The Asymmetric Power-Student-t Model for Censored and Truncated Data

TOVAR-FALÓN, ROGER; BOLFARINE, HELENO; MARTÍNEZ-FLÓREZ, GUILLERMO

doi:10.1590/0001-3765202120190920

person ROGER TOVAR-FALÓN

schoolDepartamento de Matemáticas y Estadística, Facultad de Ciencias Básicas, Universidad de Córdoba, Cra. 6^a, No. 77-306, Montería, Colombia

0000-0001-5649-532X

person HELENO BOLFARINE

schoolDepartamento de Estatística, IME, Universidade de São Paulo, Cidade Universitária, Rua da Matão, 1010, 05508-090 São Paulo, SP, Brazil

0000-0001-9195-3672

person GUILLERMO MARTÍNEZ-FLÓREZ

schoolDepartamento de Matemáticas y Estadística, Facultad de Ciencias Básicas, Universidad de Córdoba, Cra. 6^a, No. 77-306, Montería, Colombia

0000-0001-6441-5377

Correspondence to: Roger Tovar-Falón E-mail: rjtovar@correo.unicordoba.edu.co

AUTHOR CONTRIBUTIONS

The findings in this paper are part of the first author’s (Roger Tovar-Falón) doctoral thesis, which was supervised by the second author (Heleno Bolfarine) and the third author (Guillermo Matínez-Flórez). The participation of the authors in the production of the manuscript is as follows: Roger Tovar-Falón - conceptualization, characterization of the new model, mathematical properties and implementation of computational routines. Guillermo Matínez-Flórez - application, simulation studies, writing the original draft, and computational routines. Heleno Bolfarine - review and general correction of the paper.

$ν$	$min$	$max$
4	-9.3423	5.5913
5	-7.5147	3.5101
6	-7.3825	2.7668
7	-4.6073	2.3721
8	-4.1468	2.1228
9	-3.8410	1.9490
10	-3.4826	1.8202
12	-2.2654	1.6409
15	-2.0131	1.4752
20	-1.8802	1.3208
25	-1.8245	1.2329
30	-1.6372	1.1760
50	-1.1158	1.0664
100	-1.0401	0.9864
500	-0.9390	0.9264

Cens.	$R C (\cdot)$	$δ$
Level		1	2	3	4	5	6	7	8	9	10
	$R C (β_{1})$	0.00194	0.02610	0.01427	0.05336	0.08081	0.09474	0.12469	0.15654	0.15260	0.15643
10%	$R C (β_{2})$	0.00010	0.00004	0.00009	0.00022	0.00024	0.00032	0.00045	0.00060	0.00064	0.00080
	$R C (η)$	0.00046	0.01137	0.00780	0.02271	0.03341	0.03780	0.04594	0.05558	0.05101	0.04596
	$R C (β_{1})$	0.01337	0.01963	0.00945	0.06445	0.12935	0.10080	0.09735	0.07929	0.04515	0.03117
20%	$R C (β_{2})$	0.00010	0.00008	0.00030	0.00031	0.00002	0.00018	0.00040	0.00075	0.00124	0.00157
	$R C (η)$	0.00219	0.00020	0.00160	0.00849	0.00984	0.00615	0.00417	0.00024	0.00910	0.01333
	$R C (β_{1})$	0.02466	0.00670	0.10623	0.06087	0.09179	0.15043	0.05653	0.19465	0.03994	0.16353
30%	$R C (β_{2})$	0.00062	0.00190	0.00016	0.00109	0.00006	0.00066	0.00019	0.00015	0.00012	0.00123
	$R C (η)$	0.00176	0.00860	0.01234	0.00391	0.00881	0.01215	0.00324	0.01675	0.00467	0.01425

		$α = 0.40$								$α = 2.5$
		$p = 0 %$		$p = 10 %$		$p = 25 %$		$p = 45 %$		$p = 0 %$		$p = 10 %$		$p = 25 %$		$p = 45 %$
$n$	$\hat{𝛉}$	RB	MSE	RB	MSE	RB	MSE	RB	MSE	RB	MSE	RB	MSE	RB	MSE	RB	MSE
150	${\hat{β}}_{1}$	0.010	0.982	0.019	1.127	0.022	0.746	0.050	0.657	0.092	0.700	0.132	0.813	0.138	0.913	0.155	1.200
	${\hat{β}}_{2}$	0.002	0.083	0.002	0.070	0.002	0.047	0.001	0.041	0.002	0.014	0.002	0.026	0.003	0.034	0.005	0.055
	$\hat{η}$	0.030	0.278	0.035	0.260	0.028	0.225	0.033	0.215	0.022	0.207	0.018	0.231	0.015	0.248	0.004	0.275
	$\hat{α}$	0.039	0.119	0.040	0.112	0.041	0.102	0.044	0.098	0.309	1.778	0.301	1.878	0.297	1.913	0.315	2.020
	$\hat{ν}$	0.309	2.673	0.296	2.428	0.255	2.125	0.223	1.980	0.188	1.735	0.196	1.840	0.211	2.057	0.216	2.228
300	${\hat{β}}_{1}$	0.010	0.839	0.025	0.787	0.014	0.504	0.025	0.441	0.061	0.552	0.086	0.601	0.094	0.689	0.122	0.922
	${\hat{β}}_{2}$	0.001	0.062	0.001	0.051	0.001	0.033	0.001	0.029	0.001	0.011	0.001	0.018	0.001	0.024	0.002	0.038
	$\hat{η}$	0.020	0.203	0.020	0.184	0.014	0.155	0.016	0.147	0.018	0.150	0.018	0.168	0.016	0.178	0.016	0.205
	$\hat{α}$	0.035	0.091	0.031	0.080	0.026	0.072	0.026	0.068	0.195	1.223	0.184	1.343	0.206	1.472	0.262	1.715
	$\hat{ν}$	0.186	1.739	0.160	1.610	0.123	1.263	0.102	1.127	0.093	0.973	0.107	1.025	0.109	1.026	0.128	1.175
750	${\hat{β}}_{1}$	0.002	0.588	0.003	0.481	0.006	0.303	0.009	0.263	0.024	0.219	0.037	0.346	0.044	0.415	0.064	0.574
	${\hat{β}}_{2}$	0.001	0.046	0.001	0.031	0.001	0.021	0.001	0.018	0.001	0.009	0.001	0.012	0.001	0.014	0.001	0.024
	$\hat{η}$	0.008	0.136	0.007	0.111	0.005	0.097	0.006	0.092	0.005	0.077	0.006	0.099	0.007	0.108	0.008	0.128
	$\hat{α}$	0.009	0.042	0.009	0.045	0.008	0.042	0.008	0.040	0.084	0.599	0.070	0.640	0.089	0.781	0.128	1.024
	$\hat{ν}$	0.052	0.715	0.041	0.602	0.034	0.534	0.029	0.502	0.020	0.512	0.035	0.498	0.040	0.531	0.048	0.603
1000	${\hat{β}}_{1}$	0.001	0.417	0.001	0.394	0.004	0.262	0.008	0.227	0.010	0.167	0.026	0.289	0.031	0.347	0.044	0.474
	${\hat{β}}_{2}$	0.001	0.031	0.001	0.025	0.001	0.017	0.001	0.015	0.001	0.009	0.001	0.010	0.001	0.012	0.001	0.020
	$\hat{η}$	0.001	0.091	0.002	0.095	0.003	0.082	0.004	0.078	0.003	0.069	0.005	0.083	0.005	0.093	0.006	0.107
	$\hat{α}$	0.008	0.035	0.001	0.040	0.006	0.036	0.006	0.035	0.068	0.489	0.051	0.519	0.063	0.609	0.089	0.778
	$\hat{ν}$	0.035	0.605	0.036	0.513	0.027	0.444	0.004	0.036	0.013	0.293	0.028	0.401	0.031	0.435	0.038	0.487

	$n$	Mean	SE	Skewness	Kurtosis
Complete Data	753	2.3746	3.2418	2.7778	15.7967
Non Censored Data	428	4.1777	3.3103	3.0801	16.8212

	PTCR		TCR		SLCR		NCR
$𝛉$	Estimate	SE	Estimate	SE	Estimate	SE	Estimate	SE
$β_{0}$	-8.8412	0.3785	-1.0474	1.1549	-1.1445	1.4377	-2.7510	1.7334
$β_{1}$	-0.1331	0.0004	-0.1108	0.0204	-0.1084	0.0232	-0.1046	0.0276
$β_{2}$	0.4887	0.0050	0.6475	0.0658	0.6434	0.0635	0.7281	0.0831
$β_{3}$	-3.4010	0.1869	-3.1636	0.3878	-3.0958	0.3785	-3.0264	0.4406
$β_{4}$	-0.3053	0.0149	-0.2964	0.1247	-0.2946	0.1210	-0.2143	0.1527
$η$	4.1538	0.1320	3.2616	0.1731	2.5958	0.3071	4.5760	0.1697
$ν$	6.5899	0.9014	4.2000	0.8068	1.4061	0.4361	–	–
$α$	12.8760	0.9871	–	–	–	–	–	–

Criterion	PTCR	TCR	SLCR	NCR
$ℓ (𝛉)$	$- 1413.942$	-1440.145	-1436.286	-1481.655
AIC	2843.883	2894.291	2886.573	2975.311
AICc	2828.077	2880.441	2872.723	2963.424
BIC	2880.876	2926.659	2918.941	3003.055

	PTCR		TCR		SLCR		NCR
$𝛉$	Estimate	SE	Estimate	SE	Estimate	SE	Estimate	SE
$β_{0}$	-1.9692	0.0658	-1.8923	0.1482	-0.5441	0.8733	-1.6478	0.5941
$β_{1}$	0.5353	0.0021	0.5091	0.0046	0.2719	0.0262	0.4372	0.0180
$η$	0.0608	0.0002	0.0873	0.0004	0.1987	0.0254	0.3843	0.0014
$ν$	2.0891	0.4530	0.8335	0.0419	1.0567	0.1652	–	–
$α$	0.3265	0.0084	–	–	–	–	–	–

Criterion	PTCR	TCR	SLCR	NCR
$ℓ (𝛉)$	3.688	-6.635	-25.096	-38.281
AIC	2.624	21.269	58.192	82.562
AICc	-6.408	13.905	50.827	76.937
BIC	13.722	30.148	67.069	89.220