A competitive family to the Beta and Kumaraswamy generators: Properties, Regressions and Applications

CORDEIRO, GAUSS M.; VASCONCELOS, JULIO CEZAR S.; ORTEGA, EDWIN M.M.; MARINHO, PEDRO RAFAEL D.

doi:10.1590/0001-3765202220201972

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MATHEMATICAL SCIENCES • An. Acad. Bras. Ciênc. 94 (2) • 2022 • https://doi.org/10.1590/0001-3765202220201972 copy

A competitive family to the Beta and Kumaraswamy generators: Properties, Regressions and Applications

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

We define two new flexible families of continuous distributions to fit real data by compoun-ding the Marshall–Olkin class and the power series distribution. These families are very competitive to the popular beta and Kumaraswamy generators. Their densities have linear representations of exponentiated densities. In fact, as the main properties of thirty five exponentiated distributions are well-known, we can easily obtain several properties of about three hundred fifty distributions using the references of this article and five special cases of the power series distribution. We provide a package implemented in R software that shows numerically the precision of one of the linear representations. This package is useful to calculate numerical values for some statistical measurements of the generated distributions. We estimate the parameters by maximum likelihood. We define a regression based on one of the two families. The usefulness of a generated distribution and the associated regression is proved empirically.

Key words
generating function; Marshall–Olkin family; maximum likelihood; moment; distribution

	$n = 100$			$n = 250$
Parameter	AE	Bias	MSE	AE	Bias	MSE
$λ$	3.001	0.001	0.005	2.996	-0.00	0.005
$β$	0.998	-0.002	0.013	1.004	0.004	0.012
$θ$	1.585	0.085	0.081	1.567	0.0667	0.064
$α$	0.569	-0.130	0.090	0.563	-0.137	0.089

	$n = 500$			$n = 1, 000$
Parameter	AE	Bias	MSE	AE	Bias	MSE
$λ$	2.994	-0.006	0.004	2.995	-0.006	0.003
$β$	1.006	0.006	0.008	1.007	0.007	0.005
$θ$	1.546	0.046	0.035	1.526	0.026	0.016
$α$	0.597	-0.103	0.077	0.632	-0.068	0.063

scenario 1
	$n = 100$			$n = 500$			$n = 1, 000$
Parameter	AE	Bias	MSE	AE	Bias	MSE	AE	Bias	MSE
$γ_{10}$	0.614	0.114	0.074	0.557	0.057	0.029	0.527	0.027	0.017
$γ_{11}$	0.785	-0.015	0.031	0.792	-0.008	0.006	0.798	-0.002	0.002
$γ_{20}$	0.256	-0.044	0.031	0.271	-0.029	0.012	0.285	-0.015	0.007
$γ_{21}$	0.101	0.001	0.031	0.101	0.001	0.006	0.102	0.002	0.003
$θ$	0.734	0.134	0.164	0.651	0.051	0.093	0.621	0.021	0.070
$α$	0.477	0.077	0.089	0.440	0.040	0.072	0.413	0.013	0.067
scenario 2
	$n = 100$			$n = 500$			$n = 1, 000$
Parameter	AE	Bias	MSE	AE	Bias	MSE	AE	Bias	MSE
$γ_{10}$	0.684	0.184	0.149	0.567	0.067	0.045	0.528	0.028	0.025
$γ_{11}$	0.779	-0.021	0.044	0.790	-0.009	0.007	0.798	-0.002	0.004
$γ_{20}$	0.235	-0.065	0.045	0.272	-0.028	0.015	0.289	-0.011	0.009
$γ_{21}$	0.096	-0.004	0.035	0.103	0.003	0.007	0.100	0.000	0.003
$θ$	0.637	0.037	0.157	0.578	-0.023	0.086	0.558	-0.042	0.077
$α$	1.722	0.322	0.382	1.530	0.130	0.180	1.467	0.067	0.123
scenario 3
	$n = 100$			$n = 500$			$n = 1, 000$
Parameter	AE	Bias	MSE	AE	Bias	MSE	AE	Bias	MSE
$γ_{10}$	0.337	-0.161	0.079	0.483	-0.017	0.028	0.494	-0.007	0.019
$γ_{11}$	0.819	0.019	0.019	0.802	0.002	0.005	0.799	-0.001	0.002
$γ_{20}$	0.465	0.165	0.069	0.323	0.023	0.014	0.311	0.012	0.009
$γ_{21}$	0.094	-0.006	0.033	0.101	0.001	0.005	0.101	0.001	0.003
$θ$	1.349	-0.350	0.258	1.643	-0.057	0.035	1.679	-0.022	0.015
$α$	0.460	0.060	0.083	0.429	0.029	0.079	0.407	0.007	0.069
scenario 4
	$n = 100$			$n = 500$			$n = 1, 000$
Parameter	AE	Bias	MSE	AE	Bias	MSE	AE	Bias	MSE
$γ_{10}$	0.549	0.049	0.132	0.551	0.051	0.036	0.495	-0.005	0.015
$γ_{11}$	0.796	-0.004	0.038	0.795	-0.006	0.006	0.798	-0.002	0.003
$γ_{20}$	0.332	0.032	0.054	0.286	-0.014	0.012	0.307	0.007	0.006
$γ_{21}$	0.096	-0.004	0.032	0.100	0.000	0.005	0.103	0.003	0.003
$θ$	1.406	-0.294	0.222	1.643	-0.057	0.029	1.684	-0.016	0.013
$α$	1.913	0.513	0.739	1.604	0.204	0.240	1.408	0.008	0.090

Model	$log (λ)$	$log (β)$	$θ$	$α$	AIC	BIC	GD
MOTPW	-2.720	0.694	11.210	0.019	3031.288	3048.410	3023.288
	(0.141)	(0.085)	(3.020)	(0.007)
TPW	0.248	-0.541	31.100	(-)	3035.448	3048.289	3029.448
	(0.444)	(0.112)	(12.540)	(-)
Model	$log (λ)$	$a$	$b$	$log (c)$	AIC	BIC	GD
KwW	-0.601	12.124	0.317	0.060	3034.039	3051.160	3026.039
	(0.023)	(0.802)	(0.013)	(0.014)
BW	-5.453	2.327	126.000	0.216	3084.086	3101.208	3076.086
	(0.020)	(0.067)	(0.009)	(0.007)

Model	AIC	BIC	GD
MOTPW	-194.316	-170.502	-210.316
TPW	-191.726	-170.889	-205.726
KwW	-188.769	-164.955	-204.769
BW	-185.607	-161.793	-201.607

Parameter	Estimate	SE	p-Value
$η_{10}$	0.065	0.093	0.489
$η_{11}$	-0.119	0.036	0.001
$η_{12}$	-0.049	0.028	0.082
$η_{20}$	1.719	0.245	$<$ 0.001
$η_{21}$	0.373	0.140	0.009
$η_{22}$	0.131	0.141	0.355
$θ$	12.401	8.866
$α$	0.095	0.079

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[1] Correspondence to: Julio Cezar Souza Vasconcelos E-mail: juliocezarvasconcelos@hotmail.com