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

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1 Bhatti et al. Journal of Statistical Distributions and Applications (018) 5:4 RESEARCH The transmuted geometric-quadratic hazard rate distribution: development, properties, characterizations and applications Fiaz Ahmad Bhatti 1*, G. G. Hamedani, Mustafa Ç. Korkmaz 3 and Munir Ahmad 1 Open Access * Correspondence: fiazahmad7@gmail.com 1 National College of Business Administration and Economic, Lahore, Pakistan Full list of author information is available at the end of the article Abstract We propose a five parameter transmuted geometric quadratic hazard rate (TG-QHR) distribution derived from miture 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 3(), , 016). 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 maimum likelihood method. The simulation studies are performed on the basis of graphical results to illustrate the performance of maimum likelihood estimates (MLEs) of the TG-QHR distribution. The significance and fleibility of TG-QHR distribution is tested through different measures by application to two real data sets. Keywords: Quadratic hazard rate, Geometric distribution, Characterizations, Maimum likelihood estimation Introduction Generalizations and etensions of the probability distributions are more fleible 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 eponentiated family. Marshall and Olkin (1997) introduced a parameter to the family of distributions. Eugene et al. (00) established family formed from beta distribution. Jones (004) also presented a family generated from beta distribution. The transmuted family was presented by Shaw and Buckley (007). Zografos Balakrishnan (009) established family based on gamma distribution. Cordeiro and Castro (011) developed family produced from Kumaraswamy distribution. Aleander et al. (01) studied family based on McDonald distribution. Cordeiro et al. (013) studied eponentiated generalized family of distribution. Torabi and Montazari (014) studied family of distributions created from logistic distribution. Elbatal and Butt (014) developed Kumaraswamy quadratic hazard rate distribution. Alizadeh et al. (015a) developed Kumaraswamy Marshal Olkin family. Alizadeh et al. (015b) also proposed Kumaraswamy odd log-logistic family. Yousof et al. The Author(s). 018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Bhatti et al. Journal of Statistical Distributions and Applications (018) 5:4 Page of 3 (015) studied transmuted eponentiated generalized-g family of distributions. Afify et al. (016a) developed a family of distributions called Kumaraswamy transmuted-g family. Afify et al. (016b) presented transmuted geometric-g family (TG-G). Cordeiro et al. (016) presented beta odd log-logistic family of distributions. Nofal et al. (017) studied transmuted geometric Weibull distribution in terms of mathematical properties, characterizations and regression models. Cordeiro et al. (017) studied generalized odd log-logistics family of distributions in terms of various characteristics and applications. Alizadeh et al., (018) proposed odd power Cauchy family of distributions and studied its properties and regression models. Yousof et al. (018) developed a family of distributions on the basis of Burr Hatke differential equation. The TG-G family (Afify et al.; 016b) has been developed on the basis of the T-X idea (Alzaatreh et al.; 013). Let g()=1+λ λ, 0< < 1 andwg ¼ θg 1 1 θg, where W(G()) is non-decreasing function of X and G() is a base line cumulative distribution function (cdf) of X [see Alzaatreh et al.; 013 for W(G())]. Then, the cdf of TG-G family is given by F ¼ θg 1 1 θg Z 0 1 þ λ λzdz: Another definition of TG-G family has been given by Afify et al. (016b) as follows; Let X 1 and X be independent and identically distributed (i.i.d.) random variables from θg 1 1 θg. Then, the cdf of TG-G family is F ¼ θg 1 1 θg 1 þ λg 1 1 θg Proof Consider the following order statistics: and X 1: ¼ minx 1 ; X and X : ¼ max 1 ; X let X ¼ d X 1: ; with probability λþ1, X ¼ d X :, with probability 1 λ, where λ 1. The cdf of X is F ¼ λ þ 1 PX 1: þ 1 λ ; jλj 1; θ 0; 1; 0: PrX : ; or F ¼ 1 λ " # θg 1 1 þ 1 1 θg θg F; θ; λ ¼ 1 1 θg 1 þ λg 1 1 θg 1 λ θg ; 1 1 θg ; jj 1; λ θ 0; 1; 0; 1 The probabilty density function (pdf) of TG-G family is

3 Bhatti et al. Journal of Statistical Distributions and Applications (018) 5:4 Page 3 of 3 f ; θ; λ ¼ θg ½1 þ θ 1G 1 þ λ λθg 1 þ θ 1G ; jλj 1; θ 0; 1; > 0; where g() 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 eplore 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 eamine 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 Maimum likelihood estimation, estimates of the parameters of TG-QHR distribution are obtained via maimum 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 fleibility 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 miture 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 A ¼αþβþγ p ; > 0; α > 0; γ > 0; β > ffiffiffiffiffi αγ : 3 The cdf of the random variable X with QHR distribution and parametersα, β and γ is F QHR ¼1 e Qjα;β;γ p ; α > 0; γ > 0; β > ffiffiffiffiffi αγ ; 0; 4 where Qjα; β; γ ¼α þ β þ γ 3 3 : The pdf of the random variable X with QHR distribution and parameters α, β and γ is

4 Bhatti et al. Journal of Statistical Distributions and Applications (018) 5:4 Page 4 of 3 f QHR ¼Ajα; β; γe Qjα;β;γ ; > 0: 5 The pdf and cdf of a random variable X with TG-QHR distribution are obtained by inducting (4) and (5) in (1) and () as follows " θ Ajα; β; γ f¼ # jα;β;γ e Q λθ 1 e Qjα;β;γ ½1 þ θ 11 e Qjα;β;γ 1 þ λ ; > 0; 6 1 þ θ 11 e Qjα;β;γ θ 1 e Qjα;β;γ λe Qjα;β;γ F ¼ 1 þ ; 0; 7 ½1 þ θ 11 e Qjα;β;γ 1 þ θ 11 e Qjα;β;γ where α > 0; β > p ffiffiffiffiffi αγ ; γ > 0; jλj 1 and θ 0; 1 are parameters. Useful epansions and miture representation The density function for TG-QHR can be written as " θ Ajα; β; γ f¼ # jα;β;γ e Q λθ 1 e Qjα;β;γ ½1 þ θ 11 e Qjα;β;γ 1 þ λ ; > 0; 1 þ θ 11 e Qjα;β;γ f¼ 1 þ z a ¼ X 1 k Γa þ k Γa k! z k k¼0 " 1 þ λθ X Γ þ kθ 1 k X k 1 λθ jþk k X Γ3 þ kθ 1 k X kþ1 # jþk kþ1 j 1 j Γ k! Γ3 k! k¼0 j¼0 k¼0 j¼0 Ajα; β; γ e jþ1qjα;β;γ : The pdf of TG-QHR distribution can be written as the miture of ep-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 θ 1 e Qjα;β;γ λe Qjα;β;γ S ¼1 1 þ ; ½1 þ θ 11 e Qjα;β;γ 1 þ θ 11 e Qjα;β;γ 8 h ¼Ajα; β; γ e Qjα;β;γ 1 e ; Qjα;β;γ 1 λ1 e Qjα;β;γ 9 r ¼Ajα; β; γe 1 1 e Qjα;β;γ λ ; 1 þ λe Qjα;β;γ 10 ( θ 1 e Qjα;β;γ λe Qjα;β;γ H ¼ ln 1 1 þ ; ½1 þ θ 11 e Qjα;β;γ 1 þ θ 11 e Qjα;β;γ and m ¼e Qjα;β;γ Ajα; β; γ Ajα; þ β; γ λ 1 : 1 e Qjα;β;γ 1 λ1 e Qjα;β;γ d ln S The generalized hazard g 1 ¼ d ln ¼ h of TG-QHR distribution is 11

5 Bhatti et al. Journal of Statistical Distributions and Applications (018) 5:4 Page 5 of 3 g 1 ¼A jα; β; γ e Qjα;β;γ 1 e Qjα;β;γ þ λ : 13 1 λ1 e Qjα;β;γ The elasticity e ¼ dlnf d ln e ¼Ajα; β; γ ¼ rof TG-QHR distribution is e Qjα;β;γ 1 1 e Qjα;β;γ λ 1 þ λe Qjα;β;γ : 14 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 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 33 α q þ β q þ γ 3 λ þ 1 λ þ 1 4λq q 3 þ ln61 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0: 15 λθ þ 1 θ λ þ 1 λ þ 1 4λq Median of TG-QHR distribution is a b Fig. 1 a Plots of pdf of TG-QHR Distribution. b Plots of hrf of TG-QHR Distribution

6 Bhatti et al. Journal of Statistical Distributions and Applications (018) 5:4 Page 6 of 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 33 α Med: þ β Med: þ γ 3 λ þ 1 λ þ 1 λ Med: 3 þ ln61 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0: λθ þ 1 θ λ þ 1 λ þ 1 λ 16 The random number generator of TG-QHR distribution is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 33 αx þ β X þ γ λ þ 1 λ þ 1 4λZ 3 X3 þ ln61 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0; 17 λθ þ 1 θ λ þ 1 λ þ 1 4λZ 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 Table 1 Sub-models of TG-QHR distribution Sr.No. θ λ α β γ Name of Distribution 1 θ λ α β γ Transmuted Geometric Quadratic Hazard Rate(TG-QHR) 1 λ α β γ Transmuted Quadratic Hazard Rate(T-QHR) α β γ Quadratic Hazard Rate(QHR) 4 θ 0 α β γ Quadratic Hazard Rate- Geometric (QHR-G) [Okasha et al.; 015] 5 θ λ 0 0 γ Transmuted Geometric Weibull(TG-W) (Nofal et al.; 017) 6 θ λ 0 β 0 Transmuted Geometric Rayleigh(TG-R) 7 θ λ α 0 0 Transmuted Geometric Eponential (TG-E) 8 θ λ α β 0 Transmuted Geometric Linear failure rate (TG-LFR) 9 1 λ 0 0 γ Transmuted Weibull (T-W) [Khan et al.; 017] 10 1 λ 0 β 0 Transmuted Rayleigh(T-R)[Merovci, F.; 013] 11 1 λ α 0 0 Transmuted Eponential (T-E) 1 1 λ α β 0 Transmuted Linear failure rate (T-LFR) 13 θ γ Weibull Geometric (W-G)[Barreto-Souza et al.; 011] 14 θ 0 0 β 0 Rayleigh Geometric(R-G) 15 θ 0 α 0 0 Eponential Geometric(E-E) 16 θ 0 α β 0 Linear failure rate Geometric(LFR-G) γ Weibull (Weibull; 1951) β 0 Rayleigh α 0 0 Eponential α β 0 Linear failure rate(lfr)

7 Bhatti et al. Journal of Statistical Distributions and Applications (018) 5:4 Page 7 of 3 μ = r ¼ EXr ¼ μ = r ¼ Z 0 r Z 0 r fd; e Qjα;β;γ θajα; β; γ ½1 þ θ 11 e Qjα;β;γ 1 þ λ " # λθ 1 e Qjα;β;γ d; 1 þ θ 11 e Qjα;β;γ Z μ = r ¼ EXr ¼ 0 μ = r ¼ EXr ¼ μ = r ¼ EXr r Z 0 1 þ λ 6 4 θ X k¼0 Γ þ kθ 1 Γ þ k k! λθ X k¼0 k X k j¼0 Γ3 þ kθ 1 Γ3 þ k k! 1 j k j α þ β þ γ e jþ1 k X kþ1 j¼0 Qjα;β;γ 1 j kþ1 α þ β þ γ e jþ1 j Qjα;β;γ d; 3 1 þ λθ Xk X X X Γ þ kθ 1 k 1 jþlþm k j j þ 1 lþm Γ þ k k!l!m! j¼0 k¼0 l¼0 m¼0 β l γ m α lþ3mþr þ β lþ3mþrþ1 þ γ lþ3mþrþ e αjþ1 3 λθ Xkþ1 X X X Γ3 þ kθ 1 k d; jþlþm kþ1 1 j j þ 1 lþm β l γ m Γ3 þ k k!l!m! 3 j¼0 k¼0 l¼0 m¼ α lþ3mþr þ β lþ3mþrþ1 þ γ lþ3mþrþ e αjþ1 3 1 þ λθ Xk X X X Γ þ kθ 1 k 1 jþlþm k j j þ 1 lþm Γ þ k k!l!m! j¼0 k¼0 l¼0 m¼0 β l γ m αγr þ l þ 3m þ 1 βγ þ r þ l þ 3m þ γγ þ r þ l þ 3m þ 3 3 ½αj þ 1 rþlþ3mþ1 ½βj þ 1 rþlþ3mþ ½γj þ 1 rþlþ3mþ3 ¼ ; λθ Xkþ1 X X X Γ3 þ kθ 1 k jþlþm kþ1 1 j j þ 1 lþm Γ3 þ k k!l!m! j¼0 k¼0 l¼0 m¼0 β 6 l γ m αγr þ l þ 3m þ 1 βγ r þ l þ 3m þ þ γγ r þ l þ 3m þ 3 þ ½αj þ 1 rþlþ3mþ1 ½βj þ 1 rþlþ3mþ ½γj þ 1 rþlþ3mþ3 where W j;k;l;m ¼ P k P P P j¼0 k¼0 l¼0 m¼0 W j;kþ1;l;m ¼ Xkþ1 X X X j¼0 k¼0 l¼0 m¼0 Γ3 þ k Γ3 þ k Γþk Γþk θ 1 k k!l!m! 1 jþlþm k j j þ 1lþm β l γ 3 m and θ 1 k k!l!m! 1 jþlþm kþ1 j j þ 1 lþm β l γ 3 m : αγr þ l þ 3m þ 1 βγ 1 þ λθw j;k;l;m þ r þ l þ 3m þ ½αj þ 1 rþlþ3mþ1 ½βj þ 1 rþlþ3mþ μ = r ¼ γ Γ r þ l þ 3m þ 3 EXr ¼ þ ½γj þ 1 αγr þ l þ 3m þ 1 rþlþ3mþ3 λθ W j;kþ1;l;m ½αj þ 1 rþlþ3mþ1 6 4 βγ r þ l þ 3m þ þ γγ r þ l þ 3m þ 3 þ ½βj þ 1 rþlþ3mþ ½γj þ 1 rþlþ3mþ3; μ = r ¼ EXr ¼ 1 þ λθw j;k;l;m λθ αγr þ l þ 3m þ 1 W j;kþ1;l;m ½αj þ 1 rþlþ3mþ1 βγ r þ l þ 3m þ þ γ Γ þ r þ l þ 3m þ 3 ½βj þ 1 rþlþ3mþ ½γj þ 1 : 18 rþlþ3mþ3 Mean and Variance of TG-QHR distribution

8 Bhatti et al. Journal of Statistical Distributions and Applications (018) 5:4 Page 8 of 3 EX ¼ VarX ¼ 1 þ λθw j;k;l;m λθ αγ l þ 3m þ W j;kþ1;l;m ½ βγ þ l þ 3m þ 3 ½αj þ 1 lþ3mþ ½βj þ 1 lþ3mþ3 γγ l þ 3m þ 4 þ ½γj þ 1 ; lþ3mþ4 8 1 þ λθw j;k;l;m λθ αγ l þ 3m þ 3 W j;kþ1;l;m ½ βγ þ l þ 3m þ 4 ½αj þ 1 lþ3mþ3 ½βj þ 1 lþ3mþ4 >< γγ l þ 3m þ 5 þ ½γj þ 1 lþ3mþ5 1 þ λθw j;k;l;m λθ W j;kþ1;l;m ½ αγ l þ 3m þ ½αj þ 1 lþ3mþ >: βγ l þ 3m þ 3 þ ½βj þ 1 lþ3mþ3 γγ l þ 3m þ 4 þ ½γj þ 1 lþ3mþ4 : The fractional positive moments about the origin of the random variable X with TG-IW distribution are μ = r s ¼ EX r s ¼ 1 þ λθw j;k;l;m λθ αγ r þ l þ 3m þ 1 W s j;kþ1;l;m ½αj þ 1 r s þlþ3mþ1 βγ r r þ l þ 3m þ γ Γ þ l þ 3m þ 3 þ s ½βj þ 1 r s þlþ3mþ þ s ½γj þ 1 r s þlþ3mþ3 : 19 The factorial moments of X with TG- QHR distribution are E½X m ¼ P m r¼1 φ r EXr ; EX ½ m ¼ Xm r¼1 φ r 1 þ λθw j;k;l;m λθ αγ r þ l þ 3m þ 1 W j;kþ1;l;m ½ ½αj þ 1 rþlþ3mþ1 βγ r þ l þ 3m þ þ ½βj þ 1 rþlþ3mþ γ Γ r þ l þ 3m þ 3 þ ½γj þ 1 ; rþlþ3mþ3 0 where [X] i = X(X + 1)(X +) (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 Μff ; sg ¼f s ¼ R 0 f s 1 d:

9 Bhatti et al. Journal of Statistical Distributions and Applications (018) 5:4 Page 9 of 3 Z Μff; sg ¼ 0 s 1 e Qjα;β;γ θajα; β; γ ½1 þ θ 11 e Qjα;β;γ 1 þ λ " # λθ 1 e Qjα;β;γ d; 1 þ θ 11 e Qjα;β;γ Μff; sg ¼ 1 þ λθw j;k;l;m λθ αγs þ l þ 3m W j;kþ1;l;m ½αj þ 1 sþlþ3m βγ s þ l þ 3m þ 1 þ γ Γ þ s þ l þ 3m þ ½βj þ 1 sþlþ3mþ1 ½γj þ 1 : sþlþ3mþ 1 The qth central moments, Pearson s measure of skewness and Kurtosis and cumulants of X with TG-QHR distribution are determined from the relationships. μ q ¼ Xq v¼1 q v 1 v μ 0 v μ0 v q ; γ 1 ¼ μ 3 μ 3 ; β ¼ μ 4 μ and k r ¼ μ 0 r Xr 1 c¼1 r 1 c 1 kc μ 0 r c : The graphical displays to describe the parameter that controls skewness and kurtosis measures of the TG-QHR distribution are added (Fig. ). The moment generating function of the random variable X with TG-QHR distribution is Z M ¼Ee t ½ t ¼ 0 e t fd ¼ Xr t r r! μ= r ; r¼0 Fig. skewness and kurtosis measures of the TG-QHR distribution

10 Bhatti et al. Journal of Statistical Distributions and Applications (018) 5:4 Page 10 of 3 M ¼ t Xr t r r! r¼0 1 þ λθw j;k;l;m λθ W j;kþ1;l;m ½ ½αj þ 1 αγ r þ l þ 3m þ 1 rþlþ3mþ1 βγ r þ l þ 3m þ þ ½βj þ 1 rþlþ3mþ γ Γ r þ l þ 3m þ 3 þ ½γj þ 1 : rþlþ3mþ3 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 Z t φ r ¼ t 0 r fd φ r ¼½ t 1 þ λθw j;k;l;m λθ W j;kþ1;l;m αγ ; r þ l þ 3m þ 1 ½αj þ 1 rþlþ3mþ1 þ βγ ; r þ l þ 3m þ ½βj þ 1 rþlþ3mþ γ Γ ; r þ l þ 3m þ 3 þ ½γj þ 1 : rþlþ3mþ3 3 where Γ( ) is the upper incomplete gamma function. The mean deviation about mean is MD ¼ EjX μ 1 X 1 j¼μ1 1 Fμ1 1 μ1 1 φ 1 and mean deviation about median is MD M = E X M =MF(M) Mφ 1 whereμ = 1 ¼ EXand M ¼ Q1. Bonferroni and Lorenz curves defined for a specified probability p by Bp ¼φ 1 q=pμ 1 1 and Lp ¼φ 1 q=μ1 1, whereq = Q p. Residual life functions The residual life, say m n (t), of X with TG-QHR distribution is m n ¼E t ½X t n jx > t; m n ¼ t 1 Z Sz m n ¼ t t λ 1 1 F t t s fd; X n r¼0 t n r φ r ; t n r

11 Bhatti et al. Journal of Statistical Distributions and Applications (018) 5:4 Page 11 of 3 m n ¼ t 1 1 F t X n r¼0 t n r n r 1 þ λθw j;k;l;m λθ W j;kþ1;l;m αγ; r þ l þ 3m þ 1 ½αj þ 1 rþlþ3mþ1 þ βγ ; r þ l þ 3m þ ½βj þ 1 rþlþ3mþ γ Γ ; r þ l þ 3m þ 3 þ ½γj þ 1 rþlþ3mþ3 The life epectancy or mean residual life function at a specified time t, saym 1 (t), quantifies the epected left over lifetime of an individual of age t and is given by. m 1 ¼ t 1 1 F t X 1 r¼0 t 1 r 1 r 1 þ λθw j;k;l;m λθ W j;kþ1;l;m αγ; r þ l þ 3m þ 1 ½αj þ 1 rþlþ3mþ1 þ βγ ; r þ l þ 3m þ ½βj þ 1 rþlþ3mþ γ Γ ; r þ l þ 3m þ 3 þ ½γj þ 1 rþlþ3mþ3 : : The reverse residual life, say M n (t), of X with TG-QHR distribution is M n ¼E t ½t X n =X t; M n ¼ t 1 Ft M n ¼ t 1 Ft M n ¼ t 1 Ft Z t 0 X n r¼0 X n r¼0 t n fd; 1 r n r t n r φ r ; t 1 r n r t n r 1 þ λθw j;k;l;m λθ W j;kþ1;l;m αγ; r þ l þ 3m þ 1 ½αj þ 1 rþlþ3mþ1 þ βγ ; r þ l þ 3m þ ½βj þ 1 rþlþ3mþ γγ ; r þ l þ 3m þ 3 þ ½γj þ 1 rþlþ3mþ3 ; 4 where γ( ) is the lower incomplete gamma function.

12 Bhatti et al. Journal of Statistical Distributions and Applications (018) 5:4 Page 1 of 3 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 M 1 ¼ t 1 Ft X 1 r¼0 1 r 1 r t 1 r 1 þ λθw j;k;l;m λθ W j;kþ1;l;m αγ; r þ l þ 3m þ 1 ½αj þ 1 rþlþ3mþ1 þ βγ ; r þ l þ 3m þ ½βj þ 1 rþlþ3mþ γ ; r þ l þ 3m þ 3 þγ ½γj þ 1 rþlþ3mþ3 : 5 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 Appendi A. Proportion 5.1.1: Let X : Ω (0, ) be a continuous random variable. Let " h i # 1 h 1 ¼θ 1 1 þ θ 1 1 e Qjα;β;γ λθ 1 e Qjα;β;γ 1 þ λ 1 þ θ 11 e Qjα;β;γ and " h i # 1 h ¼θ 1 e Qjα;β;γ 1 þ θ 1 1 e Qjα;β;γ λθ 1 e Qjα;β;γ 1 þ λ ; > 0: 1 þ θ 11 e Qjα;β;γ The pdf of X is (6) if and only if p()(in Theorem 1) has the formp()=e Q( α, β, γ). Proof. If X has pdf (6), then and 1 F Eh 1 XjX ¼ e Qjα;β;γ ; > 0; 1 F Eh XjX ¼ e Q jα;β;γ ; > 0: Conversely, if p()=e Q( α, β, γ), then p ()=A( α, β, γ) e Q( α, β, γ) and the differential equation.

13 Bhatti et al. Journal of Statistical Distributions and Applications (018) 5:4 Page 13 of 3 s 0 ¼ p0 h ph h 1 ¼ Ajα; β; γ has solution s()=q( α, β, γ). Therefore in the light of theorem 1, X has pdf (6). Corollary 5.1.1: Let X : Ω (0, )be a continuous random variable and let h ¼e Qjα;β;γ 1 þ θ 1 1 e Qjα;β;γ " # 1 λθ 1 e Qjα;β;γ 1 þ λ ; > 0: θ 1 þ θ 11 e Qjα;β;γ The pdf of X is (6) if and only if there eist functions p() and h 1 ()satisfy the equation p 0 h ¼ θ Ajα; β; γ p h h 1 jα;β;γ eq 1 þ θ 1 1 e Qjα;β;γ " # λθ 1 e Qjα;β;γ 1 þ λ : 1 þ θ 11 e Qjα;β;γ Remark 5.1.1: The general solution of (6) is. Z p ¼e Qjα;β;γ θ Ajα; β; γe Qjα;β;γ h 1 ½1 þ θ 11 e Qjα;β;γ " #! λθ 1 e Qjα;β;γ 1 þ λ d þ D; 1 þ θ 11 e Qjα;β;γ i 6 where D is a constant. Characterization via hazard function Here we characterize TG-QHR distribution via hazard function of X. Definition 5..1: Let X : Ω (0, ) be a continuous random variable with cdf F()and pdf f().the hazard function,h F (), of a twice differentiable distribution function satisfies the differential equation d d ½ logf ¼ h0 F h F h F: Proposition 5..1 Let X : Ω (0, ) be continuous random variable. The pdf of X is (6), if and only if its hazard function, h F (), satisfies the first order differential equation e Qjα;β;γ h 0 þ Ajα; β; γ e Qjα;β;γ h " ¼ β þ γ Ajα; β; γ λβþ þ # jα;β;γ γ 1 λ 1 e Q þ λ Ajα; β; γ e Qjα;β;γ e Qjα;β;γ 1 λ1 e Qjα;β;γ ; > 0: Proof: If the pdf of X is (6), then the above differential equation holds. Now, if the differential equation holds, then or d n d h F e Qjα;β;γ o ¼ d d Ajα; β; γ 1 þ λ 1 λ1 e Qjα;β;γ e Qjα;β;γ ;

14 Bhatti et al. Journal of Statistical Distributions and Applications (018) 5:4 Page 14 of 3 h ¼Ajα; β; γ e Qjα;β;γ 1 e Qjα;β;γ þ λ ; 1 λ1 e Qjα;β;γ 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()and pdf f().the reverse hazard function,r F (), of a twice differentiable distribution function satisfies the differential equation d d ½ logf ¼ r0 F r F þ r F: Proposition 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. jα;β;γ eq r 0 F þ Ajα; β; γ e Qjα;β;γ r F ¼ ½ β þ γ jα;β;γ 1 e Q Ajα; β; γ e Qjα;β;γ 1 e Qjα;β;γ λβþγ jα;β;γ 1 þ λe Q þ λ Ajα; β; γ e Qjα;β;γ 1 þ λe Qjα;β;γ : Proof: If the pdf of X is (6), then the above differential equation holds. Now, if the differential equation holds, then or d n d r F e Qjα;β;γ r F ¼Ajα; β; γ o ¼ d Ajα; β; γ λa jα; β; γ ; d 1 e Qjα;β;γ 1 þ λe Qjα;β;γ λ ; > 0 1 þ λe Qjα;β;γ e Qjα;β;γ 1 1 e Qjα;β;γ 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() and pdf f().the elasticity function, e F (), of a twice differentiable distribution function satisfies the differential equation d d ½ ln f ¼ e= e þ e 1 : Proposition Let X : Ω (0, ) be continuous random variable. The pdf of X is (6) if and only if its elasticity,e F (), satisfies the first order differential equation.

15 Bhatti et al. Journal of Statistical Distributions and Applications (018) 5:4 Page 15 of 3 jα;β;γ eq e 0 F þ Ajα; β; γ e Qjα;β;γ e F ¼ ½ α 1 þ β þ 3γ 1 e Qjα;β;γ þ Ajα; β; γ e Qjα;β;γ 1 e Qjα;β;γ λα 1 þ β þ 3γ 1 þ λe Qjα;β;γ þ λ Ajα; β; γ e Qjα;β;γ 1 þ λe Qjα;β;γ : Proof: If the pdf of X is (6), then the above differential equation holds. Now, if the differential equation holds, then d n d e F e Qjα;β;γ o ¼ d Ajα; β; γ d 1 1 e Qjα;β;γ λ 1 þ λe Qjα;β;γ ; or e F ¼A jα; β; γ λ ; > 0; 1 þ λe Qjα;β;γ e Qjα;β;γ 1 1 e Qjα;β;γ which is the elasticity function of the TG-QHR distribution. Maimum likelihood estimation In this section, parameters estimates are derived using maimum likelihood method. The log likelihood function for TG-QHR with the vector of parameters Φ =(α, β, θ, λ) is L i ; Φ ¼ n lnθ þ X lna i jα; β; γ X Q i jα; β; γ X h i ln 1 þ θ 1 1 e Q ijα;β;γ þ X " # λθ 1 e Qijα;β;γ ln 1 þ λ : 1 þ θ 11 e Q iα;β;γ 7 In order to compute the estimates of parameters of TG-QHR distribution, the following nonlinear equations must be solved simultaneously: α L i; Φ ¼ X A i jα; β; γ 1 X i þ X " # θ i 1 þ 1 e Q ijα;β;γ þ λθ X i θ 1þ 1 e Q ijα;β;γ 1 1 e Q ijα;β;γ þ λ λθ θ 1þ1 e Q iα;β;γ 1 1 e Q ijα;β;γ β L i; Φ ¼ n lnθ þ X i A i jα; β; γ 1 X i θ 1 X e Qijα;β;γ 1 þ θ 1 þλθ X i θ 1þ 1 e Qijα;β;γ 1 1 e Q ijα;β;γ þ λ λθ θ 1þ1 e Q ijα;β;γ ¼ 0; 1 e Qijα;β;γ 3 e Q iα;β;γ ¼ 0; 9

16 Bhatti et al. Journal of Statistical Distributions and Applications (018) 5:4 Page 16 of 3 γ L i; Φ ¼ X i A i jα; β; γ 1 X 3 i 3 3 θ 1X 3 i e Qiα;β;γ 1 þ θ 11 e Qijα;β;γ þ λθ 3 X i θ 1þ 1 e Qijα;β;γ 1 1 e Q ijα;β;γ þ λ λθ θ 1þ1 e Qijα;β;γ 1 1 " δ # 1 1 δλ L i; Φ ¼ 1 θ θ 1þ 1 e Q ijα;β;γ " # þ λ λθ θ 1þ 1 e Q iα;β;γ ¼ 0; δ δθ L i; Φ ¼ n θ X 1 1 θ 1þ 1 e Qiα;β;γ X λ iα;β;γ 1 e Q 1 1 λθ þ θ 1 1 e Qiα;β;γ e Qijα;β;γ 4 1 þ λ λθ 1 e Qijα;β;γ 1 1 þ θ 1 e Q ijα;β;γ 1 þ θ ¼ 0; ¼ 0: 3 Simulation studies In this Section, we perform two simulation studies based on graphical results by using selected TG-QHR distributions. To see the performance of MLE s ofthese distributions, we generate N = 1000 samples of sizes n = 0,30,,1000 from TG-QHR distribution. For the first simulation study, we take true values of α, β, γ, θparameters as 5,3,,0.5 respectively. For the second simulation study, we take true values of α, β, γ, θparameters as,,1,0. respectively. Note that we assume λas known for both simulation studies. The random number generation is obtained with inverse of its cdf. The MLEs, say ^α i ; ^β i ; ^γ i ; ^θ i for i = 1,,...,N, have been obtained by CG routine in R programme. We also calculate the mean, standard deviations (sd), bias and mean square error (MSE) of the MLEs. The bias and MSE are calculated by (for h = α, β, γ, θ). Bias h ¼ X 1000 i¼1 ^h i h and MSE h ¼ X 1000 i¼1 : ^h i h The results of the simulations are reported in Figs. 3 and 4. From these Figures, we can say that for four parameters the empirical means approach the true values when sample size increases whereas the bias, MSE and sd s decrease as the sample size increases, as epected. Applications The potentiality of TG-QHR distribution is demonstrated by its application to real data sets. The TG-QHR distribution is compared with G-QHR, T-QHR, QHR distributions. Goodness of fit of this distribution through different methods is studied. Different goodness fit measures such as Cramer-von Mises (W), Anderson Darling (A), Kolmogorov-Smirnov (K-S) statistics with p-values and likelihood ratio

17 Bhatti et al. Journal of Statistical Distributions and Applications (018) 5:4 Page 17 of 3 Fig. 3 First simulation results with λ = 0.5 statistics are computed using R-package for strengths of 1.5 cm glass fibers and breaking stress of carbon fibers. The maimum likelihood estimates (MLEs) of unknown parameters and values of goodness of fit measures are computed for TG-QHR distribution and its sub-models. The better fit corresponds to smaller W, A, K-S and l value. Fig. 4 Second simulation results with λ = 0.5

18 Bhatti et al. Journal of Statistical Distributions and Applications (018) 5:4 Page 18 of 3 Table MLEs (standard errors) and goodness-of-fit statistics for data set I Model α β γ θ λ TG-QHR ( ) ( ) ( ) G-QHR T-QHR ( ) ( ) ( ) QHR ( ) ( ) ( )

19 Bhatti et al. Journal of Statistical Distributions and Applications (018) 5:4 Page 19 of 3 Table 3 Goodness-of-fit statistics for data set I Model W A K-S (p-value) l TG-QHR (0.5688) G-QHR (0.5517) T-QHR (8.51e-05) QHR (8.51e-05) Boldface entries indicates that the proposed distribution is best fitted to data sets Application I: Strengths of glass fibers The values of data about strengths of 1.5 cm glass fibers (Smith and Naylor; 1987) are: 0.55, 0.74, 0.77, 0.81, 0.84, 0.93, 1.04, 1.11, 1.13, 1.4, 1.5, 1.7, 1.8, 1.9, 1.30, 1.36, 1.39, 1.4, 1.48, 1.48, 1.49, 1.49, 1.50, 1.50, 1.51, 1.5, 1.53, 1.54, 1.55, 1.55, 1.58, 1.59, 1.60, 1.61, 1.61, 1.61, 1.61, 1.6, 1.6, 1.63, 1.64, 1.66, 1.66, 1.66, 1.67, 1.68, 1.68, 1.69, 1.70, 1.70, 1.73, 1.76, 1.76, 1.77, 1.78, 1.81, 1.8, 1.84, 1.84, 1.89,.00,.01,.4. The MLEs (standard errors) are given in Table. 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 I because good of fit measures are smaller and graphical plots such as estimated pdf, cdf and pp. plots are closer to data set I (Fig. 5). a b c Fig. 5 Estimated pdf for TG-QHR and Estimated cdf for TG-QHR for data set I. PP plot for data Set I

20 Bhatti et al. Journal of Statistical Distributions and Applications (018) 5:4 Page 0 of 3 Table 4 MLEs (standard errors) and goodness-of-fit statistics for data set II Model α β γ θ λ TG-QHR ( ) ( ) ( ) ( ) ( ) G-QHR ( ) ( ) ( ) ( ) T-QHR ( ) ( ) ( ) QHR ( ) ( ) ( )

21 Bhatti et al. Journal of Statistical Distributions and Applications (018) 5:4 Page 1 of 3 Table 5 Goodness-of-fit statistics for data set II Model W A K-S l TG-QHR (0.9689) G-QHR (0.9458) T-QHR (0.353) QHR (0.353) Boldface entries indicates that the proposed distribution is best fitted to data sets Application II: Breaking stress of carbon fibers The values of data about breaking stress of carbon fibers (Nichols and Padgett; 006) are: 3.70,.74,.73,.50, 3.60, 3.11, 3.7,.87, 1.47, 3.11, 3.56, 4.4,.41, 3.19, 3., 1.69, 3.8, 3.09, 1.87, 3.15, 4.90, 1.57,.67,.93, 3., 3.39,.81, 4.0, 3.33,.55, 3.31, 3.31,.85, 1.5, 4.38, 1.84, 0.39, 3.68,.48, 0.85, 1.61,.79, 4.70,.03, 1.89,.88,.8,.05, 3.65, 3.75,.43,.95,.97, 3.39,.96,.35,.55,.59,.03, 1.61,.1, 3.15, 1.08,.56, 1.80,.53. The MLEs (standard errors) are given in Table 4. Table 5 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). a b c Fig. 6 Estimated pdf for TG-QHR and Estimated cdf for TG-QHR for data set II. PP Plots for data set II

22 Bhatti et al. Journal of Statistical Distributions and Applications (018) 5:4 Page of 3 Conclusions We have developed a more fleible distribution which is suitable for applications in survival analysis, reliability and actuarial science. The important properties of the proposed TG-QHR distribution such as survival function, hazard function, reverse hazard function, cumulative hazard function, mills ratio, elasticity, quantile function, moments about the origin, incomplete moments and inequality measures are presented. The proposed distribution is characterized via different techniques. Maimum Likelihood estimates are computed. The simulation studies are performed on the basis of graphical results to illustrate the performance of maimum likelihood estimates of this distribution. Applications of our model to real data sets (strength of glass fibers and stress of carbon fibers) are given to show its significance and fleibility. Goodness of fit shows that our distribution is a better fit. We have demonstrated that the proposed distribution is empirically better for lifetime applications. Appendi A Theorem 1: Let (Ω,F,P)be a probability space and let [d 1,d ] be an interval with d 1 < d d 1 =, d = ). Also suppose that a continuous random variable X : Ω [d 1, d ] has distribution function F. Let h 1 () and h () be two real functions continuous on [d 1,d ] such that E½h 1XjX E½h XjX ¼ p where p() is real function and should be in simple form. Assume that, h 1 (), h ()ε C([d 1,d ])p() ε C ([d 1,d ]) and F is strictly monotone function and twofold continuously differentiable on interval[d 1,d ]. Assume that the equality h ()p()=h 1 () has no real result inside of [d 1,d ]. Then cdf F is obtained from h 1 (), h () andp()functions as F ¼ R 0 Kj p 0 t pth t h 1 t j ep stdt, where s(t) is obtained from equation s 0 t ¼ make R d d 1 df ¼ 1: p0 th t pth t h 1 t and K is a constant, selected to Acknowledgments The authors are grateful to the Editor-in-Chief, the Associate Editor and anonymous referees for many of their constructive comments and suggestions which lead to remarkable improvement on the revised version of the paper. Funding GGH (co-author of the manuscript) is an Associate Editor of JSDA, 100% discount on Article Processing Charge (APC) for accepted article). Authors contributions FAB proposed the TG-QHR model and wrote the initial draft of the manuscript. MCK wrote graphs of skewness and kurtosis (Fig. ), simulation Section Applications and graphs about applications (Figs. 5 and 6). Especially GGH did much work in order to improve the motivation, characterizations and revise the article as a whole. The authors, viz. FAB, GGH, MCK and MA with the consultation of each other finalized this work. All authors read and approved the final manuscript. Competing interests The authors declare that they have no competing interests. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Author details 1 National College of Business Administration and Economic, Lahore, Pakistan. Marquette University, Milwaukee, WI , USA. 3 Department of Measurement and Evaluation, Artvin Çoruh University, Artvin, Turkey.

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