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

ARAÚJO, MARIANA C.; CYSNEIROS, AUDREY H.M.A.; MONTENEGRO, LOURDES C.

doi:10.1590/0001-3765202220200568

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MATHEMATICAL SCIENCES • An. Acad. Bras. Ciênc. 94 (suppl 3) • 2022 • https://doi.org/10.1590/0001-3765202220200568 linkcopy

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

Authorship SCIMAGO INSTITUTIONS RANKINGS

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

Authorship

person MARIANA C. ARAÚJO

developed the present paper · derived all analytical results and performed the Monte Carlo simulations

schoolUniversidade 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

0000-0002-5682-520X

person AUDREY H.M.A. CYSNEIROS

developed the present paper · introduced the idea, checked, refined the notations and edited the improved test inference section

schoolUniversidade 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

0000-0002-8920-4004

person LOURDES C. MONTENEGRO

developed the present paper · carried out the empirical application, and wrote the abstract and the introduction section

schoolUniversidade 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

0000-0003-0867-3931

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

SCIMAGO INSTITUTIONS RANKINGS

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

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

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

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.

image 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. open_in_new

imageFigure 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. open_in_new

table_chartTable 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_chartTable 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_chartTable 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

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