The T–R {Y} power series family of probability distributions

A new family of univariate probability distributions called the T − R {Y} power series family of probability distributions is introduced in this paper by compounding the T − R {Y} family of distributions and the power series family of discrete distributions. A treatment of the general mathematical properties of the new family is carried out and some sub-families of the new family are specified to depict the broadness of the new family. The maximum likelihood method of parameter estimation is suggested for the estimation of the parameters of the new family of distributions. A special member of the new family called the Gumbel–Weibull–{logistic}–Poisson (GUWELOP) distribution is defined and found to exhibit both unimodal and bimodal shapes. The GUWELOG distribution is further applied to a real multi-modal data set to buttress its applicability.

The transformed-transformer method previously called the T-X family of distributions (Alzaatreh [15]) and later renamed the T-R{Y} family of distributions (Alzaatreh et al. [16]) has been thought of as the largest family of univariate distributions, in that it includes several families of univariate distributions as special cases. Alzaatreh et al. [16] defined the T-R{Y} system using the following arguments: Suppose T, R, and Y are random variables with respective cumulative distribution function (cdf) F T (x) = P(T ≤ x), F R (x) = P(R ≤ x) and F Y (x) = P(Y ≤ x). Let the corresponding quantile functions be Q T (p), Q R (p) and Q Y (p), where the quantile function is defined as Q W (p) = inf {w : F W (w) ≥ p }, 0 < p < 1. Suppose the corresponding densities of T, R and Y exist and denote them by f T (x), f R (x) and f Y (x). Assume that Tϵ(a, b) and Yϵ(c, d)for − ∞ ≤ a < b ≤ ∞ and− ∞ ≤ c < d ≤ ∞ ,then the T-R{Y} family of distributions was defined by the cdf The corresponding probability density function (pdf) of the cdf in (1) was given by The discrete counterpart of univariate probability distributions has also received some attention over the years in the literature. One of the most common families of discrete univariate distributions is the power series family of discrete univariate distributions (Kosambi [18]; Noack [19]; Patil [20]; Patil [21]) defined by the probability mass function (pmf) where a n ≥ 0 depends only on n, CðθÞ ¼ P ∞ n¼1 a n θ n and θ > 0 is such that C(θ) is finite and its first, second and third derivatives are defined and shown by C ′ (θ), C ′′ (θ), and C ′ ′ ′ (θ). Observe that the pmf in (3) is truncated at zero and could be generalized to a zero-inflated one (Patil,[21]). In Table 1, some members of the power series family of distributions (truncated at zero) defined by (3) such as the Poisson, geometric, binomial and logarithmic distributions are presented alongside their respective a n , C(θ), C ′ (θ), C ′′ (θ), and C ′ ′ ′ (θ).
In this paper, the compounding of the T-R {Y} family of univariate distributions and the power series family of discrete univariate distributions is carried out. We shall present how the new family is constructed, examine the general mathematical properties of the new family, show how parameters of the new family can be estimated using the maximum likelihood method as well as define and apply a special member of the new family to a real data set. Table 1 Useful quantities for some power series distributions Distribution a n C(θ)

Construction of the T-R {Y} power series family of distributions
Let X 1 , X 2 , …, X n be independent and identically distributed (iid) random variables constituting a sample of size n from the T-R {Y} family of distributions as defined in (1). Let X (1) , X (2) , …, X (N) be the corresponding order statistic of the random sample. From the theory of order statistics, the cdf of first order statistic X (1) for a given N = n is expressed as Suppose N is a discrete random variable and follows the power series distribution in (3), the marginal cdf of X (1) can be written as Thus, the cdf of the T-R {Y}-power series (T-R {Y}-PS) family of distributions is given by A physical interpretation of the family of models in (4) is as follows: consider that the failure of a system, device, product, or component occurs due to the presence of an unknown number, say N, of initial defects of the same kind, which can be identifiable only after causing the failure and repaired perfectly. If X i denotes the time to the failure of the device due to the ith defect, for i ≥ 1, such that each X i follows the T-R {Y} distribution in (1), suppose N is discrete and follows a power series distribution in (3), then the distribution of the random variable X (1) which is the time of first failure is the distribution in (4).
The pdf corresponding to (4) is obtained by differentiating (4) w.r.t x and it is given by The survival and hazard functions of the T-R {Y}-PS family of distributions are given respectively by  Table 2 by their cdfs. In Table 3, five standardized distributions of the random variable Y are presented alongside their various quantile functions Q Y (p) and the corresponding support of the random variable T which is needed to make (1) a valid cdf. These standardized distributions include the standard exponential, logistic, extreme value, log logistic, and uniform distributions. The use of standardized distributions is to reduce the number of parameters in the T-R{Y}-PS distributions. For practical purposes and when highly necessary, these standardized distributions can be replaced with their non-standardized versions.

General mathematical properties of the T-R {Y} power series family of distributions
Some useful statistical properties of the new family are presented. We begin by looking at some limiting distributions as contained in Propositions 1 and 2.

Proposition 1:
The T-R{Y} distribution defined by (1) is a limiting case of the T -R{Y} − PS family of distributions defined in (4) when θ → 0 + . Proof: Applying CðθÞ ¼ P ∞ n¼1 a n θ n , one readily obtains Evaluating using standard procedure gives which is the cdf of the T -R {Y} distribution defined by (1).  Proof: The proof follows directly and explicitly from substituting x for Q Y (F R (x)) in (1) and the proof of Proposition 1.

Proposition 3:
The pdf of the T -R{Y} − PS family of distributions can be expressed as linear combination of density of the first order statistic of the T -R{Y} distribution as Proof: na n θ n−1 . Using (5), one readily obtains Hence, the proof.

Quantiles and moments
The quantile function and moments of a probability distribution provide the theoretical base upon which many statistical properties of a distribution are assessed with. The quantile function in particular is very useful in Monte Carlo simulations since it helps in producing simulated random variates for any distribution, especially when it is in closed form. Table 3 Some distributions of Y with corresponding Q Y (p) and support of T Theorem 1: The quantile function Q(p) of the T -R{Y} − PS family of distributions is given by The result in (8) is obtained by solving the equation

Corollary 1:
Random samples can be simulated from the T -R{Y} − PS family of distributions by making use of the relation where X is a T -R{Y} − PS random variable and U, a uniform random variable on the interval (0, 1). Proof: The proof follows by substituting U for p in (8), where U is a uniform random variable on the interval (0, 1).
An expression for the rth non-central moments of the T -R{Y} − PS family of distributions random variable follows from Proposition 3. The rth non-central moments of the T -R{Y} − PS family of distributions random variable X is given by Using Proposition 3, the mgf can be expressed as Thus the mgf of the T -R{Y} − PS family of distributions can be expressed as a linear combination of the mgf of the first order statistics of the T -R{Y} distribution.

Order statistics
Order statistics are among the most essential tools in non-parametric statistics and inference. Their importance is highly visible in the problems of estimation and hypotheses tests in a variety of ways. Their moments play an important role in quality control testing and reliability, where an analyst needs to predict the failure of future components or items based on the times of a few observed early failures. These predictors are most of the time based on moments of order statistics.
Theorem 2: Let X 1 , X 2 , …, X m be a random sample of size m from the T -R{Y} − PS family of distributions and suppose X 1 : m < X 2 : m < … < X m : m denote the corresponding order statistics. The pdf of the k th order statistic can be expressed as where B(., .) is the complete beta function.

Proof:
From definition, the pdf of the kth order statistic of the T -R{Y} − PS family of distributions can be written as Using the binomial expansion formula, one readily obtains Substituting into (13) gives (4) and (5) into (14) gives where b 0 = 1, b n = a n + 1 /a 1 for n = 1, 2, 3, ….
Using the identity (see. Gradshteyn and Ryzhik [23]) for a positive integer m + j − k, one can write Consequently, where d m + j − k, 0 = 1 and the coefficients for t ≥ 1 can be obtained from the recurrence equation ))] can also be defined. In particular, Thus, where b 0 = 1, b r = a r + 1 /a 1 for r = 1, 2, 3, … Inserting (16) and (17) in (15) gives where δ r;n;m;k; and f X x ð1Þ ðx; n þ m þ j−k þ r þ 1Þ denote the pdf of X ð1Þ ¼ minf X i g nþmþ j−kþrþ1 i¼1 : One readily observes that the pdf of the T -R{Y} − PS family order statistics is an infinite linear combination of the density of X ð1Þ ¼ minf , where the quantities δ r, n, m, k, j depend only on the power series family.
The sth moment of the T -R{Y} − PS family kth order statistics is given as Thus, A characterization for the new family Following a dual concept in statistical mechanics, Shannon [24] introduced the probabilistic definition of entropy. The Shannon entropy which is sometimes referred to as a measure of uncertainty plays an essential role in information theory. To measure randomness or uncertainty, the entropy of a random variable comes handy since it can be defined in terms of its probability distribution. Suppose X is a continuous random variable with density function f. Then, the Shannon entropy of X is defined by Another powerful method often employed in the field of probability and statistics and closely related to the Shannon entropy is the "maximum entropy method" pioneered by Jaynes [25]. The method considers a family of density functions where T 1 (X), …, T m (X) are absolutely integrable functions with respect to f, and T 0 (X) = α 0 = 1. In the continuous case, the maximum entropy principle suggests deriving the unknown density function of the random variable X by the model that maximizes the Shannon entropy (19) subject to the information constraints defined in the family F (see. Shore and Johnson [26]). The maximum entropy method has been used for the characterization of several standard probability distributions; see for example, Zografos and Balakrishnan [27]. The maximum entropy distribution is the density of the family F, denoted f ME , obtained as the solution of the optimization problem As demonstrated by Jaynes [25], the maximum entropy distribution f ME determined by the constrained maximization problem depicted above "is the only unbiased assignment we can make; to use any other would amount to arbitrary assumption of information which by hypothesis we do not have" To provide a maximum entropy characterization for the T -R{Y} − PS family, a derivation of important constraints is undertaken.

Proposition 4:
If X is a random variable with density (5) and Z follows a T -R{Y} distribution with density given by (2), the following constraints hold Proof: The proof is trivial and hence it is omitted. The density function f T -R{Y} − PS (.) given in (5) for the random variable X following the T -R{Y} − PS family of distributions, is the unique solution of the optimization problem under the constraints C1 and C2 given in Proposition 4.

Proof:
Suppose v(.) is a pdf which satisfies the constraints C1 and C2. The Kullback-Leibler divergence between the densities v and f T -R{Y} − PS is Following Cover and Thomas [28], one obtains Let Z have the pdf given by (2). From the definition of f T -R{Y} − PS and based on the constraints C1 and C2, the following result holds: Since the density v satisfies the constraints C1 and C2. Thus, hence, with equality if and only if v(x) = f T -R{Y} − PS (x) for all x except for a null measure set. This proves Theorem 3.

Corollary 2:
The Shannon entropy of the T -R{Y} − PS family of distributions is given by Proof:

Maximum likelihood estimation of the parameters of the new family
Suppose ξ is a p × 1 vector containing all the parameters of the T -R{Y} distribution, for a complete random sample x 1 , x 2 , …, x n of size n from the T -R{Y} − PS family, the total log-likelihood function is given by Let Θ = (θ ξ) T be the unknown parameter vector of the T -R{Y} − PS family, the associated score function is given by where ∂ℓ ∂θ and ∂ℓ ∂ξ are given by The maximum likelihood estimate of Θ,Θ; can be obtained by solving the non-linear systems of equations, U(Θ) = 0. Since the resulting systems of equations are not in closed form, the solutions can be found numerically using some specialized numerical iterative scheme such as the Newton-Raphson type algorithms, which can be implemented on several computing software like R, SAS, MATHEMATICA, and MATLAB.
For interval estimation of the parameters of the T -R{Y} − PS family, one would require the Fisher information matrix (FIM) given by the (1 + p) × (1 + p) symmetric matrix where p is the number of parameter(s) in the T -R{Y} distribution and The total FIM, I(Θ), can be approximated by For real data, J ðΘÞ is obtained after the maximum likelihood estimate of Θ is gotten, which implies the convergence of the iterative numerical procedure involved in finding such estimate.
Given thatΘ is the maximum likelihood estimate of Θ and under the conditions that are fulfilled for the parameters Θ in the interior of the parameter space but not on the boundary, it follows that Taking T, R, and Y as random variables following the Gumbel, Weibull and logistic distributions, respectively, Al-Aqtash et al. [29] defined the Gumbel-Weibull {logistics} (GW) distribution by the cdf and pdf expressed respectively as x > 0; α; β; λ; γ > 0: Taking the power series distribution as the Poisson distribution with properties as specified in Table 1 and substituting (26) and (27) into (4) and (5), we define the Gumbel -Weibull {logistic} Poisson (GUWELOP) distribution by the cdf and pdf given respectively by x > 0; α; β; λ; γ > 0; θϵℝ: A graph of the pdf of the GUWELOP distribution is shown in Fig. 1. The graph of the pdf reveals that the GUWELOP density can be right-skewed, left-skewed, almost symmetric, and bimodal. To buttress the applicability of members of the new family in modeling complex real life data, the GUWELOP distribution is used to fit a multimodal data set. The data set represents Kevlar 49/epoxy strands failure times data (pressure at 70%) reported in Al-Aqtash et al. [29] The data are multimodal, platykurtic, and approximately symmetric. (Skewness = 0.1, kurtosis = − 0.79). The data set is given in Table 8. The maximum likelihood method is used to fit the GUWELOP distribution, GW distribution, and the beta-normal (BN) distribution (Eugene et al. [11] to the data set. The results of the fit and other summary statistics are presented in Table 9. The graph of the fitted densities alongside the histogram of the data set is shown in Fig. 2.
Results from Table 9 show that the three distributions provided good fits to the data set since all the distributions have high p values of the K-S statistics. However, The GUWELOP distribution has the highest p value and hence provided the best fit to the data. This application suggests the adequacy of the GUWELOP distribution in fitting multi-modal data sets.