Bartlett and Bartlett-type corrections in heteroscedastic symmetric nonlinear regression models

Abstract

This paper provides general expressions for Bartlett and Bartlett-type correction factors for the likelihood ratio and gradient statistics to test the dispersion parameter vector in heteroscedastic symmetric nonlinear models. This class of regression models is potentially useful to model data containing outlying observations. Furthermore, we develop Monte Carlo simulations to compare size and power of the proposed corrected tests to the original likelihood ratio, score, gradient tests, corrected score test, and bootstrap tests. Our simulation results favor the score and gradient corrected tests as well as the bootstrap tests. We also present an empirical application.

Key words
Bartlett corrections; Bartlett-type corrections; bootstrap; gradient test; large-sample test statistics

INTRODUCTION

The symmetric class of models has received increasing attention in the literature. Besides the normal distribution, the symmetric family covers both light and heavy tailed distributions including the Cauchy, Student$-t$, generalized Student$-t$ and power exponential, among others. The symmetric models provide a very useful extension of the normal model, since using a heavy tailed distribution for the error component reduces the influence of extreme observations and enables carrying out a more robust statistical analysis (Lange et al. 1989). An extensive range of practical applications considering symmetric distributions can be found in various fields, such as engineering, biology and economics, among others. Symmetric regression models have been the subject of several studies (e.g., Cysneiros et al. 2010, Lemonte 2012, Lin et al., 2009, Maior & Cysneiros 2018).

Constant dispersion is often a standard assumption when symmetric data are fitted. However, in many practical situations this condition is not satisfied, requiring verification, since the inference strategies change when one observes variable dispersion of the observations. The likelihood ratio (LR), Wald and score are the large-sample tests commonly used for this purpose. The gradient test proposed by Terrel 2002, whose statistic shares the same first-order asymptotic properties with the LR, Wald and score statistics (Lemonte & Ferrari 2012a), has been the subject of many studies in the past few years (e.g., Lemonte 2011, 2013, Lemonte & Ferrari 2012b, Medeiros & Ferrari 2017). Compared to the Wald and score statistics, the gradient statistic does not depend on the information matrix, either expected or observed, and is also simpler to compute.

The four statistics for testing hypothesis in regression models have a null asymptotic ${\chi }_q^2$ distribution, where $q$ is the difference between the dimensions of the parameter space under the two hypotheses being tested, up to an error order of $n^{-1}$. Relying on inference in tests based on such statistics has less justification when dealing with small and moderately sized samples. A strategy to improve the ${\chi }^2$ approximation for the exact distributions of the LR, score and gradient statistics is to multiply them by a correction factor. For the LR statistic, Bartlett 1937 proposed a correction factor known as the Bartlett correction, which was put into a general framework later by Lawley 1956, while for the score statistic, Cordeiro & Ferrari 1991 proposed a Bartlett-type correction. Based on the results of Cordeiro & Ferrari 1991, Bartlett-type correction for the Wald statistic was derived by Cribari-Neto & Ferrari 1995 in the heteroscedastic linear models. Furthermore, Santos & Cordeiro 1999 obtained corrected Wald test statistics for one-parameter exponential family models. Gradient statistic was proposed in a general framework by Vargas et al. 2013. The corrected versions of the test statistics have the same ${\chi }_q^2$ null distribution with approximation error of order $n^{-2}.$ Cordeiro & Cribari-Neto 2014 provided additional details on Bartlett and Bartlett-type corrections. Improved tests have been discussed in some recent articles, in particular Lemonte et al. 2012, Bayer & Cribari-Neto 2013, Vargas et al. 2014 and Medeiros et al. 2017.

Considering the class of heteroscedastic symmetric nonlinear models (HSNLM) proposed by Cysneiros et al. 2010, Cysneiros 2011 derived a Bartlett-type correction for the score statistic, and carried out a numerical study to test the regression coefficients in the dispersion parameter. In this paper, our main goal is to derive Bartlett and Bartlett-type corrections to improve inference of the dispersion parameter based on the LR and gradient statistics, respectively, for the class of HSNLM considering the parameterization presented in Cysneiros et al. 2010. In other words, we deal only with one aspect of the high-order asymptotic theory which aims to obtain adjustments of test statistics. Furthermore, we consider a partition of the dispersion parameter, which is an advantage since in some cases we are not interested in making inferences of all parameters of the model. One of the main results presented in this paper, the Bartlett correction factor for the $LR$ statistic, is not the same as presented in Araújo & L. C. Montenegro 2020, since the aforementioned dealt with two aspects of the high-order asymptotic theory: first we obtained the adjustment for the profiled likelihood (first aspect) and then we obtained the adjustment for the test statistic based on the profiled likelihood (second aspect).

In order to achieve our aim, we adopt a regression structure to model the dispersion parameter vector so that under the null hypothesis the dispersion is constant. In other words, the null hypothesis delivers the symmetric nonlinear regression model. Our results provide a new class of tests that can be used in practical applications, mainly those involving small datasets.

We perform a Monte Carlo simulation study to evaluate the performance of the proposed tests. For comparison purposes, besides the proposed tests and the usual score and gradient tests, we also consider in the Monte Carlo experiment the improved score test (Cysneiros 2011), the modified score tests proposed by Kakisawa 1996 and Cordeiro et al. 1998 and bootstrap-based tests. Our simulation results show that the improved gradient test proposed in this paper is an interesting alternative to the classic large-sample tests, delivering accurate inferences, mainly when dealing with small datasets. We are unaware of any simulation study in the literature drawing a comparison between the performance of the proposed tests in the class of models considered, so this paper fills this gap.

The remainder of this paper is organized as follows. In the next Section we present the class of HSNLM, explaining inferential aspects. After that, we derive Bartlett and Bartlett-type corrections to improve the LR and gradient tests for investigating varying dispersion in the model class of interest. We also conduct a Monte Carlo study to evaluate and compare the performance of the proposed tests. An application to real data is presented. Some concluding remarks are given in the last Section.

MODEL SPECIFICATION

Let $y$ be a random variable with symmetric distribution. Its density function is given by

$$\pi (y;\mu ,\varphi )=\frac{1}{\sqrt{\varphi }}g\left(u\right),y,\mu \in ℝ,\varphi >0,$$ (1)

where $\mu$ is a location parameter, $\varphi$ is a dispersion parameter, $u={(y-\mu )}^2/\varphi$, $g:ℝ\to [0,\infty )$ is the density generator (see, for example, Fang et al. 1990). We then denote $y\sim S(\mu ,\varphi ,g).$ Cysneiros et al. 2005 presented the density generator function $g(\cdot )$ for some symmetric distributions. In some symmetric distributions the density generator function, $g(\cdot ),$ depends on an additional shape parameter, say $\nu$, which controls the kurtosis. This parameter can be estimated from the data or can be kept fixed. Villegas et al. 2013 presented aspects of the symmetric distributions relating the issue of robustness to the estimation of the parameter $\nu$. Thus, in this work we keep $\nu$ fixed.

Assume $y_1,\dots ,y_n$ are independent random variables where each $y_{\ell }$ has a symmetric distribution (1) with location parameter ${\mu }_{\ell }$ and dispersion parameter ${\varphi }_{\ell }.$ Also, consider that the components of $𝛍={({\mu }_1,\dots ,{\mu }_n)}^{\top }$ and $𝛟={({\varphi }_1,\dots ,{\varphi }_n)}^{\top }$ vary across observations through nonlinear regression structures. The heteroscedastic symmetric nonlinear regression model $y_{\ell }\sim S({\mu }_{\ell },{\varphi }_{\ell },g),\ell =1,\dots ,n,$ proposed by Cysneiros et al. 2010 is defined by (1) and by the systematic components for the mean vector response $𝛍$ and the dispersion parameter vector $𝛟$ described as follows:

$${\mu }_{\ell }=f({𝐱}_{\ell };𝛃)\mathrm{\text{and}}{\varphi }_{\ell }=h\left({\tau }_{\ell }\right),$$

where $f(\cdot ;\cdot )$ is a possibly nonlinear function in the second argument which is continuous and differentiable in $𝛃,$ where $𝛃={({\beta }_1,\dots ,{\beta }_p)}^{\top }$ ($p<n$ and $𝛃\in {ℝ}^p$) is a vector of unknown parameters to be estimated, and ${𝐱}_{\ell }={(x_{\ell 1},\dots ,x_{\ell m})}^{\top }$ is an $m\times 1$ vector of known explanatory variables associated with the $\ell$th observation. The matrix of derivatives of $𝛍$ with respect to $𝛃,$ $\widetilde{𝐗}=\partial 𝛍/\partial 𝛃,$ is assumed to have full rank for all $𝛃$. Moreover, $h(\cdot )$ is a known continuous bijective function and differentiable in $𝛅.$ Furthermore, ${\tau }_{\ell }={{𝛚}_{\ell }}^{\top }𝛅$ is a linear predictor where ${𝛚}_{\ell }={(1,{\omega }_{\ell 1},\dots ,{\omega }_{\ell k-1})}^{\top }$ is a vector of explanatory variables whose components are not necessarily different from ${𝐱}_{\ell }$ and $𝛅={({\delta }_0,\dots ,{\delta }_{k-1})}^{\top }$ $(𝛅\in {ℝ}^k)$ is a vector of unknown parameters.

It is further assumed that if a value ${𝛅}_0$ of $𝛅$ exists, then $h\left({{𝛚}_{\ell }}^{\top }{𝛅}_0\right)=1$ for all $\ell$, so the terms $y_{\ell }\prime s$ have constant dispersion if $𝛅={𝛅}_0.$ The function $h(\cdot )$ should be positively valued, and a possible choice is $h(\cdot )=exp(\cdot ),$ which is adopted in several papers (e.g., Barroso & Cordeiro 2005, Cook & Weisberg, 1983, Simonoff & Tsai 1994, Verbyla 1993). Furthermore, considering $h\left({\tau }_{\ell }\right)=exp\left({\tau }_{\ell }\right)=exp\left({{𝛚}_{\ell }}^{\top }𝛅\right),$ it is not necessary impose any restriction on the components of ${𝛚}_{\ell }$ Cook & Weisberg 1983, Lin et al. 2009. It is important to note that the meaning of heteroscedasticity we use in this work refers to varying dispersion, that is, when ${\varphi }_1={\varphi }_2=\dots ={\varphi }_n$ we have a homoscedastic model; without this we have a heteroscedastic model.

Let $l\left(𝛉\right)$ denote the total log-likelihood function for the parameter of vector $𝛉={({𝛃}^{\top },{𝛅}^{\top })}^{\top }$ given $y_1,\dots ,y_n.$ We have $l\left(𝛉\right)=-\frac{1}{2}{\sum }_{\ell =1}^{n}log\left({\varphi }_{\ell }\right)+{\sum }_{\ell =1}^{n}t\left(z_{\ell }\right),$ with $t\left(z_{\ell }\right)=logg\left(z_{\ell }^2\right)$ and $z_{\ell }=\sqrt{u_{\ell }}=\frac{(y_{\ell }-{\mu }_{\ell })}{\sqrt{{\varphi }_{\ell }}}.$ We assume that the function $l\left(𝛉\right)$ is regular (Cox & Hinkley 1974, Chap 9) with respect to all $𝛃$ and $𝛅$ derivatives to the fourth order. The total Fisher information matrix for $𝛉$ has a block diagonal structure, i.e., ${𝐊}_{𝛉}=\mathrm{\text{diag}}\{{𝐊}_{𝛃},{𝐊}_{𝛅}\},$ where ${𝐊}_{𝛃}=-{\alpha }_{2,0}\widetilde{𝐗}{𝚲}^{-1}\widetilde{𝐗}$ and ${𝐊}_{𝛅}={𝐖}^{\top }𝐕𝐖,$ with $𝚲=\mathrm{\text{diag}}\{1/{\varphi }_1,\dots ,1/{\varphi }_n\},$ $𝐖=\partial 𝛕/\partial 𝛅$ and $𝐕=\mathrm{\text{diag}}\{v_1,\dots ,v_n\},$ such that $v_{\ell }=\left((1-{\alpha }_{2,0}){h\prime }_{\ell }^2\right)/4{\varphi }_{\ell }^2,$ where $h\prime =\partial {\varphi }_{\ell }/\partial {\tau }_{\ell }$ and ${\alpha }_{r,s}=E\{t{\left(z_{\ell }\right)}^{\left(r\right)}{z}_{\ell }^s\}$ for $r,s\in \{1,2,3,4\}$ and $t{\left(z_{\ell }\right)}^{\left(k\right)}={\partial }^{k}t\left(z_{\ell }\right)/\partial z_{\ell }^k,$ for $k=1,2,3,4$ and $\ell =1,\dots ,n.$ For some symmetric distributions, the quantities ${\alpha }_{r,s}$ are given in Uribe-Opazo et al. 2008. The parameters $𝛃$ and $𝛅$ are globally orthogonal, so their respective maximum likelihood estimators (MLEs), $\widehat{𝛃}$ and $\widehat{𝛅},$ are asymptotically independent. In order to obtain the MLEs $\widehat{𝛃}$ and $\widehat{𝛅}$ iteratively, the scoring method can be applied. This procedure is described in detail in Cysneiros et al. 2010.

Our interest is to test heteroscedasticity in symmetric nonlinear regression models. The null and alternative hypotheses considered are, respectively, $H_0:{𝛅}_1={𝛅}_1^{\left(0\right)}$ and $H_1:{𝛅}_1\ne {𝛅}_1^{\left(0\right)},$ where $𝛅$ is partitioned as $𝛅={({\delta }_0,{{𝛅}_1}^{\top })}^{\top },$ with ${\delta }_0$ a scalar and ${𝛅}_1={({\delta }_1,\dots ,{\delta }_{k-1})}^{\top }.$ Here, ${𝛅}_1^{\left(0\right)}$ is a fixed column vector of dimension $k-1$ such that $h\left({𝛚}_{\ell }^{\top }{𝛅}_1^{\left(0\right)}\right)=1$ and ${\delta }_0$ and $𝛃$ are considered nuisance parameters. Actually, we are testing the dispersion parameters in the HSNLM, considering that under the null hypothesis this model boils down to the symmetric nonlinear regression model. The partition previously considered for $𝛅$ induces the corresponding partitions: $𝐖=({𝐖}_0,{𝐖}_1),$ where ${𝐖}_0$ is an $n\times 1$ vector with all ones and ${𝐖}_1=\partial 𝛕/\partial {\delta }_1,$

$$K_{\delta }{K}_{\delta }=\left[\begin{array}{cc}{K}_{{\delta }_0{\delta }_0}& K{K}_{{\delta }_0{\delta }_1{\delta }_0{\delta }_1}\\ K{K}_{{\delta }_1{\delta }_0{\delta }_1{\delta }_0}& K{K}_{{\delta }_1{\delta }_1{\delta }_1{\delta }_1}\end{array}\right],$$

with $K_{{\delta }_0{\delta }_0}={𝐖}_0^{\top }𝐕{𝐖}_0,$ ${𝐊}_{{𝛅}_0{𝛅}_1}^{\top }={𝐊}_{{𝛅}_1{𝛅}_0}={𝐖}_1^{\top }𝐕{𝐖}_0$ e ${𝐊}_{{𝛅}_1{𝛅}_1}={𝐖}_1^{\top }𝐕{𝐖}_1.$ The likelihood ratio ($S_{LR}$), score ($S_r$) and gradient ($S_g$) statistics for testing $H_0$ can be expressed, respectively, as

$$\begin{array}{ccc}\hfill S_{LR}& =\hfill & \hfill 2\{l({\widehat{𝛅}}_1,{\hat{\delta }}_0,\widehat{𝛃})-l({𝛅}_1^{\left(0\right)},{\tilde{\delta }}_0,\widetilde{𝛃})\},\\ \hfill S_r& =\hfill & \hfill \frac{1}{4}{\left[{𝐖}_1\widetilde{𝚲}(\widetilde{𝐒}\widetilde{{𝐅}_1}\widetilde{𝐮}-\widetilde{{𝐅}_1}𝛊)\right]}^{\top }{\left({\widetilde{𝐑}}^{\top }\widetilde{𝐕}\widetilde{𝐑}\right)}^{-1}\left[{𝐖}_1\widetilde{𝚲}(\widetilde{𝐒}\widetilde{{𝐅}_1}\widetilde{𝐮}-\widetilde{{𝐅}_1}𝛊)\right]\mathrm{\text{and}}\\ \hfill S_g& =\hfill & \hfill \frac{1}{2}{\left[{𝐖}_1\widetilde{𝚲}(\widetilde{𝐒}\widetilde{{𝐅}_1}\widetilde{𝐮}-\widetilde{{𝐅}_1}𝛊)\right]}^{\top }({\widehat{𝛅}}_1-{𝛅}_1^{\left(0\right)}),\end{array}$$

where $(\widehat{𝛃},\widehat{{\delta }_0},\widehat{{𝛅}_1})$ and $(\widetilde{𝛃},\widetilde{{\delta }_0},{𝛅}_1^{\left(0\right)})$ are, respectively, the unrestricted and restricted (under $H_0$) MLEs of $(𝛃,{\delta }_0,{𝛅}_1),$ $𝛊$ is an $n\times 1$ vector of ones and $𝐑={𝐖}_1-{𝐖}_0𝐂,$ with $𝐂={\left({{𝐖}_0}^{\top }𝐕{𝐖}_0\right)}^{-1}\left({{𝐖}_0}^{-1}𝐕{𝐖}_1\right).$ Under the null hypothesis, these statistics have an asymptotic ${\chi }_{k-1}^2$ distribution up to an error of order $n^{-1}.$

IMPROVED TEST INFERENCE

In order to obtain a more accurate inference when dealing with small and moderately sized samples, some procedures based on second-order asymptotic theory have been developed in the literature. For the HSNLM, a Bartlett-type correction factor for the score statistic was derived by Cysneiros 2011. To provide another improved statistics to test varying dispersion in the HSNLM class, we derive Bartlett and Bartlett-type correction factors for the LR and gradient statistics, respectively, considering the general procedures developed by Lawley 1956 and Vargas et al. 2014. The Bartlett and Bartlett-type correction factors are very general and need to be obtained for every model of interest, since they involve complex functions of the moments of log-likelihood derivatives up to the fourth order. Details about the derivation of the Bartlett and Bartlett-type correction factors are given in the supplementary material.

To test $H_0:{𝛅}_1={𝛅}_1^{\left(0\right)}$ in HSNLM considering $h\left({{𝛚}_{𝐥}}^{\top }𝛅\right)=exp\left({{𝛚}_{𝐥}}^{\top }𝛅\right),$ i.e., the case of heteroscedasticity with multiplicative effects, the Bartlett-corrected $LR$ statistic is given by

$$S_{{LR}^{*}}=\frac{S_{LR}}{1+c/(k-1)},$$

where $c=ϵ\left(𝛅\right)+ϵ(𝛃,𝛅)-ϵ\left({\delta }_0\right)-ϵ(𝛃,{\delta }_0),$

$$\begin{array}{ccc}\hfill ϵ\left(𝛅\right)& =\hfill & \hfill N_{1}tr\{{{𝐙}_{{𝛅}_{𝐝}}}^{\left(2\right)}\}+N_2{𝛊}^{\top }{{𝐙}_{𝛅}}^{\left(3\right)}𝛊+N_3{𝛊}^{\top }𝚲{{𝐙}_{𝛅}}^{\left(3\right)}𝛊+N_4{𝛊}^{\top }{𝚲{𝐙}_{𝛅}}^{\left(3\right)}𝚲𝛊\\ \hfill & +\hfill & \hfill N_5{𝛊}^{\top }{{𝐙}_{{𝛅}_{𝐝}}}^{\left(2\right)}{𝐙}_{𝛅}𝛊+N_6{𝛊}^{\top }{{𝐙}_{{𝛅}_{𝐝}}}^{\left(2\right)}{𝐙}_{𝛅}𝚲𝛊+(N_7+N_8){𝛊}^{\top }𝚲{{𝐙}_{{𝛅}_{𝐝}}}^{\left(2\right)}{𝐙}_{𝛅}𝛊,\\ \hfill \\ \hfill ϵ(𝛃,𝛅)& =\hfill & \hfill -N_{15}tr\{𝚲{𝐙}_{{𝛃}_{𝐝}}{𝐙}_{{𝛅}_{𝐝}}\}-(N_{10}+N_{12}){𝛊}^{\top }𝚲{𝐙}_{{𝛃}_{𝐝}}{𝐙}_{𝛅}{𝐙}_{{𝛅}_{𝐝}}𝛊\\ \hfill & +\hfill & \hfill N_{14}{𝛊}^{\top }𝚲{𝐙}_{{𝛃}_{𝐝}}{𝐙}_{𝛅}{𝐙}_{{𝛃}_{𝐝}}𝚲𝛊-(N_{11}+N_{13}){𝛊}^{\top }𝚲{𝐙}_{{𝛃}_{𝐝}}{𝐙}_{𝛅}{𝐙}_{{𝛅}_{𝐝}}𝚲𝛊\\ \hfill & +\hfill & \hfill N_9{𝛊}^{\top }𝚲{𝐙}_{𝛅}{{𝐙}_{𝛃}}^{\left(2\right)}𝚲𝛊,\\ \hfill \\ \hfill ϵ\left({\delta }_0\right)& =\hfill & \hfill N_{1}tr\{{{𝐙}_{{𝛅}_{0𝐝}}}^{\left(2\right)}\}+N_2{𝛊}^{\top }{{𝐙}_{{𝛅}_0}}^{\left(3\right)}𝛊+N_3{𝛊}^{\top }𝚲{{𝐙}_{{𝛅}_0}}^{\left(3\right)}𝛊+N_4{𝛊}^{\top }{𝚲{𝐙}_{{𝛅}_0}}^{\left(3\right)}𝚲𝛊\\ \hfill & +\hfill & \hfill N_5{𝛊}^{\top }{{𝐙}_{{𝛅}_{0𝐝}}}^{\left(2\right)}{𝐙}_{{𝛅}_0}𝛊+N_6{𝛊}^{\top }{{𝐙}_{{𝛅}_{0𝐝}}}^{\left(2\right)}{𝐙}_{{𝛅}_0}𝚲𝛊\\ \hfill & +\hfill & \hfill (N_7+N_8){𝛊}^{\top }𝚲{{𝐙}_{{𝛅}_{0𝐝}}}^{\left(2\right)}{𝐙}_{{𝛅}_0}𝛊\mathrm{\text{and}}\\ \hfill \\ \hfill ϵ(𝛃,{\delta }_0)& =\hfill & \hfill -N_{15}tr\{𝚲{𝐙}_{{𝛃}_{𝐝}}{𝐙}_{{𝛅}_{0𝐝}}\}-(N_{10}+N_{12}){𝛊}^{\top }𝚲{𝐙}_{{𝛃}_{𝐝}}{𝐙}_{{𝛅}_0}{𝐙}_{{𝛅}_{0𝐝}}𝛊\\ \hfill & +\hfill & \hfill N_{14}{𝛊}^{\top }𝚲{𝐙}_{{𝛃}_{𝐝}}{𝐙}_{{𝛅}_0}{𝐙}_{{𝛃}_{𝐝}}𝚲𝛊-(N_{11}+N_{13}){𝛊}^{\top }𝚲{𝐙}_{{𝛃}_{𝐝}}{𝐙}_{{𝛅}_0}{𝐙}_{{𝛅}_{0𝐝}}𝚲𝛊\\ \hfill & +\hfill & \hfill N_9{𝛊}^{\top }𝚲{𝐙}_{𝛅}{{𝐙}_{𝛃}}^{\left(2\right)}𝚲𝛊,\end{array}$$ (2)

where ${𝐙}_{𝛃}=\widetilde{𝐗}{\left({\widetilde{𝐗}}^{\top }𝚲\widetilde{𝐗}\right)}^{-1}{\widetilde{𝐗}}^{\top },$ ${𝐙}_{𝛅}=𝐖{\left({𝐖}^{\top }𝐕𝐖\right)}^{-1}{𝐖}^{\top },$ ${𝐙}_{{𝛅}_0}={𝐖}_0$ ${\left({{𝐖}_0}^{\top }𝐕{𝐖}_0\right)}^{-1}{{𝐖}_0}^{\top },$ ${{𝐙}_{𝛃}}^{\left(2\right)}={𝐙}_{𝛃}\odot {𝐙}_{𝛃},$ ${{𝐙}_{𝛅}}^{\left(2\right)}={𝐙}_{𝛅}\odot {𝐙}_{𝛅},$ ${{𝐙}_{{𝛅}_0}}^{\left(2\right)}={𝐙}_{{𝛅}_0}\odot {𝐙}_{{𝛅}_0},$ ${{𝐙}_{𝛅}}^{\left(3\right)}={{𝐙}_{𝛅}}^{\left(2\right)}\odot {𝐙}_{𝛅},$ ${{𝐙}_{{𝛅}_0}}^{\left(3\right)}={{𝐙}_{{𝛅}_0}}^{\left(2\right)}\odot {𝐙}_{{𝛅}_0},$ $\odot$ denotes the Hadamard (elementwise) product of matrices, and ${(\cdot )}_d$ indicates that the off-diagonal elements of the matrix are set equal to zero. For the sake of brevity, the elements $N_i,i=1,\dots ,15$ are presented in the supplementary material.

The improved gradient statistic is obtained by multiplying its original statistic by a polynomial in the original statistic itself. The corrected gradient statistic continues to have a chi-squared distribution under the null hypothesis but its asymptotic approximation error decreases from $n^{-1}$ to $n^{-2},$ providing a more accurate inference. To test $H_0:{𝛅}_1={𝛅}_1^{\left(0\right)}$ in HSNLM when $h\left({{𝛚}_{\ell }}^{\top }𝛅\right)=exp\left({{𝛚}_{\ell }}^{\top }𝛅\right),$ the corrected gradient statistic is given by

$$S_{g^{*}}=S_g\{1-(c_g+b_g{S}_g+a_g{S}_g^2)\},$$

where $a_g=\frac{A_3^g}{12(k-1)((k-1)+2)((k-1)+4)},b_g=\frac{A_2^g-2A_3^g}{12(k-1)((k-1)+2)},$ $$ $c_g=\frac{A_1^g-A_2^g+A_3^g}{12(k-1)},$ with

$$\begin{array}{ccc}\hfill A_1^g& =\hfill & \hfill 12{\alpha }_{2,0}{Q}_2{𝛊}^{\top }𝚲{{𝐙}_{𝛃}}^{\left(2\right)}\odot ({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})𝚲𝛊+3Q_2^2{𝛊}^{\top }𝚲{𝐙}_{{𝛃}_{𝐝}}({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0}){𝐙}_{{𝛃}_{𝐝}}𝚲𝛊\\ \hfill & +\hfill & \hfill 6Q_2^2{𝛊}^{\top }𝚲({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})\odot {{𝐙}_{𝛃}}^{\left(2\right)}𝚲𝛊+3Q_1{Q}_2{𝛊}^{\top }𝚲{𝐙}_{{𝛃}_{𝐝}}({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0}){𝐙}_{{𝛅}_{0𝐝}}𝛊\\ \hfill & +\hfill & \hfill 3Q_1{Q}_2{𝛊}^{\top }{𝐙}_{{𝛅}_{0𝐝}}({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0}){𝐙}_{{𝛃}_{𝐝}}𝚲𝛊+3Q_1{Q}_2{𝛊}^{\top }{({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})}_d({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0}){𝐙}_{{𝛃}_{𝐝}}𝚲𝛊\\ \hfill & +\hfill & \hfill 6Q_1{Q}_2{𝛊}^{\top }{({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})}_d{𝐙}_{{𝛅}_0}{𝐙}_{{𝛃}_{𝐝}}𝚲𝛊+3Q_1^2{𝛊}^{\top }{({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})}_d({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0}){𝐙}_{{𝛅}_{0𝐝}}𝛊\\ \hfill & +\hfill & \hfill 6Q_1^2{𝛊}^{\top }{({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})}_d{𝐙}_{{𝛅}_0}{𝐙}_{{𝛅}_{0𝐝}}𝛊+3Q_1^2{𝛊}^{\top }{𝐙}_{{𝛅}_{0𝐝}}({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0}){𝐙}_{{𝛅}_{0𝐝}}𝛊\\ \hfill & +\hfill & \hfill 6Q_1^2{𝛊}^{\top }({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})\odot {{𝐙}_{{𝛅}_0}}^{\left(2\right)}𝛊+6Q_{3}tr\{{𝐙}_{{𝛅}_{0𝐝}}{({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})}_d\}\\ \hfill & -\hfill & \hfill 12Q_{5}tr\{𝚲{({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})}_d{𝐙}_{{𝛃}_{𝐝}}\}+6Q_{4}tr\{𝚲{({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})}_d{𝐙}_{{𝛃}_{𝐝}}\},\\ \hfill \\ \hfill A_2^g& =\hfill & \hfill -3Q_1{Q}_3{𝛊}^{\top }{({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})}_d({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0}){𝐙}_{{𝛃}_{𝐝}}𝚲𝛊\\ \hfill & -\hfill & \hfill 3Q_1^2{𝛊}^{\top }{({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})}_d({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0}){𝐙}_{{𝛅}_{0𝐝}}𝛊\\ \hfill & -\hfill & \hfill 3Q_1^2{𝛊}^{\top }{({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})}_d{𝐙}_{{𝛅}_0}{({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})}_d𝛊-6Q_1^2𝛊{({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})}^{\left(2\right)}\odot {𝐙}_{{𝛅}_0}𝛊\\ \hfill & -\hfill & \hfill \frac{9}{4}{Q}_1^2{𝛊}^{\top }{({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})}_d({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0}){({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})}_d𝛊-\frac{3}{2}{Q}_1^2{𝛊}^{\top }{({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})}^{\left(3\right)}𝛊\\ \hfill & -\hfill & \hfill 3Q_{3}tr\{{({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})}_d^{\left(2\right)}\}\mathrm{\text{and}}\\ \hfill \\ \hfill A_3^g& =\hfill & \hfill \frac{3}{4}{Q}_1^2{𝛊}^{\top }{({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})}_d({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0}){({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})}_d𝛊+\frac{1}{2}{Q}_1^2{𝛊}^{\top }{({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})}^{\left(3\right)}𝛊,\end{array}$$ (3)

where ${({𝐙}_{𝛅}-{𝐙}_{{𝛅}_0})}_d={𝐙}_{{𝛅}_{𝐝}}-{𝐙}_{{𝛅}_{0𝐝}}$ and $Q_i,i=1,\dots ,5$ are given in the supplementary material.

The correction factors that improve the LR and gradient statistics are not easy to interpret, although they involve only simple matrix operations and can be easily implemented in any programming environment that performs linear algebra operations, such as MAPLE, Ox, R, etc. Also, they depend on the distribution in (1) only through the $\alpha$’s and also depend on the number of nuisance parameters, the dimension of the hypothesis tested and the matrices $𝐗$ and $𝐖$ of covariates. Finally, all unknown parameters in the correction factors are replaced by their restricted MLEs.

NUMERICAL EVIDENCE

The simulation experiments are based on the heteroscedastic symmetric nonlinear regression model

$$y_{\ell }={\beta }_0+exp\{{\beta }_1{x}_{\ell 1}\}+\sum _{s=2}^p{\beta }_s{x}_{s\ell }+{ϵ}_{\ell },\ell =1,\dots ,n,$$

where ${ϵ}_{\ell }\sim S(0,exp\{{{𝛚}_{\ell }}^{\top }𝛅\},g).$ The response variable was generated assuming that ${\beta }_0=\dots ={\beta }_{p-1}=1,$ ${\delta }_0=0.1,{\delta }_2=0.3,{\delta }_3=0.5$ and ${\delta }_4={\delta }_5={\delta }_6=1$ and different values of $p$ and $k$ were considered. The covariates $x_1,\dots ,x_{p-1}$ and ${\omega }_1,\dots ,{\omega }_k$ were generated as random samples of the $U(0,1)$ distribution and were kept fixed throughout the simulations. The null hypothesis tested is $H_0:{\delta }_1=\dots ={\delta }_{k-1}=0,$ i.e., $exp\{{{𝛚}_{\ell }}^{\top }𝛅\}=exp\{{\delta }_0\},$ so that under $H_0$ we have constant dispersion. We report the null rejection rates of the tests based on the following statistics: the original likelihood ratio, score and gradient statistics ($S_{LR},S_r,S_g$), their respective Bartlett and Bartlett-type corrected versions ($S_{{LR}^{*}},S_{r^{*}},S_{g^{*}}$) and the monotonic versions of the corrected score statistic proposed by Kakisawa 1996 and Cordeiro et al. 1998 ($S_{{r_1}^{*}},S_{{r_2}^{*}}),$ respectively. The bootstrap versions of the of the LR, score and gradient tests, with $S_{LR}^{boot}$, $S_r^{boot}$ and $S_g^{boot}$ being their respective test statistics, were also included. For the bootstrap-based tests, we followed the steps described as in Araújo & L. C. Montenegro 2020.

The simulation results are based on the Student-$t$ (with $\nu =5$) and power exponential (with $\nu =0.3$) models. The following nominal levels and sample sizes were considered: $\alpha =1\%,5\%$ and $10\%,$ and $n=20,30,\mathrm{\text{and}}40,$ respectively. All results were obtained using 10,000 Monte Carlo replications and 500 bootstrap samples. The bootstrap sampling was performed parametrically under the null hypothesis. The simulations were carried out using the Ox matrix programming language (Doornik 2006). All entries are percentages.

Tables I and II show results for different sample sizes while keeping fixed (varying) the number of nuisance (interest) parameters. The results clearly show that the LR test is notably liberal (i.e., it over-rejects the null hypotheses), especially when the numbers of interest parameters and nuisance parameters increase (the results with varying number of nuisance parameters are not shown to save space). It also can be noted that the gradient test behaves similarly to the LR test, but is less size distorted, while the usual score test performs much better than the other two uncorrected ones, although it is a bit liberal in a few cases. Considering $\alpha =1\%$ and $n=30$ for the Student-$t$ model (see Table I), the null rejection rates of the $LR$ test are $3.2\%(k=3),$ $5.0\%(k=4)$ and $6.5\%(k=5),$ while for the gradient test they are $2.4\%(k=3),$ $3.7\%(k=4)$ and $5.3\%(k=5),$ and for the score test they are $0.8\%(k=3),$ $1.0\%(k=4)$ and $1.0\%(k=5).$

Table I
Null rejection rates $\left(\%\right)$ for $H_0:{\delta }_1=\dots ={\delta }_k=0$ with $p=3;$ $t_5$ model.

n		$Stat$		$\alpha =10\%$				$\alpha =5\%$				$\alpha =1\%$
				$k$				$k$				$k$
				$3$	$4$	$5$		$3$	$4$	$5$		$3$	$4$	$5$
$20$		$S_{LR}$		$26.7$	$37.1$	$41.4$		$17.7$	$26.6$	$30.2$		$6.5$	$11.8$	$14.7$
		$S_{{LR}^{*}}$		$14.7$	$18.1$	$18.3$		$8.2$	$10.7$	$10.7$		$2.0$	$3.1$	$3.0$
		$S_r$		$11.0$	$11.3$	$12.2$		$5.7$	$5.7$	$6.1$		$0.9$	$1.1$	$1.0$
		$S_{r^{*}}$		$10.0$	$10.3$	$11.2$		$5.5$	$5.2$	$5.8$		$1.1$	$1.0$	$1.0$
		$S_{{r_1}^{*}}$		$10.1$	$10.3$	$11.2$		$5.5$	$5.2$	$5.8$		$1.1$	$1.0$	$1.0$
		$S_{{r_2}^{*}}$		$10.1$	$10.3$	$11.2$		$5.5$	$5.2$	$5.8$		$1.1$	$1.0$	$1.0$
		$S_g$		$24.1$	$33.1$	$36.5$		$15.2$	$22.6$	$26.1$		$5.3$	$9.2$	$11.4$
		$S_{g^{*}}$		$10.6$	$16.9$	$18.7$		$5.6$	$10.2$	$11.5$		$1.6$	$3.2$	$4.3$
		$S_{LR}^{boot}$		$10.1$	$10.0$	$9.8$		$4.8$	$4.8$	$5.0$		$0.9$	$1.1$	$1.1$
		$S_r^{boot}$		$9.5$	$10.1$	$9.9$		$4.8$	$4.7$	$4.8$		$1.1$	$1.0$	$0.9$
		$S_g^{boot}$		$10.1$	$10.3$	$9.6$		$5.2$	$4.9$	$4.6$		$1.0$	$1.1$	$0.9$
$30$		$S_{LR}$		$18.4$	$23.0$	$26.0$		$11.0$	$14.1$	$17.3$		$3.2$	$5.0$	$6.5$
		$S_{{LR}^{*}}$		$12.0$	$12.7$	$13.6$		$6.4$	$6.7$	$7.8$		$1.3$	$1.7$	$2.0$
		$S_r$		$10.4$	$10.2$	$11.0$		$5.0$	$5.2$	$5.5$		$0.8$	$1.0$	$1.0$
		$S_{r^{*}}$		$9.8$	$9.9$	$10.2$		$4.8$	$5.1$	$5.1$		$0.8$	$1.1$	$1.0$
		$S_{{r_1}^{*}}$		$9.8$	$9.9$	$10.2$		$4.8$	$5.1$	$5.1$		$0.8$	$1.1$	$1.0$
		$S_{{r_2}^{*}}$		$9.8$	$9.9$	$10.2$		$4.8$	$5.1$	$5.1$		$0.8$	$1.1$	$1.0$
		$S_g$		$17.1$	$20.1$	$24.0$		$9.7$	$11.9$	$15.5$		$2.4$	$3.7$	$5.3$
		$S_{g^{*}}$		$10.5$	$11.1$	$14.6$		$5.4$	$6.0$	$8.3$		$1.0$	$1.5$	$2.6$
		$S_{LR}^{boot}$		$10.4$	$10.0$	$10.6$		$5.4$	$4.8$	$5.5$		$1.4$	$0.9$	$1.3$
		$S_r^{boot}$		$10.3$	$10.4$	$9.9$		$5.7$	$5.3$	$5.0$		$1.2$	$0.9$	$0.9$
		$S_g^{boot}$		$10.2$	$10.2$	$9.8$		$5.4$	$5.0$	$5.2$		$1.1$	$0.9$	$1.0$
$40$		$S_{LR}$		$15.8$	$17.6$	$19.1$		$9.9$	$10.7$	$11.1$		$2.6$	$3.1$	$3.2$
		$S_{{LR}^{*}}$		$11.7$	$11.2$	$10.9$		$6.1$	$6.0$	$5.2$		$1.2$	$1.3$	$1.2$
		$S_r$		$10.8$	$10.5$	$10.6$		$5.5$	$5.2$	$5.1$		$0.9$	$0.9$	$0.9$
		$S_{r^{*}}$		$10.2$	$10.0$	$9.9$		$5.3$	$5.1$	$4.8$		$0.9$	$0.9$	$0.9$
		$S_{{r_1}^{*}}$		$10.2$	$10.1$	$10.0$		$5.3$	$5.1$	$4.8$		$0.9$	$0.9$	$0.9$
		$S_{{r_2}^{*}}$		$10.2$	$10.0$	$9.9$		$5.3$	$5.1$	$4.8$		$0.9$	$0.9$	$0.9$
		$S_g$		$15.2$	$16.6$	$17.8$		$8.8$	$9.9$	$10.1$		$2.1$	$2.5$	$2.5$
		$S_{g^{*}}$		$9.9$	$10.8$	$10.6$		$4.9$	$5.5$	$5.4$		$1.1$	$1.1$	$1.2$
		$S_{LR}^{boot}$		$9.9$	$10.2$	$9.6$		$4.7$	$5.1$	$4.8$		$1.0$	$1.0$	$1.1$
		$S_r^{boot}$		$9.8$	$9.9$	$10.1$		$5.1$	$4.9$	$5.2$		$0.9$	$1.0$	$1.0$
		$S_g^{boot}$		$9.9$	$9.9$	$9.7$		$5.0$	$4.9$	$4.8$		$0.9$	$0.9$	$0.9$

Table II
Null rejection rates (%) for $H_0:{\delta }_1=\dots ={\delta }_k=0$ with $p=3;$ power exponential $\nu =0.3$ model.

n		$Stat$		$\alpha =10\%$				$\alpha =5\%$				$\alpha =1\%$
				$k$				$k$				$k$
				$3$	$4$	$5$		$3$	$4$	$5$		$3$	$4$	$5$
$20$		$S_{LR}$		$25.2$	$34.0$	$35.9$		$16.3$	$23.7$	$25.3$		$6.2$	$9.8$	$10.7$
		$S_{{LR}^{*}}$		$16.7$	$21.0$	$20.0$		$9.6$	$12.5$	$12.0$		$2.9$	$3.5$	$3.5$
		$S_r$		$9.7$	$9.9$	$10.3$		$4.8$	$5.2$	$5.7$		$0.9$	$1.7$	$1.6$
		$S_{r^{*}}$		$10.3$	$9.6$	$10.0$		$5.4$	$3.6$	$4.5$		$1.2$	$1.1$	$0.6$
		$S_{{r_1}^{*}}$		$10.3$	$10.1$	$10.1$		$5.4$	$3.6$	$4.5$		$1.2$	$1.1$	$0.6$
		$S_{{r_2}^{*}}$		$10.3$	$10.0$	$10.1$		$5.4$	$4.3$	$4.7$		$1.2$	$1.1$	$0.6$
		$S_g$		$23.0$	$30.1$	$31.5$		$14.6$	$20.4$	$21.4$		$5.1$	$7.6$	$8.7$
		$S_{g^{*}}$		$11.3$	$15.7$	$15.4$		$6.2$	$8.5$	$9.0$		$1.3$	$2.2$	$2.7$
		$S_{LR}^{boot}$		$10.1$	$10.4$	$10.1$		$4.8$	$5.3$	$5.4$		$1.0$	$1.1$	$1.3$
		$S_r^{boot}$		$9.8$	$10.4$	$10.1$		$4.9$	$5.0$	$5.2$		$1.0$	$0.9$	$1.3$
		$S_g^{boot}$		$10.4$	$10.1$	$10.0$		$5.1$	$5.1$	$5.2$		$0.9$	$0.9$	$1.0$
$30$		$S_{LR}$		$18.6$	$21.3$	$22.8$		$11.1$	$13.1$	$14.2$		$3.4$	$4.2$	$4.5$
		$S_{{LR}^{*}}$		$13.9$	$14.3$	$14.3$		$7.7$	$8.0$	$7.5$		$1.9$	$2.0$	$1.6$
		$S_r$		$9.9$	$10.3$	$10.1$		$5.2$	$5.3$	$5.4$		$1.4$	$1.4$	$1.3$
		$S_{r^{*}}$		$10.2$	$10.2$	$10.0$		$5.5$	$4.5$	$4.8$		$1.4$	$1.2$	$0.7$
		$S_{{r_1}^{*}}$		$10.2$	$10.3$	$10.0$		$5.5$	$4.7$	$4.8$		$1.4$	$1.2$	$0.7$
		$S_{{r_2}^{*}}$		$10.2$	$10.3$	$10.0$		$5.5$	$4.7$	$4.8$		$1.4$	$1.3$	$0.7$
		$S_g$		$18,0$	$20.2$	$21.5$		$10.5$	$12.2$	$13.0$		$3.2$	$3.5$	$3.7$
		$S_{g^{*}}$		$11.4$	$12.1$	$12.4$		$6.0$	$6.4$	$6.2$		$1.2$	$1.4$	$1.4$
		$S_{LR}^{boot}$		$10.2$	$10.7$	$10.5$		$5.3$	$5.4$	$5.5$		$1.3$	$1.0$	$1.2$
		$S_r^{boot}$		$10.3$	$10.3$	$9.6$		$5.2$	$5.1$	$5.0$		$0.9$	$1.1$	$1.1$
		$S_g^{boot}$		$10.3$	$10.3$	$9.7$		$5.2$	$5.2$	$4.9$		$0.9$	$1.0$	$0.9$
$40$		$S_{LR}$		$14.4$	$16.4$	$18.3$		$8.2$	$9.3$	$10.9$		$2.3$	$2.9$	$3.4$
		$S_{{LR}^{*}}$		$12.6$	$11.7$	$12.7$		$6.6$	$6.4$	$7.3$		$1.8$	$1.7$	$1.8$
		$S_r$		$10.4$	$9.3$	$10.1$		$5.4$	$4.9$	$5.3$		$1.2$	$1.1$	$1.4$
		$S_{r^{*}}$		$10.2$	$9.4$	$10.7$		$4.8$	$4.8$	$5.7$		$0.5$	$0.8$	$1.5$
		$S_{{r_1}^{*}}$		$10.3$	$9.4$	$10.7$		$4.9$	$4.8$	$5.7$		$0.8$	$0.9$	$1.5$
		$S_{{r_2}^{*}}$		$10.3$	$9.4$	$10.7$		$4.9$	$4.8$	$5.7$		$0.8$	$0.9$	$1.5$
		$S_g$		$13.9$	$15.5$	$17.3$		$7.8$	$8.7$	$10.1$		$2.1$	$2.5$	$2.8$
		$S_{g^{*}}$		$9.9$	$10.5$	$11.7$		$5.0$	$5.6$	$6.4$		$1.2$	$1.5$	$1.5$
		$S_{LR}^{boot}$		$9.7$	$10.5$	$9.7$		$5.5$	$5.0$	$4.9$		$1.1$	$0.9$	$1.2$
		$S_r^{boot}$		$10.2$	$10.1$	$10.2$		$5.0$	$4.8$	$5.3$		$1.1$	$0.8$	$1.2$
		$S_g^{boot}$		$10.2$	$9.9$	$10.1$		$5.1$	$4.8$	$5.3$		$1.1$	$0.9$	$1.2$

The simulation results also showed that the corrected tests based on the $S_{{LR}^{*}},$ $S_{r^{*}}$ and $S_{g^{*}}$ statistics outperformed their uncorrected versions, regardless of the sample size and the number of interest or nuisance parameters. Additionally, as shown in Tables I and II, the corrected versions of the LR and gradient tests are very sensitive to increasing the number of parameters in the model, whether they are interest or nuisance parameters. Otherwise, the corrected score test is not influenced by the increase in the number of parameters in the model and among the improved tests, the one based on the $S_{r^{*}}$ statistic presents the best performance, exhibiting null rejection rates very close to the nominal level in most cases. For example, considering the power exponential model (Table II), if $k=3,n=20$ and $\alpha =10\%,$ the null rejection rate of the tests based on $S_{{LR}^{*}},$ $S_{g^{*}}$ and $S_{r^{*}}$ are, respectively, $16.7\%,$ $11.3\%$ and $10.3\%,$ while considering the same scenario with $k=4,$ the null rejection rates for the tests based on $S_{{LR}^{*}},$ $S_{g^{*}}$ and $S_{r^{*}}$ are, respectively, $21.0\%,$ $15.7\%$ and $9.6\%.$ Now considering the tests based on the monotonic versions of the corrected score statistics $S_{r_1^{*}}$ and $S_{r_2^{*}}$ proposed by Kakisawa (1996) and Cordeiro et al. (1998), the simulation results show that the performance of the tests based on those statistics are very similar to the corrected score test, presenting the same null rejection rate in most cases.

Tables I and II also show that the bootstrap-based tests are less size distorted than the corresponding uncorrected tests. Also, for the LR and gradient tests, their bootstrap versions outperform the corrected ones. On the other hand, the bootstrap score test behaves, in general, similarly to the monotonic and non-monotonic corrected ones. For small samples ($n=20$), the bootstrap score test presents null rejection rates closer to the nominal level than the monotonic and non-monotonic corrected versions. For example, considering $p=3,k=3,n=20$ and $\alpha =5\%$, the null rejection rates are $17.7\%\left(S_{LR}\right),$ $8.2\%\left(S_{{LR}^{*}}\right)$, $4.8\%(S_{LR}^{boot},S_r^{boot})$, $5.7\%\left(S_r\right),$ $5.5\%(S_{r^{*}},S_{r_1^{*}},S_{r_2^{*}}),$ $15.2\%\left(S_g\right),$ $5.6\%\left(S_{g^{*}}\right)$ and $5.2\%\left(S_g^{boot}\right)$ for the Student-$t$ model and $16.3\%\left(S_{LR}\right),$ $9.6\%\left(S_{{LR}^{*}}\right),$ $4.8\%(S_{LR}^{boot},S_r),$ $5.4\%(S_{r^{*}},S_{r_1^{*}},S_{r_2^{*}}),$ $14.6\%\left(S_g\right),$ $6.2\%\left(S_g^{*}\right),$ $4.9\%\left(S_r^{boot}\right)$ and $5.1\%\left(S_g^{boot}\right)$ for the power exponential model.

Finally, we also point out that all corrected and uncorrected tests present null rejection rates very close to the corresponding nominal level as the sample size increases, as expected.

Completing our simulations, we also obtained rejection rates under the alternative hypothesis (heteroscedasticity) for $n=30,$ $p=3,$ different values of ${\delta }_1={\delta }_2={\delta }_3=\delta$ and at the $10\%$ nominal level. It is noteworthy that these power simulations correspond to the setting in Tables I and II for $n=30$. We did not include the likelihood ratio test, corrected likelihood ratio test, and gradient test in the power comparison since they are too liberal to be recommended. All other tests studied in this paper were considered. It should be noted in Figure 1 that the powers of the tests are similar, although there is an ordering, being the bootstrap-based tests and the corrected gradient test the ones which were slightly more powerful. As expected, the power tends to one when $|\delta |$ grows.

Power functions of the tests for $H_0:{\delta }_1=\dots ={\delta }_3=\delta$ with $p=3,$ $n=30,$ $\alpha =10\%;$ $\left(a\right)$ $t_5$ and $\left(b\right)$ power exponential $\nu =0.3$ models.

In summary, the simulation results presented in this section show that the $LR$ and gradient tests are considerably oversized (liberal) and the analytical Bartlett and Bartlett-type corrections for these tests are effective in reducing the size distortion. Also, their bootstrap versions outperformed the corrected ones, presenting results closer to the nominal levels considered. The score test is the best performing uncorrected test. Its (monotonic or not) corrected versions perform the same, as well its bootstrap-based test which, in some cases, performed better than the corrected versions.

REAL DATA APPLICATION

In this section, we consider a dataset analysed by Rawlings 1988 and Brazzale 2000. The dataset comes from a study executed by the Botany Department of North Carolina State University, where an experiment was carried out with the aim of performing biochemical analysis of intracellular storage and calcium transport across the plasma membrane. For this purpose, the cells studied were suspended in a solution of radioactive calcium for a certain period of time $x$ (in minutes), after which was measured the amount of calcium absorbed by the cells $y$ (in nmol/mg). The studied sample contains 27 observations and the model used is given by

$$y_{\ell }={\beta }_0\{1-exp(-{\beta }_1{x}_{\ell })\}+u_{\ell },\ell =1,\dots ,27,$$ (4)

where $Cov(u_{\ell },u_m)=0$ for all $\ell \ne m.$ Initially, we assume that $u_{\ell }\sim N(0,exp\{{\delta }_0\})$. The maximum likelihood parameter estimates and their asymptotic standard errors in brackets are ${\hat{\beta }}_0=4.3094\left(0.2901\right)$ ${\hat{\beta }}_1=0.2084\left(0.0376\right)$ and ${\hat{\delta }}_0=-1.286\left(0.2722\right).$

Residual analysis of the fitted homoscedastic normal model suggest an evidence of heteroscedasticity. In addition, we detect the presence of two observations $\#21$ and $\#22$ with large residuals (in absolute value) outside range [-2,2] (see Figure 2a). Due to the evidence of heteroscedasticity, we assume that $u_{\ell }\sim N(0,exp\{{\delta }_0+{\delta }_1{x}_{\ell }\}),$ named here, heteroscedastic normal model.

Our main interest lies in testing $H_0:{\delta }_1=0$ (homoscedasticity) against $H_1:{\delta }_1\ne 0.$ Rejection of the null hypothesis would suggest that the nonconstant response variance should be modeled as well. Table III presents the observed values for the test statistics and their respective $p-$values. Considering the $10\%$ nominal level, the score, the standard likelihood ratio tests (and their corrected versions) and the gradient test, lead to rejecting $H_0,$ suggesting the presence of heteroscedasticity. While the tests based on $S_g^{*},$ $S_{LR}^{boot},$ $S_r^{boot}$ and $S_g^{boot}$ statistics may not reject the null hypothesis at the same nominal level.

The studentized residual plot versus fitted values for the fitted heteroscedasticity normal model (see Figure 3b) still presents a slight indication of heteroscedasticity and residuals in the limit of the range $[-2,2].$ Furthermore, we may suspect the presence of aberrant points, which we propose a symmetrical model with heavy tails as Student-$t$ model. Lange et al. 1989 suggest that the degrees of freedom should be fixed for small-sized samples and indicate that $\nu =4$ has worked well for some applications.

Figure 2
Calcium data. Plots of the studentized residual versus fitted values (a) homoscedastic normal model, (b) heteroscedastic normal model (c) homoscedastic Student-t model, (d) heteroscedastic Student-t with 𝜈 = 4 model.

Figure 2c from the fitted homoscedastic Student-$t$ model does not highlight aberrant points but still suggests little evidence of heteroscedasticity. Finally, Figure 2d from the heteroscedastic Student-$t$ model does not present evidence of heteroscedasticity and does not highlight aberrant points. It follows that all proposed tests lead to the same decision of homoscedasticity at the nominal level $\alpha =10\%$ (see Table III). Therefore, based on the parsimony principle we suggest the Student-$t$ model.

Table III
Test statistics and $p-$ values for testing $H_0:{\delta }_1=0$ in normal model and t₄ model.

Stat	Normal model			t4 model
	Observed value	$p-$value		Observed value	$p-$value
SLR	3.097	0.078		2.283	0.131
SLR*	2.908	0.088		2.174	0.140
Sr	3.029	0.082		2.634	0.105
S*r	3.456	0.063		2.515	0.113
S*r1	3.491	0.063		2.518	0.113
S*r2	3.473	0.062		2.517	0.113
Sg	3.030	0.082		2.277	0.131
Sg*	2.710	0.100		2.038	0.153
SLRboot		0.108			0.183
Srboot		1.000			0.945
Sgboot		0.398			0.276
Sgboot		0.398			0.276

CONCLUSIONS

In this paper we derive Bartlett and Bartlett-type corrections to improve hypothesis testing of the dispersion parameters for the HSNLM class of proposed by Cysneiros et al. 2010 and compare in a simulation study the performance of the proposed tests with the score test, its Bartlett-type corrected version and the uncorrected LR and gradient tests. We also consider for the simulation study monotonic versions of the Bartlett-type corrected score test and bootstrapped tests.

The numerical evidence suggests that the usual LR and gradient tests have similar performance, being oversized, mainly if the sample size is small or even moderate. The Bartlett and Bartlett-type corrections attenuate this tendency, although the Bartlett corrected LR test still presents very liberal behavior while the Bartlett-type corrected gradient test produces results comparable to those of the usual and (monotonic or not) Bartlett-type corrected score tests. Additionally, the corrected score test and the bootstrapped tests perform the best overall. An advantage of the analytically corrected tests in relation to the bootstrapped tests is that they do not demand as much computational burden. Moreover, it is important to note that the corrected tests deliver more reliable inferences than their uncorrected versions when dealing with small or even moderate sized samples. We hence encourage practitioners to use the Bartlett-type corrected score and gradient or bootstrapped tests.

ACKNOWLEDGMENTS

We are thankful for the financial support of Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and FACEPE. The research of Lourdes C. Montenegro was supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES, Grant 6796/14-1).

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Publication Dates

• Publication in this collection
  21 Nov 2022
• Date of issue
  2022

History

• Received
  16 Apr 2020
• Accepted
  12 Apr 2021