The transmuted geometric-quadratic hazard rate distribution: development, properties, characterizations and applications

We propose a five parameter transmuted geometric quadratic hazard rate (TG-QHR) distribution derived from mixture of quadratic hazard rate (QHR), geometric and transmuted distributions via the application of transmuted geometric-G (TG-G) family of Afify et al.(Pak J Statist 32(2), 139-160, 2016). Some of its structural properties are studied. Moments, incomplete moments, inequality measures, residual life functions and some other properties are theoretically taken up. The TG-QHR distribution is characterized via different techniques. Estimates of the parameters for TG-QHR distribution are obtained using maximum likelihood method. The simulation studies are performed on the basis of graphical results to illustrate the performance of maximum likelihood estimates (MLEs) of the TG-QHR distribution. The significance and flexibility of TG-QHR distribution is tested through different measures by application to two real data sets.


Introduction
Generalizations and extensions of the probability distributions are more flexible and suitable for many real data sets as compared to the classical distributions.Azzalini (1985) derived Skewed Family with additional skewing parameter.Gupta et al. (1998) developed exponentiated family.Marshall and Olkin (1997) introduced a parameter to the family of distributions.Eugene et al. (2002) established family formed from beta distribution.Jones (2004) also presented a family generated from beta distribution.The transmuted family was presented by Shaw and Buckley (2007).Zografos Balakrishnan (2009) established family based on gamma distribution.Cordeiro and Castro (2011) developed family produced from Kumaraswamy distribution.Alexander et al. (2012) studied family based on McDonald distribution.Cordeiro et al. (2013) studied exponentiated generalized family of distribution.Torabi and Montazari (2014) studied family of distributions created from logistic distribution.Elbatal and Butt (2014) developed Kumaraswamy quadratic hazard rate distribution.Alizadeh et al. (2015a) developed Kumaraswamy Marshal Olkin family.Alizadeh et al. (2015b) also proposed Kumaraswamy odd log-logistic family.Yousof et al. (2015) studied transmuted exponentiated generalized-G family of distributions.Afify et al. (2016a) developed a family of distributions called Kumaraswamy transmuted-G family.Afify et al. (2016b) presented transmuted geometric-G family (TG-G).Cordeiro et al. (2016) presented beta odd log-logistic family of distributions.Nofal et al. (2017) studied transmuted geometric Weibull distribution in terms of mathematical properties, characterizations and regression models.Cordeiro et al. (2017) studied generalized odd log-logistics family of distributions in terms of various characteristics and applications.Alizadeh et al., (2018) proposed odd power Cauchy family of distributions and studied its properties and regression models.Yousof et al. (2018) developed a family of distributions on the basis of Burr Hatke differential equation.
The TG-G family (Afify et al.;2016b) has been developed on the basis of the T-X idea (Alzaatreh et al.;2013).Let g(x) = 1 + λ − 2λx, 0 < x < 1 andW ðGðxÞÞ ¼ θGðxÞ 1−ð1−θÞGðxÞ , where W(G(x)) is non-decreasing function of X and G(x) is a base line cumulative distribution function (cdf) of X [see Alzaatreh et al.;2013 for W(G(x))].Then, the cdf of TG-G family is given by Another definition of TG-G family has been given by Afify et al. (2016b) as follows; Let X 1 and X 2 be independent and identically distributed (i.i.d.) random variables from θGðxÞ 1−ð1−θÞGðxÞ .Then, the cdf of TG-G family is Proof Consider the following order statistics: The probabilty density function (pdf ) of TG-G family is where g(x) is the baseline pdf.
The basic motivations for proposing the TG-QHR distribution are: (i) to generate distributions with arc, positively skewed, negatively skewed and symmetrical shaped; (ii) to obtain increasing, decreasing and inverted bathtub hazard rate function; (iii) to serve as the best alternative model for the current models to explore and modeling real data in economics, life testing, reliability, survival analysis manufacturing and other areas of research and (iv) to provide better fits than other sub-models.
Our interest is to study the TG-QHR distribution along with its properties, applications and examine the usefulness of this distribution for modeling phenomena compared to the sub-models.
This article is composed as follows.In Section "TG-QHR distribution", TG-QHR distribution is introduced.In Section "Structural properties of TG-QHR distribution", TG-QHR distribution is studied in terms of various structural properties, plots, sub-models and descriptive measures on the basis of quantiles.In Section "Moments and inequality measures", moments, incomplete moments, residual life functions and inequality measures and some other properties are theoretically derived.In Section "Characterizations", TG-QHR distribution is characterized via (i) ratio of truncated moments; (ii) hazard function; (iii) reverse hazard rate function and (iv) elasticity function.In Section "Maximum likelihood estimation", estimates of the parameters of TG-QHR distribution are obtained via maximum likelihood method.In Section "Simulation studies", simulation studies are performed on the basis of graphical results to illustrate the performance of MLEs.In Section "Applications", the significance and flexibility of TG-QHR distribution is tested through different measures by application to two real data sets.Goodness of fit of TG-QHR distribution is checked via different methods.Conclusion is given in Section "Conclusions".

TG-QHR distribution
The goal of this article is to propose a five parameter transmuted geometric quadratic hazard rate (TG-QHR) distribution from mixture of QHR, geometric and transmuted distributions by the application of TG-G family.Bain (1974) developed quadratic hazard rate (QHR) distribution from the following quadratic function The cdf of the random variable X with QHR distribution and parametersα, β and γ is where Qðxjα; β; γÞ The pdf of the random variable X with QHR distribution and parameters α, β and γ is The pdf and cdf of a random variable X with TG-QHR distribution are obtained by inducting (4) and ( 5) in ( 1) and (2) as follows where α > 0; β > −2 ffiffiffiffiffi ffi αγ p ; γ > 0; jλ j ≤1 and θ ∈ ð0; 1Þ are parameters.

Useful expansions and mixture representation
The density function for TG-QHR can be written as The pdf of TG-QHR distribution can be written as the mixture of exp-G densities.

Structural properties of TG-QHR distribution
The survival, hazard, reverse hazard, cumulative hazard functions and Mills ratio of a random variable X with TG-QHR distribution are given, respectively, by and The generalized hazard g 1 ðxÞ ¼ − d ln SðxÞ d ln x ¼ xhðxÞ of TG-QHR distribution is The elasticity eðxÞ Shapes of the TG-QHR density and hazard rate functions The TG-QHR density is arc, positively skewed, positively skewed and symmetrical distribution (Fig. 1a).The TG-QHR hazard is increasing, decreasing and inverted bathtub hazard rate function (Fig. 1b).

Descriptive measures based on quantiles
In this sub-section, descriptive measures on the basis of quantiles are taken up.The quantile function of TG-QHR distribution is The random number generator of TG-QHR distribution is where the random variable Z has the uniform distribution on (0, 1).

Sub models of TG-QHR distribution
The TG-QHR has wide applications in life testing, survival analysis and reliability theory.The TG-QHR has the following sub models (Table 1).

Moments and inequality measures
In this section, moments about the origin, incomplete moments, inequality measures, residual life functions and some other properties are theoretically derived.

Moments about the origin
The r th moment about the origin of the random variable X with TG-QHR distribution is Mean and Variance of TG-QHR distribution The fractional positive moments about the origin of the random variable X with TG-IW distribution are The factorial moments of X with TG-QHR distribution are where [X] i = X(X + 1)(X + 2)…(X + i − 1)and φ r is Stirling number of the first kind.
The Mellin transform helps to determine the moments for a probability distribution.
The Mellin transform of X with the TG-QHR distribution is Μf f ðxÞ; The qth central moments, Pearson's measure of skewness and Kurtosis and cumulants of X with TG-QHR distribution are determined from the relationships.
The graphical displays to describe the parameter that controls skewness and kurtosis measures of the TG-QHR distribution are added (Fig. 2).
The moment generating function of the random variable X with TG-QHR distribution is Fig. 2 skewness and kurtosis measures of the TG-QHR distribution

Incomplete moments
Incomplete moments are used in mean inactivity life, mean residual life function and other inequality measures.The r th incomplete moment about the origin of X with TG-QHR distribution is where Γ(x…) is the upper incomplete gamma function.

Residual life functions
The residual life, say m n (t), of X with TG-QHR distribution is The life expectancy or mean residual life function at a specified time t, saym 1 (t), quantifies the expected left over lifetime of an individual of age t and is given by.
The reverse residual life, say M n (t), of X with TG-QHR distribution is where γ(x…) is the lower incomplete gamma function.
The mean waiting time (MWT) or mean inactivity time signifies the waiting time passed since the failure of an item on condition that this failure had happened in the interval [0, t].The MWT of X, say M 1 (t), is defined by

Characterizations
In order to develop a stochastic function in a certain problem, it is necessary to know whether the selected function fulfills the requirements of the specific underlying probability distribution.To this end, it is required to study characterizations of the specific probability distribution.Different characterization techniques have developed.
The TG-QHR distribution is characterized via (i) ratio of truncated Moments (ii) hazard function (iii) reverse hazard rate function and (iv) elasticity function.

Characterization of TG-QHR distribution via ratio of truncated moments
The TG-QHR distribution is characterized using Theorem 1 (Glänzel, 1986) on the basis of a simple relationship between two truncated moments of functions of X. Theorem 1 is given in Appendix A.
where D is a constant.

Characterization via hazard function
Here we characterize TG-QHR distribution via hazard function of X. Definition 5.2.1:Let X : Ω → (0, ∞) be a continuous random variable with cdf F(x)and pdf f(x).The hazard function,h F (x), of a twice differentiable distribution function satisfies the differential equation Proposition 5.2.1 Let X : Ω → (0, ∞) be continuous random variable.The pdf of X is (6), if and only if its hazard function, h F (x), satisfies the first order differential equation Proof: If the pdf of X is (6), then the above differential equation holds.Now, if the differential equation holds, then which is the hazard function of the TG-QHR distribution.

Characterization via reverse hazard function
Here we characterize TG-QHR distribution via reverse hazard function of X. Definition 5.3.1:Let X : Ω → (0, ∞) be a continuous random variable with cdf F(x)and pdf f(x).The reverse hazard function,r F (x), of a twice differentiable distribution function satisfies the differential equation Proposition 5.3.1 Let X : Ω → (0, ∞) be continuous random variable.The pdf of X is (6) if and only if its reverse hazard function,r F , satisfies the first order differential equation.
Proof: If the pdf of X is (6), then the above differential equation holds.Now, if the differential equation holds, then which is the reverse hazard function of the TG-QHR distribution.Definition 5.4.1:Let X : Ω → (0, ∞) be a continuous random variable with cdf F(x) and pdf f(x).The elasticity function, e F (x), of a twice differentiable distribution function satisfies the differential equation Proposition 5.4.1 Let X : Ω → (0, ∞) be continuous random variable.The pdf of X is (6) if and only if its elasticity,e F (x), satisfies the first order differential equation.
Proof: If the pdf of X is (6), then the above differential equation holds.Now, if the differential equation holds, then which is the elasticity function of the TG-QHR distribution.
The MLEs (standard errors) are given in Table 2. Table 3 displays goodness-of-fit statistics such as W, A, K-S (p-values) and −ℓ.
We can perceive that TG-QHR model is accurate fitted to data II because good of fit measures are smaller and graphical plots such as estimated pdf, cdf and pp.plots are closer to data set II (Fig. 6).

Fig. 1 a
Fig. 1 a Plots of pdf of TG-QHR Distribution.b Plots of hrf of TG-QHR Distribution

Fig. 5
Fig. 5 Estimated pdf for TG-QHR and Estimated cdf for TG-QHR for data set I. PP plot for data Set I

Table 1
Sub-models of TG-QHR distribution

Table 2
MLEs (standard errors) and goodness-of-fit statistics for data set I

Table 3
Goodness-of-fit statistics for data set IBoldface entries indicates that the proposed distribution is best fitted to data sets

Table 4
MLEs (standard errors) and goodness-of-fit statistics for data set II

Table 5
Goodness-of-fit statistics for data set II Estimated pdf for TG-QHR and Estimated cdf for TG-QHR for data set II. PP Plots for data set II