On Topp-Leone-G Power Series: Saturation in the Hausdorff Sense and Applications

: This paper discusses the Topp-Leone-G power series class of distributions. The greatest attention is paid to the investigation of intrinsic characteristic “saturation” to the horizontal asymptote in the Hausdorff sense. Some estimates for the value of the Hausdorff distance are obtained. We present a new family of recurrence generated adaptive functions with corresponding applications. The usefulness of the obtained results is demonstrated in a simulation study of some real data sets from the medical sector and insurance. Some suitable software modules within the programming environment CAS MATHEMATICA are proposed.


Introduction
One basic and very important practical problem is the construction of adequate, accurate and sufficiently flexible approximation models.In practice, for a given data set, a reliable statistical model can be developed from an appropriate probability distribution.Over the last few years, many authors have derived new compounding distributions by mixing continuous distributions with power series distributions, such as the generalized modified Weibull power series [1], the Gompertz-power series [2], the Inverse Weibull power series [3], the Exponentiated Burr XII power series [4], the Exponentiated generalized power series family [5], the odd power generalized Weibull-G power series [6] and others.In 1950, Albert Noack [7] introduced the family of discrete univariate distributions evolving in power series, although the earliest work on this topic was written by Kosambi (1949) [8].Let us recall that the probability mass function (pmf), for a random variable distributed as a power series which excludes zero, is given by where a n ≥ 0 depends only on n, C(θ) = ∑ ∞ n=1 a n θ n and θ > 0 is chosen such that C(θ) is finite.The power series distributions are discrete and mainly used to model count data.The Topp-Leone is one of the most commonly used distributions.Its generalization, Topp-Leone-G, was proposed by Al-Shomran et al. [9] in 2016.It is very often used as a generator for numerous generalizations with the aim of increasing the versatility of new distributions such as the Topp-Leone generated family [10], the new power Topp-Leone generated family [11], the new Topp-Leone class [12], the Extended Topp-Leone family [13], the alpha power Topp-Leone-G [14] and the new extension of the Topp-Leone family [15].
In 2021, Makubate et al. [16] considered a new class of distributions called the Topp-Leone-G Power Series (TL-GPS).This model is obtained by compounding the Topp-Leone-G distribution with the power series distribution (see also [17]).Note that TL-GPS is an extension of the Topp-Leone power series distribution proposed by Roozegar and Nadarajah [18].Similar distributions that involve Topp-Leone with power series are the reflected generalized Topp-Leone power series [19], the Topp-Leone generalized exponential Power series [20], the type II exponentiated half-logistic-Topp-Leone-G power series [21], the Exponentiated half Logistic-Topp-Leone-G power series [22] and the type II Topp-Leone-G power series [23].
Definition 1. Topp-Leone-G Power Series (TL-GPS) distribution is associated with the following general form of CDF function where C(θ) is finite, θ > 0 and S TL−G(t;b,ξ) is the survival function of Topp-Leone-G that is defined by for b > 0 with the baseline CDF function G(t; ξ) depending on a parameter vector ξ.
Table 1 presents some sub-classes of Topp-Leone-G Power Series.

Distribution a n C(θ) CDF Function
The main aim of this work is investigation of intrinsic characteristic "saturation" to the horizontal asymptote in the Hausdorff sense.This research can be very useful in choosing an appropriate model for approximating specific data and be applicable in various fields such as finance, economics, actuarial sciences, biostatistics and many others.We need a metric for measuring similarity between sigmoidal (cumulative) function and a step function.The Hausdorff distance can be considered as the highest optimal path between these curves as a smaller Hausdorff distance shows a higher closeness.Definition 2. The shifted Heaviside step function is defined by We consider Euclidean space with the maximum norm namely for the points Hence we can define the Hausdorff distance [24,25].Definition 3. The Hausdorff distance (the H-distance) ρ( f , g) between two interval functions f , g on Ω ⊆ R, is the distance between their completed graphs F( f ) and F(g) considered as closed subsets of Ω × R.More precisely, The formal definition of saturation of distribution in Hausdorff sense is proposed by Zaevski and Kyurkchiev [26].
Its saturation is the Hausdorff distance between the completed graph of F(•) and the curve consisting of two lines -one vertical between the points (a, 0) and (a, 1) and another horizontal between (a, 1) and (b, F(b)).
Iliev et al. [27] investigated the Hausdorff approximation between the Heaviside step function and Topp -Leone cumulative sigmoids.Similar investigations for some modifications of Topp-Leone and their behavior in the Hausdorff sense can be found in related papers and monographs [28][29][30][31].Examination of family containing power series distribution is proposed in [32].
The article is structured as follows: In Section 2, our attention is paid to investigation of intrinsic characteristic "saturation" to the horizontal asymptote in the terms of Hausdorff metric.We present in detail approximation analysis for Topp-Leone-Weibull-Poisson (TL-WP).Numerical and real data examples demonstrate the usefulness of obtained theoretical results.Furthermore, family of recurrence generated adaptive functions with corresponding applications is defined in Section 3. We propose a simple dynamic programming module implemented within the programming environment CAS Wolfram Mathematica for computation the Hausdorff distance.

Approximation Results
Let us focus on one special case of a Topp-Leone-G Power Series (TL-GPS), namely with a baseline distribution-Weibull and power series distribution-Poisson.We obtain Topp-Leone-Weibull-Poisson (TL-WP) with the CDF function given by where α, β, b, θ > 0 and t > 0.

Hausdorff Approximation
We study the behavior of CDF in the Hausdorff sense and, more precisely, "saturation" to the horizontal asymptote a = 1.The methodology applied here can be used in a similar way for other special cases of proposed families using different baseline distributions and sub-classes of power series distribution with corresponding approximation problems.In detail, we present an examination of Topp-Leone-Weibull-Poisson CDF functions (3).
According to Definitions (3) and ( 4), we have that saturation of distribution is the side of the smallest unit square, centered at the point (0, 1) touching the graph of the cumulative function.Hence, the Hausdorff distance d between F TL−WP (t; θ, b, α, β) and the Heaviside function h t 0 (t) at the "median level" satisfies the following nonlinear equation: where Note that (4) has a unique root since the function h(d) = F(t 0 + d) − 1 + d is increasing and continuous.
The next theorem gives some estimates for the saturation in terms of the Hausdorff metric of the TL-GPS CDF function.

Theorem 1. The Hausdorff distance d between the Heaviside function h t 0 (t) and the CDF function
where and 2.1A > e 1.05 .
Proof of Theorem 1.We define function H(d) by where the CDF function F(t) ≡ F TL−WP (t; θ, b, α, β) is defined by (3).It is easy to show that H (d) > 0, i.e., H(d), is monotony increasing.Now, we will show that H(d) can be approximated by where A is defined by (6).From the Taylor expansion, we have With simple calculations, we have Hence, F (t 0 ) = A − 1 for t 0 defined by (5).Then, it follows from here that Table 2 presents some computational experiments for distinct combinations of distribution parameters θ, b, α and β.We calculate values of Hausdorff approximation d with its left d l and right d r estimations using Theorem 1. Furthermore, some graphical examples are shown in Figure 2.

Some Applications
Let us consider one practical application using modeling data from the medical sector.The data set is a subset of the data corresponding to the survival times (in years) of a group of patients who received chemotherapy alone [33].The data include the survival times (in years) for 45 patients.The data-chemotherapy can be approximated with the Topp-Leone-Weibull-Poisson (TL-WP) distribution with corresponding parameters θ = 3.125, β = 5.412, α = 0.00027 and b = 0.188.This fact is not rejected from standard goodness-of-fit statistical tests.The estimated P-P graphic is displayed in Figure 3.

Statistic p-Value
Anderson-Darling    We continue to investigate applications by considering the automobile insurance claims taken from [34].The Topp-Leone-Weibull-Poisson (TL-WP) model is applied to this data set (normalized).We obtain the following parameters: θ = 0.12, β = 0.29, α = 1.66 and b = 5.53.According to the Cramér-von Mises and Pearson χ 2 statistical tests, the null hypothesis that the data are distributed according to the TL-WP is not rejected at the 5 percent level.The estimated CDF and the fitted TL-WP distribution as well as the estimated P-P for the automobile insurance claims data set are plotted in Figure 5.We apply Theorem 1 and obtain the saturation d = 0.300044 with corresponding estimates d l = 0.225546 and d r = 0.33589.

Family of Recurrence Generated Activation Functions
We define a family of recurrence generated activation functions based on Topp-Leone-Weibull-Poisson (TL-WP) distribution.Reader can consider a similar recurrence activation function based on TL-GPS distribution with different baseline distribution with relevant approximation models.
where i = 0, 1, 2, . .., with We can investigate the behavior of the CDF functions F i (t), i = 1, 2, . . . in the Hausdorff sense.Hence, for the Hausdorff distance d we have where t 0 is a positive solution of the equation F i (t 0 ) = 1 2 .
The Hausdorff distance d for some combinations of parameters is shown in Table 3. From here, we see that the Hausdorff distance becomes smaller with how deep we go into the recursion.We propose a cloud software module implemented in the programming environment Wolfram Cloud for the computation of the Hausdorff distance for recurrence generated adaptive functions from family (7) (see Figure 6).This web (cloud) version of the module only requires browser and internet connection.The user defines the parameters of distribution and the number of recursions.The result is the calculated Hausdorff distance in table view with graphical representation of the generated recurrence adaptive functions.

Conclusions
This article is dedicated to an investigation of the characteristics of "saturation" in the Hausdorff sense for Topp-Leone-G power series.This research can help scientists in adequate construction and accurate and sufficiently flexible approximation models.We explore in detail the asymptotic behavior of the Hausdorff distance between the Heaviside step function and the Topp-Leone-Weibull-Poisson CDF function.For the considered Hausdorff approximation, some estimates are proved.Suitable numerical examples are shown.Additionally, we present modeling data from the medical sector and insurance.Furthermore, we discuss the application of new families of recurrence generated adaptive functions.The proposed methodology can be successfully applied to other sub-models of Topp-Leone-G power series as well as other commonly used CDFs in practice.Some related results can be found in [35][36][37][38][39] .Several dynamic modules implemented in CAS Wolfram Mathematica are included.A cloud version that only requires a browser and internet connection is offered for some of them.The proposed modules can be upgraded as well as adapted for other distributions and data sets.

Figure 3 .
Figure 3.Estimated P-P graphic for data-chemotherapy.From Theorem 1, for the considered parameters we obtain the values of Hausdorff distance d = 0.365606 with estimates d l = 0.335175 and d r = 0.366381, respectively.The obtained result are presented by the dynamic module in Figure 4.

Figure 4 .
Figure 4.Estimated CDF for TL-WP distribution for chemotherapy data.

Figure 5 .
Figure 5.Estimated CDF and P-P graphics for TL-WP distribution for automobile insurance claims.

Table 1 .
Sub -classes of Topp-Leone-G Power Series.

Table 2 .
Values of Hausdorff distance d and its estimations.