A generalisation of the exponential distribution and its applications on modelling skewed data

Figures & data

Figure 1. Density function plots of the WEE distribution.

Figure 2. Hazard rate plots of the WEE distribution.

Figure 3. Skewness (a) and kurtosis (b) of X based on quantiles when .

Table 1. Monte Carlo simulation results: Bias, MSE, CP, LB and UB.

Parameter	n	Bias	MSE	CP	LB	UB
			I
c	25	0.322	0.178	1.00	0.775	2.400
	50	0.227	0.098	1.00	0.329	1.699
	75	0.180	0.059	1.00	0.218	1.388
	100	0.148	0.041	0.98	0.212	1.196
	200	0.096	0.018	0.97	0.314	0.887
α	25	−0.141	0.049	0.99	0.402	1.113
	50	−0.115	0.031	0.97	0.212	0.955
	75	−0.112	0.025	0.97	0.134	0.837
	100	−0.101	0.020	0.96	0.122	0.782
	200	−0.076	0.012	0.95	0.189	0.670
λ	25	−0.343	0.615	0.88	1.833	3.128
	50	−0.323	0.315	0.88	1.119	2.434
	75	−0.312	0.221	0.92	0.770	2.077
	100	−0.279	0.177	0.94	0.599	1.910
	200	−0.215	0.096	0.96	0.308	1.566
			II
c	25	0.400	0.498	1.00	0.849	2.503
	50	0.200	0.137	1.00	0.343	1.321
	75	0.145	0.052	1.00	0.323	1.012
	100	0.121	0.029	1.00	0.356	0.891
	200	0.082	0.011	0.99	0.427	0.737
α	25	−0.411	0.673	0.88	0.600	2.765
	50	−0.354	0.340	0.94	0.214	2.286
	75	−0.333	0.243	0.95	0.305	2.072
	100	−0.322	0.196	0.96	0.422	1.946
	200	−0.264	0.120	0.96	0.708	1.769
λ	25	−0.536	1.333	0.87	1.078	3.967
	50	−0.453	0.704	0.89	0.380	3.234
	75	−0.403	0.463	0.90	0.383	2.946
	100	−0.384	0.358	0.93	0.517	2.763
	200	−0.311	0.202	0.95	0.910	2.469
			III
c	25	0.346	0.995	0.86	2.827	6.460
	50	0.343	0.921	0.87	1.979	5.522
	75	0.305	0.790	0.89	1.502	4.879
	100	0.305	0.756	0.90	1.357	4.652
	200	0.267	0.599	0.88	1.028	4.002
α	25	0.305	1.019	1.00	0.935	2.544
	50	0.135	0.279	0.99	0.479	1.737
	75	0.090	0.160	0.99	0.338	1.483
	100	0.063	0.121	0.97	0.282	1.344
	200	0.026	0.071	0.96	0.225	1.109
λ	25	0.439	2.191	0.99	2.105	4.934
	50	0.212	1.120	0.96	1.406	3.692
	75	0.141	0.774	0.96	1.078	3.198
	100	0.091	0.620	0.94	0.937	2.917
	200	0.026	0.397	0.94	0.732	2.440

Table 2. MLEs and their standard errors (in parentheses) for the guinea pigs data.

Distribution
WEE	–	–	0.6244	–	7.6518	0.0301
	–	–	(0.0983)	–	(3.0734)	(0.0057)
GEE	–	–	–	0.2177	4.7533	0.0645
	–	–	–	(0.1115)	(2.4874)	(0.0284)
BE	5.1575	0.2159	–	–	–	0.0649
	(2.5306)	(0.1085)	–	–	–	(0.0280)

Table 3. MLEs and their standard errors (in parentheses) for the glass fibres data.

Distribution
WEE	–	–	14.5240	–	0.2564	0.1122
	–	–	(12.9062)	–	(0.2777)	(0.2384)
GEE	–	–	–	18.8149	24.6684	0.4336
	–	–	–	(20.3751)	(7.0427)	(0.3939)
BE	17.5097	59.6132	–	–	–	0.1721
	(3.1108)	(103.9638)	–	–	–	(0.2683)

Table 4. The statistics AIC, BIC, , , K–S and K–S p-value for the guinea pigs data.

Distribution	AIC	BIC			K–S	K–S p-value
WEE	785.1445	791.9745	0.6372	0.1145	0.0966	0.5126
GEE	787.5205	794.3505	1.0316	0.1924	0.1088	0.3617
BE	787.5252	794.3552	1.0309	0.1922	0.1089	0.3600

Table 5. The statistics AIC, BIC, , , K–S and K–S p-value for the glass fibres data.

Distribution	AIC	BIC			K–S	K–S p-value
WEE	35.6519	42.0814	1.1974	0.2171	0.1478	0.1277
GEE	55.0189	61.4483	3.2183	0.5870	0.2186	0.0049
BE	53.9716	60.4010	3.1236	0.5695	0.2164	0.0055

Figure 4. QQ-plot of data set 1.

Figure 5. QQ-plot of data set 2.

Figure 6. TTT plots (a) and (b) data sets 1 and 2.