A New Class of Exponentiated Beta-Skew-Laplace Distribution

TOVAR-FALÓN, ROGER; MARTÍNEZ-FLÓREZ, GUILLERMO

doi:10.1590/0001-3765202220191597

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MATHEMATICAL SCIENCES • An. Acad. Bras. Ciênc. 94 (4) • 2022 • https://doi.org/10.1590/0001-3765202220191597 copy

A New Class of Exponentiated Beta-Skew-Laplace Distribution

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

This paper introduce two new families of distributions that allow fitting unimodal, bimodal or trimodal data sets. Statistical properties such as distribution function, moments, moment generating function and stochastic representation of these new families are studied in details. The problem of estimating parameters is addressed by considering the maximum likelihood method and Fisher information matrices are derived. A small Monte Carlo simulation study is conducted to examine the performance of the obtained estimators. The methodology developed is illustrated with three real data applications.

Key words
Asymmetry; beta-skew-Laplace distribution; information matrix; maximum likelihood estimation; multimodality

𝛽	$n$	Estimates of $μ$		Estimates of $σ$		Estimates of $β$
𝛽	$n$	Bias	MSE	Bias	MSE	Bias	MSE
−1.50	100	−0.0816	0.2139	−0.0121	0.0935	0.0375	0.0827
	200	−0.0533	0.1522	−0.0101	0.0717	0.0109	0.0561
	500	−0.0149	0.0813	−0.0084	0.0309	0.0085	0.0406
	1,000	−0.0081	0.0412	−0.0055	0.0092	0.0024	0.0205
−0.15	100	−0.0567	0.1893	−0.0053	0.0848	0.0158	0.0619
	200	−0.0381	0.1066	−0.0051	0.0629	0.0048	0.0302
	500	−0.0051	0.0450	−0.0038	0.0114	0.0038	0.0265
	1,000	−0.0029	0.0283	−0.0027	0.0097	0.0009	0.0194
0	100	0.0316	0.1401	−0.0858	0.0759	0.0071	0.0230
	200	0.0118	0.0574	−0.0460	0.0503	0.0034	0.0096
	500	0.0007	0.0121	−0.0177	0.0111	0.0012	0.0039
	1.000	0.0019	0.0045	−0.0085	0.0041	0.0005	0.0010
0.15	100	0.0212	0.1716	0.0011	0.0917	−0.0058	0.0736
	200	0.0091	0.0916	0.0007	0.0678	−0.0011	0.0313
	500	0.0042	0.0434	0.0002	0.0213	−0.0004	0.0114
	1,000	0.0034	0.0209	0.0001	0.0086	−0.0003	0.0093
2.00	100	0.0973	0.2315	0.0132	0.1055	−0.0410	0.0882
	200	0.0575	0.1589	0.0097	0.0721	−0.0179	0.0441
	500	0.0238	0.0721	0.0082	0.0294	−0.0075	0.0209
	1,000	0.0097	0.0294	0.0035	0.0108	−0.0017	0.0109

Estimates	Distribution
Estimates	ASL	BSL	ABSL	EBSL
$\hat{μ}$	69.001(0.682)	68.345(0.740)	68.996(0.623)	64.000(0.529)
$\hat{σ}$	4.295(0.160)	1.942(0.046)	2.130(0.089)	2.166(0.053)
$\hat{α}$	-4.009(1.150)	–	-1.216(0.428)	1.772(0.123)
$\hat{β}$	–	-0.169(0.022)	-0.175(0.041)	0.169(0.021)
$\hat{ℓ}$	-1057.236	-1054.877	-1046.016	-1038.073
AIC	2120.472	2115.754	2100.032	2084.146
BIC	2131.289	2126.571	2112.206	2098.569
CAIC	2139.451	2134.733	2125.338	2109.452

Estimates	Distribution
Estimates	CASL	CBSL	CABSL	CEBSL
$\hat{μ}$	0.7493(0.0003)	0.4495(0.0028)	0.5520(0.0066)	1.3891(0.0003)
$\hat{σ}$	0.1552(0.0003)	0.0873(0.0001)	0.0792 (0.0001)	0.0068(0.0000)
$\hat{α}$	-1.4180(0.0786)	–	-0.5165(0.2665)	0.0146(0.0000)
$\hat{β}$	–	-0.1974 0.0043	-0.1708(0.0098)	0.2682(1.2871)
$\hat{ℓ}$	-32.330	-38.577	-38.749	-23.493
AIC	70.660	83.154	85.499	54.986
BIC	77.319	89.813	94.378	63.864
CAIC	80.319	92.813	98.377	67.864

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