Ahmed Sayed M Metwally1, Amal S Hassan2, Ehab M Almetwally3, B M Golam Kibria4, Hisham M Almongy5. 1. Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia. 2. Department of Mathematical Statistics, Faculty of Graduate Studies for Statistical Research, Cairo University, Giza 12613, Egypt. 3. Department of Statistics, Faculty of Business Administration, Delta University of Science and Technology, Gamasa 11152, Egypt. 4. Department of Mathematics and Statistics, Florida International University (FIU), 11200 SW 8th St, Miami, FL 33199, USA. 5. Department of Applied Statistics and Insurance, Faculty of Commerce, Mansoura University, El-Mansoura 35516, Egypt.
Abstract
The inverted Topp-Leone distribution is a new, appealing model for reliability analysis. In this paper, a new distribution, named new exponential inverted Topp-Leone (NEITL) is presented, which adds an extra shape parameter to the inverted Topp-Leone distribution. The graphical representations of its density, survival, and hazard rate functions are provided. The following properties are explored: quantile function, mixture representation, entropies, moments, and stress-strength reliability. We plotted the skewness and kurtosis measures of the proposed model based on the quantiles. Three different estimation procedures are suggested to estimate the distribution parameters, reliability, and hazard rate functions, along with their confidence intervals. Additionally, stress-strength reliability estimators for the NEITL model were obtained. To illustrate the findings of the paper, two real datasets on engineering and medical fields have been analyzed.
The inverted Topp-Leone distribution is a new, appealing model for reliability analysis. In this paper, a new distribution, named new exponential inverted Topp-Leone (NEITL) is presented, which adds an extra shape parameter to the inverted Topp-Leone distribution. The graphical representations of its density, survival, and hazard rate functions are provided. The following properties are explored: quantile function, mixture representation, entropies, moments, and stress-strength reliability. We plotted the skewness and kurtosis measures of the proposed model based on the quantiles. Three different estimation procedures are suggested to estimate the distribution parameters, reliability, and hazard rate functions, along with their confidence intervals. Additionally, stress-strength reliability estimators for the NEITL model were obtained. To illustrate the findings of the paper, two real datasets on engineering and medical fields have been analyzed.
Entities:
Keywords:
Bayesian; entropy; maximum product spacing; new exponential-X; stress–strength reliability
There are several univariate continuous distributions in the present statistical literature that may be used in a range of data modeling applications. However, it appears that the many distributions that are available are insufficient to manage the diverse data encountered in fields such as medicine, engineering, demography, biology, actuarial science, economics, finance, and reliability. Statistical and applied researchers are interested in constructing new extended continuous distributions that are more effective for data modeling. Adding parameters, compounding, generating, transformation, and composition are all methods for extending well-known distributions.In the last couple of decades, the generation of new families of continuous distributions has attracted several statisticians to develop new models. Our interest is particularly in a new family proposed by Huo et al. [1] called the new exponential-X (NE-X) family. The cumulative distribution function (CDF) and probability density function (PDF) of the NE-X family are defined as:
and
where and are the PDF and CDF, respectively, for any baseline distribution with the set of parameters . The set of parameters can contain more than one parameter according to the type of distribution, and is a parameter of NE-X family where .Inverted or inverse distributions are important in many fields, including biological sciences, life test problems, chemistry data, medical sciences, and so on, because of their applicability. Inverted conformation distributions have a different structure than non-inverted conformation distributions in terms of density and hazard functions. The reader can consult Barco et al. [2], Abd AL-Fattah et al. [3], Hassan and Abd-Allah [4], Hassan and Mohamed [5], Muhammed [6], Hassan and Mohamed [7], Almetwally [8], and Hassan and Nassr [9] for discussions and applications of inverted distributions.The inverted Topp–Leone (ITL) distribution with only one shape parameter (), which was presented by Hassan et al. [10], is a recent, significant model among the well-known inverted distributions. It density and hazard functions take different shapes according to value of , including unimodal, right skewed, increasing, decreasing, and upside down. The PDF and CDF of the ITL distribution are specified, respectively, as follows:
and,Some authors studied and developed new extensions and generalizations of the ITL distribution, such as the power ITL distribution prepared by Abushal et al. [11], Kumaraswamy ITL distribution introduced by Hassen et al. [12], alpha power ITL distribution presented by Ibrahim et al. [13], modified Kies ITL distribution introduced by Almetwally et al. [14], odd Weibull ITL distribution suggested by Almetwally [15], and half logistic ITL distribution prepared by Bantan et al. [16].In this paper, a new ITL distribution based on the NE-X family is proposed. We call it the new exponential ITL (NEITL) distribution. Our motivations for presenting the NEITL distribution are as follows: (i) to increase the flexibility of the ITL distribution for modeling several types of data; (ii) to allow researches to obtain more flexible density and hazard rate functions; (iii) real-world examples from medical, engineering, and other fields demonstrate that the NEITL model outperforms other competing distributions, justifying its implementation in these domains.Another motivation for the present study was estimating the NEITL parameters, reliability function, and hazard rate function using three estimation methods to recommend the best estimates via a simulation study. The suggested procedures are maximum likelihood (ML), maximum product of spacing (MPS), and Bayesian procedures. The asymptotic and bootstrap confidence intervals are shown. Furthermore, we obtained the stress–strength (S–S) reliability estimator assuming that both the strength and stress have NEITLs with different shape parameters. In simulation research, statistical analysis was performed between these methods to assess their effectiveness and to investigate how these estimators function for various sample sizes and parameter values. Two applications showed that the NEITL distribution provides a better fit than some other distributions.The rest of this essay is presented as follows. In Section 2, we define the NEITL distribution. Some of the statistical features of the NEITL distribution are determined in Section 3. The NEITL distribution’s reliability, hazard function (HF), and S–S reliability are covered in Section 4. Section 5 considers point estimate, asymptotic, and bootstrap confidence intervals utilizing ML, Bayesian, and MPS estimation methods. A simulation experiment is presented in Section 6 to compare the performances of the estimates presented in Section 5. Data implementations are explored in Section 7. The article is closed with some conclusions.
2. NEITL Distribution
The two-parameter NEITL distribution is a special model of the NE-X family with the ITL distribution as a baseline function. We get the CDF and PDF of the NEITL distribution by replacing the CDF and PDF of the ITL model in (1) and (2), respectively.
and
where is a vector of parameters for this distribution. Figure 1 visually displays the PDF plots and 3-D plots of X using NEITL and parameters (). The NEITL distribution may be right-skewed and unimodal, according to the PDF plots.
Figure 1
PDFs plots of the NEITL distribution.
Furthermore, we obtain an explicit linear representation of the density and distribution functions by using the generalized binomial expansion. Hence, for , the k is a real non-integer, and for we use the following expansion with negative power:
Additionally, we use the binomial expansion below with positive power:
Let , and . Then, the linear representation of CDF for NE-X family (1) is given by
By applying the previous expansion (7) on ITL distribution (3), we have CDF of the NEITL distribution in an expanded form as the following:
and let , and . Then, the CDF of the NEITL distribution can be rewritten as follows:
Hence, the CDF (9) is represented as an infinite linear combination of the ITL distribution function with parameter .In addition to the expression (9), we derive PDF expression of the NEITL distribution distribution as follows:Let . Then the linear representation of PDF for NE-X family (2) is given by
Using CDF (3) and PDF (4) in (10), and binomial expansion, we obtain the following expansion:
where . It is the PDF function of the ITL distribution with parameter .
3. Mathematical Properties
Here, some structure properties of the NEITL distribution are investigated, such as ordinary and incomplete moments, the quantile function and random number generation, Rényi and -entropies, and the S–S reliability model.
3.1. Ordinal Moments
The rth moment of the NEITL distribution is given byUsing the binomial expansion in (12), we can define . Let . Then the rth moment of the NEITL distribution has the form
where and . Furthermore, the mth central moment of X is given byTable 1 gives some different statistical measures such as mean , variance , skewness (SK), and kurtosis (KU) for some values of parameters.
Table 1
Moments measures for NEITL distribution.
(δ,θ)
μ1′
σ2
SK
KU
(2,3)
0.739
0.204
2.624
23.372
(3,3)
0.534
0.083
1.871
11.412
(5,3)
0.369
0.032
1.409
7.371
(5,4)
0.293
0.016
1.07
5.572
(5,5)
0.248
0.01
0.847
4.602
(1,5)
0.787
0.192
2.089
17.382
(2,7)
0.414
0.048
1.599
8.225
(3,7)
0.295
0.018
1.222
6.201
From Table 1, we conclude that the NEITL distribution is skewed to the right and leptokurtic.
3.2. Incomplete Moments
The rth incomplete moment, say, of X, is obtained from (11) as follows:
where stands for an incomplete beta function. The first incomplete moment, for in (14), is obtained. The famous applications of the first incomplete moment are the Lorenz and Bonferroni curves which are defined, respectively, by and .
3.3. Quantile Function
The quantile function of the NEITL distribution, say, , is derived by inverting (5) as follows:
where .In particular, the first quartile, say, ; the second quartile, say, ; and the third quartile, say, are obtained by setting x = 0.25, 0.5, and 0.75, respectively, in (15). The Bowley’s skewness depends on quartiles as follows:
where Q(.) is the NEITL quantile function. The Moor’s kurtosis is given asSkewness and kurtosis plots of the NEITL distribution, based on quantiles, are exhibited in Figure 2.
Figure 2
Plots of the skewness and kurtosis of the NEITL distribution.
3.4. Rényi and Other Entropies
Here, we obtain Rényi and -entropies. The Rényi entropy, , of a random variable X, is defined by
where and . Using expansions in (6) and after some simplification, then should be written as:Again, using the binomial expansions more than one times leads to
where . Substituting (20) in (18) gives
The entropy, , is defined as follows:
The entropy of the NEITL takes the form
4. Reliability Analysis
In this section, we discus the reliability analysis in terms of hazard, survival, and S–S reliability for the NEITL distribution.
4.1. Hazard and Survival Reliability
The survival function (SF) of the NEITL distribution is given byFigure 3 gives SF plots of the NEITL distribution for specific values of parameters.
Figure 3
SF plots of the NEITL distribution.
The HF of the NEITL distribution is given byThe HF plots of the NEITL distribution are displayed in Figure 4 to control sequence for certain values of parameters. These figures show that the HF of the NEITL distribution can be increasing, decreasing, or upside-down shaped.
Figure 4
HF plots of the NEITL distribution.
4.2. Stress–Strength Reliability
The stress–strength model is extensively used in reliability estimation. The S–S model has many applications in physics and engineering, including strength failure testing, structural modeling, estimating the deterioration of rocket motors, and modeling the static fatigue of ceramic components. In the S–S model, reliability R measures the reliability of the component that has strength when it is subjected to random stress . The component fails if the applied stress exceeds its strength: . For more information about this model, see Abu El Azm et al. [17], Sabry et al. [18], Yousef and Almetwally [19], and Hassan et al. [20]. Let and be two independent random variables with NEITL and NEITL distributions, respectively. Hence, the S–S reliability is obtained, using the same expansions in (9) and (11) with different indicators, as follows:
where , , , and . Plots of S–S model for some values of parameters are given in Figure 5.
Figure 5
Stress–strength plots of the NEITL distribution.
5. Parameter Estimation
In this section, we use different point estimation methods to estimate the unknown parameters of the NEITL distribution. We use classical (ML and MPS) and non-classical (Bayesian) methods. In the last few years, parameter estimation using different methods has received great attention from many authors, such as Haj Ahmad and Almetwally [21], Basheer et al. [22], and Almetwally [15].
5.1. Maximum Likelihood Method
Let be a random sample from the NEITL distribution with parameters and . The log-likelihood function of the NEITL can be written as:
where . The log-likelihood function of the NEITL distribution isThe ML estimators are obtained by solving the following equations:
andThese equations cannot be solved explicitly; hence, a nonlinear optimization algorithm such as the Newton Raphson method is used.
5.2. Maximum Product Spacing
According to Cheng and Amin [23], the maximum product spacing method is an efficient estimation method that has proved to have some advantages with respect to other point estimation methods. Thus, we use MPS in this section to have point estimation of the unknown parameters of the NEITL distribution. This can be obtained by solving equations resulted from taking partial derivatives of logarithm of product spacing function which is written as:
and the logarithmic function ofThe MPS estimators of are obtained by differentiating the log-product equation (Equation (29)) with respect to each parameter separately. We can solve the nonlinear system of equations by using any iterative technique, such as conjugate-gradient algorithms. Over the last few years, the estimation parameters of such models have been improved under censoring schemes—for instance, by Almetwally et al. [24] and El-Sherpieny et al. [25].
5.3. Bayesian Estimation
Bayesian method provide statistical inferences that are based on the prior distribution and loss function that are chosen. All parameters are treated as random variables with certain distributions, termed the prior distribution in this method. We must choose one if prior information is not available, which is frequently the case. The independent gamma distributions are our priors of choice because prior distribution selection plays such an essential role in parameter estimation. The joint prior distribution can be written as follows:
The joint posterior density function of is obtained from (27) and (30):
where . Then, the posterior NEITL distribution isThe conditional posterior distribution is as follows:
andThe loss function, on the other hand, is crucial in Bayesian approaches. The symmetric and asymmetric loss functions are used to create the majority of Bayesian inference processes. The Bayes estimators of , say, , based on a squared error loss function, are given byIt is noted that the integrals given by (35) cannot be obtained explicitly. Due to that, we used the Markov chain Monte Carlo technique (MCMC) to find approximate values of integrals (35). Many studies have used the MCMC technique, such as El-Sherpieny et al. [26], Almongy et al. [27], Haj Ahmad et al. [28], Bantan et al. [29], Almetwally et al. [24], Al-Omari et al. [30], Al-Babtain et al. [31], and Hassan and Zaki [32].
6. Simulation
A simulation study has been conducted to examine the performances of point estimates in terms of their average estimates (AE), mean squared errors (MSE), interval estimates, and lengths of confidence interval (L.CI). The simulation study was carried out with various parameter values and sample sizes. This section is divided into two parts.For the first reliability analysis: The parameters of the NEITL distribution were = (0.5; 0.5) and (0.5; 3) for the results in Table 2 and = (3; 0.5) and (3; 3) for the results in Table 3. The sample sizes were n = 30, 80, and 150, respectively. We selected time (Q) to determine the HF and SF of the NEITL distribution where , , , and . The various simulation results are based on a total of 10,000 repetitions. The Bayes estimates are based on 10,000 samples and were derived using the MCMC approach. In Table 2 and Table 3, the AE, MSEs, and L.CI of the various approaches are displayed.
Table 2
Accuracy measures for parameters of the NEITL distribution, and reliability analysis for different periods of time for .
θ=0.5
MLE
MPS
Bayesian
δ
n
AE
MSE
L.CI
AE
MSE
L.CI
AE
MSE
L.CI
0.5
30
θ
0.0875
0.2002
1.7121
0.0813
0.2001
1.7009
0.0419
0.0635
0.8738
δ
0.1646
0.2306
1.7411
0.0731
0.1905
1.6706
0.0400
0.0429
0.7872
Q = 0.25
R1
0.7500
0.0035
0.2262
0.7498
0.0032
0.2173
0.7359
0.0053
0.2714
H1
0.0043
1.29 × 10−6
0.0039
0.0041
9.11 × 10−7
0.0038
0.0042
1.49 × 10−6
0.0048
Q = 0.35
R2
0.6369
0.0051
0.2735
0.6489
0.0047
0.2651
0.6336
0.0078
0.3286
H2
0.0021
2.11 × 10−7
0.0016
0.0018
1.75 × 10−7
0.0017
0.0021
3.09 × 10−7
0.0021
80
θ
0.1093
0.1784
1.6000
0.1621
0.1722
1.5607
0.0177
0.0215
0.5545
δ
0.0822
0.1284
1.3681
0.0518
0.1230
1.2400
0.0243
0.0203
0.5598
Q = 0.25
R1
0.7458
0.0011
0.1265
0.7494
0.0010
0.1258
0.7415
0.0014
0.1317
H1
0.0040
3.51 × 10−7
0.0022
0.0039
3.13 × 10−7
0.0021
0.0040
3.84 × 10−7
0.0022
Q = 0.35
R2
0.6423
0.0016
0.1526
0.6472
0.0015
0.1516
0.6392
0.0021
0.1612
H2
0.0020
7.01 × 10−8
0.0010
0.0019
6.27 × 10−8
0.0010
0.0020
8.26 × 10−8
0.0010
150
θ
0.1014
0.1627
1.5310
0.1412
0.1521
1.4619
0.0038
0.0068
0.3083
δ
0.0627
0.0951
1.1839
0.0498
0.0910
1.1253
0.0103
0.0062
0.3008
Q = 0.25
R1
0.7472
0.0006
0.0966
0.7494
0.0006
0.0967
0.7469
6.11 × 10−4
0.0940
H1
0.0040
2.06 × 10−7
0.0016
0.0039
1.93 × 10−7
0.0016
0.0039
1.61 × 10−7
0.0016
Q = 0.35
R2
0.6440
0.0009
0.1147
0.6468
8.65 × 10−4
0.1146
0.6461
9.27 × 10−4
0.1181
H2
0.0020
4.39 × 10−8
0.0008
0.0019
4.07 × 10−8
0.0008
0.0019
3.49 × 10−8
0.0007
3
25
θ
0.0128
0.2007
1.7037
0.0140
0.0135
0.4524
0.0197
0.0086
0.3640
δ
−0.0015
0.2298
1.7140
−0.0693
0.1306
1.3911
−0.0178
0.0681
1.0029
Q = 0.25
R1
0.7429
0.0032
0.2212
0.7535
0.0030
0.2148
0.7390
0.0037
0.2296
H1
0.2805
0.0043
0.2533
0.2671
0.0038
0.2412
0.2841
0.0049
0.2588
Q = 0.35
R2
0.6427
0.0047
0.2688
0.6562
0.0045
0.2625
0.6384
0.0054
0.2772
H2
0.2851
0.0036
0.2344
0.2722
0.0033
0.2234
0.2882
0.0042
0.2387
80
θ
0.0233
0.0136
0.4482
0.0007
0.0029
0.2101
0.0067
0.0028
0.2001
δ
−0.0012
0.1313
1.2926
−0.0189
0.0360
0.7405
−0.0068
0.0308
0.6843
Q = 0.25
R1
0.7475
0.0009
0.1203
0.7529
0.0009
0.1185
0.7466
0.0011
0.1252
H1
0.2740
0.0012
0.1339
0.2673
0.0011
0.1318
0.2745
0.0014
0.1406
Q = 0.35
R2
0.6473
0.0014
0.1464
0.6542
0.0014
0.1457
0.6464
0.0016
0.1538
H2
0.2791
0.0010
0.1257
0.2732
0.0010
0.1228
0.2800
0.0012
0.1319
150
θ
0.0034
0.0035
0.2332
−0.0017
0.0018
0.1679
0.0032
0.0016
0.1496
δ
0.0063
0.0746
1.0707
−0.0065
0.0224
0.5869
−0.0042
0.0150
0.4773
Q = 0.25
R1
0.7494
0.0006
0.0971
0.7528
0.0006
0.0961
0.7484
0.0007
0.0962
H1
0.2716
0.0008
0.1083
0.2675
0.0008
0.1069
0.2724
0.0009
0.1075
Q = 0.35
R2
0.6495
0.0009
0.1189
0.6538
0.0009
0.1182
0.6484
0.0010
0.1185
H2
0.2774
0.0007
0.1012
0.2736
0.0007
0.0996
0.2782
0.0008
0.1013
Table 3
Accuracy measures for parameters of the NEITL distribution, and reliability analysis for different periods of time for .
θ=3
MLE
MPS
Bayesian
δ
n
AE
MSE
L.CI
AE
MSE
L.CI
AE
MSE
L.CI
0.5
30
θ
0.0044
0.4520
2.6372
−0.3054
0.4210
2.2450
−0.0197
0.0670
0.9916
δ
0.0215
0.1230
1.3740
0.1181
0.0898
1.0801
0.0240
0.0115
0.3713
Q = 0.25
R1
0.7461
0.0038
0.2451
0.7479
0.0033
0.2276
0.7375
0.0046
0.2388
H1
0.4116
0.0115
0.4159
0.4015
0.0097
0.3852
0.4261
0.0140
0.4222
Q = 0.35
R2
0.6466
0.0056
0.2970
0.6508
0.0051
0.2775
0.6367
0.0065
0.2943
H2
0.4372
0.0112
0.4083
0.4220
0.0095
0.3776
0.4495
0.0138
0.4233
80
θ
−0.0571
0.2683
2.0189
−0.1001
0.0945
1.1397
−0.0118
0.0279
0.6472
δ
0.0404
0.0381
0.7487
0.0270
0.0103
0.3833
0.0098
0.0029
0.2039
Q = 0.25
R1
0.7470
0.0010
0.1241
0.7505
0.0009
0.1192
0.7455
0.0011
0.1232
H1
0.4056
0.0028
0.2067
0.3995
0.0027
0.2031
0.4099
0.0033
0.2120
Q = 0.35
R2
0.6472
0.0015
0.1515
0.6517
0.0014
0.1477
0.6449
0.0017
0.1532
H2
0.4296
0.0027
0.2046
0.4232
0.0027
0.2019
0.4345
0.0033
0.2125
150
θ
0.0134
0.0735
1.0617
−0.0557
0.0382
0.7353
−0.0086
0.0117
0.4165
δ
0.0038
0.0044
0.2605
0.0118
0.0028
0.2036
0.0040
0.0014
0.1387
Q = 0.25
R1
0.7493
0.0005
0.0885
0.7510
0.0005
0.0877
0.7486
0.0006
0.0927
H1
0.4029
0.0015
0.1520
0.3993
0.0015
0.1506
0.4042
0.0017
0.1597
Q = 0.35
R2
0.6494
0.0008
0.1103
0.6518
0.0008
0.1095
0.6486
0.0009
0.1161
H2
0.4279
0.0015
0.1521
0.4237
0.0015
0.1506
0.4290
0.0017
0.1601
3
25
θ
0.7823
3.2109
6.3225
−0.1464
0.3771
2.3388
−0.0035
0.0587
0.9477
δ
0.0619
3.2238
7.0377
0.2793
0.6266
2.9050
−0.0072
0.0667
1.0296
Q = 0.25
R1
0.7490
0.0031
0.2172
0.7527
0.0026
0.1984
0.7515
0.0017
0.1578
H1
2.3694
0.3144
2.1952
2.3050
0.2756
2.0555
2.3288
0.1847
1.6417
Q = 0.35
R2
0.6495
0.0046
0.2665
0.6554
0.0040
0.2485
0.6528
0.0027
0.2000
H2
2.8692
0.4192
2.5313
2.7668
0.3667
2.3667
2.8072
0.2489
1.9051
80
θ
0.3595
1.3131
4.2674
−0.0903
0.1382
1.4145
−0.0080
0.0269
0.6306
δ
0.0021
1.2808
4.4386
0.1369
0.2231
1.7731
−0.0052
0.0276
0.6449
Q = 0.25
R1
0.7513
0.0010
0.1262
0.7519
0.0008
0.1126
0.7519
0.0007
0.0992
H1
2.3306
0.0959
1.2146
2.3116
0.0863
1.1484
2.3194
0.0739
1.0318
Q = 0.35
R2
0.6518
0.0016
0.1545
0.6532
0.0013
0.1420
0.6528
0.0011
0.1260
H2
2.8178
0.1231
1.3759
2.7823
0.1157
1.3265
2.7978
0.1002
1.2029
150
θ
0.2715
0.9255
3.6196
−0.0411
0.0771
1.0772
−0.0064
0.0098
0.3843
δ
0.0304
1.1950
4.2856
0.0627
0.1185
1.3276
−0.0006
0.0108
0.3920
Q = 0.25
R1
0.7514
0.0006
0.0982
0.7515
0.0004
0.0829
0.7510
0.0003
0.0653
H1
2.3255
0.0533
0.9045
2.3185
0.0465
0.8429
2.3267
0.0303
0.6728
Q = 0.35
R2
0.6519
0.0009
0.1180
0.6523
0.0007
0.1044
0.6514
0.0005
0.0825
H2
2.8103
0.0657
1.0047
2.7942
0.0620
0.9720
2.8067
0.0409
0.7801
Secondly, we estimated the reliability of the S–S model. The parameters of the NEITL distribution were = (0.6; 0.75; 0.65; 2.5) is case 1 and (0.6; 0.75; 2.65; 2.5)—see Table 4; and = (2; 1.75; 2.5; 2.5) is case 3 and (0.6; 2.75; 2.65; 2.5) is case 4—see Table 5. The sample sizes of S–S model were (n, m) = (25, 30), (80, 70), and (150, 120), respectively.
Table 4
Accuracy measures for parameters of the NEITL distribution, and reliability analysis for different periods of time for case 1 and case 2.
Case
MLE
MPS
Bayesian
n, m
AE
MSE
L.CI
AE
MSE
L.CI
AE
MSE
L.CI
1
25, 30
θ1
0.0616
0.1473
1.4864
0.1520
0.1840
1.5742
0.0335
0.0437
0.7637
δ1
0.1833
0.3414
2.1768
0.0562
0.2808
2.0675
0.0211
0.0481
0.8245
θ2
0.0443
0.0724
1.0414
0.0726
0.0649
0.9579
0.0079
0.0134
0.4490
δ2
0.1103
0.7321
3.3293
−0.1033
0.5550
2.8949
−0.0034
0.0622
0.9261
R
0.8257
0.0008
0.1057
0.8188
0.0009
0.1048
0.8198
0.0026
0.1168
80, 70
θ1
0.0510
0.1108
1.2910
0.0995
0.1361
1.3942
0.0248
0.0156
0.4902
δ1
0.1207
0.2031
1.7038
0.0646
0.1908
1.6953
0.0008
0.0223
0.5705
θ2
0.0097
0.0211
0.5689
0.0351
0.0307
0.6739
0.0091
0.0053
0.2729
δ2
0.0776
0.2878
2.0828
−0.0305
0.3172
2.2068
−0.0074
0.0334
0.7208
R
0.8316
0.0003
0.0692
0.8281
0.0003
0.0697
0.8300
0.0003
0.0694
150, 120
θ1
0.0364
0.0790
1.0937
0.0613
0.0888
1.1441
0.0034
0.0051
0.2641
δ1
0.0905
0.1390
1.4189
0.0609
0.1388
1.4424
0.0072
0.0071
0.3124
θ2
0.0146
0.0185
0.5302
0.0266
0.0235
0.5929
0.0011
0.0024
0.1825
δ2
0.0424
0.2233
1.8467
−0.0125
0.2486
1.9558
0.0088
0.0104
0.4007
R
0.8333
0.0002
0.0528
0.8310
0.0002
0.0534
0.8319
0.0002
0.0459
2
25, 30
θ1
0.0439
0.1438
1.4778
0.1211
0.1711
1.5518
0.0366
0.0404
0.7013
δ1
0.1944
0.3157
2.0686
0.0782
0.2577
1.9680
0.0241
0.0491
0.8161
θ2
0.2202
0.5042
2.6489
−0.1283
0.2027
1.6934
−0.0028
0.0540
0.8770
δ2
−0.0569
0.4444
2.6064
0.1952
0.3302
2.1209
0.0008
0.0512
0.8629
R
0.9413
0.0006
0.0964
0.9350
0.0009
0.1055
0.9419
0.0014
0.0843
80, 70
θ1
0.0483
0.1344
1.4775
0.1200
0.1689
1.5422
0.0326
0.0373
0.6871
δ1
0.1839
0.3006
2.0270
0.0759
0.2500
1.9394
0.0248
0.0469
0.8096
θ2
0.2154
0.4842
2.5963
−0.1161
0.1949
1.6714
−0.0023
0.0509
0.8596
δ2
−0.0524
0.4238
2.5462
0.1866
0.3088
2.0538
0.0008
0.0469
0.8333
R
0.9420
0.0006
0.0915
0.9362
0.0008
0.0996
0.9423
0.0013
0.0827
150, 120
θ1
0.0203
0.0837
1.1326
0.0384
0.0833
1.1222
0.0276
0.0364
0.6734
δ1
0.1220
0.1737
1.5636
0.0861
0.1475
1.4688
0.0382
0.0469
0.8254
θ2
0.0302
0.0668
1.0074
−0.1569
0.2685
1.9378
0.0046
0.0487
0.8573
δ2
−0.0076
0.0753
1.0761
0.2513
0.4349
2.3923
−0.0051
0.0533
0.8637
R
0.9456
0.0001
0.0351
0.9436
0.0001
0.0364
0.9495
0.0001
0.0314
Table 5
Accuracy measures for parameters of the NEITL distribution, and reliability analysis for different periods of time for case 3 and case 4.
Case
MLE
MPS
Bayesian
n, m
AE
MSE
L.CI
AE
MSE
L.CI
AE
MSE
L.CI
3
25, 30
θ1
0.7656
3.1827
6.3231
0.0110
0.7518
3.4021
0.0212
0.0454
0.8269
δ1
0.1439
1.7960
5.2283
0.2880
0.7248
3.1438
−0.0114
0.0521
0.8734
θ2
0.9499
4.4971
7.4397
−0.0677
0.5883
2.9978
0.0097
0.0520
0.8664
δ2
0.1369
3.2851
7.0917
0.3426
1.0036
3.6939
−0.0015
0.0648
0.9911
R
0.7228
0.0037
0.2406
0.7202
0.0036
0.2337
0.7198
0.0034
0.2146
80, 70
θ1
0.3765
1.3337
4.2840
0.0197
0.3368
2.2759
0.0025
0.0171
0.4865
δ1
0.0930
0.9049
3.7147
0.1322
0.3505
2.2643
0.0035
0.0202
0.5197
θ2
0.3805
1.4584
4.4973
−0.0547
0.3464
2.2996
0.0025
0.0249
0.6042
δ2
0.0725
1.2650
4.4042
0.2302
0.5551
2.7805
−0.0022
0.0294
0.6962
R
0.7256
0.0013
0.1402
0.7219
0.0012
0.1354
0.7202
0.0013
0.1402
150, 120
θ1
0.2757
0.9253
3.6162
0.0430
0.2072
1.7780
0.0036
0.0088
0.3603
δ1
0.0914
0.7308
3.3352
0.0487
0.1931
1.7136
−0.0055
0.0083
0.3559
θ2
0.3190
1.1487
4.0149
−0.0215
0.2164
1.8236
0.0057
0.0095
0.3781
δ2
0.0404
1.0387
3.9961
0.1224
0.3149
2.1490
−0.0092
0.0106
0.4029
R
0.7243
0.0008
0.1076
0.7217
0.0007
0.1031
0.7218
0.0006
0.0964
4
25, 30
θ1
0.0873
0.1165
1.2946
0.1136
0.1002
1.1594
0.0204
0.0163
0.4498
δ1
0.0713
1.1009
4.1077
−0.1877
0.8136
3.4619
−0.0133
0.0745
0.9809
θ2
0.5909
2.0654
5.1406
−0.0731
0.4676
2.6679
−0.0043
0.0520
0.8618
δ2
−0.0785
1.3511
4.5506
0.2558
0.6836
3.0852
0.0041
0.0580
0.9599
R
0.9248
0.0005
0.0878
0.9246
0.0005
0.0888
0.9207
0.0009
0.0962
80, 70
θ1
0.0788
0.0696
0.9878
0.0535
0.0363
0.7175
0.0081
0.0040
0.2443
δ1
−0.0753
0.6112
3.0534
−0.1039
0.3955
2.4339
−0.0091
0.0302
0.6873
θ2
0.2045
0.5337
2.7519
−0.0618
0.2220
1.8330
−0.0037
0.0227
0.5760
δ2
−0.0210
0.4744
2.7013
0.1593
0.3110
2.0972
−0.0036
0.0235
0.5869
R
0.9266
0.0002
0.0554
0.9259
0.0002
0.0557
0.9245
0.0002
0.0552
150, 120
θ1
0.0477
0.0317
0.6736
0.0271
0.0158
0.4820
0.0040
0.0020
0.1743
δ1
−0.0626
0.3421
2.2817
−0.0505
0.2179
1.8211
0.0010
0.0113
0.4141
θ2
0.0683
0.2062
1.7614
−0.0761
0.1350
1.4107
0.0043
0.0095
0.3757
δ2
0.0123
0.2282
1.8738
0.1348
0.1905
1.6290
−0.0017
0.0111
0.4077
R
0.9260
0.0001
0.0419
0.9257
0.0001
0.0421
0.9258
0.0001
0.0395
Table 2, Table 3, Table 4 and Table 5 present the results, which highlight some interesting facts. As the sample size gets larger, the estimates get more accurate, demonstrating that they are asymptotically unbiased. Furthermore, the MSE decreases as the sample size increases in all cases, demonstrating that the various estimates are consistent. When comparing the various estimates, we can observe that in the majority of cases, the Bayes estimates have the lowest MSE. MPS estimate is a good alternative for ML estimate (MLE). The L-CI for the estimates approach zero as the sample size (n) increases, indicating that the CI for the largest sample size is the shortest CI. The greater the time we tested , the lower the HF and SF values. When estimating the reliability of the S–S model in most cases, we received large values close to one, which indicates the quality of the model used.
7. Application of Real Data
To demonstrate the NEITL model’s flexibility and applicability in practice, two real life datasets are analyzed in this section. The NEITL distribution is compared to the ITL, exponentiated Lomax (EL), exponentiated exponential (ExEx), Weibull (W), Kumaraswamy Weibull (KW), modified Kies ITL (MKITL), and odd Weibull ITL (OWITL) distributions for the first batch of data. The NEITL distribution is compared to the ITL, EL, ExEx, W, KW, Kumaraswamy ITL (KIT), MKITL, and OWITL distributions for the second dataset. The approach of maximum likelihood was used to estimate the unknown parameters of the specified models for the two real datasets. To compare all of the models, the following statistics are used: Kolmogorov–Smirnov (KS), Cramer–von Mises (CVM), Anderson–Darling (AD), Akaike information criterion (AIC), and Bayesian information criterion (BIC).
7.1. Survival Times
Bjerkedal [33] observed and reported the survival periods (in days) of 72 guinea pigs infected with virulent tubercle bacilli in the first dataset. These data are as follows: 0.1, 0.33, 0.44, 0.56, 0.59, 0.59, 0.72, 0.74, 0.92, 0.93, 0.96, 1, 1, 1.02, 1.05, 1.07, 1.07, 1.08, 1.08, 1.08, 1.09, 1.12, 1.13, 1.15, 1.16, 1.2, 1.21, 1.22, 1.22, 1.24, 1.3, 1.34, 1.36, 1.39, 1.44, 1.46, 1.53, 1.59, 1.6, 1.63, 1.63, 1.68, 1.71, 1.72, 1.76, 1.83, 1.95, 1.96, 1.97, 2.02, 2.13, 2.15, 2.16, 2.22, 2.3, 2.31, 2.4, 2.45, 2.51, 2.53, 2.54, 2.54, 2.78, 2.93, 3.27, 3.42, 3.47, 3.61, 4.02, 4.32, 4.58, and 5.55.MLEs, SE, KS, CVM, AD, AIC, and BIC values for the first dataset are summarized in Table 6. The NEITL model has the least values for the statistical measures among all fitted models, as shown in the table.
Table 6
MLE with SE and other metrics: Survival Times.
Estimation
SE
KS
CVM
AD
AIC
BIC
ITL
δ
2.0225
0.2384
0.2989
0.0942
0.6663
229.6917
231.9684
NEITL
θ
60.9983
19.8002
0.0902
0.0776
0.4946
193.1635
197.7168
δ
0.0299
0.0549
EL
α
3.7415
0.8152
0.0978
0.0766
0.4949
195.2402
202.0702
β
37.0309
60.7818
λ
31.2893
54.2759
ExEx
α
15.4717
20.7674
0.2194
0.2209
1.2910
210.8807
215.4340
β
0.0240
0.0334
W
α
1.8173
0.1583
0.7439
0.0865
0.5852
195.8812
200.4345
β
0.2856
0.0544
KW
α
0.7474
0.6138
0.0917
0.0878
0.5351
196.6326
205.7393
β
0.9899
1.0882
λ
3.0474
3.9283
θ
1.7871
6.0095
MKITL
α
1.4212
0.1359
0.1015
0.1272
0.7577
194.5589
199.1122
β
1.1937
0.0725
OWITL
α
1.8048
0.2146
0.0969
0.0873
0.5415
195.0995
201.9295
β
25.9044
64.9941
λ
0.2721
0.3106
As a result, the NEITL model might be the best option. Figure 6 shows the estimated CDF, estimated PDF, and PP plot of the fitted NEITL model, respectively.
Figure 6
PDF, CDF, and PP plot of the NEITL distribution:Survival Times.
7.2. Example of Reliability of the S–S Model
Nelson [34] (Ch. 10, Table 4.1) calculated the time it takes for an insulating fluid to break down under high voltage stress in minutes. The failure times were observed in groups of ten insulating fluids, with each group reporting data on ten of them. Consider the following two sets of failure time data samples presented as follows for the purpose of showing the methods of inference outlined in the preceding sections:Group 1: 0.31, 0.66, 1.54, 1.70, 1.82, 1.89, 2.17, 2.24, 4.03, and 9.99.Group 2: 0.49, 0.64, 0.82, 0.93, 1.08, 1.99, 2.06, 2.15, 2.57, and 4.75.MLEs, SE, KS, CVM, AD, AIC, and BIC values for the data of Group 1 and Group 2 are summarized in Table 7 and Table 8. The NEITL model resulted in the best values for the statistical measures among all fitted models, as shown in theses tables. Table 9 provided MLE, MPS, and Bayesian estimates for reliability of the S–S model.
Table 7
MLE with SE and other metrics: Group 1.
Estimation
SE
KS
CVM
AD
AIC
BIC
ITL
δ
1.5750
0.4981
0.3141
0.0984
0.5089
41.4301
41.7327
NEITL
θ
0.4244
0.4225
0.2129
0.0924
0.4806
41.4921
42.0973
δ
5.6463
5.1060
EL
α
2.6559
2.0932
0.2181
0.0928
0.4831
43.5499
44.4577
β
3.4714
4.0843
λ
3.4753
6.7246
ExEx
α
1.0925
0.7304
0.2348
0.1133
0.6265
43.3593
43.9644
β
0.3309
0.3558
W
α
1.1585
0.2641
0.5557
0.0968
0.5082
42.9958
43.6010
β
0.3042
0.1512
KW
α
1.7884
2.5890
0.2187
0.0959
0.4999
45.6162
46.8265
β
0.4544
1.7933
λ
8.9793
52.9562
θ
1.3099
10.1035
KITL
α
1.5511
0.5350
0.2242
0.0944
0.4895
43.5807
44.4885
β
10.1497
9.1324
λ
0.3556
1.1931
MKITL
α
1.0353
0.2709
0.2523
0.1036
0.5577
42.1597
42.7649
β
0.9040
0.1948
OWITL
α
1.3963
0.3356
0.2173
0.0943
0.4919
43.5921
44.4998
β
55.1455
193.5675
λ
0.0780
0.1781
Table 8
MLE with SE and other metrics: Group 2.
Estimates
SE
KS
CVM
AD
AIC
BIC
ITL
δ
2.0929
0.6618
0.2130
0.0520
0.2973
32.7512
33.0537
NEITL
θ
0.8260
1.5968
0.2018
0.0405
0.2400
31.6464
32.2516
δ
3.8521
8.1014
EL
α
7.0316
22.8576
0.2174
0.0527
0.2983
33.3300
34.2377
β
3.1443
7.0486
λ
1.1988
6.2228
ExEx
α
6.3206
23.1429
0.1789
0.0621
0.4414
33.7072
34.3124
β
0.0589
0.2375
W
α
1.5527
0.3675
0.7012
0.0505
0.3002
32.4143
33.0195
β
0.3519
0.1683
KW
α
7.7574
0.0025
0.2124
0.0411
0.2415
33.5910
34.8013
β
0.9910
0.0025
λ
70.4798
59.0513
θ
0.1124
0.0366
KITL
α
6.3918
26.4265
0.2233
0.0501
0.2878
33.0906
33.9984
β
0.2378
0.8625
λ
12.4011
40.8768
Table 9
MLE, MPS, and Bayeisan for reliability of the S–S model.
MLE
MPS
Bayesian
Estimates
SE
R
Estimates
SE
R
Estimates
SE
R
θ1
0.4246
0.4226
0.6123
0.4143
0.4163
0.6345
0.5423
0.3812
0.6743
δ1
5.6440
5.1041
5.4284
4.9402
6.6363
4.5001
θ2
0.8312
1.6372
0.8312
1.2626
1.1196
0.8512
δ2
3.8267
8.2125
3.8267
7.4954
4.8956
3.8509
As a result, the NEITL model might be the best option. Figure 7 and Figure 8 show the estimated CDFs, estimated PDFs, and PP plot of the fitted NEITL model, respectively.
Figure 7
PDF, CDF and PP plot of the NEITL distribution: Group 1.
Figure 8
PDF, CDF, and PP plot of the NEITL distribution: Group 2.
Figure 9 and Figure 10 show convergence plots of MCMC for parameter estimates of the NEITL distribution.
Figure 9
Trace, proposed distribution, and convergence of MCMC results for .
Figure 10
Trace, proposed distribution, and convergence of MCMC results for .
8. Conclusions
A new two-parameter lifetime model, named “new exponential inverted Topp–Leone”, is introduced in this paper. The new distribution gives more flexibility and wide applicability compared to the existing models. It appears that the shape of the distribution depends on the values of the parameters. Some of the novel hazard rates that can be used are: decreased, constant hazard rate, increasing hazard rate, upside down (reversed bathtub shape), and increasing-constant hazard rate. Several mathematical and distributional properties, such as ordinal moments, incomplete moments, quantile function, Renyi, and entropies, were described in detail. The new density is a linear combination of the well-known inverted Topp–Leone density. The reliability of stress strength was calculated. Using Bayesian and non-Bayesian estimation methods, the parameters of the NEITL distribution were estimated. In simulation research, statistical analysis was used to compare these methods in order to evaluate their effectiveness and investigate how these estimates perform for different sample sizes and parameter values. The simulation results indicate that the Bayes estimate performed the best in the smaller MSE sense. In most cases, we received large values near to one when calculating the S–S model’s reliability, indicating that the model is of good quality. Furthermore, we propose using MPS estimation instead of ML estimation. To demonstrate the use of the novel distribution, two real-life datasets from the engineering and medical fields were studied. In addition, the use of these data in the stress–strength model has been validated. We hope that this distribution could be used in more areas.
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10