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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 |
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-α) (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) } ) x∈Supp(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-α)
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-α)
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.
1. Ampadu CB (2020) New classes of quantile generated distributions: Statistical measures, model fit and characterizations. Lulu Press Inc.
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.
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