Abstract
The truncated Cauchy power odd Fréchet-G family of distributions is presented in this article. This family’s unique models are launched. Statistical properties of the new family are proposed, such as density function expansion, moments, incomplete moments, mean deviation, Bonferroni and Lorenz curves, and entropy. We investigate the maximum likelihood method for predicting model parameters of the new family. Two real-world datasets are used to show the importance and flexibility of the new family by using the truncated Cauchy power odd Fréchet exponential model as example of the family and compare it with some known models, and this model proves the importance and the flexibility for the new family.
1. Introduction
Many authors have lately made considerable attempts to create new families to expand well-known distributions and give flexible classes to represent data in a wide range of areas, including medical sciences, environmental sciences, engineering, demography, actuarial science, and economics. Many generic families have been developed and are used to explain a wide range of real-world events. Some examples of these families are beta-G [1], gamma-G [2], upper truncated Weibull distribution [3], and truncated Weibull-G more flexible and more reliable than beta-G distribution [4], truncated inverted Kumaraswamy-G by Bantan et al. [5], type II truncated Fréchet-G [6], type II power TL by Bantan et al. [7], odd generalized N-H by Ahmad et al. [8], Topp–Leone (TL) odd Fréchet-G by Al-Marzouki et al. [9], transmuted odd Fréchet-G by Badr et al. [10], and truncated Burr X-G by Bantan et al. [11], among others.
Haq and Elgarhy [12] proposed the odd Fréchet-G (, ) with cumulative function (cdf) as follows, for ,and probability density function (pdf) is
The Cauchy (C) distribution plays an important role and has applications in different fields such as econometrics, engineering, spectroscopy, biological analysis, reliability, queueing theory, and stochastic modeling of decreasing hazard rate life devices. There are many authors who have been displayed various generalization and extension forms of Cauchy distribution in the statistical literature, for examples, Rider [13] presented generalized C distribution, A truncated C distribution by Nadarajah et al. [14], the existence of the moments of the C distribution by Ohakwe and Osu [15], Jacob and Jayakumar[16] studied half-C distribution, Kumaraswamy-half-C distribution by Hamedani and Ghosh [17], Alshawarbeh and Famoye [18] presented properties of beta-C distribution, and the power-C negative-binomial distribution by Zubair et al. [19], among others.
Recently, Aldahlan et al. [20] proposed the truncated C power-G (TCP-G) family. The cdf of the TCP-G family is given bywhere . The corresponding pdf is
The aim of this article is to provide a new, broader, and more flexible family of distributions based on the odd Fréchet-G and truncated C power-G (TCP-G) families. The new proposed family has many wide applications in physics, medicine, engineering, and finance. We construct a new family called TCP odd Fréchet-G family of distributions by inserting equation (1) into equation (3); the cdf and pdf of the areand
Henceforward, a random variable having pdf equation (6) will be defined as . The survival and hazard rate functions for the family are
The quantile function (qf) of TCPOF-G family is given by
The median is given by
The structure of this paper is as follows. Section 2 has a useful linear explanation of the TCPOF density as well as several specific models. Section 3 looks at structural properties of the TCPOF-G. In Section 4, we discuss the suggested family’s entropy. The maximum likelihood method is used to estimate the model parameters in Section 5. Section 6 provides applications to real-world datasets to illustrate the proposed family’s flexibility.
2. Linear Representation of TCPOF
If and is a real noninteger, then the following power series hold:
Applying equation (10) in equation (6), we get
By using the power series for the exponential function, the last term in equation (11) gives
The density reduces to
Using the general binomial expansion, we can writeand
Inserting equations (14) and (15) in equation (13), the density becomeswhere denotes the pdf of the exponentiated generalized (Exp-G) distribution with power parameter , and
2.1. Four Special Models of the TCPOF Family
In this part, we presented three distinct models of the TCPOF family of distributions. When the cdf and pdf have simple analytic expressions, the pdf equation (8) will be most tractable. Based on the baseline distributions, we propose four submodels of this family: Weibull, exponential, Rayleigh, and Lomax. The cdf and pdf files for these baseline models are provided in Table 1.
2.1.1. TCPOF Weibull (TCPOFW) Distribution
The cdf and pdf of TCPOFW distribution are
2.1.2. TCPOF Exponential (TCPOFE) Distribution
The cdf and pdf of the TCPOFE model (for ) are
2.1.3. TCPOF Rayleigh (TCPOFR) Distribution
The cdf and pdf of the TCPOFR model are
2.1.4. TCPOF Lomax (TCPOFL) Distribution
The cdf and pdf of the TCPOFL model are
Plots of the TCPOFW, TCPOFE, TCPOFR, and TCPOFL densities are represented in Figure 1.
3. Structural Properties
Our study focused on ordinary moments, incomplete moments (IMs), moment generating functions (MGFs), mean deviations (MD), Lorenz and Bonferroni curves (LBCs), and residual life (RL) functions.
3.1. Ordinary Moments and Incomplete Moments Functions
A first formula for the ordinary moment of , say , is
A second formula depending on the quantile function can be written as . The IMs of defined by for any real can be expressed from equation (16) as
Equation (23) denotes the IMs of . The MDs about the mean and the MDs about the median are defined byrespectively, where median , is evaluated from equation (5), and is the first IM given by equation (23) with , where
We can determine and by two techniques, the first can be obtained from equation (16) as where is the first IM of the Exp-G distribution. The second technique is given by where
For a positive random variable the LBCs, for a given probability , are given by and , respectively, where , and is the quantile function of at .
3.2. Moment Generating Function
The MGF of iswhere denotes the MGF of .
A second alternative formula can be derived from equation (16) as follows: , where
3.3. Moments of RL and Reversed RL
The order moment of the RL is given bywhere The mean RL of , family can be obtained by setting in equation (28), defined as
The order moment of the reversed RL (or inactivity time) can be obtained by the next equation
4. Entropy
The Rényi entropy is defined by (
Using the general binomial expansion, applying the same method of the linear representation equation (16) and after some simplifications, we getwhere
Thus, Rényi entropy of , family is given by
Also, entropy can be obtained as
5. Maximum Likelihood Estimation
Let be a random sample of size from the given by equation (11). Let be vector of parameters. The log-likelihood function is
The score vector components, say, are given bywhere and The maximum likelihood estimation (MLE) of parameters is obtained by setting and solving these equations simultaneously to get the.
6. Applications
In this section, we used two datasets taken from Bjerkedal [21] and Gross and Clark [22]. The TCPOFE model as example of the new family is compared with some other known competitive models to demonstrate its importance in data modeling. The MLE method is used to estimate the parameters of the competitive models. Some information criterions (IC) are as follows: the Akaike IC (Z1), Bayesian IC (Z2), consistent Akaike IC (Z3), Hannan–Quinn IC (Z4), Cramér–Von Mises (Z5), Anderson–Darling (Z6), Kolmogorov–Smirnov (Z7) statistics with its value (Z8) model selection criteria, and goodness of fit tests are used to identify the best model.
The goodness of fits of the TCPOFE model is compared with those of other competitive models, namely, Marshall–Olkin (MO) E (MOE), Burr X-E (BrXE), Kumaraswamy E (KwE), generalized MO exponential (GMOE), beta exponential (BE), Kumaraswamy MO E (KMOE), MO Kumaraswamy E (MOKE), moment exponential (ME), and exponential (E) models.
The MLEs and standard errors (SEs) are calculated for both datasets. The numerical results are listed in Tables 2 and 3. From the numerical values provided in Tables 4 and 5 and the information shown in Figures 2 and 3, the TCPOFE model performs considerably better than the other considered extensions of the E model. Hence, the TCPOFE model is a good alternative over these models for both datasets.
7. Conclusion
In this article, we have shown how the use of the “truncated Cauchy power” scheme applied to the odd Fréchet family can lead to new flexible statistical models for the analysis of right-skewed data. Precisely, we have proceeded as follows. First, we have defined the TCPOF-G family of distributions, discussed the motivations behind it, and studied its main properties of interest. Four particular examples of the new family are proposed. Rényi entropy of the TCPOF is calculated. We used the maximum likelihood technique to estimate the parameters. Examples from real-world data demonstrate empirically the significance and promise of the proposed family. In the future, we are planning to use the new family to generate a new model and study its statistical properties and prove the flexibility of it by using more real datasets.
Data Availability
Please contact the relevant author if you would like to acquire the numerical dataset used to conduct the research described in the paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors extend their appreciation to the deputyship for research and innovation, Ministry of Education in Saudi Arabia, for funding this research work through project no. IFKSURG-1438-086.