Iris Publishers- Open access Journal of Annals of Biostatistics & Biometric Applications | The Ampadu-G Family of Distributions with Application to the T − X (W) Class of Distributions

Authored by Clement Boateng Ampadu

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 A_T − 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 A_T − 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
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 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
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
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
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 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
and the PDF is given by
Theorem 5.1. The CDF of the Ampadu-Weibull 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

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