The generalized weighted Lindley distribution: Properties, estimation, and applications

Figures & data

Figure 1. Density function shapes for GWL distribution considering different values of $\mathit{\varphi },\mathit{\lambda }$ and $\mathit{\alpha }$.

Figure 2. Hazard function shapes for GWL distribution and considering different values of $\mathit{\varphi },\mathit{\lambda }$ and $\mathit{\alpha }$.

Figure 3. Proportion of failure from N simulated samples, considering different values of n using the following estimation method 1-MLE, 2-MPS, 3-ADE, 4-RTADE, 5-LSE, 6-WLSE, 7-ME, 8-CME.

Figure 4. MREs, MSEs related from the estimates of $\mathit{\varphi }=0.5,\mathit{\lambda }=0.7$ and $\mathit{\alpha }=1.5$ for N simulated samples, considering different values of n obtained using the following estimation method 1-MLE, 2-MPS, 3-ADE, 4-RTADE.

Figure 5. (left panel) the TTT-plot, (middle panel) the fitted survival superimposed to the empirical survival function and (right panels) the hazard function adjusted by GWL distribution.

Figure 6. (left panel) the TTT-plot, (middle panel) the fitted survival superimposed to the empirical survival function and (right panels) the hazard function adjusted by GWL distribution.

Table 1. Lifetimes data (in hours) related to a device on test

0.1	0.2	1	1	1	1	1	2	3	6	7	11	12
18	18	18	18	18	21	32	36	40	45	46	47	50
55	60	63	63	67	67	67	67	72	15	79	82	82
83	84	84	84	85	85	85	85	85	86	86

Table 2. Results of AIC and AICc criteria and the p-value from the KS test for all fitted distributions considering the Aarset dataset

Criteria	Gen. WL	Gen. Gamma	Gen. Weibull	Exp. Weibull	Gen. EP
AIC	418.031	448.294	430.055	463.674	486.255
AICc	412.552	442.816	424.576	458.196	480.777
KS	0.5787	0.0115	0.0453	0.0222	0.0302

Table 3. MPS estimates, standard-error and $95\%$ CI for $\mathit{\varphi },\mathit{\lambda }$ and $\mathit{\alpha }$

$\mathit{\theta }$	${\widehat{\mathit{\theta }}}_{MPS}$	S.E($\widehat{\mathit{\theta }}$)	CI${}_{95\%}(\mathit{\theta })$
$\mathit{\varphi }$	0.0057	0.00091	( 0.0039; 0.0075)
$\mathit{\lambda }$	0.0118	0.00003	(0.0117; 0.0118)
$\mathit{\alpha }$	110.4964	6.58144	(97.5971; 123.3958)

Table 4. January average flows (m${}^3$/s) of the Cantareira system

82.0	80.9	102.5	65.3	65.5	47.1	53.0	139.4	82.4	80.2	92.5
50.0	50.4	50.2	36.2	35.9	100.0	94.2	78.1	54.8	86.9	80.1
60.3	26.9	48.5	51.0	51.1	84.5	76.9	69.4	77.3	109.2	55.3
106.3	30.5	94.2	87.3	115.0	70.0	31.3	87.1	35.9	67.7	55.1
89.9	50.1	52.6	82.0	54.1	44.3	69.2	94.4	83.4	122.7	88.1
73.3	35.9	82.4	64.9	90.8	80.4	55.3	31.4	45.7	43.6	45.8
96.8	85.8	43.6	122.3	66.5	41.0	75.4	79.4	34.8	78.8	52.4
77.1	47.0	67.4	132.8	144.9	64.1

Table 5. Results of AIC and AICc criteria and the p-value from the KS test for all fitted distributions considering the data-set related to the january average flows (m${}^3$/s) of the Cantareira system

Criteria	Gen. WL	Gen. Gamma	Gen. Weibull	Exp. Weibull	Gen. EP
AIC	775.431	775.461	777.280	780.304	778.873
AICc	769.735	769.765	771.584	774.608	773.176
KS	0.4683	0.4223	0.3935	0.1654	0.4599

Table 6. ML estimates, standard-error and $95\%$ CI for $\mathit{\varphi },\mathit{\lambda }$ and $\mathit{\alpha }$

$\mathit{\theta }$	${\widehat{\mathit{\theta }}}_{\text{MLE}}$	S.E($\widehat{\mathit{\theta }}$)	CI${}_{95\%}(\mathit{\theta })$
$\mathit{\varphi }$	7.0485	1.5425	(2.3847; 11.7124)
$\mathit{\lambda }$	0.1244	0.0557	(0.1183; 0.1305)
$\mathit{\alpha }$	0.9579	0.1173	(0.9310; 0.9849)