A new model for describing remission times: the generalized beta-generated Lindley distribution

LIMA, MARIA DO CARMO S.; CORDEIRO, GAUSS M.; NASCIMENTO, ABRAÃO D.C.; SILVA, KÁSSIO F.

doi:10.1590/0001-3765201720160455

MARIA DO CARMO S. LIMA

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

GAUSS M. CORDEIRO

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

ABRAÃO D.C. NASCIMENTO

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

KÁSSIO F. SILVA

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

Correspondence to: Abraão David Costa do Nascimento E-mail: abraao.susej@gmail.com

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Pdfs

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Hrfs

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Illustration of ramon number generator

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GBGL skewness

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GBGL skewness

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GBGL skewness

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GBGL kurtosis

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GBGL kurtosis

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GBGL kurtosis

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Bonferroni curves

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Lorenz curves

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Pdf: GBGL

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L

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Cdf: GBGL

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L

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For pdfs.

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For cdfs.

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TABLE I
Simulation results for MLEs.

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TABLE II
Simulation results for LSEs.

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TABLE III
MLEs of fitted models to Stress data and likelihood ratio statistics.

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TABLE IV
MLEs of the fitted models to the current data.

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TABLE V
Goodness-of-fit measures.

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$𝜽$	$n$	MLEs				MSEs
$𝜽$	$n$	a	b	c	$λ$	a	b	c	$λ$
(2,2,2,0.5)	50	2.018	2.031	2.014	0.502	0.060	0.061	0.045	0.001
	100	2.023	2.002	2.019	0.500	0.029	0.026	0.021	0.000
	150	2.012	2.002	2.010	0.500	0.019	0.019	0.014	0.000
(5,5,5,0.5)	50	5.011	5.042	5.007	0.501	0.177	0.183	0.107	0.000
	100	5.008	5.014	5.006	0.500	0.078	0.077	0.047	0.000
	150	5.006	5.013	5.004	0.500	0.060	0.059	0.036	0.000
(2,2,2,1)	50	2.030	2.023	2.024	1.002	0.061	0.062	0.046	0.003
	100	2.015	2.008	2.013	1.001	0.027	0.026	0.020	0.001
	150	2.014	2.000	2.012	0.999	0.019	0.019	0.014	0.001
(5,5,5,1)	50	5.011	5.040	5.007	1.001	0.171	0.174	0.103	0.001
	100	5.001	5.024	5.000	1.001	0.085	0.088	0.052	0.000
	150	5.010	5.009	5.007	1.000	0.060	0.058	0.036	0.000
(2,2,2,2)	50	2.040	2.015	2.034	2.001	0.060	0.062	0.045	0.014
	100	2.010	2.017	2.008	2.005	0.030	0.030	0.023	0.007
	150	2.002	2.015	2.001	2.006	0.019	0.020	0.014	0.005
(5,5,5,2)	50	5.016	5.027	5.012	2.001	0.172	0.168	0.104	0.002
	100	5.004	5.017	5.002	2.001	0.086	0.088	0.052	0.001
	150	5.002	5.018	5.001	2.001	0.057	0.057	0.034	0.001
(2,2,2,3)	50	2.018	2.032	2.014	3.017	0.061	0.058	0.045	0.034
	100	2.016	2.008	2.013	3.002	0.030	0.027	0.022	0.016
	150	2.018	1.998	2.015	2.996	0.019	0.018	0.014	0.011
(5,5,5,3)	50	4.986	5.066	4.987	3.009	0.171	0.176	0.103	0.006
	100	5.013	5.019	5.009	3.001	0.087	0.087	0.053	0.003
	150	5.000	5.020	5.000	3.003	0.059	0.058	0.035	0.002
(2,2,2,4)	50	2.020	2.027	2.016	4.017	0.056	0.059	0.042	0.062
	100	2.006	2.018	2.004	4.015	0.027	0.029	0.020	0.031
	150	2.009	2.008	2.008	4.005	0.019	0.019	0.014	0.021
(5,5,5,4)	50	5.016	5.044	5.010	4.006	0.169	0.179	0.102	0.011
	100	5.017	5.016	5.012	4.001	0.094	0.090	0.057	0.006
	150	5.019	4.994	5.014	3.997	0.060	0.059	0.036	0.004

$θ$	$n$	LSEs				MSEs
$θ$	$n$	a	b	c	$λ$	a	b	c	$λ$
(2,2,2,0.5)	50	1.972	1.996	1.969	0.491	0.049	0.011	0.061	0.003
	100	1.992	1.995	1.991	0.498	0.021	0.009	0.026	0.001
	150	1.987	1.982	1.983	0.498	0.017	0.008	0.020	0.001
(5,5,5,0.5)	50	5.011	5.011	5.013	0.501	0.046	0.032	0.063	0.000
	100	5.041	5.036	5.048	0.502	0.060	0.040	0.083	0.000
	150	4.936	4.945	4.923	0.497	0.030	0.021	0.042	0.000
(2,2,2,1)	50	2.002	1.999	2.002	0.994	0.066	0.032	0.081	0.013
	100	1.998	1.998	1.999	1.000	0.034	0.035	0.041	0.008
	150	1.996	2.004	1.997	0.998	0.021	0.008	0.026	0.004
(5,5,5,1)	50	4.998	5.007	4.998	0.999	0.114	0.077	0.156	0.002
	100	5.037	5.026	5.043	1.003	0.116	0.079	0.157	0.001
	150	4.990	4.993	4.988	0.999	0.023	0.014	0.032	0.000
(2,2,2,2)	50	1.986	2.025	2.000	1.980	0.069	0.061	0.082	0.045
	100	2.001	1.993	2.004	2.004	0.031	0.040	0.038	0.024
	150	1.998	2.002	2.002	1.993	0.021	0.021	0.025	0.016
(5,5,5,2)	50	4.950	4.972	4.940	1.989	0.196	0.120	0.274	0.011
	100	4.942	4.955	4.929	1.986	0.130	0.081	0.181	0.008
	150	4.980	4.986	4.975	1.996	0.053	0.030	0.074	0.003
(2,2,2,3)	50	1.993	2.000	2.003	2.993	0.068	0.087	0.077	0.078
	100	2.005	1.968	2.004	3.063	0.037	0.107	0.036	0.122
	150	1.998	1.994	2.002	3.008	0.019	0.043	0.023	0.031
(5,5,5,3)	50	4.893	4.910	4.867	2.960	0.287	0.190	0.403	0.039
	100	5.010	5.014	5.011	2.998	0.132	0.085	0.183	0.016
	150	4.940	4.951	4.927	2.976	0.074	0.051	0.106	0.009
(2,2,2,4)	50	2.004	1.995	2.007	4.018	0.062	0.120	0.074	0.072
	100	1.983	2.011	2.004	4.001	0.039	0.096	0.037	0.109
	150	1.997	1.998	1.999	4.001	0.019	0.040	0.023	0.024
(5,5,5,4)	50	4.949	4.949	4.933	3.967	0.323	0.257	0.454	0.060
	100	5.004	5.012	5.004	3.992	0.169	0.123	0.234	0.033
	150	4.944	4.943	4.930	3.972	0.118	0.089	0.165	0.025

Model	MLEs (SEs)
Model	a	b	c	$λ$	$H_{0} : G B G L \times L (a = b = c = 1)$
GBG-L	0.096 (0.014)	0.065 (0.020)	47.284 (1.383)	10.859 (0.384)	8.311 ( $\hat{α} = 0.040$ )
GBG-L	L	0.177 (0.007)	-	-	-

Model	MLEs (SEs)
L	$\hat{λ} = 0.19 (0.0123)$
LE	$\hat{α} = 1.55 (0.1647)$ , $\hat{λ} = 0.11 (0.0136)$
GL	$\hat{α} = 0.73 (0.0917)$ , $\hat{λ} = 0.16 (0.0165)$
GBGL	$\hat{λ} = 0.24 (0.0335)$ , $\hat{a} = 0.03 (0.0077)$ ,
GBGL	$\hat{b} = 0.31 (0.0807)$ , $\hat{c} = 34.95 (6.8937)$
TL	$\hat{λ} = - 0.38 (0.1494)$ , $\hat{θ} = 0.03 (0.0028)$ ,
TL	$\hat{δ} = 0.73 (0.1347)$ , $\hat{α} = 0.73 (0.1833)$

Model	Dependent on the pdf	Dependent on the cdf
Model	AIC	KS	p-value
L	814.3574	0.1075	0.1163
LE	802.3400	0.0575	0.8105
GL	809.6681	0.1982	$1.273 \times 10^{- 4}$
GBGL	801.9561	0.0282	0.9999
TL	813.6681	0.088731	0.2875

= - 1 - \frac{1}{λ} - \frac{1}{λ} W (\frac{[1 + λ]}{e^{1 + λ}} [Q_{β (a, b)} {(u)}^{1 / c} - 1]),

= \frac{[Q_{GBGL} (7 / 8) - Q_{GBGL} (5 / 8)] + [Q_{GBGL} (3 / 8) - Q_{GBGL} (1 / 8)]}{Q_{GBGL} (6 / 8) - Q_{GBGL} (2 / 8)} .

τ_{i, k} = \sum_{l = 0}^{\infty} ζ_{l} w_{i, k} (l), w_{i, k} (l) = \frac{{(- 1)}^{i} v_{i, k}}{{(1 + λ)}^{i + 2} {(1 + i)}^{k + 1}} (\frac{(a + l) c - 1}{i}),

μ_{n}^{'} = E (X^{n}) = \sum_{i = 0}^{\infty} \sum_{k = 0}^{i + 1} τ_{i, k} {[(i + 1) λ]}^{n} {(k + 1)}^{[n]},

μ_{n}^{'} = \sum_{i = 0}^{\infty} \sum_{k = 0}^{i + 1} τ_{i, k} \int_{0}^{1} Q_{L} {(u)}^{n} u^{a (i + c) - 1} 𝑑 u .

\int_{0}^{y} x^{n} \sum_{i = 0}^{\infty} \sum_{k = 0}^{i + 1} w_{i, k} \frac{(λ + λ i)}{k!} {[(k + 1) x]}^{λ + λ i - 1} \exp [- (λ + λ i) x] 𝑑 x

\int_{0}^{1} \exp [s Q_{L} (u)] u^{a (i + c) - 1} 𝑑 u = \sum_{j, k = 0}^{\infty} \frac{s^{k} d_{k, j}}{[a (i + c) + j] k!},

𝕌 (𝜽) = (U_{a}, U_{b}, U_{c}, U_{λ}) = (\frac{\partial ℓ (𝜽; x)}{\partial a}, \frac{\partial ℓ (𝜽; x)}{\partial b}, \frac{\partial ℓ (𝜽; x)}{\partial c}, \frac{\partial ℓ (𝜽; x)}{\partial λ})

- ψ (b) + ψ (a + b) + \log {1 - {1 - e^{- λ x} [1 + \frac{λ x}{1 + λ}]}^{c}},

\frac{(b - 1) {[1 - e^{- λ x} (1 + \frac{λ x}{1 + λ})]}^{c} \log [1 - e^{- λ x} (1 + \frac{λ x}{1 + λ})]}{1 - {[1 - e^{- λ x} (1 + \frac{λ x}{1 + λ})]}^{c}}

\frac{(a c - 1) {x e^{- λ x} (1 + \frac{λ x}{1 + λ}) - e^{- λ x} [\frac{x}{1 + λ} - \frac{λ x}{{(1 + λ)}^{2}}]}}{1 - e^{- λ x} (1 + \frac{λ x}{1 + λ})}

\frac{c (b - 1) {x e^{- λ x} (1 + \frac{λ x}{1 + λ}) - e^{- λ x} [\frac{x}{1 + λ} - \frac{λ x}{{(1 + λ)}^{2}}]}}{1 - {[1 - e^{- λ x} (1 + \frac{λ x}{1 + λ})]}^{c}}

J = J (𝜽) \equiv (\begin{matrix} J_{a a} & J_{a b} & J_{a c} & J_{a λ} \\ J_{b a} & J_{b b} & J_{b c} & J_{b λ} \\ J_{c a} & J_{c b} & J_{c c} & J_{c λ} \\ J_{λ a} & J_{λ b} & J_{λ c} & J_{λ λ} \end{matrix}),

Q (a, b, c, λ) = \sum_{i = 1}^{n} {I ({[1 - e^{- λ x_{(i)}} (1 + \frac{λ x_{(i)}}{1 + λ})]}^{c}; a, b) - \frac{i}{n + 1}}^{2},

\frac{\partial Q (a, b, c, λ)}{\partial a} = \frac{\partial Q (a, b, c, λ)}{\partial b} = \frac{\partial Q (a, b, c, λ)}{\partial c} = \frac{\partial Q (a, b, c, λ)}{\partial λ} = 0,

2 {I ({[1 - e^{- λ x_{(i)}} (1 + \frac{λ x_{(i)}}{1 + λ})]}^{c}; a, b) - \frac{i}{n + 1}}

\times {- \frac{B^{(a)} (a, b)}{B (a, b)} I ({[1 - e^{- λ x_{(i)}} (1 + \frac{λ x_{(i)}}{1 + λ})]}^{c}; a, b)

+ \frac{1}{B (a, b)} \int_{0}^{{1 - e^{- λ x_{(i)}} [1 + \frac{λ x_{(i)}}{1 + λ}]}^{c}} \log (w) ω^{a - 1} {(1 - ω)}^{b - 1} d ω},

2 {I ({[1 - e^{- λ x_{(i)}} (1 + \frac{λ x_{(i)}}{1 + λ})]}^{c}; a, b) - \frac{i}{n + 1}}

\times {- \frac{B^{(b)} (a, b)}{B (a, b)} I ({[1 - e^{- λ x_{(i)}} (1 + \frac{λ x_{(i)}}{1 + λ})]}^{c}; a, b)

+ \frac{1}{B (a, b)} \int_{0}^{{1 - e^{- λ x_{(i)}} [1 + \frac{λ x_{(i)}}{1 + λ}]}^{c}} \log (1 - w) ω^{a - 1} {(1 - ω)}^{b - 1} d ω},

2 {I ({[1 - e^{- λ x_{(i)}} (1 + \frac{λ x_{(i)}}{1 + λ})]}^{c}; a, b) - \frac{i}{n + 1}}

{[1 - e^{- λ x_{(i)}} (1 + \frac{λ x_{(i)}}{1 + λ})]}^{c} \log [1 - e^{- λ x_{(i)}} (1 + \frac{λ x_{(i)}}{1 + λ})]}

2 {I ({[1 - e^{- λ x_{(i)}} (1 + \frac{λ x_{(i)}}{1 + λ})]}^{c}; a, b) - \frac{i}{n + 1}}

\int_{0}^{{[1 - e^{- λ x_{(i)}} (1 + \frac{λ x_{(i)}}{1 + λ})]}^{c}} \log (w) w^{a - 1} {(1 - w)}^{b - 1} d w =

\times H_{q}^{p} ({a, a, 1 - b}, {1 + a, 1 + a}, {[1 - e^{- λ x_{(i)}} (1 + \frac{λ x_{(i)}}{1 + λ})]}^{c})

+ c I ({[1 - e^{- λ x_{(i)}} (1 + \frac{λ x_{(i)}}{1 + λ})]}^{c}; a, b) \log [1 - e^{- λ x_{(i)}} (1 + \frac{λ x_{(i)}}{1 + λ})]

f (x) = \frac{c λ^{2} (1 + x)}{(1 + λ) B (a, b)} e^{- λ x} {[1 - e^{- λ x} (1 + \frac{λ x}{1 + λ})]}^{ac - 1} A (x; λ, b, c),

A (x; λ, b, c) = \sum_{n = 0}^{\infty} {(- 1)}^{n} (\frac{b - 1}{n}) {[1 - e^{- λ x} (1 + \frac{λ x}{1 + λ})]}^{n c} .

J (x; c, λ) = \int_{0}^{x} (1 + s) e^{- λ s} {[1 - \frac{1 + λ + λ s}{1 + λ} e^{- λ s}]}^{c - 1} ds .

\sum_{i = 0}^{\infty} \frac{{(- 1)}^{i} (\frac{c - 1}{i})}{{(1 + λ)}^{i}} \int_{0}^{x} (1 + s) {(1 + λ + λ s)}^{i} \exp (- λ i s - λ s) ds

\sum_{i = 0}^{\infty} \frac{{(- 1)}^{i} (\frac{c - 1}{i})}{{(1 + λ)}^{i}} \sum_{j = 0}^{i} (\frac{i}{j}) λ^{j} \int_{0}^{x} {(1 + s)}^{j + 1} \exp [- λ s (1 + i)] ds

\sum_{i = 0}^{\infty} \sum_{j = 0}^{i} \sum_{k = 0}^{j + 1} \frac{{(- 1)}^{i} λ^{j} (\frac{c - 1}{i}) (\frac{i}{j})}{{(1 + λ)}^{i}} \int_{0}^{x} s^{(k + 1) - 1} \exp [- (1 + i) λ s] ds .

\frac{c λ^{2}}{1 + λ} J (x; c, λ) = \sum_{i = 0}^{\infty} \sum_{j = 0}^{i} \sum_{k = 0}^{j + 1} \frac{{(- 1)}^{i} λ^{j - k + 1} k!}{{(1 + λ)}^{i + 2} {(1 + i)}^{k + 1}}

F (x) = \sum_{i = 0}^{\infty} \sum_{k = 0}^{i + 1} \sum_{j = δ_{k}}^{i} \frac{{(- 1)}^{i} λ^{j - k + 1} k!}{{(1 + λ)}^{i + 2} {(1 + i)}^{k + 1}} (\frac{c - 1}{i}) H_{i, k} (x)

F (x) = \sum_{i = 0}^{\infty} \sum_{k = 0}^{i + 1} \frac{{(- 1)}^{i} v_{i, k}}{{(1 + λ)}^{i + 2} {(1 + i)}^{k + 1}} (\frac{c - 1}{i}) H_{i, k} (x),

f (x) = \sum_{l, i = 0}^{\infty} \sum_{k = 0}^{i + 1} ζ_{l} w_{i, k} (l) h_{i, k} (x) \sum_{i = 0}^{\infty} \sum_{k = 0}^{i + 1} τ_{i, k} h_{i, k} (x),

τ_{i, k} = \sum_{l = 0}^{\infty} ζ_{l} w_{i, k} (l), w_{i, k} (l) = \frac{{(- 1)}^{i} v_{i, k}}{{(1 + λ)}^{i + 2} {(1 + i)}^{k + 1}} (\frac{(a + l) c - 1}{i})

w + \frac{w^{2}}{a + 1} + \frac{(3 a + 5) w^{3}}{2 {(a + 1)}^{2} (a + 2)} + \frac{[a (8 a + 33) + 31] w^{4}}{3 {(a + 1)}^{3} (a + 2) (a + 3)}

\frac{{a (a [a (125 a + 1179) + 3971] + 5661) + 2888} w^{5}}{24 {(a + 1)}^{4} {(a + 2)}^{2} (a + 3) (a + 4)} + 𝑂 (w^{6}),

\frac{1}{λ (- 1 + \frac{1}{1 + λ})} \sum_{i = 0}^{\infty} ϱ_{i} x^{i} = (\frac{1 + λ}{λ^{2}}) \sum_{i = 0}^{\infty} {\bar{ϱ}}_{i} x^{i},

{\frac{d^{k - 1}}{{dx}^{k - 1}} {{[ψ (x)]}^{k}} |}_{x = x_{0}} = \frac{ν_{k, k - 1} {(1 + λ)}^{k} (k - 1)!}{λ^{2 k}},

π_{k} = {\frac{1}{k!} \frac{d^{k - 1}}{{dx}^{k - 1}} {{[ψ (x)]}^{k}} |}_{x = x_{0}} = \frac{ν_{k, k - 1} {(1 + λ)}^{k}}{k λ^{2 k}} .

= \frac{{x e^{- λ x} (1 + \frac{λ x}{1 + λ}) - e^{- λ x} [\frac{x}{1 + λ} - \frac{λ x}{{(1 + λ)}^{2}}]}}{1 - e^{- λ x} (1 + \frac{λ x}{1 + λ})},

= \frac{{[1 - e^{- λ x} (1 + \frac{λ x}{1 + λ})]}^{c} \log [1 - e^{- λ x} (1 + \frac{λ x}{1 + λ})]}{1 - {[1 - e^{- λ x} (1 + \frac{λ x}{1 + λ})]}^{c}},

- \frac{c [x e^{- λ x} (1 + \frac{λ x}{1 + λ}) - e^{- λ x} (\frac{x}{1 + λ} - \frac{λ x}{{(1 + λ)}^{2}})]}{1 - {[1 - e^{- λ x} (1 + \frac{λ x}{1 + λ})]}^{c}}

- \frac{(b - 1) {[1 - e^{- λ x (\frac{λ x}{λ + 1} + 1)}]}^{c} {[1 - e^{- λ x} (\frac{λ x}{λ + 1} + 1)]}^{c}}{{1 - {[1 - e^{- λ x (\frac{λ x}{λ + 1} + 1)}]}^{c}}^{2}}

\frac{(b - 1) {[1 - e^{- λ x} (\frac{λ x}{λ + 1} + 1)]}^{c} \log [1 - e^{- λ x (\frac{λ x}{λ + 1} + 1)}]}{1 - {[1 - e^{- λ x (\frac{λ x}{λ + 1} + 1)}]}^{c}}

- \frac{a e^{- λ x (\frac{λ x}{λ + 1} + 1)} [- λ x (\frac{x}{λ + 1} - \frac{λ x}{{(λ + 1)}^{2}}) - x (\frac{λ x}{λ + 1} + 1)]}{1 - e^{- λ x (\frac{λ x}{λ + 1} + 1)}}

\frac{(b - 1) e^{- λ x (\frac{λ x}{λ + 1} + 1)} [- λ x (\frac{x}{λ + 1} - \frac{λ x}{{(λ + 1)}^{2}}) - x (\frac{λ x}{λ + 1} + 1)]}{[1 - e^{- λ x (\frac{λ x}{λ + 1} + 1)}] {1 - {[1 - e^{- λ x (\frac{λ x}{λ + 1} + 1)}]}^{c}}}

\frac{c (b - 1) e^{- λ x (\frac{λ x}{λ + 1} + 1)} [- λ x (\frac{x}{λ + 1} - \frac{λ x}{{(λ + 1)}^{2}}) - x (\frac{λ x}{λ + 1} + 1)]}{{1 - {[1 - e^{- λ x (\frac{λ x}{λ + 1} + 1)}]}^{c}}^{2}}

{[1 - e^{- λ x} (\frac{λ x}{λ + 1} + 1)]}^{c} {[1 - e^{- λ x (\frac{λ x}{λ + 1} + 1)}]}^{c - 1} \log [1 - e^{- λ x (\frac{λ x}{λ + 1} + 1)}]

\frac{(b - 1) c [x e^{- λ x} (\frac{λ x}{λ + 1} + 1) - e^{- λ x} (\frac{x}{λ + 1} - \frac{λ x}{{(λ + 1)}^{2}})]}{1 - {[1 - e^{- λ x (\frac{λ x}{λ + 1} + 1)}]}^{c}}

{[1 - e^{- λ x} (\frac{λ x}{λ + 1} + 1)]}^{c - 1} \log [1 - e^{- λ x (\frac{λ x}{λ + 1} + 1)}] .

A new model for describing remission times: the generalized beta-generated Lindley distribution

Abstract