Short Communication
The Quantile Zeid-G Family of Distributions by a Lambert-W Type Weight with Illustration to Bladder Cancer Patients Data
Clement Boateng Ampadu*
Corresponding Author: Clement Boateng Ampadu, 31 Carrolton Road, Boston, MA 02132-6303, USA
Received: May 28th, 2020; Revised: June 08th, 2020; Accepted: June 10th, 2020
Citation: Ampadu CB. (2020) The Quantile Zeid-G Family of Distributions by a Lambert-W Type Weight with Illustration to Bladder Cancer Patients Data. J Clin Trials Res, 3(3): 203-205.
Copyrights: ©2020 Ampadu CB, Alarcon SE, Puertolas SM, Bravo OM, Valero VL & Arnaiz SJA. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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Some contributions to quantile distribution theory appeared in [1-2]. In this short note, the class of quantile Zeid-G statistical distributions are introduced. A sub-model of this family is shown to be practically significant in fitting real-life data. The researchers are asked to further develop the properties and applications of this new class.

Keywords:   (1/e)α PT-G family distributions, Zeid-G, quantile generated family of distributions, cancer patient’s data, heavy-tailed Ampadu-G

 

INTRODUCTION

Firstly, we recall the following

e

Definition 1.1. [3].  Let α ≠(1¦e) ,α > (1¦e) and ξ > 0 where α is the rate scale parameter and ξ is a vector of parameters in the baseline distribution all of whose entries are positive. A random variable Z is said to follow (1/e)α the power transform family of distributions if the Cumulative Distribution Function (CDF) is given by
F (x; α,ξ)=  (1- e-αG (x; ξ))/(1-e )    x ϵ R,
where the baseline distribution has CDF G (x; ξ).
 

Proposition 1.2 [3]. The PDF of the (1/e)α power transform family distribution is given by

ƒ (x; α, ξ) =1/(1-e)  (αg(x,ξ))e-αg(x,ξ)

where α ≠(1¦e) ,α > (1¦e) and ξ > 0  ,  and  and G(x;ξ) is the baseline cumulative distribution with probability density function g(x; ξ).

Remark 1.3. The parameter space for α have been relaxed to α R, α ≠ 0. The relaxation has been employed in several papers, and for example, see [4]

Using this relaxation, we introduced the following, inspired by the structure of the Chen-G CDF [5].

Definition 1.4 [6]. A random variable Y is called a Zeid random variable, if the CDF is given by Zeid-G, that is

F(x;α,β,ξ)=(1-e-α{1-e-βG(x;ξ) } )/(1-e-α{1-e^(-β) } )                 xSupp(G),

where the baseline distribution has CDF G (x; ξ), ξ is a vector of parameters in the baseline CDF whose support depends on the chosen baseline CDF. α, β R, and α, β ≠ 0

The Zeid-Normal distribution was shown to be a good fit to Table 2 [7]. On the other hand, in [8], we introduced the following:

 

Theorem 1.5 [8]. Let X follow Ampadu- {Standard Uniform}, and put

Y=QG (e1-X)

where QG = G−1 (.) is the quantile function of the distribution with baseline CDF G, then the CDF of Y is

FY (y;α,ξ)=1-(1-e-α(1-log(G(y;ξ)))2 )/(1-e)

Moreover, for any t>0, we have

lim(y)ety (1-FY (y;α,ξ))=∞

Thus, Y is a heavy-tailed Ampadu-G random variable.

In particular, we showed the Heavy-tailed Ampadu-Weibull distribution was a good fit to the bladder cancer patients data recorded in [9]. Motivated by these developments, this paper introduces a so-called quantile Zeid-G family of distributions. Moreover, in the special case G follows the Heavy-tailed Ampadu-Weibull distribution, we show the quantile Zeid-{Heavy-tailed Ampadu-Weibull} distribution is a good fit to the bladder cancer patients data recorded [9]. The reader is recommended to explore further properties and applications of the new class of distributions presented in this  paper.

The New Family Defined

Recall that the generic form of the quantile CDF in the sense of [1] and [2] is Q [V (F (x))] where Q is a quantile, V is an appropriate weight, and F (x) is some baseline distribution.

Remark 2.1. If in Definition 1.4, α = β = 1, and G is the CDF of the uniform distribution on [0, 1], then we say Y is a standard Zeid random variable.

 

Our “Q” is a special solution obtained by solving the following equation for y

 

x=(1-ee-y-1)/1-e1/e  -1

In particular, the special solution, our “Q”, is given by

 

Q(x)=log(1/(1-log(e/(e(1/e) x-ex+e))

Our “V” is obtained by solving the following equation for y

 Y * e1-y=  (1- ee-x-1)/(1- e1/e-1 )

In particular, our “V” is given by,

 

V(x)=-W((e-ee-x))/(e(e(1/e)-e))

Where W(z) gives the principal solution for m in . Thus, we introduce the following

 

Definition 2.2. The CDF of the quantile Zeid-G family of distributions is given by

 

K(x;ξ)=log(-1/(log(1/((e-e1/e )W ((e-ee-G(x;ξ) ))/e(e1/e-e) )+e))

Where G is some baseline distribution, x Supp (G), ξ is a vector of parameters in the baseline distribution whose support depends on G, and W (z) gives the principal solution for m in z = mem.

 

 PRACTICAL ILLUSTRATION

Assume the baseline distribution is given by the heavy-tailed Ampadu-Weibul distribution, that is, in Definition 2.2 we set

 

G(x;ξ)=1-(1-e-αe-2(x/b)a)/(1-e)

Where x, a, b > 0, 0

Remark 3.1. We write QZHT AW (a, b, α) for short to represent the Quantile Zeid- {heavy-tailed Ampadu Weibull} distribution

 

The quantile Zeid- {heavy-tailed Ampadu Weibull} distribution appears a good fit to the bladder cancer patient’s data recorded in [9] as shown in Figure 1.

CONCLUSION

 

In this paper, we introduced the quantile Zeid-G family of distributions, and showed the quantile Zeid- {Heavy-tailed Ampadu-Weibull} distribution is a good fit to real life data. We hope the researchers will further develop the properties and applications of this new class of statistical distributions.

 

 

 

 

 

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2.   Ampadu C (2018) Results in distribution theory and its applications inspired by quantile generated probability distributions. Lulu Press Inc.

 

3. Anafo AY (2019) The New Alpha Power Transform: Properties and Applications, Master of Science in Mathematical Sciences Essay, African Institute for Mathematical Sciences (Ghana). Unpublished Manuscript.

 

4.  Ampadu CB (2019) A New T-X Family of Distributions. Annal Biostat Biomed Appl 2: 1-3.

 

5.   Anzagra L, Sarpong S, Nasiru S (2020) Chen-G class of distributions. Cogent Math Stat 7.

 

6.  Ampadu CB (2020) The Zeid-G Family of Distributions. Biomed J Sci Tech Res 27: 20940-20942.

 

7.   Alzaatreh A, Lee C, Famoye F (2014) T-normal family of distributions: A new approach to generalize the normal distribution. J Stat Distr Appl 1: 16.

 

8.   Ampadu CB (2020) The Heavy-Tailed Ampadu-G Family of Distributions. -Unpublished Manuscript.

 

9.  Ahmad Z, Elgarhy M, Hamedani GG (2018) A new Weibull-X family of distributions: Properties, characterizations and applications. J Stat Distrib App.