Some contributions to quantile distribution theory appeared in [12]. In this short note, the class of quantile ZeidG statistical distributions are introduced. A submodel of this family is shown to be practically significant in fitting reallife data. The researchers are asked to further develop the properties and applications of this new class.
Keywords: (1/e)^{α }PTG family distributions, ZeidG, quantile generated family of distributions, cancer patient’s data, heavytailed AmpaduG
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/(1e^{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 ChenG CDF [5].
Definition 1.4 [6]. A random variable Y is called a Zeid random variable, if the CDF is given by ZeidG, that is^{}
F(x;α,β,ξ)=(1e^{α{1eβG(x;ξ) }} )/(1e^{α{1e^(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 ZeidNormal 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=Q_{G} (e^{1X})
where Q_{G} = G^{−1} (.) is the quantile function of the distribution with baseline CDF G, then the CDF of Y is
F_{Y} (y;α,ξ)=1(1eα(1log(G(y;ξ)))^{2} )/(1e^{α})
Moreover, for any t>0, we have
lim┬(y⇢∞)e^{ty} (1F_{Y} (y;α,ξ))=∞
Thus, Y is a heavytailed AmpaduG random variable.
In particular, we showed the Heavytailed AmpaduWeibull distribution was a good fit to the bladder cancer patients data recorded in [9]. Motivated by these developments, this paper introduces a socalled quantile ZeidG family of distributions. Moreover, in the special case G follows the Heavytailed AmpaduWeibull distribution, we show the quantile Zeid{Heavytailed AmpaduWeibull} 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=(1e^{ey}_{1})/1e^{1/e 1}
In particular, the special solution, our “Q”, is given by
Q(x)=log(1/(1log(e/(e^{(1/e)} xex+e))
Our “V” is obtained by solving the following equation for y
^{ }Y * e^{1y}= (1 e^{ex1})/(1 e^{1/e1} )
In particular, our “V” is given by,
V(x)=W((ee^{ex}))/(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 ZeidG family of distributions is given by
K(x;ξ)=log(1/(log(1/((ee^{1/e} )^{W} ((ee^{eG(x;}^{ξ)} ))/e(e^{1/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 = me^{m}.
PRACTICAL ILLUSTRATION
Assume the baseline distribution is given by the heavytailed AmpaduWeibul distribution, that is, in Definition 2.2 we set
G(x;ξ)=1(1e^{αe2(x/b)a})/(1e^{α})
Where x, a, b > 0, 0
Remark 3.1. We write QZHT AW (a, b, α) for short to represent the Quantile Zeid {heavytailed Ampadu Weibull} distribution
The quantile Zeid {heavytailed 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 ZeidG family of distributions, and showed the quantile Zeid {Heavytailed AmpaduWeibull} 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 TX Family of Distributions. Annal Biostat Biomed Appl 2: 13.
5. Anzagra L, Sarpong S, Nasiru S (2020) ChenG class of distributions. Cogent Math Stat 7.
6. Ampadu CB (2020) The ZeidG Family of Distributions. Biomed J Sci Tech Res 27: 2094020942.
7. Alzaatreh A, Lee C, Famoye F (2014) Tnormal family of distributions: A new approach to generalize the normal distribution. J Stat Distr Appl 1: 16.
8. Ampadu CB (2020) The HeavyTailed AmpaduG Family of Distributions. Unpublished Manuscript.
9. Ahmad Z, Elgarhy M, Hamedani GG (2018) A new WeibullX family of distributions: Properties, characterizations and applications. J Stat Distrib App.
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