The Odd Exponentiated Half-Logistic-G Family: Properties, Characterizations and Applications

Transcription

1 Chilean Journal of Statistics Vol 8, No 2, September 2017, The Odd Eponentiated Half-Logistic-G Family: Properties, Characterizations and Applications Ahmed Z Afify 1, Emrah Altun 2,, Morad Alizadeh 3, Gamze Ozel 2 and GG Hamedani 4 1 Department of Statistics, Mathematics and Insurance, Benha University, Egypt 2 Department of Statistics, Hacettepe University, Ankara, Turkey 3 Department of Statistics, Faculty of Sciences, Persian Gulf University, Bushehr, Iran 4 Department of Mathematics, Statistics and Computer Science, Marquette University, USA (Received: February 15, 2017 Accepted in final form: July 26, 2017 Abstract We introduce a new class of continuous distributions called the odd eponentiated halflogistic-g family Some special models of the new family are provided These special models are capable of modeling various shapes of aging and failure criteria Some of its mathematical properties including eplicit epressions for the ordinary and incomplete moments, generating function, Rényi and Shannon entropies, order statistics, probability weighted moments and characterizations are obtained The maimum likelihood method is used for estimating the model parameters The fleibility of the generated family is illustrated by means of three applications to real data sets Keywords: Generating Function Maimum Likelihood Order Statistic T-X Family Characterization Mathematics Subject Classification: 62E10 60E05 1 Introduction In many practical situations, classical distributions do not provide adequate fits to real data Therefore, there has been an increased interest in developing more fleible distributions through etending the classical distributions via introducing additional shape parameters to the baseline model Many generalized families of distributions have been proposed and studied over the last two decades for modeling data in many applied areas such as economics, engineering, biological studies, environmental sciences, medical sciences and finance Some well-known families are the Marshall-Olkin-G by Marshall and Olkin (1997, the beta-g by Eugene et al (2002, odd log-logistic-g by Gleaton and Lynch (2004, 2006, the transmuted-g by Shaw and Buckley (2009, the gamma-g by Zografos and Balakrishnan (2009, the Kumaraswamy-G by Cordeiro and de Castro (2011, the logistic-g by Torabi and Montazeri (2014, eponentiated generalized-g by Cordeiro et al (2013, the McDonald-G by Aleander et al (2012, T-X family by Alzaatreh et al (2013, the Weibull-G by Bourguignon et al (2014, the eponentiated half-logistic generated family by Cordeiro et al (2014, the beta odd log-logistic generalized by Cordeiro et al (2015, the generalized transmuted-g by Nofal et al (2017, the Kumaraswamy transmuted-g by Afify Corresponding author emrahaltun@hacettepeedutr ISSN: (print/issn: (online c Chilean Statistical Society Sociedad Chilena de Estadística

2 66 Afify, Altun, Alizadeh, Ozel and Hamedani et al (2016a, the beta transmuted-h by Afify et al (2016b, generalized odd log-logistic- G by Cordeiro al (2017 Several mathematical properties of the etended distributions may be easily eplored using miture forms of eponentiated-g (ep-g distributions One of the probability distributions which is a member of the family of the logistic distribution is the half logistic distribution The probability density function (pdf and cumulative density function (cdf of the half-logistic (HL distribution are, respectively, given by f ( = 2 λ ep ( λ (1 + ep ( λ 2, F ( = 1 ep ( λ 1 + ep ( λ, where > 0 and λ > 0 is a shape parameter The HL distribution has not receive much attention from researchers in terms of generalization Furthermore, the density function of HL distribution has unimodal or reversed J-shaped and this property is a disadvantage of the HL distribution since the empirical approaches to real data are often non-monotone hazard rate function shapes such as unimodal, bathtub and various shaped, specifically in the lifetime applications Hence, in this research work, we present a new generalization of the HL distribution The goal of this study is to propose a new fleible family of distributions called the odd eponentiated half logistic-g (OEHL-G for short family of distributions using the HL distribution as the generator and study its mathematical properties This way, we will utilize the fleibility of the baseline distribution for modelling the data The cdf and pdf of the OEHL-G family are given, respectively, by and 1 ep F (; α, λ, ξ = 1 + ep λ G(;ξ Ḡ(;ξ λ G(;ξ Ḡ(;ξ ] ] α, R, (1 f(; α, λ, ξ = 2αλ g(, ξ ep ] { λ G(;ξ Ḡ(;ξ Ḡ(; ξ 2 {1 + ep 1 ep λ G(;ξ Ḡ(;ξ ]} α 1 λ G(;ξ Ḡ(;ξ ]} α+1, R, (2 where g(; ξ = dg(; ξ/d, α and λ are positive shape parameters and ξ is the vector of parameters for the baseline cdf G Henceforth, a random variable with density (2 is denoted by X OEHL-G(α, λ, ξ The reliability function, hazard rate function (hrf and cumulative hazard rate function (chrf of X are, respectively, given by 1 ep R(; α, λ, ξ = ep λ G(;ξ Ḡ(;ξ λ G(;ξ Ḡ(;ξ ] α ],

3 and h(; α, λ, ξ = 2αλ g(, {1 ξḡ(; ξ 2 + ep ] ({ ep 1 + ep λ G(;ξ Ḡ(;ξ Chilean Journal of Statistics 67 λ G(;ξ Ḡ(;ξ λ G(;ξ Ḡ(;ξ ]} 1 { 1 ep ]} α { 1 ep ]} α 1 λ G(;ξ Ḡ(;ξ ]} λ G(;ξ Ḡ(;ξ (3 1 ep H(; α, λ, ξ = log ep λ G(;ξ Ḡ(;ξ λ G(;ξ Ḡ(;ξ ] α ] An interpretation of the OEHL-G family can be given as follows: Let T be a random variable with cdf G ( describing a stochastic system Let the random variable X represent the odds, the risk that the system following the lifetime T will be not working at time is given by G(; ξ/1 G(; ξ] If we are interested in modeling ] the randomness of the odds by the eponentiated half-logistic cdf Π(t = 1 e λ t α 1+e (for t > 0, the cdf of X is λ t given by ( G(; ξ 1 e P r(x = Π = 1 G(; ξ λ G(;ξ Ḡ(;ξ λ G(;ξ 1 + e Ḡ(;ξ ] α Furthermore, the basic motivations for using the OEHL-G family in practice are the following: (1 to make the kurtosis more fleible compared to the baseline model; (2 to produce a skewness for symmetrical distributions; (3 to construct heavy-tailed distributions that are not longer-tailed for modeling real data; (4 to generate distributions with symmetric, left-skewed, right-skewed and reversed-j shaped; (5 to define special models with all types of the hrfs; (6 to provide consistently better fits than other generated models under the same baseline distribution The rest of the paper is organized as follows In Section 2, we give a very useful linear representation for the density function of the family In Section 3, we present three special models and plots of their pdfs and hrfs In Section 4, we obtain some of its general mathematical properties including asymptotics, ordinary and incomplete moments, skewness and kurtosis, quantile and generating functions, quantile power series, entropies, order statistics and probability weighted moments (PWMs Section 5 is devoted to characterizations of the OEHL-G family The maimum likelihood estimation of the model parameters is addressed in Section 6 In Section 7, we provide two applications to real data to illustrate the fleibility of the new family Simulation results to assess the performance of the maimum likelihood estimation method are reported in Section 8 Finally, we give some concluding remarks in Section 9

4 68 Afify, Altun, Alizadeh, Ozel and Hamedani 2 Linear representation In this section, we provide a useful representation for the OEHL-G pdf The pdf (2 can be epressed as f( = 2αλg( Ḡ( 2 ]} α 1 { e λ G( Ḡ( λg( 1 ep Ḡ( }{{} Using the generalized binomial series, we have A ]} α 1 { λg( 1 + ep (4 Ḡ( }{{} B A = ( ] ( 1 j α 1 λjg( ep j Ḡ( j=0 and B = ( α 1 ep i i=0 λig( Ḡ( ] (5 Combining equations (4 and (5, we obtain f( = 2αλg( Ḡ( 2 j,i=0 ( 1 j ( α 1 i ( α 1 j ep λ (j + i + 1 G( Ḡ( ] From the eponential series, we can write f( = 2αλg( j,i,k=0 Consider the power series ( 1 j+k k! λ (j + i + 1] k ( α 1 i (1 z q = ( α 1 j G( k Ḡ( k 2 (6 ( q ( 1 k z k (7 k k=0 Using the power series in (7, equation (6 can be epressed as f( = 2αλg( ( 1 j+k+l j,i, k! λ (j + i + 1] k ( α 1 j or, equivalently, we can write f( = ( α 1 i ( k 2 l G( k+l, a k,l h k+l+1 (, (8 where ( 1 j+k+l λ (j + i + 1] k ( ( ( α 1 α 1 k 2 a k,l = 2αλ k! (k + l + 1 i j l j,i=0 and h k+l+1 ( = (k + l + 1 g(g( k+l is the Ep-G density with power parameter (k + l + 1 Thus, several mathematical properties of the OEHL-G family can be obtained simply from those properties of the Ep-G family

5 Chilean Journal of Statistics 69 The cdf of the OEHL-G family can also be epressed as a miture of Ep-G cdfs By integrating (8, we obtain the same miture representation F ( = a k,l H k+l+1 (, where H k+l+1 ( is the cdf of the Ep-G family with power parameter (k + l Special models In this section, we provide three special models of the OEHL-G family The pdf (2 will be most tractable when the cdf G(; ξ and the pdf g(; ξ have simple analytic epressions 31 The OEHL-Weibull (OEHL-W distribution By taking G(; ξ and g(; ξ in (2 to be the cdf and pdf of the Weibull (W distribution with cdf and pdf G(; a, b = 1 e (/ba and g(; a, b = ab a a 1 e (/ba, respectively, where a > 0 is a shape parameter and b > 0 is a scale parameter The pdf of the OEHL-W reduces (for > 0 to f( = 2αλab a a 1 ep ] ( { ]} α 1 (/b a 1 ep λ e (/ba 1 ep { λ e (/ba 1 ]} ( 1 + ep { λ e (/ba 1 ]} α+1 For b = 1, we have the OEHL-eponential (OEHL-E distribution and for b = 2, we obtain the OEHL-Rayleigh (OEHL-R distribution Plots of the pdf and hrf of OEHL-W distribution for selected parameter values are shown in Figure 1 Figure 1 demonstrates that the OEHL-W ensures rich shaped distributions with various shapes for modeling It brings unimodal, bathtub and decreasing shape properties Figure 1 also reveals that this family can produce fleible hrf shapes such as decreasing, increasing, bathtub, upside-down bathtub, firstly unimodal Other shapes can be obtained using another distribution These shape properties show that the OEHL-W family can be very useful to fit different data sets with various shapes f( α = 15 λ = 075 a=15 b=1 α = 075 λ = 10 a=2 b=1 α = 2 λ = 2 a = 15 b = 1 α = 25 λ = 175 a=05 b=1 α = 075 λ = 15 a=05 b=1 α = 075 λ = 075 a=175 b=1 α = 4 λ = 075 a=25 b=1 h( α = 075 λ = 02 a=13 b=1 α = 015 λ = 075 a=025 b=1 α = 35 λ = 2 a=03 b=1 α = 25 λ = 175 a=05 b=1 α = 025 λ = 15 a=09 b=1 α = 4 λ = 175 a=03 b=1 α = 05 λ = 1 a=075 b= Figure 1 Plots of the pdf and hrf of the OEHL-W distribution for the selected parameter values

6 70 Afify, Altun, Alizadeh, Ozel and Hamedani 32 The OEHL-gamma (OEHL-Ga distribution By taking G(; ξ and g(; ξ in (2 to be the cdf G(; a, b = γ (a, /b /Γ (a and the pdf g(; a, b = a 1 e /b /b a Γ (a of the gamma distribution, where a > 0 is a shape parameter and b > 0 is a scale parameter, the pdf of the OEHL-Ga (for > 0 reduces to f( = 2αλ a 1 ep ( /b ep b a Γ (a λγ(a,/b/γ(a 1 γ(a,/b/γ(a 1 λγ(a,/b/γ(a 1 γ(a,/b/γ(a ] 2 { 1 + ep ] { 1 ep ]} α 1 λγ(a,/b/γ(a 1 γ(a,/b/γ(a λγ(a,/b/γ(a 1 γ(a,/b/γ(a ]} α+1 Plots of the pdf and hrf of the OEHL-Ga distribution for selected parameter values are shown in Figure 2 Figure 2 shows that the pdf of the OEHL-Ga is right-skewed and nearly symmetric As seen in Figure 2, a characteristic of the OEHL-Ga distribution is that its hrf can be monotonically increasing or bathtub depending basically on the parameter values f( α = 15 λ = 075 a=075 b=1 α = 075 λ = 10 a=2 b=1 α = 2 λ = 2 a = 15 b = 1 α = 25 λ = 175 a=05 b=1 α = 075 λ = 15 a=05 b=1 α = 075 λ = 075 a=075 b=1 α = 4 λ = 075 a=25 b=1 h( α = 275 λ = 3 a=175 b=1 α = 015 λ = 075 a=125 b=1 α = 15 λ = 05 a=11 b=1 α = 25 λ = 075 a=175 b=1 α = 175 λ = 05 a=075 b=1 α = 4 λ = 5 a = 35 b = 1 α = 05 λ = 1 a=075 b= Figure 2 Plots of the pdf and hrf of the OEHL-Ga distribution for the selected parameter values 33 The OEHL-normal (OEHL-N distribution The OEHL-N distribution is defined from (2 by taking G(; µ, σ 2 = Φ ( µ σ and g(; µ, σ 2 = σ 1 φ ( µ σ for the cdf and pdf of the normal (N distribution with a location parameter µ R and a scale positive parameter σ 2, where Φ ( and φ ( are the pdf and cdf of the standard N distribution, respectively The OEHL-N pdf is given (for R by f( = 2αλφ ( µ σ ep λφ( µ σ 1 Φ ( µ σ σ σ 1 Φ( µ ] 2 { 1 + ep ] { 1 ep λφ( µ 1 Φ( µ λφ( µ 1 Φ( µ σ σ σ σ ]} α 1 ]} α+1 For µ = 0 and σ = 1, we obtain the standard OEHL-N distribution Plots of the OEHL- N density for selected parameter values are shown in Figure 3 Figure 3 reveals that the density function of OEHL-N has left-skewed, nearly symmetric and non-monotonic shapes

7 Chilean Journal of Statistics 71 f( α = 003 λ = 01 µ = 0 σ = 1 α = 01 λ = 003 µ = 0 σ = 1 α = 025 λ = 005 µ = 0 σ = 1 α = 005 λ = 025 µ = 0 σ = 1 α = 001 λ = 01 µ = 0 σ = 1 α = 009 λ = 009 µ = 0 σ = 1 f( α = 025 λ = 02 µ = 0 σ = 1 α = 015 λ = 03 µ = 0 σ = 1 α = 015 λ = 04 µ = 0 σ = 1 α = 02 λ = 05 µ = 0 σ = 1 α = 03 λ = 06 µ = 0 σ = 1 α = 015 λ = 009 µ = 0 σ = f( α = 075 λ = 025 µ = 0 σ = 1 α = 02 λ = 025 µ = 0 σ = 1 α = 04 λ = 025 µ = 0 σ = 1 α = 06 λ = 025 µ = 0 σ = 1 α = 08 λ = 025 µ = 0 σ = 1 α = 1 λ = 025 µ = 0 σ = 1 f( α = 15 λ = 05 µ = 0 σ = 1 α = 15 λ = 075 µ = 0 σ = 1 α = 15 λ = 1 µ = 0 σ = 1 α = 15 λ = 15 µ = 0 σ = 1 α = 15 λ = 2 µ = 0 σ = 1 α = 15 λ = 3 µ = 0 σ = Figure 3 Plots of the density function of the OEHL-N distribution for the selected parameter values 4 Properties In this section, we obtain some general properties of the OEHL-G family including the ordinary and incomplete moments, generating function, entropies, order statistics and probability weighted moments (PWMs 41 Asymptotics Let a = inf { G( > 0}, then, the asymptotics of equations (1, (2 and (3 as a are given by F ( λα 2 α G(α as a, f ( αλα 2 α g(g(α 1 as a, h ( αλα 2 α g(g(α 1 as a

8 72 Afify, Altun, Alizadeh, Ozel and Hamedani The asymptotics of equations (1, (2 and (3 as are given by 1 F ( 2αe λ Ḡ( as, f ( 2αλg(e λ Ḡ( Ḡ( 2 as, h ( 2αλg( Ḡ( 2 as 42 Ordinary and incomplete moments Henceforth, T k+l+1 denotes the Ep-G random variable with power parameter k + l + 1 The rth moment of X, say µ r, follows from (8 as µ r = E (X r = The nth central moment of X is given by µ n = n r=0 ( n ( µ n r r 1 E (X r = a k,l E ( Tk+l+1 r (9 n r=0 The cumulants (κ n of X follow recursively from r=0 ( n a k,l r n 1 ( n 1 κ n = µ n κ r µ r 1 n r, ( µ 1 n r E ( T r k+l+1 where κ 1 = µ 1, κ 2 = µ 2 µ 2 1, κ 3 = µ 3 3µ 2 µ 1 + µ 3 1, etc The measures of skewness and kurtosis can be calculated from the ordinary moments using well-known relationships The sth incomplete moment of X can be epressed from (8 as ϕ s (t = t s f ( d = t a k,l s h k+l+1 ( d (10 ϕ 1 (t can be applied to construct Bonferroni and Lorenz curves defined for a given probability π by B(π = ϕ 1 (q /(πµ 1 and L(π = ϕ 1 (q /µ 1, respectively, where µ 1 given by (9 with r = 1 and q = Q(π is the qf of X at π These curves are very useful in economics, reliability, demography, insurance and medicine Now, we provide two ways to determine ϕ 1 (t First, a general equation for ϕ 1 (t can be derived from (10 as ϕ 1 (t = a k,l J k+l+1 (t, where J k+l+1 (t = t h k+l+1 ( d is the first incomplete moment of the Ep-G distribution

9 Chilean Journal of Statistics 73 A second general formula for ϕ 1 (t is given by ϕ 1 (t = a k,l v k+l+1 (t, where v k+l+1 (t = (k + l + 1 G(t 0 Q G (u u k+l du which can be computed numerically and Q G (u is the qf corresponding to G (; ξ, ie, Q G (u = G 1 (u; ξ Figures 4 and 5 display the mean and variance measures of OEHL-N and OEHL-W distributions Based on these figures, we conclude that: when parameter α increases, mean increases and variance decreases; when parameter λ increases, mean and variance decrease Mean Variance lambda alpha lambda alpha Figure 4 Plots of the mean and variance of the OEHL-N distribution for µ = 0, σ = 1 Mean Variance lambda alpha lambda alpha Figure 5 Plots of the mean and variance of the OEHL-W distribution for a = 2, b = 2 43 Quantile and generating functions The quantile function (qf of the OEHL-G distribution follows, by inverting (1, as X U = Q G log(1 u 1 α + log(1 + u 1 α λ log(1 u 1 α + log(1 + u 1 α ], (11

10 74 Afify, Altun, Alizadeh, Ozel and Hamedani where Q G ( denote the qf of X and u U (0, 1 Here, we provide two formulae for the moment generating function (mgf M X (t = E ( e t X of X which can be derived from equation (8The first one is given by M X (t = a k,l M k+l+1 (t, where M k+l+1 (t is the mgf of T k+l+1 Hence, M X (t can be determined from the Ep-G generating function The second formula for M X (t can be epressed as M X (t = a k,l τ (t, k, where τ (t, k = 1 0 ep t Q G (u] u k+l du The skewness and kurtosis plots of the OEHL-Ga and OEHL-W are given in Figure 4 These plots indicate that the members of the this family can model various data types in terms of skewness and kurtosis Figure 6 Plots of skewness and kurtosis of OEHL-Ga (left panel and OEHL-W (right panel distribution for several values of parameters 44 Quantile power series In this subsection, we derive a power series for the qf = Q(u = F 1 (u of X by epanding (11 If Q G (u does not have a closed-form epression, it can epressed as a power series Q G (u = a i u i, (12 i=0

11 Chilean Journal of Statistics 75 where the coefficients a is are suitably chosen real numbers They depend on the parameters of the baseline G distribution For several important distributions, such as the normal, Student t, gamma and beta distributions, Q G (u does not have eplicit epressions but it can be epanded as in equation (12 As a simple eample, for the normal N(0, 1 distribution, a i = 0 for i = 0, 2, 4, and a 1 = 1, a 3 = 1/6, a 5 = 7/120 and a 7 = 127/7560, Throughout the paper, we use a result of Gradshteyn and Ryzhik (2000 for a power series raised to a positive integer n (for n 1 ( n Q G (u n = a i u i = i=0 c n,i u i, (13 i=0 where c n,0 = a n 0 and the coefficients c n,i (for i = 1, 2, are determined from the recurrence equation c n,i = (i a 0 1 i m(n + 1 i] a m c n,i m (14 m=1 Net, we derive an epansion for the argument of Q G ( in (11, namely A = log(1 u 1 α + log(1 + u 1 α λ log(1 u 1 α + log(1 + u 1 α First, we have where a k = log(1 u 1 α = j=k Also, we can write where b k = = ( 1 ( j+k i ( α j i j k u i α i = j=0 k=0 j=0 j ( 1 j+k ( 1 i u i log(1 + u 1 α α = = i j=k = j=0 k=0 ( 1 ( i+j+k+1 i ( α j i j k j ( 1 j (1 u j i ( i α j j=0 ( 1 i+j+k+1 i i ( j u k = k ( i α j a k u k, k=0 ( 1 i+j+1 (1 u j i ( i ( α j u k = j k ( i α j b k u k, k=0

12 76 Afify, Altun, Alizadeh, Ozel and Hamedani Then, using the ratio of two power series we can write A = α k u k k=0 = β k u k k=0 δ k u k, (15 k=0 where α k = a k + b k for k 0, β 0 = λ + α 0 and β k = α k for k 1 and δ 0 = α 0 /β 0 and for k 1, we have δ k = 1 β 0 α k 1 β 0 Then, the qf of X can be epressed using (11 as ] k β r δ k r r=1 ( Q(u = Q G δ k u k (16 For any baseline G distribution, we combine (12 and (16 to obtain k=0 ( Q(u = Q G δ m u m = m=0 ( i a i δ m u m i=0 m=0 Then using (13 and (14, we have Q(u = e m u m, (17 m=0 where e m = i=0 a i d i,m, and, for i = 0, 1,, d i,0 = δ0 i and (for m > 1 d i,m = (m δ 0 1 m n=1 n(i + 1 m] δ n d i,m n Equation (17 reveals that the qf of the OEHL-G family can be epressed as a power series Then, several mathematical quantities of X can be reduced to integrals over (0, 1 based on this power series Let W ( be any integrable function on the real line We can write W ( f(d = 1 0 ( W e m u m du (18 m=0 Equations (17 and (18 are the main results of this section since we can obtain various OEHL-G mathematical properties based on them In fact, they can follow by using the integral on the right-hand side for special W ( functions, which are usually simpler than if they were based on the left-hand integral For the great majority of these quantities, we can adopt twenty terms in this power series The formulae derived throughout the paper can be easily handled in most symbolic computation platforms such as Maple, Mathematica and Matlab

13 Chilean Journal of Statistics Entropies The Rényi entropy of a random variable X represents a measure of variation of the uncertainty The Rényi entropy is given by ( I θ (X = (1 θ 1 log f ( θ d, θ > 0 and θ 1 Using the pdf (2, we can write f ( θ = (2αλ θ g( θ ep ] { λg( Ḡ( Ḡ( 2θ {1 + ep ]} θ(α 1 λg( 1 ep Ḡ( ]} θ(α+1 λg( Ḡ( Applying the binomial series to the last term, the last equation reduces to f ( θ = (2αλθ g( θ Ḡ( 2θ j,i=0 ( 1 j ( θ (α 1 j ( θ (α + 1 i ep Using the eponential series and then the power series (7, we have f ( θ = j,i, λ (j + i + 1 G( Ḡ( ( 1 j+k+l (2αλ θ g( θ ( ( ( θ (α 1 θ (α + 1 k 2θ k! λ (j + i + 1] k G( k+l j i l Then, the Rényi entropy of the OEHL-G class is given by where υ k,j = (2αλ θ I θ (X = (1 θ 1 log υ k,l g( θ G( k+l d, j,i=0 ( 1 j+k+l k! The θ-entropy can be obtained as H θ (X = (1 θ 1 log 1 ( ( λ (j + i + 1] k θ (α 1 θ (α + 1 j i υ k,l g( θ G( k+l d ( k 2θ The Shannon entropy of a random variable X is a special case of the Rényi entropy when θ 1 The Shannon entropy, say SI, is defined by SI = E { log f (X]}, which follows by taking the limit of I θ (X as θ tends to 1 l ] 46 Order statistics Order statistics make their appearance in many areas of statistical theory and practice Let X 1,, X n be a random sample from the OEHL-G family The pdf of X i:n can be

14 78 Afify, Altun, Alizadeh, Ozel and Hamedani written as f i:n ( = f ( B (i, n i + 1 n i j=0 ( ( 1 j n i F ( j+i 1, (19 j where B(, is the beta function Using equations (1 and (2, we have f ( F ( j+i 1 = 2αλg( ] { λg( λg( Ḡ( 2 ep 1 ep Ḡ( Ḡ( { ]} λg( α(j+i ep Ḡ( After applying the generalized binomial and eponential series, we obtain f ( F ( j+i 1 = ]} α(j+i 1 ( 1 s+k+l ( 2αλg( α (j + i 1 k! λ (s + w + 1] k s s,w, ( ( α (j + i 1 k 2 G( k+l (20 w l Substituting (20 in equation (19, the pdf of X i:n can be epressed as f i:n ( = b k,l h k+l+1 (, where h k+l+1 ( is the Ep-G density with power parameter (k + l + 1 and n i 2αλ ( 1 j+s+k+l λ (s + w + 1] k ( n i b k,l = k! (k + l + 1 B (i, n i + 1 j j=0 s,w=0 ( ( ( α (j + i 1 α (j + i 1 k 2 s w l Then, the density function of the OEHL-G order statistics is a miture of Ep-G densities Based on the last equation, we note that the properties of X i:n follow from those of T k+l+1 The qth moments of X i:n can be epressed as E (X q i:n = b k,l E (T k+l+1 (21 Based on the moments in equation (21, we can derive eplicit epressions for the L- moments of X as infinite weighted linear combinations of the means of suitable OEHL-G order statistics The rth L-moments is given by λ r = 1 r r 1 ( ( 1 d r 1 E (X r d:r, r 1 d d=0

15 Chilean Journal of Statistics PWMs The PWM is the epectation of certain function of a random variable whose mean eists A general theory for the PWMs covers the summarization and description of theoretical probability distributions and observed data samples, nonparametric estimation of the underlying distribution of an observed sample, estimation of parameters, quantiles of probability distributions and hypothesis tests The PWM method can generally be used for estimating parameters of a distribution whose inverse form cannot be epressed eplicitly The (j, ith PWM of X following the OEHL-G distribution, say ρ j,i, is formally defined by ρ j,i = E { X j F (X i} = From equation (20, we can write f ( F ( i = 2αλg( s,w, The last equation can be epressed as where Then, the PWM of X is given by ρ j,i = ( 1 s+k+l k! j f ( F (X i d λ (s + w + 1] k ( ( ( α (i α (i k 2 G( k+l s w l f ( F (X i = m k,l h k+l+1 (, ( 1 s+k+l λ (s + w + 1] k m k,l = 2αλ k! (k + l + 1 s,w=0 ( ( ( α (i α (i k 2 s w l m k,l j h k+l+1 ( d = ( m k,l E T j k+l+1 5 Characterizations This section deals with certain characterizations of OEHL-G distribution These characterizations are based on: (i a simple relation between two truncated moments; (ii the hazard function ; (iii the reverse hazard function and (iv conditional epectation of a function of the random variable One of the advantages of characterization (i is that the cdf is not required to have a closed form We present our characterizations (i (iv in four subsections

16 80 Afify, Altun, Alizadeh, Ozel and Hamedani 51 Characterizations based on ratio of two truncated moments In this subsection we present characterizations of OEHL-G distribution in terms of a simple relationship between two truncated moments This characterization result employs a theorem due to Glänzel (1987, see Theorem A1 of Appendi A Note that the result holds also when the interval H is not closed Moreover, as mentioned above, it could also be applied when the cdf F does not have a closed form As shown in Glänzel (1990, this characterization is stable in the sense of weak convergence λg(;ξ G(;ξ λg(;ξ G(;ξ Proposition 51 Let X : Ω R be a continuous random variable and let q 1 ( = ( ] α+1 ( ] α 1 + ep and q2 ( = q 1 ( 1 ep for R The random variable X has pdf (2 if and only if the function η defined in Theorem A1 has the form η ( = 1 2 { ( ep λg (; ξ G (; ξ ] α }, R Proof Let X be a random variable with pdf (2, then { ( (1 F ( E q 1 ( X ] = ep λg (; ξ G (; ξ ] α }, R, and { ( ] } λg (; ξ 2α (1 F ( E q 2 ( X ] = 1 1 ep, R, G (; ξ and finally η ( q 1 ( q 2 ( = 1 { ( ] 2 q λg (; ξ α } 1 ( 1 1 ep > 0 for R G (; ξ Conversely, if η is given as above, then s ( = ( η ( q 1 ( αλg (; ξ ep η ( q 1 ( q 2 ( = λg(;ξ G(;ξ G (; ξ 2 { 1 ( 1 ep ( 1 ep λg(;ξ G(;ξ ] α 1 λg(;ξ G(;ξ ] α }, R, and hence { ( s ( = log 1 1 ep λg (; ξ G (; ξ ] α }, R Now, in view of Theorem A1, X has density (2 Corollary 52 Let X : Ω R be a continuous random variable and let q 1 ( be as in Proposition 51 The pdf of X is (2 if and only if there eist functions q 2 and η defined

17 Chilean Journal of Statistics 81 in Theorem A1 satisfying the differential equation ( η ( q 1 ( αλg (; ξ ep η ( q 1 ( q 2 ( = λg(;ξ G(;ξ G (; ξ 2 { 1 ( 1 ep ( 1 ep λg(;ξ G(;ξ The general solution of the differential equation in Corollary 52 is η ( = { ( λg (; ξ 1 1 ep G (; ξ ] α } 1 1 ep ] α 1 λg(;ξ G(;ξ ( αλg (; ξ ep ( λg(;ξ G(;ξ ] α }, R λg(;ξ G(;ξ ] α 1, (q1 ( 1 q 2 ( + D where D is a constant Note that a set of functions satisfying the above differential equation is given in Proposition 51 with D = 1 2 However, it should be also noted that there are other triplets (q 1, q 2, η satisfying the conditions of Theorem A1 52 Characterization based on hazard function It is known that the hazard function, h F, of a twice differentiable distribution function, F, satisfies the first order differential equation f ( f ( = h F ( h F ( h F ( For many univariate continuous distributions, this is the only characterization available in terms of the hazard function The following characterization establish a non-trivial characterization of OELH-G distribution, when α = 1, which is not of the above trivial form Proposition 53 Let X : Ω R be a continuous random variable The pdf of X for α = 1, is (2 if and only if its hazard function h F ( satisfies the differential equation h F ( g (; ξ g (; ξ h F ( = λg (; ξ d ( ] } {G (; ξ 2 λg (; ξ ep, R d G (; ξ Proof If X has pdf (2, then clearly the above differential equation holds differential equation holds, then Now, this d { } g (; ξ 1 h F ( = λ d ( ] } {G (; ξ 2 λg (; ξ ep, R d d G (; ξ from which, we obtain h F ( = λg (; ξ ( G (; ξ ep λg(;ξ G(;ξ ], R, which is the hazard function of OELH-G distribution for α = 1

18 82 Afify, Altun, Alizadeh, Ozel and Hamedani 53 Characterizations in terms of the reverse hazard function The reverse hazard function, r F, of a twice differentiable distribution function, F, is defined as r F ( = f (, support of F F ( This subsection deals with the characterizations of OEHL-G distribution based on the reverse hazard function Proposition 54 Let X : Ω R be a continuous random variable The random variable X has pdf (2 if and only if its reverse hazard function r F ( satisfies the following differential equation { r F ( g (; ξ g (; ξ r F ( = 2αλg (; ξ d G (; ξ 2 d ( ( ] } λg (; ξ λg (; ξ 1 ep ep, R G (; ξ G (; ξ Proof If X has pdf (2, then clearly the above differential equation holds If this differential equation holds, then d { } g (; ξ 1 r F ( = 2αλ d ( ( ] } {G (; ξ 2 λg (; ξ λg (; ξ 1 ep ep, d d G (; ξ G (; ξ from which, we have r F ( = ( 2αλg (; ξ ep ( G (; ξ 2 1 ep λg(;ξ G(;ξ 2 λg(;ξ G(;ξ ], R 54 Characterization based on the conditional epectation of certain functions of the random variable In this subsection we employ a single function ψ of X and characterize the distribution of X in terms of the truncated moment of ψ (X The following proposition has already appeared in Hamedani s previous work (2013, so we will just state it as a proposition here, which can be used to characterize OELH-G distribution Proposition 55 Let X : Ω (d, e be a continuous random variable with cdf F Let ψ ( be a differentiable function on (d, e with lim e ψ ( = 1 Then, for δ 1, implies E ψ (X X ] = δψ (, (d, e ψ ( = (F ( 1 δ 1, (d, e

19 Chilean Journal of Statistics 83 Remark 51 For (d, e = R, ψ ( = ( 1 ep ( 1 + ep λg(;ξ G(;ξ λg(;ξ G(;ξ and δ = α α+1, Proposition 55 provides a characterization of OELH-G distribution 6 Maimum likelihood estimation Several approaches for parameter estimation were proposed in the literature but the maimum likelihood method is the most commonly employed The maimum likelihood estimators (MLEs enjoy desirable properties and can be used to obtain confidence intervals for the model parameters The normal approimation for these estimators in large samples can be easily handled either analytically or numerically Here, we consider the estimation of the unknown parameters of the new family from complete samples only by maimum likelihood method Let X 1,, X n be a random sample from the OEHL-G family with parameters α, λ and ξ Let θ =(α, λ, ξ be the p 1 parameter vector To obtain the MLE of θ, the log-likelihood function is given by l(θ = n log (2α + n log g ( i ; ξ 2 n log Ḡ( i ; ξ ] n G( i ; ξ n { ]} λ Ḡ( i ; ξ + (α 1 λg(i ; ξ log 1 ep Ḡ( i ; ξ +n log (λ (α + 1 n { log 1 + ep λg(i ; ξ Ḡ( i ; ξ ]} Then, the score vector components, U (θ = l θ = ( U α, U λ, U ξk, are U α = n α + n { log 1 ep λg(i ; ξ Ḡ( i ; ξ ]} n { log 1 + ep λg(i ; ξ Ḡ( i ; ξ ]}, U λ = n λ n + (α + 1 G( i ; ξ n Ḡ( i ; ξ + (α 1 n G( i ; ξep Ḡ( i ; ξ G( i ; ξep { Ḡ( i ; ξ 1 ep ] λg(i;ξ Ḡ( i;ξ λg(i;ξ Ḡ( i;ξ { 1 + ep ]} ] λg(i;ξ Ḡ( i;ξ λg(i;ξ Ḡ( i;ξ ]}

20 84 Afify, Altun, Alizadeh, Ozel and Hamedani and U ξk = n g ( i ; ξ n g ( i ; ξ 2 Ḡ ( i ; ξ Ḡ( i ; ξ n Ḡ( i ; ξg ( i ; ξ G ( i ; ξ λ Ḡ ( i ; ξ Ḡ( i ; ξ 2 Ḡ(i n ; ξg ( i ; ξ G ( i ; ξ Ḡ ( i ; ξ ] ] ep λg(i;ξ Ḡ( i;ξ +λ (α 1 ]} Ḡ( i ; ξ {1 2 ep λg(i;ξ Ḡ( i;ξ Ḡ(i n ; ξg ( i ; ξ G ( i ; ξ Ḡ ( i ; ξ ] ] ep λg(i;ξ Ḡ( i;ξ +λ (α + 1 ]}, Ḡ( i ; ξ {1 2 + ep λg(i;ξ Ḡ( i;ξ where g ( i ; ξ = g ( i ; ξ / ξ k, G ( i ; ξ = G ( i ; ξ / ξ k and Ḡ ( i ; ξ = Ḡ ( i; ξ / ξ k Setting the nonlinear system of equations U α = U λ = 0 and U ξk = 0 and solving them simultaneously yields the MLE θ = ( α, λ, ξ To do this, it is usually more convenient to adopt nonlinear optimization methods such as the quasi-newton algorithm to maimize l numerically For interval estimation of the parameters, we obtain the p p observed information matri J(θ = { 2 l r s } (for r, s = α, λ, ξ, whose elements are given in appendi A and can be computed numerically Under standard regularity conditions when n, the distribution of θ can be approimated by a multivariate normal N p (0, J( θ 1 distribution to obtain confidence intervals for the parameters Here, J( θ is the total observed information matri evaluated at θ The method of the re-sampling bootstrap can be used for correcting the biases of the MLEs of the model parameters Good interval estimates may also be obtained using the bootstrap percentile method Improved MLEs can be obtained for the new family using second-order bias corrections However, these corrected estimates depend on cumulants of log-likelihood derivatives and will be addressed in future research 7 Applications In this section, we provide applications to three real data sets to illustrate the importance of the OEHL-W and OEHL-Ga distributions presented in Section 2 The goodness-of-fit statistics for these distributions and other competitive distributions are compared and the MLEs of their parameters are provided In order to compare the fitted distributions, we consider goodness-of-fit measures including l, Anderson-Darling statistic (A and Cramér-von Mises statistic (W, where l denotes the maimized log-likelihood, Generally, the smaller these statistics are, the better the fit 71 Relief times of twenty patients The first real data set is taken from Gross and Clark (1975, p105, which gives the relief times of 20 patients receiving an analgesic The data are as follows: 11, 14, 13, 17, 19, 18, 16, 22, 17, 27, 41, 18, 15, 12, 14, 3, 17, 23, 16, 2 We compare the fits of the OEHL-W distribution with other competitive distributions, namely: the Weibull (W, odd log-logistic Weibull (OLL-W, generalized odd log-logistic Weibull (GOLL-W,

21 Chilean Journal of Statistics 85 Kumaraswamy Weibull (Kum-W, eponentiated generalized Weibull (EG-W and eponentiated half-logistic Weibull (EHL-W distributions The MLEs of the model parameters, their corresponding standard errors (in parentheses and the values of l, A and W are given in Table 1 Table 1 compares the fits of the OEHL-W distribution with the EHL-W, EG-W, Kum-W, GOLL-W, OLL-W and W distributions The results in Table 1 show that the OEHL-W distribution has the lowest values for the l, A and W statistics among the fitted models So, the OEHL-W distribution could be chosen as the best model Table 1 MLEs, their standard errors and goodness-of-fit statistics for the relief times data Model α λ a b l A W W (0427 (0182 OLL-W (19915 (0034 ( GOLL-W (032 (0003 (00008 (0368 Kum-W (0565 (0034 (0002 (0002 EG-W (0041 (1769 (0002 (0002 EHL-W (1986 (0038 (0002 (0002 OEHL-W (4750 (0121 (00009 (5295 (a (b Fitted density Fitted hrf f( OLL W GOLL W EHL W OEHL W f( S(t KM versus Fitted Survival h(t Epected Probability t PP plot t Observed Probability Figure 7 (a Estimated densities of the best four models and (b fitted functions of the OEHL-W model for strengths of glass fibers data set 72 Time to failure (10 3 h of turbocharger The second data set (n = 40 below is from Xu et al (2003 and it represents the time to failure (10 3 h of turbocharger of one type of engine The data are: 16, 35, 48, 54, 60,

22 86 Afify, Altun, Alizadeh, Ozel and Hamedani 65, 70, 73, 77, 80, 84, 20, 39, 50, 56, 61, 65, 71, 73, 78, 81, 84, 26, 45, 51, 58, 63, 67, 73, 77, 79, 83, 85, 30, 46, 53, 60, 87, 88, 90 For these data, we compare the fits of the OEHL-Ga distribution with the gamma (Ga, odd log-logistic gamma (OLL- Ga, generalized odd log-logistic gamma (GOLL-Ga, Kumaraswamy gamma (Kum-Ga, eponentiated generalized gamma (EG-Ga and eponentiated half-logistic gamma (EHL- Ga distributions Table 2 lists the MLEs of the model parameters, their corresponding standard errors (in parentheses and the values of l, A and W Table 2 compares the fits of the OEHL-Ga distribution with the EHL-Ga, EG-Ga, Kum-Ga, GOLL-Ga, OLL-Ga and Ga distributions The OEHL-Ga distribution has the lowest values for goodness-of-fit statistics among all fitted models So, the OEHL-Ga distribution can be chosen as the best model Table 2 MLEs, their standard errors and goodness-of-fit statistics for time to failure data Model α λ a b l A W Ga (1689 (0279 OLL-Ga (2942 (1888 (0298 GOLL-Ga (0139 (0024 ( (25720 Kum-Ga (1104 (51630 (9102 (0301 EG-Ga (0236 (0025 (0002 (0003 EHL-Ga (0026 (14270 (0002 (0002 OEHL-Ga (0189 (0095 (11466 (1819 (a (b Fitted density Fitted hrf f( GOLL Ga EG Ga EHL Ga OEHL Ga f( S(t KM versus Fitted Survival h(t Epected Probability t PP plot t Observed Probability Figure 8 (a Estimated densities of the best four models and (b fitted functions of the OEHL-Ga model for time to failure (10 3 h of turbocharger data set

23 Chilean Journal of Statistics Strengths of 15 cm glass fibers The third data set (n = 63 consists of 63 observations of the strengths of 15 cm glass fibers, originally obtained by workers at the UK National Physical Laboratory The data are: 055, 074, 077, 081, 08,4 093, 104, 111, 113, 124, 125, 127, 128, 129, 13, 136, 139, 142, 148, 148, 149, 149, 15, 15, 151, 152, 153, 154, 155, 155, 158, 159, 16, 161, 161, 161, 161, 162, 162, 163, 164, 166, 166, 166, 167, 168, 168, 169, 17, 17, 173, 176, 176, 177, 178, 181, 182, 184, 184, 189, 2, 201, 224 Table 3 lists the MLEs of the model parameters, and corresponding standard errors (in parentheses with estimated l, A and W statistics Based on the figures in Table 3, OEHL-Ga distribution provides the best fit among others Table 3 MLEs, their standard errors and goodness-of-fit statistics for strengths of 15 cm glass fibers Models α λ a b A W l Ga (3078 (2072 OLL-Ga (4794 (1452 (0925 GOLL-Ga (2171 (0117 (0935 (1725 Kum-Ga (0537 (6974 (4758 (2118 EG-Ga (12492 (0235 (12454 (6130 EHL-Ga (0342 (16298 (5181 (2813 OEHL-Ga (0253 (12088 (4659 (1959 (a (b Fitted density Fitted hrf f( Kum Ga EG Ga EHL Ga OEHL Ga f( S(t KM versus Fitted Survival h(t Epected Probability t PP plot t Observed Probability Figure 9 (a Estimated densities of the best four models and (b fitted functions of the OEHL-Ga model for strengths of glass fibers data set Moreover, Table 4 shows the Akaike Information Criteria (AIC for all fitted models and for three data sets Table 4 reveals that the OEHL-G family provides better fits for

24 88 Afify, Altun, Alizadeh, Ozel and Hamedani these three data sets than other all competitive models It is clear from Tables 1, 2, 3 and 4 that the OEHL-W and OEHL-Ga distributions provide the best fits among others The histograms of the fitted distributions for the OEHL-W and OEHL-Ga models are displayed in Figures 7(a, 8(a and 9(a, respectively Figures 7(b, 8(b and 9(b display the fitted pdf, estimated hrf, fitted survival functions and probability-probability (P-P plots for the OEHL-W and OEHL-Ga models, respectively It is evident from these plots that the OEHL-G distributions provide superior fit to the three data sets Table 4 AIC values of fitted models for all used data sets First data set Second data set Third data set Models AIC Models AIC Models AIC W Ga Ga OLL-W OLL-Ga OLL-Ga GOLL-W GOLL-Ga GOLL-Ga Kum-W Kum-Ga Kum-Ga EG-W EG-Ga EG-Ga EHL-W EHL-Ga EHL-Ga OEHL-W OEHL-Ga OEHL-Ga Simulation study In this section, we evaluate the performance of the maimum likelihood method for estimating the OEHL-N parameters using a Monte Carlo simulation study with 10, 000 replications We calculate biases, the mean square errors (MSEs of the parameter estimates, estimated average lengths (ALs and coverage probabilities (CPs using the R package The MSEs, ALs and CPs can be calculated by using following equations: Bias ɛ (n = 1 N N (ˆɛ i ɛ, MSE ɛ (n = 1 N N (ˆɛ i ɛ 2, CP ɛ (n = 1 N N I(ˆɛ i s ˆɛi, ˆɛ i s ˆɛi, AL ɛ (n = N N s ˆɛi for ɛ = α, λ, µ, σ We generate N = 1, 000 samples of sizes n = 50, 55,, 1000 from the OEHL-N distribution with α = λ = 05 and µ = σ = 2 The numerical results for the above measures are shown in the plots of Figure 9 It is noted, from Figure 9, that the estimated biases

25 Chilean Journal of Statistics 89 decrease when the sample size n increases Further, the estimated MSEs decay toward zero as n increases This fact reveals the consistency property of the MLEs The CPs are near to 095 and approach to the nominal value when the sample size increases Moreover, if the sample size increases, the ALs decrease in each case α λ CP α λ µ σ Estimated Bias Estimated Bias n µ Estimated Bias Estimated Bias n σ n n n α λ α λ Estimated MSE Estimated MSE n n µ Estimated MSE Estimated MSE AL AL n n σ µ σ AL AL n n n n Figure 10 Estimated CPs, biases, MSEs and ALs of the selected parameter vector 9 Conclusions We proposed a new odd eponentiated half-logistic-g (OEHL-G family of distributions with two etra shape parameters Many well-known distributions emerge as special cases of the OEHL-G family The mathematical properties of the new family including eplicit epansions for the ordinary and incomplete moments, quantile and generating functions, entropies, order statistics and probability weighted moments have been provided The model parameters have been estimated by the maimum likelihood estimation method It has been shown, by means of two real data sets, that special cases of the OEHL-G family can provide better fits than other competitive models generated using well-known families A graphical simulation to assess the performance of the maimum likelihood estimators is provided We hope that the OEHL-G family may be etensively used in statistics

26 90 Afify, Altun, Alizadeh, Ozel and Hamedani Appendi A Theorem A1 Let (Ω, F, P be a given probability space and let H = a, b] be an interval for some d < b (a =, b = might as well be allowed Let X : Ω H be a continuous random variable with the distribution function F and let q 1 and q 2 be two real functions defined on H such that E q 2 (X X ] = E q 1 (X X ] η (, H, is defined with some real function η Assume that q 1, q 2 C 1 (H, η C 2 (H and F is twice continuously differentiable and strictly monotone function on the set H Finally, assume that the equation ηq 1 = q 2 has no real solution in the interior of H Then F is uniquely determined by the functions q 1, q 2 and η, particularly F ( = a C η (u η (u q 1 (u q 2 (u ep ( s (u du, where the function s is a solution of the differential equation s = η q 1 ηq 1 q 2 normalization constant, such that H df = 1 and C is the References Afify A Z, Cordeiro, G M, Yousof, H M, Alzaatreh, A and Nofal, Z M (2016a The Kumaraswamy transmuted-g family of distributions: properties and applications Journal of Data Science 14, Afify A Z, Yousof, H M and Nadarajah, S (2016b The beta transmuted-h family for lifetime data Statistics and Its Interface, forthcoming Aleander, C, Cordeiro, G M, Ortega, E M M and Sarabia, J M (2012 Generalized beta generated distributions Computational Statistics and Data Analysis 56, Alzaatreh, A, Famoye, F and Lee, C (2013 A new method for generating families of continuous distributions Metron 71, Bourguignon, M, Silva, RB, Cordeiro, GM (2014 The Weibull-G family of probability distributions Journal of Data Science 12, Cordeiro, G M and de Castro, M (2011 A new family of generalized distributions Journal of Statistical Computation and Simulation 81, Cordeiro, G M, Ortega, E M, and da Cunha, D C (2013 The eponentiated generalized class of distributions Journal of Data Science 11, 1-27 Cordeiro, G M, Alizadeh, M and Ortega, E M M (2014 The eponentiated halflogistic family of distributions: properties and applications Journal of Probability and Statistics 81, 1-21 Cordeiro, G M, Alizadeh, M, Tahir, M H, Mansoor, M, Bourguignon, M and Hamedani, G G (2015 The beta odd log-logistic family of distributions Hacettepe Journal of Mathematics and Statistics, forthcoming Cordeiro, GM, Alizadeh, M, Ozel, G, Hosseini, B, Ortega, EMM and Altun, E (2017 The generalized odd log-logistic family of distributions: properties, regression models and applications, Journal of Statistical Computation and Simulation 87, Eugene, N, Lee, C and Famoye, F (2002 Beta-normal distribution and its applications Communications in Statistics - Theory and Methods 31,

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