Abstract
In this paper, we present a new family of continuous distributions known as the type I half logistic Burr X-G. The proposed family’s essential mathematical properties, such as quantile function (QuFu), moments (Mo), incomplete moments (InMo), mean deviation (MeD), Lorenz (Lo) and Bonferroni (Bo) curves, and entropy (En), are provided. Special models of the family are presented, including type I half logistic Burr X-Lomax, type I half logistic Burr X-Rayleigh, and type I half logistic Burr X-exponential. The maximum likelihood (MLL) and Bayesian techniques are utilized to produce parameter estimators for the recommended family using type II censored data. Monte Carlo simulation is used to evaluate the accuracy of estimates for one of the family’s special models. The COVID-19 real datasets from Italy, Canada, and Belgium are analysed to demonstrate the significance and flexibility of some new distributions from the family.
1. Introduction
Statistical researchers have been encouraged in recent years to propose new broad families of continuous univariate distributions and to focus their efforts on improving their desired characteristics. For the time being, there is still a need for providing wider classes of distributions in order to provide them with greater flexibility and precision when fitting data. Some of the more recent generators sounding in the literature are the beta–G [1], type I half logistic [2], odd exponentiated half logistic G [3], Marshall–Olkin Burr X-G [4], generalized odd log-logistic-G [5], beta Burr type [6], new generalized odd log-logistic-G [7], generalized Burr X–G [8], type II half logistic [9], the transmuted odd Fréchet–G family in [10], Kumaraswamy-type I half logistic [11], and Burr X-exponential-G [12], among others.
Reference [13] proposed a new simple family of distributions with cumulative distribution function (CDFu) and probability density function (PDFu) using the Burr as generator; the so-called Burr family is as follows:where and are the PDFu and CDFu of any baseline distribution based on a parameter . The type I half-logistic- G family is [2] a represented family with an additional positive parameter . The CDFu of the distribution family is
The corresponding PDFu is
The failure (hazard) rate function is
We intend to benefit from the combined features of the Burr X-G and the TIHL-G in this work by introducing a new generated family of distributions known as the type I half-logistic Burr X-G (HLBX -G). We hope that the new family will provide more flexibility and attract a broader range of applications in reliability, engineering, and other research areas. The CDFu and PDFu of the family of distributions are provided by
Henceforward, a random variable having PDFu (7) will be defined as . The hazard rate function of family is given by
The quantile function, say , can be obtained by inverting (6) as follows:where denotes the QuFu. The quantile measurements are crucial in determining the impact of form parameters on skewness and kurtosis. For information, see [14, 15]. The new suggested family is extremely adaptable and includes several additional distributions. This study will present three new models of the family: HLBX Lomax, HLBX exponential, and HLBX Rayleigh. The pdfs of these models can be symmetric, right-skewed, unimodal, and up-side-down shaped; they are novel and extremely adaptable. The HRF of these models can also be increasing, decreasing, J-shaped, or U-shaped. The remainder of the paper is structured as follows: a useful expansion for the density and some special models are investigated in Section 2. Several mathematical properties including Mos, InMos, MeD, Lo, and Bo curves; residual life (ReL) and reversed ReL (RReL) functions; probability weighted moments (PrWMo); and the En of proposed family are derived in Section 3. MLL and Bayesian estimation methods are provided based on censored sample (CS) in Section 4. Applications of COVID-19 dataset to illustrate the flexibility and potentiality of the proposed family are analysed in Section 5. Section 6 discusses simulation analysis. Section 7 concludes with closing comments.
2. Useful Expansion
The following results are useful for expansions of and . If and is a real noninteger, then the following power series holds:
When we apply (10) to the final word in (7), we obtain
The power series expansion of is
By adding (14) to (13), we get
Making use of the generalized binomial expansion to , we can write
The density function may be represented as an endless combination of Expo-G density functions by substituting (16) into (15)whereand is the expo-G PDFu with power parameter . As a result, numerous mathematical and statistical features of the distribution are evident from those of the exp-G distribution. Similarly, the family CDFu may be represented as a combination of exp-G CDFus wherewhere is the exp-G cdf with power parameter .
2.1. Some HLBX-G Family Special Models
We present three submodels of this family based on the baseline distributions: Lomax, exponential, and Rayleigh. These models’ CDFu and PDFu files are given in Table 1.
2.1.1. Half-Logistic Burr X Lomax (HLBXL) Distribution
The CDFu and PDFu of distribution are
Plots of the HLBXLo densities are represented in Figure 1.
2.1.2. Half-Logistic Burr X Exponential (HLBXE) Distribution
The CDFu and PDFu of the model (for ) are
Plots of the HLBXE densities are represented in Figure 2.
2.1.3. Half-Logistic Burr X Rayleigh (HLBXR) Distribution
The CDFu and PDFu of the model (for ) are
Plots of the HLBXR densities are represented in Figure 3.
3. Fundamental Properties
We looked at the statistical properties of the distribution; Mos, InMos, MeD, Lo, and Bo curves; ReL and RReL functions; and PrWMs in this section.
3.1. Moments and Moment Generating Functions
The ordinary Mos and Mo generating function (MoGFu) of the family are computed. The different orders for the Mos are very useful in reliability applications to compute the expected life time of a device, skewness, and kurtosis in a given set of observations.
3.1.1. Moments
The ordinary Mo of can be obtained from (17) aswhere . The exp-G random variable with the power parameter is denoted. For , the second formula for the moment follows from (17) as , which is numerically calculable in terms of the baseline QuFu, i.e., as . For most parent distributions, this integration can be calculated numerically. Skewness and kurtosis can be calculated using the central Mo, say of , where
Remark 1. If have the ordinary Mo in (23), the MoGFu of can be investigated by using two formulae. The first formula can be computed from equation (17) aswhere is the MoGFu of . As a result, may be simply calculated from the exp-G generating function. The following is a second alternative formula that may be obtained from (17):where can be computed numerically from the baseline quantile function, i.e., .
Figure 4 show the mean, variance, skewness, and kurtosis for HLBXE model.
3.2. Incomplete Moments
The MeDs, Bo, and Lo curves, and other applications rely heavily on the first InMos. These curves have a wide range of uses, including economics, demography, and medicine. This is obvious not only in econometrics research, but also in other disciplines. For every real , the InMos of specified by may be calculated from (17) as
Equation (27) denotes the InMos of . The MeDs give important information about characteristic of population and also have been applied of income fields. If has the family of distributions, the MeDs about the mean and the MeDs about the median are defined byrespectively, where , median , is evaluated from (6), and is the first InMo given by (27) with . We can determine and by two techniques; the first can be obtained from (17) as , where is the first InMo of the exp-G distribution. The second technique is given by wherewhich can be computed numerically and . For a positive random variable , the Lo and Bo curves, for a given probability , are given by and , respectively, where , and is the QuFu of at .
3.3. Residual Lives
The order Mo of the ReL is given bywhere . The mean ReL (MReL) of family s can be obtained by setting in equation (30), defined as
The well-known formula can be used to calculate the order Mo of the RReL (or inactivity time):
The proposed family’s can be calculated by setting in (32), where
3.4. Probability Weighted Moments
The PrWMos of the family is given byusing equations (6) and (7), and with a little math, we can getwhere
Therefore, the PWMs of the family can be expressed as
3.5. Entropy
The Rényi En is defined by
Using (7), applying the same procedure of the useful expansion (17) and after some simplifications, we getwhere
Thus, Rényi entropy of family is defined as
4. Statistical Inference under Type II Censored Sample
Reference [16] examined the two most prevalent censoring systems, known as Type I and Type II censoring schemes. In Type II censoring, a life test is stopped after a specific number of failures. n and r are fixed and predefined in this case, while is a random variable. See [17] for further details.
4.1. Maximum Likelihood Estimation
The MLL has desirable features and may be used to calculate confidence intervals and test statistics. In both the Type II CS and the special case (full sample if ), we compute the MLL estimates (MLE) of the parameters of the family. Let be a -sample random sample from the distribution provided by (7). We spoke about observations, where r is the number of the uncensored items. Let be vector of parameters. The likelihood function of HLBX-G family under Type II CS can be written aswhere . The components of score function arewhere and is the element of the vector of parameters . The MLE of parameters , and is obtained by setting and simultaneously solving these equations to produce the MLL estimators. These equations cannot be solved analytically; however, they can be solved numerically using iterative approaches with statistical software.
4.2. Bayesian Estimation
The prior distribution and the loss function (LoFu) are both used in the Bayesian estimation technique. All parameters are regarded as random variables with a particular distribution known as the prior distribution in this technique. We must select one if no prior knowledge is provided, which is typically the case. We picked independent gamma distributions as our priors since the prior distribution is crucial in parameter estimation. The LoFu, on the other hand, plays an important role in the Bayesian approach. Most Bayesian inference techniques are based on symmetric and asymmetric LoFus. Two of the most frequent symmetric LoFus are the squared error and the linear exponential (Linex) LoFus. The independent joint prior density function of can be written as follows:
Reference [18] discussed how to elicit the hyperparameters of the informative priors. From the MLEs , we will get these beneficial priors by multiplying the estimate and variance by the inverse of the Fisher information matrix (FIM_ij) of , say . By equating mean and variance of gamma priors, the estimated hyperparameters can be written as and , where is a variance.
The joint posterior PDFu of is obtained from LL function and joint prior function:
Then the joint posterior of HLBX-G family under Type II CS can be written as
The Bayes estimators of , say based on squared error LoFu, is given by
The Bayes estimates of the unknown parameters under the Linex LoFu may be calculated as follows: . See, for example, [16, 18] for more information on Bayesian estimation. It is worth noting that the integrals (47) cannot be obtained explicitly. As a consequence, we estimate the value of integrals using the Markov Chain Monte Carlo (MCMC) approach.
Gibbs sampling and, more generally, Metropolis within Gibbs samplers are significant MCMC subclasses. Two popular MCMC techniques are the Metropolis-Hastings (MH) algorithm and Gibbs sampling. The MH algorithm, like acceptance-rejection sampling, evaluates whether a candidate value can be created from a proposal distribution throughout each iteration of the algorithm. The following are the MH inside Gibbs sampling stages that we used to produce random samples from conditional posterior densities of the HLBX-G family in a Type II CS:where . The MH algorithm (Algorithm 1) generates a sequence of draws from this distribution.
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5. Applications
Three real-world COVID-19 data applications from different countries are presented in this section to test the goodness of the HLBX-G family distributions. The HLBXE, HLBXL, and HLBXR models are compared with other related models such as Weibull-Lomax (WL) [19], Gompertz Lomax (GL) [20], exponentiated power Lomax (EPL) [21], Kumaraswamy exponentiated Rayleigh (KER) [17], Lomax, exponential and Rayleigh distributions. Tables 2–4 show MLE and standard errors (StEr) for all parameter of models. Also, these tables provide Kolmogorov–Smirnov (D1) statistic along with its value (D2), Cramér–von Mises (D3), and Anderson–Darling (D4) for all models fitted based on three real datasets of COVID-19 data with different countries as Italy, Canada, and Belgium, where these data are formed of drought mortality rate. Furthermore, the histograms of the three datasets are shown in Figures 5–7.
The three datasets were obtained from the following electronic address: https://github.com/CSSEGISandData/COVID-19/.The first set of data represents COVID-19 data belonging to Italy of 172 days, from 1 March to 21 August 2020. The data are as follows: 0.0490 0.0601 0.0460 0.0533 0.0630 0.0297 0.0885 0.0540 0.1720 0.0847 0.0713 0.0989 0.0495 0.1025 0.1079 0.0984 0.1124 0.0807 0.1044 0.1212 0.1167 0.1255 0.1416 0.1315 0.1073 0.1629 0.1485 0.1453 0.2000 0.2070 0.1520 0.1628 0.1666 0.1417 0.1221 0.1767 0.1987 0.1408 0.1456 0.1443 0.1319 0.1053 0.1789 0.2032 0.2167 0.1387 0.1646 0.1375 0.1421 0.2012 0.1957 0.1297 0.1754 0.1390 0.1761 0.1119 0.1915 0.1827 0.1548 0.1522 0.1369 0.2495 0.1253 0.1597 0.2195 0.2555 0.1956 0.1831 0.1791 0.2057 0.2406 0.1227 0.2196 0.2641 0.3067 0.1749 0.2148 0.2195 0.1993 0.2421 0.2430 0.1994 0.1779 0.0942 0.3067 0.1965 0.2003 0.1180 0.1686 0.2668 0.2113 0.3371 0.1730 0.2212 0.4972 0.1641 0.2667 0.2690 0.2321 0.2792 0.3515 0.1398 0.3436 0.2254 0.1302 0.0864 0.1619 0.1311 0.1994 0.3176 0.1856 0.1071 0.1041 0.1593 0.0537 0.1149 0.1176 0.0457 0.1264 0.0476 0.1620 0.1154 0.1493 0.0673 0.0894 0.0365 0.0385 0.2190 0.0777 0.0561 0.0435 0.0372 0.0385 0.0769 0.1491 0.0802 0.0870 0.0476 0.0562 0.0138 0.0684 0.1172 0.0321 0.0327 0.0198 0.0182 0.0197 0.0298 0.0545 0.0208 0.0079 0.0237 0.0169 0.0336 0.0755 0.0263 0.0260 0.0150 0.0054 0.0375 0.0043 0.0154 0.0146 0.0210 0.0115 0.0052 0.2512 0.0084 0.0125 0.0125 0.0109 0.0071.
The second set of data represents COVID-19 data belonging to Canada of 142 days, from 1 April to 21 August 2020. These data are formed of rough mortality rate. The data are as follows: 0.0122 0.0198 0.0155 0.0514 0.0176 0.0326 0.0418 0.0405 0.0452 0.0477 0.0524 0.0639 0.0554 0.0654 0.0940 0.0699 0.1138 0.0551 0.1060 0.0712 0.0588 0.0923 0.0831 0.0877 0.0948 0.0975 0.0832 0.0878 0.1023 0.1051 0.0881 0.1164 0.0620 0.1775 0.1133 0.1509 0.1176 0.1270 0.1137 0.0822 0.1534 0.1206 0.1399 0.1219 0.1253 0.0825 0.0884 0.0935 0.0436 0.0977 0.1016 0.0975 0.0861 0.0866 0.0714 0.1263 0.1116 0.1329 0.1247 0.0881 0.1239 0.2551 0.0488 0.1173 0.1861 0.1733 0.1925 0.0970 0.0421 0.0642 0.1516 0.1335 0.0840 0.1332 0.1242 0.1034 0.0806 0.1187 0.1062 0.1253 0.1125 0.1641 0.0629 0.0200 0.0552 0.1075 0.0526 0.0233 0.0336 0.0275 0.0659 0.0874 0.0898 0.0658 0.0487 0.0457 0.0226 0.0776 0.0974 0.0323 0.0312 0.0633 0.0412 0.0124 0.0242 0.0350 0.0391 0.0296 0.0273 0.0118 0.0076 0.0070 0.0147 0.0093 0.0131 0.0114 0.0141 0.0160 0.0277 0.0105 0.0365 0.0117 0.0209 0.0140 0.0136 0.0145 0.0101 0.0107 0.0094 0.0254 0.0217 0.0088 0.0138 0.0355 0.0231 0.0120 0.0169 0.0101 0.0076 0.0461 0.0119.
The third set of data represents COVID-19 data belonging to Belgium of 157 days, from 15 March to 20 August 2020. The data are as follows: 0.0279 0.0280 0.0284 0.0284 0.0412 0.0393 0.0452 0.0830 0.0844 0.0587 0.0619 0.0801 0.0748 0.0810 0.1576 0.2035 0.0972 0.1112 0.1670 0.1459 0.1379 0.2962 0.3534 0.1406 0.2000 0.1984 0.1206 0.1366 0.2799 0.6527 0.5214 0.1763 0.1684 0.1499 0.1471 0.2881 0.4752 0.1552 0.1589 0.2471 0.1803 0.1825 0.3995 0.8585 0.1682 0.2018 0.1877 0.1433 0.4135 0.2552 0.6857 0.1463 0.1484 0.1359 0.1765 0.1449 0.2833 0.6186 0.1303 0.1215 0.1682 0.1508 0.1098 0.1905 0.4321 0.1104 0.1027 0.0831 0.4143 0.1532 0.1951 0.4533 0.0764 0.1980 0.1319 0.1287 0.0788 0.4098 0.3878 0.3729 0.1105 0.1258 0.0814 0.0779 0.1594 0.2264 0.0592 0.0822 0.1008 0.0737 0.1124 0.1020 0.1364 0.0544 0.0707 0.0569 0.0463 0.0331 0.0600 0.4500 0.0839 0.0364 0.0300 0.0658 0.0177 0.1951 0.2083 0.0236 0.0800 0.0148 0.0538 0.0213 0.0469 0.0833 0.0088 0.0303 0.0073 0.0161 0.0323 0.0930 0.0145 0.0192 0.0221 0.0073 0.0233 0.0154 0.0045 0.0069 0.0036 0.0046 0.0101 0.0044 0.0107 0.0067 0.0015 0.0043 0.0031 0.0044 0.0066 0.0338 0.0038 0.0062 0.0066 0.0052 0.0077 0.0066 0.0331 0.0127 0.0181 0.0180 0.0179 0.0221 0.0429 0.0522 0.0091 0.0237 0.0349.
From Tables 2–4, when compared to other distributions, the HLBXE, HLBXR, and HLBXL distributions have the lowest values for all information criterion. D2 has the highest value as well. This leads us to conclude that HLBX family is better fitting the three real sets of data from Italy, Belgium, and Canada. Estimated PDFus of models plots shown in Figures 5–7 indicate that our distribution is a good choice for modeling the above COVID-19 data.
6. Simulation Results
In this section, the Monte Carlo simulation procedure is performed for comparison between the MLEs and Bayesian estimation method under square error and Linex LoFus based on MCMC, for estimating parameters of HLBXL distribution as an example of HLBX family distribution, and this is the best distribution according to the above section of the application. We can use a different program to generate these analyses as Mathcad, Mathematica, Maple, and R packages. Algorithm 2 is used for the Monte Carlo simulation experiments.
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Algorithm 3 is used for the Monte Carlo simulation of Type II censored sample experiments.
We could define the best estimation methods as those that minimise estimate bias and mean squared error (MSE). Tables 5–7 reveal the following observations:(1)As sample size increases, the bias and MSE decrease(2)When the number of failures increases in a Type II CS, the values of the bias and MSE for HLBXL distribution parameters decrease(3)We find that the Bayesian estimates under Linex (2) LoFu perform better than other estimates of HLBXL distribution with respect to MSE and bias(4)We find that the Bayesian method under Linex (2) loss function performs better than other estimations for estimating the parameters of HLBXL distribution with respect to MSE and bias(5)As increases and the others are fixed, then the bias and MSE are increasing for , and the bias and MSE are decreasing for estimates(6)As increases and the others are fixed, then the bias and MSE are increasing for , and the bias and MSE are decreasing for estimates
7. Conclusion
A new generalized generator of the half-logistic Burr X-G family was proposed and studied in this paper. Several statistical properties, including QuFu, Mos, InMos, MeD, Lo and Bo curves, and En were derived. The HLBX Lomax, HLBX exponential, and HLBX Rayleigh distributions are discussed. MLL and Bayesian estimation methods were used to estimate the unknown parameters. The HLBXL distribution fits better than the other submodels. To distinguish the performance of estimation methods, a simulation analysis was performed using the R package. For Bayesian estimation, the MCMC method was used. Three real COVID-19 datasets from different countries, including Italy, Canada, and Belgium, were considered. Finally, we plan to use this family to generate new models from the proposed generating family and investigate their statistical properties, as well as investigating the statistical inference of the new models using various methods and demonstrating the importance of the new models using new real datasets [19–22].
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest.
Authors’ Contributions
All the authors also contributed equally to this work.
Acknowledgments
The Deanship of Scientific Research (DSR), King Abdul-Aziz University, Jeddah, supported this work, under grant no. (KEP–PhD–75–130–42). The authors, therefore, gratefully acknowledge the DSR technical and financial support.