\varvec{\omega }})_b(\varvec{\omega }_\ell - \bar{\varvec{\omega }})_c \lambda ^{cd}(\varvec{\omega }_\ell - \bar{\varvec{\omega }})_d\\&- \frac{1}{4} \varDelta _4 \varDelta _1^2 \sum _{\ell =1}^n (\varvec{\omega }_\ell - \bar{\varvec{\omega }})_a \lambda ^{ab} (\varvec{\omega }_\ell - \bar{\varvec{\omega }})_b (\varvec{\omega }_\ell - \bar{\varvec{\omega }})_c \lambda ^{cd}(\varvec{\omega }_\ell - \bar{\varvec{\omega }})_d\\&- \frac{1}{4}\varDelta ^2_1 \sum _{\ell =1}^n \sum _{s=1}^n (\varvec{\omega }_\ell - \bar{\varvec{\omega }})_a\lambda ^{ab}(\varvec{\omega }_s - \bar{\varvec{\omega }})_a (\varvec{\omega }_s - \bar{\varvec{\omega }})_c \lambda ^{cd}(\varvec{\omega }_\ell - \bar{\varvec{\omega }})_d (\varvec{\omega }_s - \bar{\varvec{\omega }})_e \lambda ^{ef}(\varvec{\omega }_s - \bar{\varvec{\omega }})_f\\&- \frac{1}{6} \varDelta _1^2 \sum _{\ell =1}^n\sum _{s=1}^n (\varvec{\omega }_\ell - \bar{\varvec{\omega }})_a \lambda ^{ab}(\varvec{\omega }_s - \bar{\varvec{\omega }})_b (\varvec{\omega }_\ell - \bar{\varvec{\omega }})_c \lambda ^{cf}(\varvec{\omega }_s - \bar{\varvec{\omega }})_f (\varvec{\omega }_\ell - \bar{\varvec{\omega }})_d \lambda ^{de}(\varvec{\omega }_s - \bar{\varvec{\omega }})_e\\&+ \varDelta _1 \varDelta _4 \sum _{s=1}^n (\varvec{\omega }_s - \bar{\varvec{\omega }})_a\lambda ^{ab}(\varvec{\omega }_s - \bar{\varvec{\omega }})_b + \varDelta _1 \varDelta _3 \varDelta _4^2 \sum _{\ell =1}^n (\varvec{\omega }_\ell - \bar{\varvec{\omega }})_a \lambda ^{ab} (\varvec{\omega }_\ell - \bar{\varvec{\omega }})_b\\&- \frac{1}{2} \varDelta _1^2 \varDelta _4 \sum _{\ell =1}^n\sum _{s=1}^n (\varvec{\omega }_s - \bar{\varvec{\omega }})_a \lambda ^{ad}(\varvec{\omega }_s - \bar{\varvec{\omega }})_d(\varvec{\omega }_\ell - \bar{\varvec{\omega }})_b\lambda ^{bc}(\varvec{\omega }_s - \bar{\varvec{\omega }})_c, \end{aligned}$$

where $(\varvec{\omega }_\ell - \bar{\varvec{\omega }})_i = \omega _{\ell i}-\bar{\omega }_i,$ with $i=a,b,c,d.$ In matrix notation, we have

$$\begin{aligned} c_m= & {} \frac{1}{4} \varDelta _2 \text{ tr } (\varvec{H_d^2}) - \frac{1}{4} \varDelta _1^2 \varDelta _4 [\text{ tr }(\varvec{H_d})]^2 + \frac{1}{4} \varDelta ^2_1 \varvec{\iota }^\top \varvec{H_d H H_d} \varvec{\iota } + \frac{1}{6} \varDelta ^2_1 \varvec{\iota }^\top \varvec{H^{(3)}}\varvec{\iota } \nonumber \\&-\, \varDelta _1 \varDelta _4 \text{ tr }(\varvec{H_d}) - \varDelta _1 \varDelta _3 \varDelta _4^2 \text{ tr }(\varvec{H_d}) - \frac{1}{2} \varDelta ^2_1 \varDelta _4 \varvec{\iota }^\top \varvec{H^{(2)}}\varvec{\iota }. \end{aligned}$$