Review Article
The Ampadu-G Family of Distributions with Application to the T − X (W) Class of Distributions
Clement Boateng Ampadu, Department of Biostatistics, USA.
Received Date: February 14, 2019; Published Date: February 22, 2019
Abstract
The T − X (W)family of distributions appeared in [1]. In this paper, inspired by the structure of the CDF in the Zubair-G class of distributions [2], we introduce a new family of distributions called the Ampadu-G class of distributions, and use it to obtain a new class of distributions which we will call the AT − X (W) class of distributions, as a further application of the T − X (W)framework. Sub-models of the Ampadu-G class of distributions and the AT − X (W) class of distributions are shown to be practically significant in modeling real-life data. The Ampadu-G class of distributions is seen to be strikingly similar in structure to the Exponentiated EP (EEP) model contained in [3], and the Zubair-G class of distributions is seen to be strikingly similar in structure to the Complementary exponentiated Weibull-Poisson (CEWP) model contained in [4].
Keywords:Zubair-G; T − X (W)family of distributions; Ampadu-G
Introduction
Background on the T − X (W)family of distributions
Definition 3.1: [1] Let r (t) be the PDF of a continuous random variable T ∈[a, b] for −∞ ≤ a ≤ b ≤ ∞ and let R(t ) be its CDF. Also let the random variable X have CDF F (x) and PDF f (x) , respectively. A random variable B is said to be T − X (W)distributed if the CDF can be written as the following integral
![Click here to view Large equation 1 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E001.png)
whereW (F (x)) satisfies the following conditions
a) W (F (x))∈[a, b]
b) W (F (x))is differentiable and monotonically nondecreasing
c) limx→−∞ W(F(x))= a and limx→−∞ W(F(x))= b
Remark 3.2: By differentiating the RHS of the above equation with respect to x, the PDF
of the T − X (W)family of distributions can be obtained.
Remark 3.3: If the continuous random variable T has support [0, 1], we can take
![Click here to view Large equation 2 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E002.png)
where α > 0 . In particular, we will say a random variable B is T − X (W)distributed of type I, if the CDF can be written as the following integral
![Click here to view Large equation 3 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E003.png)
Remark 3.4: If the continuous random variable T has support
[a,∞)with a ≥ 0we can takeW (x) = −log(1− xα ) or where
α > 0 . In particular, we will say arandom variable B T − X (W)
distributed of type II, if the CDF can be written as either one of the
following integrals
![Click here to view Large equation 4 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E004.png)
Or
![Click here to view Large equation 5 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E005.png)
Remark 3.5: If the continuous random variable T has support
(−∞,∞) we can take W ( x) = log(−log(1− xα )) or , where
α > 0 . In particular, we will say a random variable B is T − X (W)
distributed of type III, if the CDF can be written
as either one of the following integrals
![Click here to view Large equation 6 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E006.png)
or
![Click here to view Large equation 7 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E007.png)
Remark 3.6: By differentiating the RHS of the equations in Remark 3.3, Remark 3.4, and Remark 3.5, respectively, we obtain the PDF’s of the class of T − X (W)distributions of type I, II and III, respectively.
Background on Zubair-G family of distributions
Definition 3.7: [2] A random variable B* is said to be Zubair-G distributed if the CDF is given by
where
![Click here to view Large equation 8 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E008.png)
Where α ,ξ > 0, x∈R and G is the CDF of the baseline distribution by differentiating the CDF in the above definition we obtain the PDF of the Zubair-G class of distributions as
![Click here to view Large equation 9 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E009.png)
Where α ,ξ > 0, x∈R , G is the CDF of the baseline distribution, and g is the PDF of the baseline distribution
The New Family of Distributions
The Ampadu-G family of distributions
Definition 4.1: Let λ > 0,ξ > 0 be a parameter vector all of whose entries are positive, and x∈R . A random variable X will be said to follow the Ampadu-G family of distributions if the CDF is given by
![Click here to view Large equation 10 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E010.png)
and the PDF is given by
![Click here to view Large equation 11 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E011.png)
where the baseline distribution has CDF G(x,ξ ) and PDF g(x,ξ )
Generalized AT − X (W) Family of Distributions of type I
Definition 4.2: Assume the random variable T with support [0, 1] has CDF G(t;ξ ) and
AT − X (W) distributed of type I if the CDF can be expressed as the following integral
![Click here to view Large equation 12 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E012.png)
Where λ,ξ ,β > 0, and the random variable X with parameter vector ! has CDF F (x,w) and PDF f (x,w)
Remark 4.3: If β =1 in the above definition we say S is AT − X (W) distributed of type I
Generalized AT − X (W) Family of Distributions of type II
Definition 4.4: Assume the random variable T with support [a,∞) has CDF G(t,ξ ) and
PDF g(x,ξ ). We say a random variable S is generalized AT − X (W) distributed of type II if the CDF can be expressed as either one of the following integrals
![Click here to view Large equation 13 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E013.png)
Or
![Click here to view Large equation 14 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E014.png)
where λ,ξ ,β > 0 and the random variable X with parameter vector ! has CDF F (x,w) and PDF f (x,w)
Remark 4.5: If β =1 in the above definition we say S is AT − X (W) distributed of type II
Generalized AT − X (W) Family of Distributions of type III
Definition 4.6: Assume the random variable T with support (−∞,∞) has CDF G(t;ξ ) and PDF g(t;ξ ) . We say a random variable S is generalized AT − X (W) distributed of type III if the CDF can be expressed as either one of the following integrals
![Click here to view Large equation 15 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E015.png)
or
![Click here to view Large equation 16 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E016.png)
where λ,ξ ,β > 0 and the random variable X with parameter vector w has CDF F (x,w) and PDF f (x,w)
Remark 4.7. If β =1 in the above definition, we say S is AT − X (W) distributed of type III
Practical Application to Real-life Data
Illustration of Ampadu-G family of distributions
We consider the data set [5] which is on the breaking stress of carbon fibers of 50 mm in length. We assume the baseline distribution is Weibull distributed, so that for x,a,b > 0 , the CDF is given by
![Click here to view Large equation 17 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E017.png)
and the PDF is given by
![Click here to view Large equation 18 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E018.png)
Theorem 5.1. The CDF of the Ampadu-Weibull distribution is given by
![Click here to view Large equation 19 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E019.png)
Where x,a,b,λ > 0
Proof. Since the baseline distribution is Weibull distributed, then for x,a,b > 0 , the CDF is given by
![Click here to view Large equation 20 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E020.png)
So the result follows from Definition 4.1
Remark 5.2: If a random variable R is Ampadu-Weibull distributed, we write R ~ AW (a,b,λ ) (Figure 1).
![Click here to view Large Figure 1 irispublishers-openaccess-biostatistics-biometric-applications](../images/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.G001.png)
Illustration of AT − X (W) Family of Distributions of type I
The data set refers to the remission times (in months) of a random sample of 128 bladder cancer patients studied in [6]. We assume the random variable T follows the Burr X (BX) family of distributions so that for t, a, b > 0, the CDF is given by
![Click here to view Large equation 21 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E021.png)
and the PDF is given by
![Click here to view Large equation 22 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E022.png)
We assume the random variable X is Lomax distributed so the for x, c, d > 0, the CDF is given by
![Click here to view Large equation 23 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E023.png)
and the PDF is given by
![Click here to view Large equation 24 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E024.png)
Now we consider Remark 4.3 in Definition 4.2, then we get the following
Theorem 5.3: The CDF of the ABurrX−Lomax family of distributions, for x,a,b,c,d,λ > 0 is given by
![Click here to view Large equation 25 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E025.png)
Remark 5.4: If a random variable W has CDF given by the ABurrX − Lomax family of distributions, we write W ABXL(a,b,c, d,λ ) (Figure 2).
![Click here to view Large Figure 2 irispublishers-openaccess-biostatistics-biometric-applications](../images/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.G002.png)
Illustration of AT − X (W) Family of Distributions of type II
The second data set is on 30 successive March precipitation (in inches) observations obtained from [7] and recorded in Section 7 of [8]. We assume the random variable T with support [0,∞) follows the Weibull distribution, so that for t > 0, and b, c > 0, the CDF is given by
![Click here to view Large equation 26 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E026.png)
and the PDF is given by
![Click here to view Large equation 27 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E027.png)
We also assume the random variable X follows the Rayleigh distribution, so that for x, a > 0, the PDF is given by
![Click here to view Large equation 28 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E028.png)
and the CDF is given by
![Click here to view Large equation 29 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E029.png)
Considering Remark 4.5 in the first integral of Definition 4.4, we get the following Theorem 5.5. The CDF of the AWeibull − Rayleigh family of distributions of type II is given by
![Click here to view Large equation 30 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E030.png)
Remark 5.6: When a random variable J* has CDF given by Theorem 3.5, we write J* ~ AWR(a,b,c,λ) (Figure 3).
![Click here to view Large Figure 3 irispublishers-openaccess-biostatistics-biometric-applications](../images/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.G003.png)
Illustration of AT − X (W) Family of Distributions of type III
In this application we consider the data set in [9] from [10], on the breaking stress of carbon fibers of 50 mm in length. We assume the random variable T with support (−∞,∞) follows the Cauchy distribution, so that for t, a 2 R, b > 0, the CDF is given by
![Click here to view Large equation 31 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E031.png)
and the PDF is given by
![Click here to view Large equation 32 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E032.png)
We also assume the random variable X follows the Weibull distribution, so that for x > 0, and c, d > 0, the CDF is given by
![Click here to view Large equation 33 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E033.png)
and the PDF is given by now considering Remark 4.7 in the second integral of Definition 4.6, we get the following
Theorem 5.7: The CDF of the ACauchy − Weibull family of distributions of type III is given by
![Click here to view Large equation 34 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.E034.png)
Where x;a,b,c,d,λ > 0
Remark 5.8: By differentiating the CDF of the ACauchy−Weibull family of distributions of type III, the PDF can be obtained
Remark 5.9: When a random variable N* has CDF given by Theorem 5.7, we write (Figure 4) N* ~ ACW (a,b,c,d,λ)
![Click here to view Large Figure 4 irispublishers-openaccess-biostatistics-biometric-applications](../images/irispublishers-openaccess-biostatistics-biometric-applications.ID.000518.G004.png)
Acknowledgement
None.
Conflict of Interest
No conflict of interest.
References
- Ayman Alzaatreh, Carl Lee, Felix Famoye (2013) A new method for generating families of continuous distributions. METRON 71: 63-79.
- Ahmad Z (2018) The Zubair-G Family of Distributions: Properties and Applications. Ann Data Sci: 1-14.
- Ristic MM, Nadarajah S (2014) A new lifetime distribution. J Stat Comput Simul 84: 135-150.
- Mahmoudi E, Sepahdar A (2013) Exponentiated Weibull-Poisson distribution: Model, properties and applications. Math Comput Simul 92: 76-97.
- Ayman Alzaatreh, Carl Lee, Felix Famoye (2014) T-normal family of distributions: a new approach to generalize the normal distribution. Journal of Statistical Distributions and Applications 1: 16.
- Lee ET, Wang JW (2003) Statistical Methods for Survival Data Analysis (3rd ed.), Wiley, New York, USA.
- Hinkley D (1977) On quick choice of power transformation. Applied Statistics 26(1): 67-69.
- Muhammad Ahsan ul Haq (2016) Transmuted Exponentiated Inverse Rayleigh Distribution. J Stat Appl Pro 5(2): 337-343.
- Maalee Almheidat, Felix Famoye, Carl Lee (2015) Some Generalized Families of Weibull Distribution: Properties and Applications. International Journal of Statistics and Probability 4(3).Nicholas MD, Padgett WJ (2006) A bootstrap control for Weibull percentile. Quality and Reliability Engineering International 22(2): 141- 151.
- Nicholas MD, Padgett WJ (2006) A bootstrap control for Weibull percentile. Quality and Reliability Engineering International 22(2): 141- 151.
-
Clement Boateng Ampadu*. The Ampadu-G Family of Distributions with Application to the T − X (W) Class of Distributions. Annal Biostat & Biomed Appli. 1(4): 2019. ABBA.MS.ID.000518.
Distributions, Biostatistics, New family, Exponentiated EP, Complementary exponentiated Weibull-Poisson, Zubair-G; T-X(W) family of distributions; Ampadu-G
-
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.