Figure 1.
The probability mass function (PMF) of the discrete Gompertz exponential (DGzEx) distribution for different values of the parameters.
Figure 1.
The probability mass function (PMF) of the discrete Gompertz exponential (DGzEx) distribution for different values of the parameters.
Figure 2.
The hazard rate function (HRF) of the DGzEx distribution for different values of the parameters.
Figure 2.
The hazard rate function (HRF) of the DGzEx distribution for different values of the parameters.
Figure 3.
The PMF of the DGz-Weibull (DGzW) distribution for different values of the parameters.
Figure 3.
The PMF of the DGz-Weibull (DGzW) distribution for different values of the parameters.
Figure 4.
The HRF of the DGzW distribution for different values of the parameters.
Figure 4.
The HRF of the DGzW distribution for different values of the parameters.
Figure 5.
The PMF of the DGz-inverse Weibull (DGzIW) distribution for different values of the parameters.
Figure 5.
The PMF of the DGz-inverse Weibull (DGzIW) distribution for different values of the parameters.
Figure 6.
The HRF of the DGzIW distribution for different values of the parameters.
Figure 6.
The HRF of the DGzIW distribution for different values of the parameters.
Figure 7.
The bias of and versus for the DGzEx.
Figure 7.
The bias of and versus for the DGzEx.
Figure 8.
The MSE of and versus for the DGzEx.
Figure 8.
The MSE of and versus for the DGzEx.
Figure 9.
The bias of and versus for the DGzW().
Figure 9.
The bias of and versus for the DGzW().
Figure 10.
The MSE of and versus for the DGzW().
Figure 10.
The MSE of and versus for the DGzW().
Figure 11.
The bias of and versus for the DGzIW().
Figure 11.
The bias of and versus for the DGzIW().
Figure 12.
The MSE of and versus for the DGzIW().
Figure 12.
The MSE of and versus for the DGzIW().
Figure 13.
The estimated CDFs for Dataset 1.
Figure 13.
The estimated CDFs for Dataset 1.
Figure 14.
The probability-probability (P-P) plots for Dataset 1.
Figure 14.
The probability-probability (P-P) plots for Dataset 1.
Figure 15.
The estimated cumulative distribution functions (CDFs) for Dataset 2.
Figure 15.
The estimated cumulative distribution functions (CDFs) for Dataset 2.
Figure 16.
The P-P plots for Dataset 2.
Figure 16.
The P-P plots for Dataset 2.
Figure 17.
The fitted PMFs for Dataset 3.
Figure 17.
The fitted PMFs for Dataset 3.
Figure 18.
The P-P plots for Dataset 3.
Figure 18.
The P-P plots for Dataset 3.
Figure 19.
The fitted PMFs for Dataset 4.
Figure 19.
The fitted PMFs for Dataset 4.
Figure 20.
The P-P plots for Dataset 4.
Figure 20.
The P-P plots for Dataset 4.
Table 1.
Some descriptive statistics using the DGzEx model.
Table 1.
Some descriptive statistics using the DGzEx model.
Measure | | | | | | | |
---|
| | | | | | | |
Mean | | | | | | | |
| | | | | | | |
| | | | | | | |
Var | | | | | | | |
| | | | | | | |
| | | | | | | |
DsI | | | | | | | |
| | | | | | | |
| | | | | | | |
Sk | | | | | | | |
| | | | | | | |
| | | | | | | |
Ku | | | | | | | |
| | | | | | | |
Table 2.
The mean time to failure (MTTF) and entropy using the DGzEx model.
Table 2.
The mean time to failure (MTTF) and entropy using the DGzEx model.
Measure | | | | | | | |
---|
MTTF | | | | | | | |
| | | | | | | |
| | | | | | | |
Entropy | | | | | | | |
| | | | | | | |
| | | | | | | |
Table 3.
Some descriptive statistics using the DGzW model.
Table 3.
Some descriptive statistics using the DGzW model.
Measure | | | | | | | |
---|
| | | | | | | |
Mean | | | | | | | |
| | | | | | | |
| | | | | | | |
Var | | | | | | | |
| | | | | | | |
| | | | | | | |
DsI | | | | | | | |
| | | | | | | |
| | | | | | | |
Sk | | | | | | | |
| | | | | | | |
| | | | | | | |
Ku | | | | | | | |
| | | | | | | |
Table 4.
The MTTF and entropy using the DGzW model.
Table 4.
The MTTF and entropy using the DGzW model.
Measure | | | | | | | |
---|
MTTF | | | | | | | |
| 3 | | | | | | |
| 5 | | | | | | |
Entropy | | | | | | | |
| 3 | | | | | | |
| 5 | | | | | | |
Table 5.
The competitive models of the DGzW, DGzEx, and DGzIW distributions.
Table 5.
The competitive models of the DGzW, DGzEx, and DGzIW distributions.
Distribution | Abbreviation | Author(s) |
---|
Discrete Weibull | DW | Nakagawa and Osaki [37] |
Exponentiated discrete Weibull | EDW | Nekoukhou and Bidram [28] |
Discrete inverse Weibull | DIW | Jazi et al. [38] |
Discrete exponential | DEx | Gómez-Déniz [23] |
Discrete generalized exponential type II | DGEx-II | Nekoukhou et al.
[26] |
Discrete Rayleigh | DR | Roy [22] |
Discrete inverse Rayleigh | DIR | Hussain and Ahmad [39] |
Discrete Lindley | DLi | Gómez-Déniz and Calderín-Ojeda
[40] |
Exponentiated discrete Lindley | EDLi | El-morshedy et al. [41] |
Discrete Lindley type II | DLi-II | Hussain et al. [42] |
Discrete log-logistic | DLLc | Para and Jan [43] |
Discrete Lomax | DLo | Para and Jan [44] |
Two-parameter discrete Burr type XII | DB-XII | Para and Jan [44] |
Discrete Pareto | DPa | Krishna and Pundir [45] |
Negative binomial | NvBi | Dougherty [46] |
Poisson | Poi | Poisson [47] |
Table 6.
The maximum likelihood estimations (MLEs) with their corresponding standard errors (Std-er) for Dataset 1.
Table 6.
The maximum likelihood estimations (MLEs) with their corresponding standard errors (Std-er) for Dataset 1.
Model ↓ Parameter → | p | c | a | b |
---|
MLE | Std-er | MLE | Std-er | MLE | Std-er | MLE | Std-er |
---|
DGzW | | | | | | | | |
EDW | | | | | | | − | − |
DW | | | | | − | − | − | − |
DIW | | | | | − | − | − | − |
DLi-II | | | | | − | − | − | − |
EDLi | | | | | − | − | − | − |
DLLc | | | | | − | − | − | − |
DPa | | | − | − | − | − | − | − |
Table 7.
The goodness of fit statistics for Dataset 1.
Table 7.
The goodness of fit statistics for Dataset 1.
Statistic ↓ Model → | DGzW | EDW | DW | DIW | DLi-II | EDLi | DLLc | DPa |
---|
| | | | | | | | |
AIC | | | | | | | | |
CAIC | | | | | | | | |
K-S | | | | | | | | |
p-value | | | | | | | | |
Table 8.
Some statistics for Dataset 1.
Table 8.
Some statistics for Dataset 1.
Model | Mean | Var | DsI | Sk | Ku |
---|
DGzW | | | | | |
Table 9.
Some reliability measures using Dataset 1.
Table 9.
Some reliability measures using Dataset 1.
Time Measure → | RF | HRF | MTBF |
---|
2 | | | |
4 | | | |
6 | | | |
8 | | | |
10 | | | |
12 | | | |
14 | | | |
16 | | | |
18 | | | |
20 | | | |
22 | | | |
24 | | | |
26 | | | |
28 | | | |
30 | | | |
Table 10.
The MLEs with their corresponding Std-er for Dataset 2.
Table 10.
The MLEs with their corresponding Std-er for Dataset 2.
Model ↓ Parameter → | p | c | a |
---|
MLE | Std-er | MLE | Std-er | MLE | Std-er |
---|
DGzEx | | | | | | |
DEx | | | − | − | − | − |
DGEx-II | | | | | − | − |
DR | | | − | − | − | − |
DIR | | | − | − | − | − |
DIW | | | | | − | − |
DLo | | | | | − | − |
DB-XII | | | | | − | − |
DPa | | | − | − | − | − |
Table 11.
The goodness of fit statistics for Dataset 2.
Table 11.
The goodness of fit statistics for Dataset 2.
Statistic | Model |
---|
DGzEx | DEx | DGEx-II | DR | DIR | DIW | DLo | DB-XII | DPa |
---|
| | | | | | | | | |
AIC | | | | | | | | | |
CAIC | | | | | | | | | |
K-S | | | | | | | | | |
p-value | | | | | | | | | |
Table 12.
Some statistics for Dataset 2.
Table 12.
Some statistics for Dataset 2.
Model | Mean | Var | DsI | Sk | Ku |
---|
DGzEx | | | | | |
Table 13.
Some reliability measures using Dataset 2.
Table 13.
Some reliability measures using Dataset 2.
Time Measure → | RF | HRF | MTBF |
---|
2 | | | |
4 | | | |
6 | | | |
8 | | | |
10 | | | |
12 | | | |
14 | | | |
16 | | | |
18 | | | |
20 | | | |
22 | | | |
24 | | | |
26 | | | |
28 | | | |
30 | | | |
Table 14.
The MLEs with their corresponding Std-er for Dataset 3.
Table 14.
The MLEs with their corresponding Std-er for Dataset 3.
Model ↓ Parameter → | p | c | a | b |
---|
MLE | Std-er | MLE | Std-er | MLE | Std-er | MLE | Std-er |
---|
DGzW | | | | | | | | |
DW | − | − | − | − | | | | |
DIW | − | − | − | − | | | | |
DR | − | − | − | − | | | − | − |
DEx | − | − | − | − | | | − | − |
DLi | − | − | − | − | | | − | − |
DLi-II | − | − | − | − | | | | |
DLo | − | − | − | − | | | | |
Poi | − | − | − | − | | | − | − |
Table 15.
The goodness of fit statistics for Dataset 3.
Table 15.
The goodness of fit statistics for Dataset 3.
Z | Observed Frequency | Expected Frequency |
---|
DGzW | DW | DIW | DR | DEx | DLi | DLi-II | DLo | Poi |
---|
0 | 65 | | | | | | | | | |
1 | 14 | | | | | | | | | |
2 | 10 | | | | | | | | | |
3 | 6 | | | | | | | | | |
4 | 4 | | | | | | | | | |
5 | 2 | | | | | | | | | |
6 | 2 | | | | | | | | | |
7 | 2 | | | | | | | | | |
8 | 1 | | | | | | | | | |
9 | 1 | | | | | | | | | |
10 | 1 | | | | | | | | | |
11 | 2 | | | | | | | | | |
Total | 110 | 110 | 110 | 110 | 110 | 110 | 110 | 110 | 110 | 110 |
| | | | | | | | | | |
AIC | | | | | | | | | | |
CAIC | | | | | | | | | | |
| | | | | | | | | | |
d.f | | 1 | 3 | 3 | 4 | 4 | 4 | 3 | 3 | 4 |
p-value | | | | | <0.0001 | | <0.0001 | <0.0001 | | <0.0001 |
Table 16.
Some statistics for Dataset 3.
Table 16.
Some statistics for Dataset 3.
Model | Mean | Var | DsI | Sk | Ku |
---|
DGzW | | | | | |
Table 17.
The MLEs with their corresponding Std-er for Dataset 4.
Table 17.
The MLEs with their corresponding Std-er for Dataset 4.
Model ↓ Parameter → | p | c | a | b |
---|
MLE | Std-er | MLE | Std-er | MLE | Std-er | MLE | Std-er |
---|
DGzIW | | | | | | | | |
DIW | | | | | − | − | − | − |
DB-XII | | | | | − | − | − | − |
DIR | | | − | − | − | − | − | − |
DR | | | − | − | − | − | − | − |
NvBi | | | | | − | − | − | − |
DPa | | | − | − | − | − | − | − |
Poi | | | − | − | − | − | − | − |
Table 18.
The goodness of fit statistics for Dataset 4.
Table 18.
The goodness of fit statistics for Dataset 4.
X | Observed Frequency | Expected
Frequency |
---|
DGzIW | DIW | DB-XII | DIR | DR | NvBi | DPa | Poi |
---|
0 | 43 | | | | | | | | |
1 | 35 | | | | | | | | |
2 | 17 | | | | | | | | |
3 | 11 | | | | | | | | |
4 | 5 | | | | | | | | |
5 | 4 | | | | | | | | |
6 | 1 | | | | | | | | |
7 | 2 | | | | | | | | |
8 | 2 | | | | | | | | |
Total | 120 | 120 | 120 | 120 | 120 | 120 | 120 | 120 | 120 |
| | | | | | | | | |
AIC | | | | | | | | | |
CAIC | | | | | | | | | |
| | | | | | | | | |
d.f | | 1 | 3 | 3 | 4 | 4 | 3 | 4 | 4 |
P.value | | | | | | | | | |
Table 19.
Some statistics for Dataset 4.
Table 19.
Some statistics for Dataset 4.
Model | Mean | Var | DsI | Sk | Ku |
---|
DGzIW | | | | | |