Review Article
The AmpaduG 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 ZubairG class of distributions [2], we introduce a new family of distributions called the AmpaduG class of distributions, and use it to obtain a new class of distributions which we will call the A_{T} − X (W) class of distributions, as a further application of the T − X (W)framework. Submodels of the AmpaduG class of distributions and the A_{T} − X (W) class of distributions are shown to be practically significant in modeling reallife data. The AmpaduG class of distributions is seen to be strikingly similar in structure to the Exponentiated EP (EEP) model contained in [3], and the ZubairG class of distributions is seen to be strikingly similar in structure to the Complementary exponentiated WeibullPoisson (CEWP) model contained in [4].
Keywords:ZubairG; T − X (W)family of distributions; AmpaduG
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
whereW (F (x)) satisfies the following conditions
a) W (F (x))∈[a, b]
b) W (F (x))is differentiable and monotonically nondecreasing
c) lim_{x→−∞} W(F(x))= a and lim_{x→−∞} 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
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
Remark 3.4: If the continuous random variable T has support [a,∞)with a ≥ 0we can takeW (x) = −log(1− x_{α} ) 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
Or
Remark 3.5: If the continuous random variable T has support (−∞,∞) we can take W ( x) = log(−log(1− x_{α} )) 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
or
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 ZubairG family of distributions
Definition 3.7: [2] A random variable B* is said to be ZubairG distributed if the CDF is given by
where
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 ZubairG class of distributions as
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 AmpaduG 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 AmpaduG family of distributions if the CDF is given by
and the PDF is given by
where the baseline distribution has CDF G(x,ξ ) and PDF g(x,ξ )
Generalized A_{T} − 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
A_{T} − X (W) distributed of type I if the CDF can be expressed as the following integral
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 A_{T} − X (W) distributed of type I
Generalized A_{T} − 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 A_{T} − X (W) distributed of type II if the CDF can be expressed as either one of the following integrals
Or
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 A_{T} − X (W) distributed of type II
Generalized A_{T} − 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 A_{T} − X (W) distributed of type III if the CDF can be expressed as either one of the following integrals
or
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 A_{T} − X (W) distributed of type III
Practical Application to Reallife Data
Illustration of AmpaduG 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
and the PDF is given by
Theorem 5.1. The CDF of the AmpaduWeibull distribution is given by
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
So the result follows from Definition 4.1
Remark 5.2: If a random variable R is AmpaduWeibull distributed, we write R ~ AW (a,b,λ ) (Figure 1).
Illustration of A_{T} − 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
and the PDF is given by
We assume the random variable X is Lomax distributed so the for x, c, d > 0, the CDF is given by
and the PDF is given by
Now we consider Remark 4.3 in Definition 4.2, then we get the following
Theorem 5.3: The CDF of the A_{Burr}X−Lomax family of distributions, for x,a,b,c,d,λ > 0 is given by
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).
Illustration of A_{T} − 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
and the PDF is given by
We also assume the random variable X follows the Rayleigh distribution, so that for x, a > 0, the PDF is given by
and the CDF is given by
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
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).
Illustration of A_{T} − 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
and the PDF is given by
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
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
Where x;a,b,c,d,λ > 0
Remark 5.8: By differentiating the CDF of the A_{Cauchy}−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)
Acknowledgement
None.
Conflict of Interest
No conflict of interest.
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Clement Boateng Ampadu*. The AmpaduG 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 WeibullPoisson, ZubairG; TX(W) family of distributions; AmpaduG

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