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

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

Authorship

MARIANA C. ARAÚJO

developed the present paper

derived all analytical results and performed the Monte Carlo simulations

Universidade Federal do Rio Grande do Norte, Departamento de Estatística, Lagoa Nova, Av. Senador Salgado Filho, 3000, 59078-970 Natal, RN, BrazilUniversidade Federal do Rio Grande do NorteBrazilNatal, RN, BrazilUniversidade Federal do Rio Grande do Norte, Departamento de Estatística, Lagoa Nova, Av. Senador Salgado Filho, 3000, 59078-970 Natal, RN, Brazil

https://orcid.org/0000-0002-5682-520X

AUDREY H.M.A. CYSNEIROS

developed the present paper

introduced the idea, checked, refined the notations and edited the improved test inference section

Universidade Federal de Pernambuco, Departamento de Estatística, Cidade Universitária, Av. Prof. Moraes Rego, 1235, 50740-540 Recife, PE, BrazilUniversidade Federal de PernambucoBrazilRecife, PE, BrazilUniversidade Federal de Pernambuco, Departamento de Estatística, Cidade Universitária, Av. Prof. Moraes Rego, 1235, 50740-540 Recife, PE, Brazil

https://orcid.org/0000-0002-8920-4004

LOURDES C. MONTENEGRO

developed the present paper

carried out the empirical application, and wrote the abstract and the introduction section

Universidade Federal de Minas Gerais, Departamento de Estatística, Pampulha, Av. Pres. Antônio Carlos, 6627, 31270-901 Belo Horizonte, MG, BrazilUniversidade Federal de Minas GeraisBrazilBelo Horizonte, MG, BrazilUniversidade Federal de Minas Gerais, Departamento de Estatística, Pampulha, Av. Pres. Antônio Carlos, 6627, 31270-901 Belo Horizonte, MG, Brazil

https://orcid.org/0000-0003-0867-3931

Correspondence to: Mariana C. Araújo E-mail: mariana.araujo@ufrn.br

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables

Figures (2)
Tables (3)

Thumbnail

Power functions of the tests for

H_{0} : δ_{1} = \dots = δ_{3} = δ

with

p = 3,

n = 30,

α = 10 %;

(a)

t_{5}

and

(b)

power exponential

ν = 0.3

models.

Thumbnail

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.

Thumbnail

Table I
Null rejection rates

(%)

for

H_{0} : δ_{1} = \dots = δ_{k} = 0

with

p = 3;

t_{5}

model.

Thumbnail

Table II
Null rejection rates (%) for

H_{0} : δ_{1} = \dots = δ_{k} = 0

with

p = 3;

power exponential

ν = 0.3

model.

Thumbnail

Table III
Test statistics and

p -

values for testing

H_{0} : δ_{1} = 0

in normal model and t₄ model.

Power functions of the tests for $H_{0} : δ_{1} = \dots = δ_{3} = δ$ with $p = 3,$ $n = 30,$ $α = 10 %;$ $(a)$ $t_{5}$ and $(b)$ power exponential $ν = 0.3$ models.

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.

Table I Null rejection rates $(%)$ for $H_{0} : δ_{1} = \dots = δ_{k} = 0$ with $p = 3;$ $t_{5}$ model.

n	$S t a t$	$α = 10 %$			$α = 5 %$			$α = 1 %$
		$k$			$k$			$k$
		$3$	$4$	$5$	$3$	$4$	$5$	$3$	$4$	$5$
$20$	$S_{L R}$	$26.7$	$37.1$	$41.4$	$17.7$	$26.6$	$30.2$	$6.5$	$11.8$	$14.7$
	$S_{{L R}^{*}}$	$14.7$	$18.1$	$18.3$	$8.2$	$10.7$	$10.7$	$2.0$	$3.1$	$3.0$
	$S_{r}$	$11.0$	$11.3$	$12.2$	$5.7$	$5.7$	$6.1$	$0.9$	$1.1$	$1.0$
	$S_{r^{*}}$	$10.0$	$10.3$	$11.2$	$5.5$	$5.2$	$5.8$	$1.1$	$1.0$	$1.0$
	$S_{{r_{1}}^{*}}$	$10.1$	$10.3$	$11.2$	$5.5$	$5.2$	$5.8$	$1.1$	$1.0$	$1.0$
	$S_{{r_{2}}^{*}}$	$10.1$	$10.3$	$11.2$	$5.5$	$5.2$	$5.8$	$1.1$	$1.0$	$1.0$
	$S_{g}$	$24.1$	$33.1$	$36.5$	$15.2$	$22.6$	$26.1$	$5.3$	$9.2$	$11.4$
	$S_{g^{*}}$	$10.6$	$16.9$	$18.7$	$5.6$	$10.2$	$11.5$	$1.6$	$3.2$	$4.3$
	$S_{L R}^{b o o t}$	$10.1$	$10.0$	$9.8$	$4.8$	$4.8$	$5.0$	$0.9$	$1.1$	$1.1$
	$S_{r}^{b o o t}$	$9.5$	$10.1$	$9.9$	$4.8$	$4.7$	$4.8$	$1.1$	$1.0$	$0.9$
	$S_{g}^{b o o t}$	$10.1$	$10.3$	$9.6$	$5.2$	$4.9$	$4.6$	$1.0$	$1.1$	$0.9$
$30$	$S_{L R}$	$18.4$	$23.0$	$26.0$	$11.0$	$14.1$	$17.3$	$3.2$	$5.0$	$6.5$
	$S_{{L R}^{*}}$	$12.0$	$12.7$	$13.6$	$6.4$	$6.7$	$7.8$	$1.3$	$1.7$	$2.0$
	$S_{r}$	$10.4$	$10.2$	$11.0$	$5.0$	$5.2$	$5.5$	$0.8$	$1.0$	$1.0$
	$S_{r^{*}}$	$9.8$	$9.9$	$10.2$	$4.8$	$5.1$	$5.1$	$0.8$	$1.1$	$1.0$
	$S_{{r_{1}}^{*}}$	$9.8$	$9.9$	$10.2$	$4.8$	$5.1$	$5.1$	$0.8$	$1.1$	$1.0$
	$S_{{r_{2}}^{*}}$	$9.8$	$9.9$	$10.2$	$4.8$	$5.1$	$5.1$	$0.8$	$1.1$	$1.0$
	$S_{g}$	$17.1$	$20.1$	$24.0$	$9.7$	$11.9$	$15.5$	$2.4$	$3.7$	$5.3$
	$S_{g^{*}}$	$10.5$	$11.1$	$14.6$	$5.4$	$6.0$	$8.3$	$1.0$	$1.5$	$2.6$
	$S_{L R}^{b o o t}$	$10.4$	$10.0$	$10.6$	$5.4$	$4.8$	$5.5$	$1.4$	$0.9$	$1.3$
	$S_{r}^{b o o t}$	$10.3$	$10.4$	$9.9$	$5.7$	$5.3$	$5.0$	$1.2$	$0.9$	$0.9$
	$S_{g}^{b o o t}$	$10.2$	$10.2$	$9.8$	$5.4$	$5.0$	$5.2$	$1.1$	$0.9$	$1.0$
$40$	$S_{L R}$	$15.8$	$17.6$	$19.1$	$9.9$	$10.7$	$11.1$	$2.6$	$3.1$	$3.2$
	$S_{{L R}^{*}}$	$11.7$	$11.2$	$10.9$	$6.1$	$6.0$	$5.2$	$1.2$	$1.3$	$1.2$
	$S_{r}$	$10.8$	$10.5$	$10.6$	$5.5$	$5.2$	$5.1$	$0.9$	$0.9$	$0.9$
	$S_{r^{*}}$	$10.2$	$10.0$	$9.9$	$5.3$	$5.1$	$4.8$	$0.9$	$0.9$	$0.9$
	$S_{{r_{1}}^{*}}$	$10.2$	$10.1$	$10.0$	$5.3$	$5.1$	$4.8$	$0.9$	$0.9$	$0.9$
	$S_{{r_{2}}^{*}}$	$10.2$	$10.0$	$9.9$	$5.3$	$5.1$	$4.8$	$0.9$	$0.9$	$0.9$
	$S_{g}$	$15.2$	$16.6$	$17.8$	$8.8$	$9.9$	$10.1$	$2.1$	$2.5$	$2.5$
	$S_{g^{*}}$	$9.9$	$10.8$	$10.6$	$4.9$	$5.5$	$5.4$	$1.1$	$1.1$	$1.2$
	$S_{L R}^{b o o t}$	$9.9$	$10.2$	$9.6$	$4.7$	$5.1$	$4.8$	$1.0$	$1.0$	$1.1$
	$S_{r}^{b o o t}$	$9.8$	$9.9$	$10.1$	$5.1$	$4.9$	$5.2$	$0.9$	$1.0$	$1.0$
	$S_{g}^{b o o t}$	$9.9$	$9.9$	$9.7$	$5.0$	$4.9$	$4.8$	$0.9$	$0.9$	$0.9$

Table II Null rejection rates (%) for $H_{0} : δ_{1} = \dots = δ_{k} = 0$ with $p = 3;$ power exponential $ν = 0.3$ model.

n	$S t a t$	$α = 10 %$			$α = 5 %$			$α = 1 %$
		$k$			$k$			$k$
		$3$	$4$	$5$	$3$	$4$	$5$	$3$	$4$	$5$
$20$	$S_{L R}$	$25.2$	$34.0$	$35.9$	$16.3$	$23.7$	$25.3$	$6.2$	$9.8$	$10.7$
	$S_{{L R}^{*}}$	$16.7$	$21.0$	$20.0$	$9.6$	$12.5$	$12.0$	$2.9$	$3.5$	$3.5$
	$S_{r}$	$9.7$	$9.9$	$10.3$	$4.8$	$5.2$	$5.7$	$0.9$	$1.7$	$1.6$
	$S_{r^{*}}$	$10.3$	$9.6$	$10.0$	$5.4$	$3.6$	$4.5$	$1.2$	$1.1$	$0.6$
	$S_{{r_{1}}^{*}}$	$10.3$	$10.1$	$10.1$	$5.4$	$3.6$	$4.5$	$1.2$	$1.1$	$0.6$
	$S_{{r_{2}}^{*}}$	$10.3$	$10.0$	$10.1$	$5.4$	$4.3$	$4.7$	$1.2$	$1.1$	$0.6$
	$S_{g}$	$23.0$	$30.1$	$31.5$	$14.6$	$20.4$	$21.4$	$5.1$	$7.6$	$8.7$
	$S_{g^{*}}$	$11.3$	$15.7$	$15.4$	$6.2$	$8.5$	$9.0$	$1.3$	$2.2$	$2.7$
	$S_{L R}^{b o o t}$	$10.1$	$10.4$	$10.1$	$4.8$	$5.3$	$5.4$	$1.0$	$1.1$	$1.3$
	$S_{r}^{b o o t}$	$9.8$	$10.4$	$10.1$	$4.9$	$5.0$	$5.2$	$1.0$	$0.9$	$1.3$
	$S_{g}^{b o o t}$	$10.4$	$10.1$	$10.0$	$5.1$	$5.1$	$5.2$	$0.9$	$0.9$	$1.0$
$30$	$S_{L R}$	$18.6$	$21.3$	$22.8$	$11.1$	$13.1$	$14.2$	$3.4$	$4.2$	$4.5$
	$S_{{L R}^{*}}$	$13.9$	$14.3$	$14.3$	$7.7$	$8.0$	$7.5$	$1.9$	$2.0$	$1.6$
	$S_{r}$	$9.9$	$10.3$	$10.1$	$5.2$	$5.3$	$5.4$	$1.4$	$1.4$	$1.3$
	$S_{r^{*}}$	$10.2$	$10.2$	$10.0$	$5.5$	$4.5$	$4.8$	$1.4$	$1.2$	$0.7$
	$S_{{r_{1}}^{*}}$	$10.2$	$10.3$	$10.0$	$5.5$	$4.7$	$4.8$	$1.4$	$1.2$	$0.7$
	$S_{{r_{2}}^{*}}$	$10.2$	$10.3$	$10.0$	$5.5$	$4.7$	$4.8$	$1.4$	$1.3$	$0.7$
	$S_{g}$	$18, 0$	$20.2$	$21.5$	$10.5$	$12.2$	$13.0$	$3.2$	$3.5$	$3.7$
	$S_{g^{*}}$	$11.4$	$12.1$	$12.4$	$6.0$	$6.4$	$6.2$	$1.2$	$1.4$	$1.4$
	$S_{L R}^{b o o t}$	$10.2$	$10.7$	$10.5$	$5.3$	$5.4$	$5.5$	$1.3$	$1.0$	$1.2$
	$S_{r}^{b o o t}$	$10.3$	$10.3$	$9.6$	$5.2$	$5.1$	$5.0$	$0.9$	$1.1$	$1.1$
	$S_{g}^{b o o t}$	$10.3$	$10.3$	$9.7$	$5.2$	$5.2$	$4.9$	$0.9$	$1.0$	$0.9$
$40$	$S_{L R}$	$14.4$	$16.4$	$18.3$	$8.2$	$9.3$	$10.9$	$2.3$	$2.9$	$3.4$
	$S_{{L R}^{*}}$	$12.6$	$11.7$	$12.7$	$6.6$	$6.4$	$7.3$	$1.8$	$1.7$	$1.8$
	$S_{r}$	$10.4$	$9.3$	$10.1$	$5.4$	$4.9$	$5.3$	$1.2$	$1.1$	$1.4$
	$S_{r^{*}}$	$10.2$	$9.4$	$10.7$	$4.8$	$4.8$	$5.7$	$0.5$	$0.8$	$1.5$
	$S_{{r_{1}}^{*}}$	$10.3$	$9.4$	$10.7$	$4.9$	$4.8$	$5.7$	$0.8$	$0.9$	$1.5$
	$S_{{r_{2}}^{*}}$	$10.3$	$9.4$	$10.7$	$4.9$	$4.8$	$5.7$	$0.8$	$0.9$	$1.5$
	$S_{g}$	$13.9$	$15.5$	$17.3$	$7.8$	$8.7$	$10.1$	$2.1$	$2.5$	$2.8$
	$S_{g^{*}}$	$9.9$	$10.5$	$11.7$	$5.0$	$5.6$	$6.4$	$1.2$	$1.5$	$1.5$
	$S_{L R}^{b o o t}$	$9.7$	$10.5$	$9.7$	$5.5$	$5.0$	$4.9$	$1.1$	$0.9$	$1.2$
	$S_{r}^{b o o t}$	$10.2$	$10.1$	$10.2$	$5.0$	$4.8$	$5.3$	$1.1$	$0.8$	$1.2$
	$S_{g}^{b o o t}$	$10.2$	$9.9$	$10.1$	$5.1$	$4.8$	$5.3$	$1.1$	$0.9$	$1.2$

Table III Test statistics and $p -$ values for testing $H_{0} : δ_{1} = 0$ in normal model and t₄ model.

Stat	Normal model		t₄ model
Stat	Observed value	$p -$ value	Observed value	$p -$ value
S_LR	3.097	0.078	2.283	0.131
S_LR*	2.908	0.088	2.174	0.140
S_r	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
S_g	3.030	0.082	2.277	0.131
S_g^*	2.710	0.100	2.038	0.153
S_LR^boot		0.108		0.183
S_r^boot		1.000		0.945
S_g^boot		0.398		0.276
S_g^boot		0.398		0.276

How to cite

copy

Academia Brasileira de Ciências Rua Anfilófio de Carvalho, 29, 3º andar, 20030-060 Rio de Janeiro RJ Brasil, Tel: +55 21 3907-8100 - Rio de Janeiro - RJ - Brazil
E-mail: aabc@abc.org.br

Acompanhe os números deste periódico no seu leitor de RSS

Versão para download de PDF

PDF

Inglês

Versões e tradução automática

Versão original do texto

English

Tradução automática

Google Translator
Microsoft Translator

Como citar

RIS

BIBTEX

Outros formatos de citação e exportação:

SciELO - Scientific Electronic Library Online
Rua Dr. Diogo de Faria, 1087 – 9º andar – Vila Clementino 04037-003 São Paulo/SP - Brasil
E-mail: scielo@scielo.org

Leia a Declaração de Acesso Aberto

Métricas

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

PlumX

Mensagem

[1] Correspondence to: Mariana C. Araújo E-mail: mariana.araujo@ufrn.br