The Exponentiated Generalized Standardized Half-logistic Distribution

Transcription

1 International Journal of Statistics and Probability; Vol. 6, No. 3; May 27 ISSN E-ISSN Published by Canadian Center of Science and Education The Eponentiated Generalized Standardized Half-logistic Distribution Gauss M. Cordeiro, Thiago A. N. de Andrade, Marcelo Bourguignon 2 & Frank Gomes-Silva 3 Department of Statistics, Federal University of Pernambuco, Recife, Brazil 2 Department of Statistics, Federal University of Rio Grande do Norte, Natal, Brazil 3 Department of Statistics and Informatics, Rural Federal University of Pernambuco, Recife, Brazil Correspondence: Thiago A. N. de Andrade, Department of Statistics, Federal University of Pernambuco, Recife, Brazil. Tel: 55(8) thiagoan.andrade@gmail.com Received: December 23, 26 Accepted: March 9, 27 Online Published: March 9, 27 doi:.5539/ijsp.v6n3p24 Abstract URL: We study a new two-parameter lifetime model called the eponentiated generalized standardized half-logistic distribution, which etends the half-logistic pioneered by Balakrishnan in the eighties. We provide eplicit epressions for the moments, generating and quantile functions, mean deviations, Bonferroni and Lorenz curves, and order statistics. The model parameters are estimated by the maimum likelihood method. A simulation study reveals that the estimators have desirable properties such as small biases and variances even in moderate sample sizes. We prove empirically that the new distribution provides a better fit to a real data set than other competitive models. Keywords: eponentiated generalized class, half-logistic distribution, hazard rate function, lifetime distribution, moment.. Introduction It is hardly necessary to emphasize that a probabilistic model is commonly employed for practical situations in which a deterministic model is not feasible. We can verify that probabilistic models still awaken the fascination of applied scholars and researchers. This interest materializes in the great amount of works that are dedicated to the proposal of new distributions. The half-logistic (HL) distribution pioneered by Balakrishnan is the absolute value of a random variable following the logistic distribution. It has a monotonically increasing hazard rate function (hrf) for all parameter values, which is a property shared by relatively few distributions with support on the positive real line. Recently, the HL distribution has been discussed by several authors. We shall refer to the following works: (Balakrishnan, N. & Wong, K. H. T., 99) obtained approimate maimum likelihood estimates (MLEs) for the location and scale parameters with type-ii right-censoring; (Balakrishnan, N. & Chan, P., 992) presented the estimation for the scaled HL distribution under type II censoring; (Panichkitkosolkul, W. & Saothayanun, 22) investigated bootstrap confidence intervals for the process capability inde under this distribution. More recently, (Oliveira, J., 26) introduced a new etension of the HL model by considering the standardized half-logistic (SHL) distribution, which is an attractive model for statisticians and applied researchers since it does not have parameters and its mathematical properties can be easily obtained. The cumulative distribution function (cdf) and probability density function (pdf) (for t > ) of the SHL distribution are given by G(t) = e t + e t () and 2e t g(t) = ( + e t ), (2) 2 respectively. Let T be a random variable having density (2). The HL distribution is defined by a linear transformation W = µ + σ T, where µ IR + and σ >. Without loss of generality, we can work with the SHL model. The nth moment of T is E(T n ) = 2 t n e t ( + e t ) 2 dt = 2n!( 2 n )ζ(n), where ζ( ) is the Riemann zeta function. For details on the Riemann zeta function, see the Wolfram website http: //mathworld.wolfram.com/riemannzetafunction.html. In particular, the first two moments of T are E(T) = log(4) and E(T 2 ) = π 2 /3. In addition, the hrf of T is given by λ(t) = /( + e t ). The moment generating function (mgf) of T, say M T (s) = E(e st ), is given by 24

2 M T (s) = 2 e st e t ( + e t ) 2 dt = 2J ( + s, s), where J p (a, b) = p u a (a, b > ) is the type II incomplete beta function. For more properties of the HL distribution(balakrishnan, N., et al., 99; Balakrishnan, N. & Chan, P., 992; Panichkitkosolkul, W. & Saothayanun., 22; (+u) a+b Oliveira, J., et al., 26; Cordeiro, G. M., et al., 25). The addition of parameters to the SHL model may generate new distributions with great adjustment capability and, for this reason, we propose a generalization of it. The recent literature has suggested several ways of etending well-known distributions, among them, the generator approach, to provide more realistic statistical models in a great variety of applications. Some of the most important generators were recently discussed by Mansoor, M., et al., (26). For a baseline continuous cdf G(), (Cordeiro, G. M., et al., 23) defined the eponentiated generalized (EG for short) class of distributions by F() = { [ G()] a } b, (3) where a > and b > are two etra parameters whose role is to govern skewness and generate distributions with heavier/ligther tails. They are sought as a manner to furnish a more fleible distribution. Because of its tractable distribution function (3), this class can be used quite effectively even if the data are censored. The EG class is suitable for modeling continuous univariate data that can be in any interval of the real line. The pdf corresponding to (3) is given by f () = a b [ G()] a { [ G()] a } b g(), (4) where g() = dg()/d is the baseline pdf, which is a special case of (4) when a = b =. Setting a = gives the eponentiated-g ( ep-g ) class. If b =, we obtain the Lehmann type II class. So, the family (4) generalizes both Lehmann types I and II classes; that is, this method can be interpreted as a double construction of Lehmann alternatives. Note that even if g() is a symmetric density, the density f () will not be symmetric. The above properties and many others have been discussed and eplored in recent works for the EG class. We refer to the papers: (Cordeiro, G. M., et al., 24; Lemonte, A. J., 24; Elbatal, I. & Muhammed, H. Z., 24;Moors, 988; Da Silva, et al., 25; De Andrade, et al., 25; Bourguignon, M., et al., 25; Mansoor, M., et al., 26; Arya, G.& Elbata, I., 25 ; Silva, A. O., et al., 25), which used the EG class to etend the Burr III, Birnbaum-Saunders, inverse Weibull, inverted eponential, generalized gamma, Gumbel, etended eponential, Fréchet, modified Weibull and Dagum distributions, respectively. The rest of the paper is organized as follows. In Section 2, we define the eponentiated generalized standard halflogistic (EGSHL) distribution by inserting () in equation (3). In Section 3, we study the shapes of its pdf and hrf. Its hrf can take non-monotonous forms, such as bathtub and inverted bathtub, which eplain many real phenomenons. A detailed study of the quantile function (qf) and some applications is addressed in Section 4. In Section 5, we obtain a useful linear representation for the new density. Some properties of the ep-shl model are given in Section 6. Eplicit epressions for the ordinary and incomplete moments, mean deviations, Bonferroni and Lorenz curves, generating function and reliability of the EGSHL distribution are obtained in Section 7. Sections 8 and 9 are related to the probability weighted moments (PWMs) and Rényi entropy, respectively. In addition, for each important equation associated with the new model, we provide plots and numerical studies in order to illustrate its usefulness. The order statistics and their moments are investigated in Section. We discuss maimum likelihood estimation of the model parameters in Section. In Section 2, we present a simulation study. An application to real data in Section 3 shows the usefulness of the proposed distribution. Finally, concluding remarks are addressed in Section The New Distribution Let X be a random variable with support on the positive real line having the EGSHL (a, b) distribution, say X EGSHL (a, b). The cdf of X, for >, is defined by inserting () in equation (3) F() = F(; a, b) = [( + e ) a 2 a e a ] b ( + e ) ab, (5) where a > and b >. Equation (5) has a simple closed-form, which is an important aspect to generate EGSHL variables by using the inversion method. The density of X becomes f () = f (; a, b) = a b 2a e a [( + e ) a 2 a e a ] b ( + e ) ab+. (6) 25

3 For brevity of notation, we shall drop the eplicit reference to the parameters a and b unless otherwise stated. For a = b =, equation (6) reduces to the SHL density. The EGSHL model also includes the Lehmann type I and type II transformations of the SHL distribution, denoted by ESHLI and ESHLII. For eample, the eponentiated SHL distribution, say ESHLI, follows when a =. Some plots of the pdf (6) are displayed in Figures 2 and 2. These plots reveal that the pdf of X is quite fleible and can take symmetric and asymmetric forms, among others. In summary, they reinforce the importance of the proposed model. f Density f Density f Density (a) a =.4 and b = (b) a =.4 and b = 2 Figure. Plots of the EGSHL density function for some parameter values (c) a =.6 and b = 9 f Density.4 f Density f Density (d) Solid curve: a = 2, b = 5; dashed curve a = 3, b = 2; dotted curve a =.6, b = 7; (e) Solid curve: a =.8, b = 5; dashed curve a = 6, b = 2; dotted curve a = 3, b = 3; Figure 2. Plots of the EGSHL density function for some parameter values (f) Solid curve: a = 2.5, b =.5; dashed curve a = 3, b =.7; dotted curve a = 4, b =.9; The sf and hrf of X are given by and τ() = S () = ( + e ) ab [( + e ) a 2 a e a ] b ( + e ) ab a b 2 a e a [( + e ) a 2 a e a ] b ( + e ){( + e ) ab [( + e ) a 2 a e a ] b }, (7) respectively. Some plots of (7) are displayed in Figure 2. Besides monotone forms, the hrf of X can take bathtub and inverted bathtub shapes. This non-monotone form is particularly important because of its great practical applicability. The time of human life is just one of many phenomena that the bathtub shape hrf is applicable (Lee, E. T., 992). 3. Shapes Some plots of log{ f ()} using the Wolfram Mathematica software for selected parameter values are displayed in Figure 3. We can investigate the shapes of the pdf and hrf of X from their first and second derivatives. The first derivative of log{ f ()} is given by d log{ f ()} d = a + ( + ab) e η () a ( b) e v 2 () v (), where η() = + e, v () = 2 a e a + η a () and v 2 () = 2 a e ( a) η a (). Thus, the critical values of f () are the roots of the equation: a + ( + ab) e η () = a ( b) e v 2 () v (). 26

4 h Hazard h Hazard (g) bathtub a =.8 and b = (h) inverted bathtub a =.52 and b =.56 h Hazard.35 h Hazard (i) decreasing a =.89 and b =.2 (j) increasing a =.5 and b = 4.4 Figure 3. Plots of the EGSHL hazard function for some parameter values logf logf logf logf logf logf (k) a =.3 and b =.5-5 (l) a =.4 and b = (m) a =.8 and b =.2 Figure 4. Plots of log{ f ()}. The value, which solves the equation above can be a maimum, minimum or inflection point. To check this, we evaluate the sign of the second derivative of log{ f ()} at =. We have d 2 log{ f ()} = ( + ab) e [e η()] d 2 η 2 () + a ( b) e 2 v 2 () { a 2 2a e 2( a) + η 2 () [ a + a η 2a () e η() ] 2 a e a η a 2 () [ a + e η()( + a a e η()) ] }. It is often difficult to obtain an analytical solution for the critical points of this function. Therefore, it is common to obtain numerical solutions with high accuracy through optimization routines in most mathematical and statistical platforms. Some plots of the first derivative of log{ f ()} for selected parameter values are displayed in Figure 3. dlogf -thderivativeoflogf dlogf -thderivativeoflogf dlogf -thderivativeoflogf (n) a =.8 and b = (o) a =.2 and b = 2.3 Figure 5. Plots of the first derivative of log{ f ()} (p) a =.2 and b =.82 Similarly, we provide the first and second derivatives of log{h()}. The critical values of log{h()} are the roots of the 27

5 equation: d log{h()} d = a ( b) e v 2 () + e + (4 a) η() η() The second derivative of log{h()} is given by + a b e η a b () + a b e v 2 () v b () η a b () v b (). d 2 log{h()} = e 2 e η() + a e v 2 d 2 η 2 () () { ( b) [ a e v 2 2 () + v () v 3 () ] b [ η a b () + v 2 () v b () ]2 b [ η a b () v b ()] where v 3 () = 2 a a e ( a) + ( a) e η a 2 () η a (). [ η a b () ( a b) e η a b 2 () + a ( b) e v 2 2 () vb 2 () + v b () v 3 () ] }, Some plots of the first derivative of log{h()} for selected parameter values are displayed in Figure 3. dlogh -thderivativeoflogh dlogh -thderivativeoflogh dlogh -thderivativeoflogh (q) a =.8 and b = (r) a =. and b =.68 Figure 6. Plots of the first derivative of log{h()} (s) a = 4.28 and b = Quantile Function In previous sections, we provide some important functions that characterize the random variable X EGSHL (a, b). By inverting (5), we obtain the qf of X as [ ( u /b ) /a ] Q(u) = log, (8) 2 ( u /b ) /a where u (, ). The proposed distribution is easily simulated from a uniform random variable U by X = Q(U). Net, we use (8) to generate EGSHL(.5,.2) occurrences. Figure 4 displays the histogram and empirical cdf for the simulated data and also the eact pdf and cdf of X. These plots reinforce the adequacy model for practical applications. For similar studies, see (Jafari, A. A., et al., 24; Jafari, A. A. & Mahmoudi, E., 25), among others. pdf cdf (t) Histogram and eact pdf (u) Empirical and eact cdfs Figure 7. Plots of the EGSHL(.5,.2) pdf, histogram, eact and empirical cdfs for simulated data with n = As mentioned earlier, the qf practical uses are numerous. For eample, Q(/2) determines the median of the model. Table 4 gives the results of a small simulation study using the R software. The goal is to compare the empirical medians (EMed) 28

6 generated for different parameter values and random samples of size n =, 2, 4,, with their corresponding theoretical medians (Med) obtained by Q(/2). As epected, the difference between EMed and Med decreases when n increases. Table. Theoretical and empirical medians (for n =, 2, 4, ) of X for some parameter values a b Med EMed (n = ) EMed (n = 2) EMed (n = 4) EMed (n = ) Finally, we use the qf of X to determine the Bowley skewness (Kenney, J. F. & Keeping, E. S., 962) (B) and Moors kurtosis (Moors, J. J. A., 988) (M). The Bowley skewness is based on quartiles B = [Q(3/4) 2Q(/2)+ Q(/4)]/[Q(3/4) Q(/4)], whereas the Moors kurtosis is based on octiles M = [Q(7/8) Q(5/8) Q(3/8) + Q(/8)]/[Q(6/8) Q(2/8)]. These measures are fairly considered in the literature. Here, we refer to the following works: (Cordeiro, G. M. & Lemonte, A. J, 24; Silva, A. O., et al., 25; De Andrade, 25; Mansoor, M., et al., 26), among others. The effects of the additional shape parameters a and b on the skewness and kurtosis of the EGSHL model can be based on these measures. In Figures 4 and 4, we present 3D plots of B and M measures for several parameters values. These plots, obtained using the Wolfram Mathematica software, reveal that changes in these parameters have a considerable impact on the skewness and kurtosis of the EGSHL model, thus showing its greater fleibility. (a) a (.2, 8) and b (.5, 2) (b) a (.5, 2) and b (.5, 2) (c) a (.2,.5) and b (.2,.5) (d) a (.,.9) and b (.,.9) (e) a (.,.9) and b (.,.9) (f) a (6,.5) and b (.5, 6) 5. Linear Representation Figure 8. Plots of the Bowley skewness of X. We provide useful linear representations for equations (3) and (4) based on the eponentiated class of distributions. Mathematical properties of the eponentiated distributions have been published by many authors in the 9s. See, for eample, (Gupta, R. D. & Kundu, D., 2) for eponentiated eponential, (Nadarajah, S., et al., 24) for eponentiated Lindley, (Sarhan, A. M. & Kundu, D., 29) for eponentiated linear failure rate and, more recently, (Lemonte, A. J., 23) for the eponentiated Nadarajah-Haghighi and (Oliveira, J., et al., 26) for the eponentiated SHL models. For an arbitrary baseline cdf G(), a random variable Y a has the ep-g class with power parameter a >, say Y a ep- G(a), if its cdf and pdf are given by H a () = G() a and h a () = a g() G() a, respectively. For a comprehensive discussion about the eponentiated class, see a recent paper by Tahir, M. H. and Nadarajah, S. (25). Here, we consider the generalized binomial epansion ( z) b = ( ) b ( ) k z k, (9) k k= 29

7 (a) a (.2, 8) and b (.5, 2) (b) a (.5, 2) and b (.5, 2) (c) a (.2,.5) and b (.2,.5) (d) a (., 2) and b (.2, 2) (e) a (., 3) and b (.5,.5) (f) a (., 3) and b (.5,.5) Figure 9. Plots of the Moors kurtosis of X. which holds for any real non-integer b and z <. Using (9) twice in equation (4), the EG density function can be epressed as f () = w j+ h j+ (), () j= where w j+ = m= ( ) ( ) ( j+m+ b m a m j+) and h j+ () = ( j + ) g() G() j is the ep-g pdf with power parameter j +. Equation () reveals that the EG density is a linear combination of ep-g densities. We can derive some structural properties of the EG class from those ep-g properties. The cdf F() comes from () by simple integration, namely F() = w j+ H j+ (), () j= where H j+ () = G() j+ is the ep-g cdf with power parameter j +. Equations () and () were obtained by Cordeiro, G. M. and Lemonte, A. J. (24). They hold for any baseline distribution G. It is not difficult to show numerically that j= w j+ =. Moreover, for most practical purposes, we can set the upper limits equal to 2. We can adopt () for the EGSHL distribution and obtain its mathematical properties from those of the ep-shl distribution. Let Y j+ be a random variable having the ep-shl density with power parameter j + ( j ) given by h j+ () = 2 ( j + ) e ( e ) j ( + e ) j+2. (2) Clearly, several mathematical properties of X can be determined from the linear representation () and those of the ep-shl distribution given by Oliveira, J. et al. (26) and report in the net section. 6. Properties of the Ep-SHL Distribution Henceforth, let Y j+ ep-shl( j + ) have the density function (2). We use throughout an equation for a power series raised to an integer j =, 2,... j a i i = c j,i i, i= i= 3

8 where a, c j, = a j and the coefficients c j,i (for i ) are determined recursively by c j,i = ia The nth moment of Y j+ derived by epanding the binomial terms is given by E(Y n j+ ) = 2( j + ) n e i [m( j + ) i] a m c j,i m. (3) m= ( e ) j ( + e ) d = ( j + ) j+2 i= c n,i ( j + + i + n), (4) where the quantities c n,i s are obtained from equation (3) by taking a i = [ + ( ) i ]/(i + ) (for i ). For empirical purposes, the shape of many distributions can be usefully described by the incomplete moments. They form natural building blocks for measuring inequality: for eample, the Lorenz and Bonferroni curves depend upon the first incomplete moment of an income distribution. The nth incomplete moment of Y j+ is given by m j+ (n; z) = z n h j+ () d = 2( j + ) where tanh( ) is the hyperbolic tangent function. z n The mgf of Y j+, say M j+ (s) = E(e sy j+ ), can be epressed as ( e ) j ( + e ) d = ( j + ) j+2 i= c n,i tanh(z/2) j++i+n, (5) ( j + + i + n) M j+ (s) = ( j + )! Γ( s) 2 F [ j +, s; j + 2 s; ], (6) where 2 F is the regularized hypergeometric function defined by 2 F [a, b; c; z] = k= (a) k (b) k Γ(c + k) z k, z <, k! (a) k = a(a )... (a k + ) (for k > ) is the falling factorial, (a) =, and Γ(a) = a e d is the gamma function. For z < and arbitrary parameters a, b and c, the above infinite sum is convergent. For more, see (Oliveira, J., et al., 26). 7. Properties of the EGSHL Distribution In this section, we obtain eplicit epressions for some quantities of the EGSHL distribution. The formulae derived can be handled in most symbolic computation platforms such as Mathematica and Maple more efficiently than computing them directly by numerical integration of the density function (6). The infinity limits can be substituted by a large positive integer such as 2 or 3 for most practical purposes. 7. Moments The statistical relevance for calculating moments, especially in applied research, is widely know in the literature. Net, we provide two ways to compute the nth moment of X with density (6). The first formula follows as µ n = E(X n ) = n a b 2a e a [( + e ) a 2 a e a ] b ( + e ) ab+ d. (7) Although we do not have a closed-form for this integral, it is very simple to evaluate it computationally. For illustrative purposes, we provide a small numerical study by computing E(X n ) and the variance of X from (7) numerically. We consider several parameters values and n =, 2, 3, 4, 5 and the results are given in Table 7. with five decimal digits. All computations are performed using Wolfram Mathematica platform. Plots of the moments for some parameter values are display in Figure 7.. Based on the values in Table 7. and the plots in Figure 7., we conclude that the additional parameters a and b have large impact on the moments of X. Theses values and plots reveals that, in general, for fied a parameter value, the moments and the variance increases when b increase. The inverse happens when we set values for b and the parameter a increases. Alternatively, the nth moment of X can be obtained from equations () and (4) as µ ( j + ) w j+ c n,i n = ( j + + i + n), (8) where the quantities c n,i are defined in (4). j, i= 3

9 Table 2. First five moments and variance of X for several a and b values a b E(X) E(X 2 ) E(X 3 ) E(X 4 ) E(X 5 ) E(X 6 ) Var(X) (a) n = (b) n = 2 (c) n = 3 (d) n = (e) n = 2 (f) n = 3 Figure. Plots of the moments of X for some parameter values 7.2 Incomplete Moments and Their Applications The nth incomplete moment of X, say m(n; y) = y n f () d, can be determined from () and (5) as m(n; y) = ( j + ) w j+ c n,i tanh(z/2) j++i+n. (9) ( j + + i + n) j,i= 32

10 Generally, there has been a great interest in obtaining the first incomplete moment of a distribution. The mean residual function follows from (9) with n = as µ m(; y) y. Further, we can obtain mean deviations from the mean and the median given by δ = E( X µ ) = 2µ F(µ ) 2m(; µ ) and δ 2 = E( X M ) = µ 2m(; M), where the mean µ and the median M follow from (8) and (8), respectively. Equation (9) with n = is also useful to derive the Bonferroni and Lorenz curves defined (for a given probability π) by B(π) = m(; q)/(π µ ) and L(π) = m(; q)/µ, respectively, where q = Q(π) follows from (8). 7.3 Generating Function The mgf of X, say M(s) = E(e sx ), can be obtained from () and (6) (for s,, 2,...) as 7.4 Reliability M(s) = ( j + ) w j+ Γ( s) 2 F [ j +, s; j + 2 s; ]. j= Here, we derive the reliability, say R, when X EGSHL(a, b ) and X 2 EGSHL(a 2, b 2 ) are two independent random variables. Let f () denote the pdf of X and F 2 () denote the cdf of X 2. The reliability can be epressed as R = P(X > X 2 ) = f () F 2 () d and using equations () and () gives where I j,k = m,n= ( ) ( j+k+m+n+2 b m Thus, the reliability of X reduces to R = R = ) ( m a j+ I j,k h j+ () H k+ () d, j,k= ) ( b2 n ) ( na2 k+). I j,k j,k= 2 ( j + ) e ( e ) j+k+ ( + e ) j+k+3 d and then R = j,k= ( j + ) I j,k ( j + k + 2). (2) Table 7.4 gives some values of R for different parameter values. Clearly, for a = a 2 and b = b 2, we obtain R = P(X > X 2 ) = /2. All computations are done using Wolfram Mathematica software by taking the upper limits equal to 3 in (2). Table 3. The reliability of X for (a = 2, a 2 = 2) and some values of b and b 2 8. Probability Weighted Moments b b The PWMs are used to derive estimators of the parameters and quantiles of generalized distributions. The moment method of estimation is formulated by equating the population and sample PWMs. These moments have low variances and no severe biases, and they compare favorably with estimators obtained by maimum likelihood. The (s, r)th PWM of X is defined by δ s,r = E[X s F() r ]. Clearly, the ordinary moments follow as δ s, = E(X s ). Net, we derive simple epressions for the PWMs of X defined by δ s,r = s F() r f () d. (2) 33

11 Inserting (5) and (6) in equation (2), the PWMs of X can be epressed in a simple form δ s,r = s a b 2a e a [( + e ) a 2 a e a ] b(r+) ( + e ) ab(r+)+ d. (22) Table 8 gives from (22) the values of δ s,r for X EGSHL (2, 3) and some values of s and r. All computations are performed using Wolfram Mathematica software. Based on the values in Table 8, we conclude that, for fied r, the PWMs increase when s increases. The opposite happens when we fi the parameter s and r increases. Table 4. The PWMs of X for some values of s and r r s We now present a simpler epression for the PWMs of X. Under simple algebraic manipulation, we can write δ s,r as δ s,r = (r + ) s f [; a, (r + )b] d. (23) where f [; a, (r + )b] is the EGSHL density with parameters a and (r + )b. Equation (23) revels that the PWMs of X can be epressed in terms of the ordinary moments of X r EGSHL [a, (r + )b]. 9. Rényi Entropy Given a certain random phenomenon under study, it is important to quantify the uncertainty associated with the random variable of interest. One of the most popular measures used to quantify the variability of X is the Rényi entropy. See, for eample, Da Silva et al (23) for the gamma etended Fréchet model (Alshangiti, 24), for the Marshall-Olkin etended modified Weibull distribution and (Castellares, F. & Santos, M. A. C., 25) for an etended logistic distribution. The Rényi entropy of X with density (6), say I R (ρ), is given by ( I R (ρ) = ( ρ) log where ρ > and ρ. ) f () ρ d, By inserting (6) in this equation, we obtain ( [ a b 2 a I R (ρ) = ( ρ) log e a [( + e ) a 2 a e a ] b ] ρ d). (24) ( + e ) ab+ Equation (24) can be easily implemented computationally and the values of I R (ρ) are obtained in a few seconds. Table 9 gives some values of I R (ρ) for different parameter values. All computations use the Wolfram Mathematica software. Based on the figures in Table 9, we note that, independently of a and b, I R (ρ) decreases when ρ increases. For fied ρ, the Rényi entropy is larger for a < b. Table 5. Rényi entropy of X for some values of ρ, a and b a b ρ = 2 ρ = 4 ρ = 6 ρ = 8 ρ = Although, equation (24) is easily manageable computationally, we provide an epression in closed-form to compute I R (ρ). Using (9) twice in equation (4), we can write ( )( ) ρ(b ) ak + ρ(a ) f () ρ = (ab) ρ ( ) k+l g() ρ G() l. (25) k l k, l= 34

12 Substituting () and (2) in equation (25), we have I R (ρ) = ( ρ) log π (a b) ρ Γ( + ρ) 2 where ( ) [ + l K(l, ρ) = l Γ 3 2 F 2 2 ( )( ) ρ(b ) ak + ρ(a ) ( ) k+l 2 l K(l, ρ) k l, k, l= + l l, l, 2 ; 3 2, 3 + l ] + ρ; + Γ 2 can be evaluated from the regularized hypergeometric function defined by 3 F (a) k (b) k (c) k z k 2 [a, b, c; d, e; z] =, z <. Γ(d + k) Γ(e + k) k!. Order Statistics k= ( ) [ 3 2 F 2 2 l, l, 2 ; 2, + ρ + l ] 2 ; The importance of order statistics and their applications is widely disseminated in the literature. As define by Balakrishnan, N. and Cohen, A. C. (24), the main objective of the order statistics is the investigation of properties and applications of ordered random variables, as well as functions of these variables. The density function f i:n () of the ith order statistic, say X i:n, based on a random sample X,..., X n, can be epressed as (for i =,..., n) f i:n () = f i:n () = B(i, n i + ) B(i, n i + ) j= n i j= ( ) j ( n i j ( ) j ( n i j ) f () F() i+ j. By inserting (5) and (6) in the above epression, the density function of the EGSHL order statistics follow as n i ) a b 2 a e a [( + e ) a 2 a e a b(i+ j) ] ( + e ) ab(i+ j)+. (26) There are many practical applications in which we can employ the above equation. Perhaps, the most important of these refers to the moments of X i:n. The r-th moment of X i:n comes from (26) as E(X r i:n ) = B(i, n i + ) n i ( ) n i ( ) j j j= r a b 2a e a [( + e ) a 2 a e a b(i+ j) ] d. (27) ( + e ) ab(i+ j)+ The r-th moment of X i:n can be easily obtained numerically using (27) through any symbolic computing platform. In Table, we present a small illustration, in which we calculate the first five moments of X i: for a = b = 2 and some values of r and i. All computations are performed using the Wolfram Mathematica platform. For a similar study, readers may see a paper by Barreto-Souza, W. et al. (2), who evaluated E(Xi:n r ) numerically for the Weibull-geometric distribution. Table 6. The first five moments of X i: for a = b = 2 and some values of r and i r i Finally, we provide a linear representation for f i:n (). After a simple algebraic manipulation, we can write f i:n () = B(i, n i + ) n i j= ξ i, j f [; a, (i + j)b], (28) where ξ i, j = [( ) j /(i + j)] ( ) n i j and f [; a, (i + j)b] is the EGSHL density with parameters a and (i + j)b. Equation (28) revels that the pdf of X i:n is a linear combination of EGSHL densities. So, the moments, incomplete moments and other quantities for the EGSHL order statistics can be determined from the above epression. 35

13 . Estimation and Inference 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 for constructing confidence intervals and also in test statistics. The normal approimation for these estimators in large sample distribution theory is easily handled either analytically or numerically. So, we consider the estimation of the unknown parameters for the EGSHL distribution from complete samples only by maimum likelihood. Let,..., n be observed values from this distribution with parameters a and b. The log-likelihood function for the vector of parameters θ = (a, b), say l(θ), can be epressed as l(θ) = n log(ab2 a ) a n i + (b ) n log[( + e i ) a 2 a e a i ] (ab + ) n log( + e i ). (29) Equation (29) can be maimized either directly by using the R (optim function), SAS (PROC NLMIXED) or O program (sub-routine MaBFGS) or by solving the nonlinear likelihood equations obtained by differentiating (29). The components of the score function are: l(θ) a l(θ) b n = i b + (b ) n log( + e i ) + n a [ + a log(2)] n 2 a i e a i 2 a log(2) e a i + ( + e i ) a log( + e i ) ( + e i )a 2 a e a, i = n n b a log( + e i ) + The negative elements of the observation matri J(θ) are given by 2 l(θ) a 2 = (b ) + (b ) (b ) n log[( + e i ) a 2 a e a i ]. n 2 a+ log(2) i e a i 2 a i 2 e a i 2 a [log(2)] 2 e a i ( + e i )a 2 a e a i n ( + e i ) a [log( + e i )] 2 ( + e i )a 2 a e a i n [2 a i e a i 2 a [log(2)] 2 e a i + ( + e i ) a log( + e i )] 2 [( + e i )a 2 a e a i ] 2 + n a {2 log(2) + a [log(2)] 2 } n a 2 [ + a log(2)] n a log(2) [ + a log(2)], 2 l(θ) = n b 2 b, 2 2 l(θ) n ab = log( + e i ) + n 2 a i e a i 2 a log(2) e a i + ( + e i ) a log( + e i ) ( + e i )a 2 a e a. i For large n, the distribution of (ˆθ θ) can be approimated to a bivariate normal distribution with zero means and variancecovariance matri J(θ). Some asymptotic properties of ˆθ can be based on this normal approimation. 2. Simulation Study In this section, we verify if the parameter estimates are obtained with precision since the inferences and the decision processes will depend directly on the quality of the estimates. In this contet, one of the most used simulation methods to evaluate the performance of estimators is by Monte Carlo simulation, see, for eample, the following works: Lemonte, A. J. (23), Cordeiro, G. M. and Lemonte, A. J. (24), Alshangiti, A. M., et al. (24), Silva, A. O., et al. (24), Jafari, A. A., et al. (24) and De Andrade, et al. (25). We investigate the behavior of the MLEs for the parameters of the EGSHL model by generating from (8) samples sizes n = 2, 4, 8, 2 with selected values for a and b and, replications. The simulation process is performed in the R software using the simulated-annealing (SANN) maimization method in the malik script. To ensure the reproducibility of the eperiment, we use the seed for the random number generator: set.seed (3). Initial kicks are taken as equal to half of the true values of the parameters in each scenario. 36

14 Table 7. MLEs for several a and b parameter values (variances in parentheses) n = 2 n = 4 n = 8 n = 2 a b â b â b â b â b (.29) (.86) (.533) (.68) (.234) (.244) (.52) (.56) (.86) (.979) (.369) (.3524) (.66) (.336) (.8) (.843) (.74) (3.278) (.35) (.75) (.42) (.373) (.93) (.236) (.67) (6.77) (.286) (2.58) (.3) (.7694) (.85) (.4777) (.535) (8.683) (.267) (3.7875) (.22) (.3586) (.8) (.84) (.465) (.8839) (.25) (5.649) (.7) (2.77) (.76) (.3332) (.56) (.85) (.23) (.68) (.938) (.244) (.66) (.56) (.3442) (.85) (.476) (.3523) (.663) (.336) (.433) (.844) (.2844) (3.276) (.259) (.63) (.569) (.37) (.372) (.237) (.2433) (5.6459) (.43) (2.459) (.59) (.7688) (.339) (.4774) (.285) (8.286) (.66) (3.759) (.488) (.3598) (.39) (.8398) (.784) (.7) (.2) (5.7568) (.467) (2.72) (.36) (.3345) (.64) (.86) (.4793) (.68) (.2) (.244) (.364) (.56) (.7739) (.783) (.3322) (.3523) (.493) (.336) (.973) (.843) (.645) (3.2347) (.2832) (.68) (.28) (.37) (.836) (.238) (.5486) (5.6934) (.2575) (2.6) (.68) (.7687) (.764) (.4779) (.478) (7.9364) (.2393) (3.7228) (.99) (.362) (.79) (.8397) (.46) (9.737) (.228) (5.4756) (.5) (2.678) (.687) (.3336) (2.64) (.85) (.852) (.67) (.3753) (.244) (.2424) (.56) (.375) (.782) (.597) (.3527) (.2654) (.336) (.729) (.843) (.352) (3.64) (.532) (.54) (.2275) (.37) (.486) (.236) (.9799) (5.7925) (.4573) (2.44) (.277) (.7686) (.359) (.478) (.8448) (8.26) (.4245) (3.74) (.953) (.362) (.278) (.8398) (.793) (9.926) (.3944) (5.4947) (.866) (2.674) (.22) (.337) The results of the simulations are presented in Tables 2 and 2, which contain the estimates and their estimated asymptotic variances in parentheses. These results reveal that the EGSHL estimates have desirable properties even for small to moderate sample sizes. In general, the biases and variances decrease as the sample size increases, as epected. 37

15 Table 8. MLEs for several a and b parameter values (variances in parentheses) n = 2 n = 4 n = 8 n = 2 a b â b â b â b â b (3.2234) (.85) (.337) (.67) (.5864) (.244) (.3787) (.56) (2.444) (.62) (.9228) (.3524) (.447) (.336) (.272) (.844) (.7733) (3.96) (.7866) (.63) (.3555) (.372) (.2323) (.237) (.5275) (5.778) (.749) (2.53) (.3245) (.7685) (.222) (.4777) (.3238) (8.323) (.6646) (3.792) (.35) (.3594) (.996) (.84) (.379) (.558) (.69) (5.559) (.295) (2.674) (.99) (.3327) (4.6354) (.85) (.966) (.67) (.8443) (.244) (.5458) (.56) (3.857) (.553) (.3287) (.3524) (.5973) (.336) (.3893) (.844) (2.533) (3.65) (.329) (.58) (.52) (.372) (.3342) (.236) (2.763) (5.5942) (.288) (2.487) (.4672) (.7686) (.355) (.4777) (.8878) (8.698) (.958) (3.762) (.4393) (.3597) (.2875) (.8395) (.628) (9.93) (.899) (5.75) (.497) (2.642) (.2749) (.3334) (6.2724) (.8) (2.687) (.68) (.489) (.244) (.7424) (.56) (4.999) (.63) (.874) (.3523) (.83) (.336) (.5296) (.844) (3.428) (3.3) (.543) (.65) (.6967) (.37) (.455) (.238) (2.9346) (5.5863) (.3999) (2.457) (.636) (.7687) (.459) (.4778) (2.533) (7.8692) (.343) (3.7473) (.5979) (.36) (.392) (.8395) (2.695) (9.6662) (.278) (5.622) (.575) (2.686) (.3742) (.333) (8.944) (.8) (3.479) (.68) (.53) (.244) (.9699) (.56) (5.4792) (.425) (2.3624) (.3523) (.68) (.336) (.699) (.844) (4.4672) (2.9786) (2.37) (.62) (.92) (.373) (.5945) (.236) (3.7735) (5.2886) (.8292) (2.457) (.834) (.7687) (.5433) (o.4778) (3.863) (7.355) (.722) (3.7422) (.789) (.3598) (.59) (.8396) (2.76) (9.39) (.5899) (5.689) (.7459) (2.677) (.4886) (.332) 3. Application to Real Data In this section, we present an application to a real data set. The objective is to prove empirically that the EGSHL distribution can be used in practical situations for real data modeling. We consider the set of data presented by Oliveira, 38

16 J. et al. (26) referring to the soil fertility influence and the characterization of the biologic fiation of N2 for the Dimorphandra wilsonii rizz growth. The phosphorus concentration, in the leaves, for 28 plants are:.22,.7,.,.,.5,.6,.5,.7,.2,.9,.23,.25,.23,.24,.2,.8,.,.2,.,.6,.2,.7,.2,.,.6,.9,.,.2,.2,.,.9,.7,.9,.2,.8,.26,.9,.7,.8,.2,.24,.9,.2,.22,.7,.8,.8,.6,.9,.22,.23,.22,.9,.27,.6,.28,.,.,.2,.2,.5,.8,.2,.9,.4,.7,.9,.5,.6,.,.6,.2,.25,.6,.3,.,.,.,.8,.22,.,.3,.2,.5,.2,.,.,.5,.,.5,.7,.4,.2,.8,.4,.8,.3,.2,.4,.9,.,.3,.9,.,.,.4,.7,.7,.9,.7,.8,.6,.9,.5,.7,.9,.7,.,.8,.5,.2,.6,.8,.,.6,.8,.2,.3. Figure 3 shows the dispersion of the data, while Table 3 provides some descriptive statistics. It is possible to observe, for eample, that the mean and the variance are.48 and.3, respectively. We adopt the maimum likelihood method to estimate the model parameters and all computations are performed using the NLMied subroutine in SAS. Dispersion of the 28 data units Figure. Dispersion of the 28 data units of the phosphorus concentration in the leaves Table 9. Descriptives statistics for phosphorus concentration in leaves data Statistic n 28 Mean.48 Median.3 Variance.3 Minimum.5 Maimum.28 In addition to the EGSHL model and its sub-models ESHLI and ESHLII, we consider the McDonald half-logistic (MCSHL) and Kumaraswamy half-logistic (KWSHL) models introduced by Oliveira, J. et al. (26). Table lists the MLEs of the model parameters (with the corresponding standard errors in parentheses) for all fitted models and also the values of the Akaike information criterion (AIC), Bayesian information criterion (BIC) and consistent Akaike information criterion (CAIC). In general, it is considered that lower values of these statistics indicate the better fit to the data. The figures in Table reveal that the EGSHL model has the lowest AIC, BIC and CAIC values among all fitted models. Thus, the proposed distribution is the best model to eplain these data. Finally, Figure 3 displays the histogram of the data and the estimated pdf and cdf of the EGSHL model. These plots reveal that the proposed model is quite suitable for these data. 4. Conclusions In this paper, we introduce a univariate continuous distribution with two parameters that govern the asymmetry and kurtosis, named the eponentiated generalized standard half-logistic model, say EGSHL. We provide a comprehensive mathematical treatment and show by numerical studies that the formulas related to the new model are computationally manageable. In particular, the maimum likelihood estimators are easily estimated. These estimators have desirable properties, such as low biases and variances, even in small or moderate sample sizes. A study using real data shows that the new distribution can be used in practical situations due to its great power of adjustment when compared to other competitive models. We hope that the proposed model can be useful for applied statisticians and other researchers who refer to a model with few parameters but fleible to accommodate supported data in real positives. For future research, we will study bias correction via bootstrap for estimators in small samples. 39

17 Table. MLEs (and their standard errors in parentheses), AIC, BIC and CAIC for phosphorus concentration in leaves data Distribution â b AIC BIC CAIC EGSHL (3.28) (.828) ESHLI ( ) (.3234) ESHLII (.27) ( ) â b ĉ MCSHL (2.78) (.682) (25.48) KWSHL (8.975) ( ) (.369) Density EGHL cdf EGHL (v) Estimated pdf of the EGSHL model (w) Estimated cdf of the EGSHL model Figure 2. Estimated pdf and cdf of the EGSHL model for phosphorus concentration in leaves data Acknowledgements We thank two anonymous referees and the associate editor for their valuable suggestions, which certainly contributed to the improvement of this paper. Additionally, Thiago A. N. de Andrade is grateful the financial support from CAPES (Brazil). References Alshangiti, A. M., Kayid, M., & Alarfaj, B. (24). A new family of MarshallCOlkin etended distributions. Journal of Computational and Applied Mathematics, 27, 369C Arya, G., & Elbata, I. (25). On the eponentiated generalized modified Weibull distribution. Communications for Statistical Applications and Methods, 22, 333C348. Balakrishnan, N., & Chan, P. (992). Estimation for the scaled half logistic distribution under type II censoring. Computional Statistics and Data Analysis, 3, 23C4. Balakrishnan, N., & Cohen, A. C. (24). Order Statistics and Inference: Estimation Methods. Academic Press. Balakrishnan, N., & Wong, K. H. T. (99). Approimate MLEs for the location and scale parameters of the halfclogistic distribution with typecii rightccensoring. IEEE Transactions on Reliability, 4, 4C45. Barreto-Souza, W., de Morais, A. L., & Cordeiro, G. M. (2). The WeibullCgeometric distribution. Journal of Statisti- 4

18 cal Computation and Simulation, 8, 645C Castellares, F., Santos, M. A. C.,..., & Cordeiro, G. M. (25). A gammacgenerated logistic distribution: properties and inference. American Journal of Mathematical and Management Sciences, 34, 4C39. Cordeiro, G. M., Alizadeh, M., & Marinho, P. R. D. (25). The type I halfclogistic family of distributions. Journal of Statistical Computation and Simulation, 86, 77C728. Cordeiro, G. M., Gomes, A. E., & da-silva, C. Q. (24). Another etended Burr III model: some properties and applications. Journal of Statistical Computation and Simulation, 84, 2524C Cordeiro, G. M., & Lemonte, A. J. (24). The eponentiated generalized Birnbaum-Saunders distribution. Applied Mathematics and Computation, 247, 762C Cordeiro, G. M., Ortega, E. M. M., & Cunha, D. C. C. (23). The eponentiated generalized class of distributions. Journal of Data Science,, C27. Da Silva, L. C. M.,..., & Cordeiro, G. M. (23). A new lifetime model: the gamma etended Frchet distribution. Journal of Statistical Theory and Applications, 2, 39C54. da Silva, R. V., Gomes-Silva, F.,..., & Cordeiro, G. M. (25). A new etended gamma generalized model. International Journal of Pure and Applied Mathematics,, 39C De Andrade, T. A. N., Bourguignon, M., & Cordeiro, G. M. (26). The eponentiated generalized etended eponential distribution. Journal of Data Science, 4, 393C44. De Andrade, T. A. N., Rodrigues, H.,..., & Cordeiro, G. M. (25). The eponentiated generalized Gumbel distribution. Revista Colombiana de Estadstica, 38, 23C43. Elbatal, I., & Muhammed, H. Z. (24). Eponentiated generalized inverse Weibull distribution. Applied Mathematical Sciences, 8, 3997C42. Gupta, R. D., & Kundu, D. (2). Eponentiated eponential family: an alternative to gamma and Weibull distributions. Biometrical Journal, 43, 7C3. Jafari, A. A., & Mahmoudi, E. (25). Beta-linear failure rate distribution and its applications. Journal of The Iranian Statistical Society, 4, 89C5. Jafari, A. A., Tahmaseb, S., & Alizadeh, M. (24). The betacgompertz distribution. Revista Colombiana de Estadstica, 37, 39C56. Kenney, J. F., & Keeping, E. S. (962). Mathematics of Statistics. 3rd ed., Part, New Jersey. Lee, E. T. (992). Statistical Methods For Survival Data Analysis. Wiley and Sons, Inc. Lemonte, A. J. (23). A new eponential-type distribution with constant, decreasing, increasing, upside-down bathtub and bathtub-shaped failure rate function. Computational Statistics and Data Analysis, 62, 49C7. Mansoor, M., Tahir, M. H., Alzaatreh, A.,..., & Ghazali, S. S. A. (26). An etended Frchet distribution: Properties and applications. Journal of Data Science, 4, 67C88. Moors, J. J. A. (988). A quantile alternative for kurtosis. Journal of the Royal Statistical Society. Series D (The Statistician), 37, 25C32. Nadarajah, S., Bakouch, H. S., & Tahmasbi, R. (2). A generalized Lindley distribution. Sankhya B, 73, 33C Oguntunde, P. E., Adejumo, A. O., & Balogun, O. S. (24). Statistical properties of the eponentiated generalized inverted eponential distribution. Applied Mathematics, 4, 47C55. 8 Oliveira, J., Santos, J., Xavier, C.,..., & Cordeiro, G. M. (26). The McDonald half-logistic distribution: Theory and practice. Communications in Statistics - Theory and Methods, 45, 25C Panichkitkosolkul, W., & Saothayanun. (22). Bootstrap confidence intervals for the process capability inde under halfclogistic distribution. Maejo International Journal of Science and Technology, 6, 272C28. Sarhan, A. M., & Kundu, D. (29). Generalized linear failure rate distribution. Communications in Statistics - Theory 4

19 and Methods, 38, 642C66. Sharma, V. K., Singh, S. K., & Singh, U. (24). A new upside-down bathtub shaped hazard rate model for survival data analysis. Applied Mathematics and Computation, 239, 242C Silva, A. O., da Silva, L. C. M., & Cordeiro, G. M. (25). The etended Dagum distribution: Properties and application. Journal of Data Science, 3, 53C72. Tahir, M. H., & Nadarajah, S. (25). Parameter induction in continuous univariate distributions: Well-established G families. Anais da Academia Brasileira de Cincias, C Copyrights Copyright for this article is retained by the author(s), with first publication rights granted to the journal. This is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license ( 42