A new class of quantile functions useful in reliability analysis

Journal of Statistical Theory and Practice ISSN: 1559-8608 (Print) 1559-8616 (Online) Journal homepage: http://www.tandfonline.com/loi/ujsp20 A new class of quantile functions useful in reliability analysis P. G. Sankaran & Dileep Kumar To cite this article: P. G. Sankaran & Dileep Kumar (2018): A new class of quantile functions useful in reliability analysis, Journal of Statistical Theory and Practice, DOI: 10.1080/15598608.2018.1448732 To link to this article: https://doi.org/10.1080/15598608.2018.1448732 Accepted author version posted online: 08 Mar 2018. Submit your article to this journal View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=ujsp20 A New Class of Quantile Functions Useful in Reliability Analysis P. G. SANKARAN AND DILEEP KUMAR M Department of Statistics, Cochin University of Science and Technology, Cochin 682022, Kerala, India. us c rip t Abstract The present paper introduces a new flexible family of distributions, defined by means of a quantile function. The quantile function proposed is the sum of quantile functions of the half logistic and exponential geometric distributions. Various distributional properties and reliability characteristics are discussed. The estimation of the parameters of the model using L-moments is studied. The model is applied to a real life dataset. Keywords: Exponential geometric distribution, Half logistic distribution, Hazard quantile function, L-moments Quantile density function, Quantile function. an 1. Introduction M A probability distribution can be specified either in terms of its distribution function or by the quantile function. Although both convey the same information about the distribution with different interpretations, the concepts and methodologies based on distribution functions are more popular in most forms of theory and practice. For a non-negative random variable X with distribution function F ( x) , the quantile function Q(u ) is defined by, Q ( u ) = F −1 ( x ) = inf{x : F ( x ) ≥ u}, 0 ≤ u ≤ 1. (1.1) Ac ce pt ed The derivative of Q(u ) is the quantile density function denoted by q(u ) . If F ( x) is right continuous and strictly increasing we have, F (Q(u )) = u, (1.2) so that F ( x) = u implies x = Q(u ) . When f ( x ) is the probability density function(p.d.f.) of X , we have from (1.2) q(u ) f (Q(u )) = 1. (1.3) Quantile functions have several properties that are not shared by distribution functions. For example, the sum of two quantile functions is again a quantile function. Further, the product of two positive quantile functions is again a quantile function in the non-negative setup. There are explicit general distribution forms for the quantile function of order statistics. It is easier to generate random numbers from the quantile function. A major development in portraying quantile functions to model statistical data is given by Hastings et al. [7], who introduced a family of distributions by a quantile function. This was refined later by Tukey [23] future to form a symmetric distribution, called Tukey lambda distribution. Email address: [email protected] (P. G. SANKARAN AND DILEEP KUMAR M) Preprint submitted to Elsevier 1 UJSP_A_1448732 −1   , β > 0.  an x  G ( x) = 2 1 + e β   us c rip t This model was generalized in different ways referred as lambda distributions. These include various forms of quantile functions discussed in Ramberg and Schmeiser [19], Ramberg [17], Ramberg et al. [18], and Freimer et al. [3]. Govindarajulu [5] introduced a new quantile function by taking the weighted sum of quantile functions of two power distributions. Hankin and Lee [6] new presented power-Pareto distribution by taking the product of power and Pareto quantile functions. Van Staden and Loots [24] developed a four parameter distribution, using weighted sum of generalized Pareto and its reflection quantile functions. Sankaran et al. [20] developed a new quantile function based on the sum of quantile functions of generalized Pareto and Weibull quantile functions. The density and distribution functions for these models are not available in closed forms except for certain special cases. The great advantage of these models is that the simple forms of the quantile functions make it extremely straightforward to simulate random values, which is useful in inference problems. The aim of the present work is to introduce a new quantile function which is useful in reliability analysis. The proposed quantile function is derived by taking sum of quantile functions of half logistic and exponential geometric distributions. Balakrishnan [2] considered the folded form of the standard logistic distribution and termed it as the half logistic distribution. The survival function and quantile function of this distribution are respectively given by, and (1.4) ed M  1+ u  (1.5) Q1 (u ) = β log   , β > 0.  1− u  The model (1.4) is a possible life-time model, which has several recurrence relations for the single and the product moments of order statistics. Adamidis and Loukas [1] introduced the exponential geometric(EG) distribution with applications to reliability modelling in the context of decreasing failure rate data. The survival function and quantile function of the EG distribution are given by, pt F ( x) = 1 − F ( x) = (1 − p)e and 1 − x α (1 − pe 1 − x α )−1 , α > 0and 0 < p < 1. (1.6) Ac ce  1 − pu  (1.7) Q2 (u ) = α log   , α > 0and 0 < p < 1.  1− u  We now propose a new class of distributions defined by a quantile function, which is the sum of quantile functions of half logistic and exponential geometric distributions. The proposed class gives a wide variety of distributional shapes for various choices of the parameters. The rest of the article is organized as follows. In Section 2 we present a family of distributions and study its basic properties. Section 3 presents some well known distributions which are either a member of the proposed class of distributions or obtained by applying some suitable transformations on the proposed quantile function. The distributional properties such as measures of location and scale, L moments,etc., are given in Section 4. In Section 5, we present various reliability characteristics of the class. Section 6 focuses on the inference procedures. We then provide application of this class of distributions in a real life situation. Finally, Section 7 provides major conclusions of the study. 2 UJSP_A_1448732 2. Half logistic - exponential geometric (HLEG) quantile function Let X and Y be two non-negative random variables with distribution functions F ( x) and G( x) with quantile functions Q1 (u ) and Q2 (u ) respectively. Then us c rip t Q(u ) = Q1 (u ) + Q2 (u ), (2.1) is also a quantile function with quantile density function satisfying (1 − u )q(u ) = (1 − u )q1 (u ) + (1 − u )q2 (u ). (2.2) We now introduce a class of distributions given by the quantile function  1 − pu   u +1  (2.3) Q (u ) = α log   + β log   , 0 ≤ p ≤ 1, α ≥ 0, β ≥ 0.  1− u   1− u  Thus Q(u ) is the sum of (1.5) and (1.7). The support of the proposed class of distributions (2.3) is (0, ∞) . The quantile density function is obtained as, 2 β + α ((1 − p ))(u + 1) − 2 β pu (2.4) q (u ) = . ( u 2 − 1) ( pu − 1) an The quantile function (2.3) represents a family of distributions with neither the density nor the distribution function is available in closed form. However, these can be calculated by numerical inversion of the quantile function. For the proposed class of distributions, the density function f ( x) can be written in terms of the distribution function as, ed M (1 − pF ( x))(1 − ( F ( x) 2 ) (2.5) . f ( x) = α (1 − p )(1 + F ( x)) + 2(1 − pF ( x)) β For all values of the parameters, the density is strictly decreasing in x and it tends to zero as x → ∞ . Plots of the density function for different combinations of parameters are shown in Figure1. 1 . 2β + α (1 − p) pt The mode of the distribution is at zero and the modal value is 3. Members of the family Ac ce The proposed family of distributions (2.3) includes several well known distributions for various values of the parameters. We can derive some well known distributions from the proposed model by making use of various transformations described in Gilchrist [4]. Case 1. β = 0 , p = 0 and α > 0 . The quantile function of the proposed class of distributions reduces to the quantile function, Q(u ) = α (− log(1 − u )), (3.1) which is the exponential distribution with mean α . We can apply the power transformation of the form T ( x) = x K on (3.1) to form the Weibull distribution with parameters α and K . Case 2. α = β and p = 1 . The quantile function of the proposed class of distributions becomes,  1+ u  (3.2) Q (u ) = α log  ,  1− u  3 UJSP_A_1448732 us c rip t which belongs to the class of distributions with linear hazard quantile functions defined by Midhu et al. [10], with quantile function 1 1+θu  (3.3) Q (u ) = log  , a (1 + θ )  1− u  1 . with θ = 1 and a = 2α Case 3. β = 0, α > 0and 0 < p < 1 . The quantile function of the proposed class of distributions reduces to the quantile function,  1 − pu  (3.4) Q (u ) = α log  ,  1− u  this also belongs to the class of distributions (3.3), with parameters 1 θ = − p, (−1 < θ < 0) and a = . α (1 − p) p = 0, α > 0and β > 0 . The quantile function of the proposed class of distributions is obtained as, ( A − B) log(1 + Au ) − A( B + 1) log(1 − u ) (3.5) Q(u ) = , A( A + 1) K 1 α , A = 1 and B = . The quantile function (3.5) where K = α + 2β α + 2β corresponds to the family of distributions with bi-linear hazard quantile function, given in Sankaran et al. [21]. In the construction of our family, the sum of two quantile functions are involved. In the following theorems, we derive the random variable associated with the proposed quantile function (2.3).  Z   (1 + p ) + (1 − p)exp    β  Theorem 3.1 If Z  HL( β ) , then the random variable X = Z + α log    2     has HLEG(α , β , p) distribution. Proof. Consider two random variables S and T with quantile functions QS (u ) and QT (u ) and distribution functions FS ( x) and FT ( x) respectively. Ac ce pt ed M an Case 4. Now suppose, Q* (u ) is defined by, Q* (u ) = QS (u ) + QT (u ). Then the random variable corresponds to the quantile function Q* (u ) is S + QT ( FS ( S )) or T + QS ( FT (T )) ( Sankaran et al. [20]). Now take Y  EG(α , p) and Z  HL( β ) , then we have Z + QY ( FZ ( Z )) has HLEG(α , β , p) distribution. 4 UJSP_A_1448732 −1 t (3.6) rip   Z   1 − pu  Since QY (u ) = α log   and FZ ( Z ) = 1 − 2 1 + exp    , we get,  1− u   β     Z   (1 + p ) + (1 − p)exp     β  , Z + QY ( FZ ( Z )) = Z + α log    2     which completes the proof. □ us c  p − 2exp ( x / α ) + 1  Theorem 3.2 If Y  EG(α , p) , then the random variable X = Y + β log   p −1   has HLEG(α , β , p) distribution. Proof. The proof is similar to that of Theorem 3.1, and therefore the details are omitted. 4. Distributional characteristics □ M an The quantile based measures of the distributional characteristics of location, dispersion, skewness and kurtosis are popular in statistical analysis. These measures are also useful for estimating parameters of the model by matching population characteristics with corresponding sample characteristics. For the model (2.3), basic descriptive measures such as median ( M ), inter-quartile-range(IQR), Galton’s coefficient of skewness(S) and Moor’s coefficient of kurtosis(T) are obtained as; M = α log(2 − p) + β log(3), (4.1)  12 − 9 p   21  IQR = α log   + β log   ,  5  4− p  ed 1.43β − 2(1.09β + α log(2.(1 − 0.5 p))) − α log(1.33(1 − 0.25 p)) + α log(4.(1 − 0.75 p)) , 1.43β − α log(1.33(1 − 0.25 p)) + α log(4.(1 − 0.75 p)) pt S= (4.2) (4.3) Ac ce α (−0.69 log(1.14 − 0.14 p ) + 0.7 log(1.6− 0.6 p) − 0.7 log(2.67 − 1.7 p ) + 0.7 log(8 − 7 p)) + 1.24 β . −0.7α log(1.34− 0.34 p ) + 0.7α log(4− 3 p ) + β (4.4) The L-moments are often found to be more desirable than the conventional moments in describing the characteristics of the distributions as well as for inference. A unified theory and a systematic study on L-moments have been presented by Hosking [8]. The L-moments have generally lower sampling variances and robust against outliers. See Hosking [8] and Sankarasubramanian and Srinivasan [22] for details. The rth L moment is given by, 1 r −1  r − 1  r − 1 + k  k (4.5) Lr =  (−1) r −1− k    u Q(u )du.  k  k  0 k =0 For the model (2.3), first four L moments are obtained as follows; T= 5 UJSP_A_1448732 L1 = β log(4) + α ( p − 1) log(1 − p) p L2 = α + 2β − β log(4) + . (4.6) α ( p − 1) log(1 − p) α − . p2 p α ( p − 2)( p − 1) log(1 − p) 2α ( p − 1) L3 = −4 β + β log(64) − . + p3 p2 (4.8) p ( 4β p 3 (23 − 33log(2)) + α ( p − 1)(( p − 15) p + 30) ) + 6α ( p − 1)(( p − 5) p + 5) log(1 − p) t L4 = (4.7) rip . (4.9) 6 p4 For the model (2.3), L-coefficient of variation ( τ 2 ), L-coefficient of skewness( τ 3 ) and L- (4.10) (4.11) M an us c coefficient of kurtosis ( τ 4 ) have the following expressions; α ( p − 1) log(1 − p) α − α + 2 β − β log(4) + L2 p2 p τ2 = = , α ( p − 1) log(1 − p) L1 β log(4) + p α ( p − 2)( p − 1) log(1 − p ) 2α ( p − 1) −4 β + β log(64) − + L3 p3 p2 , τ3 = = α ( p − 1) log(1 − p ) α L2 α + 2β − β log(4) + − p2 p ed 3 L4 p ( 4 β p (23 − 33log(2)) + α ( p − 1)(( p − 15) p + 30) ) + 6α ( p − 1)(( p − 5) p + 5) log(1 − p ) . τ4 = = 6 p 2 ( p (α ( p − 1) − β p (log(4) − 2)) + α ( p − 1) log(1 − p )) L2 (4.12) Figures 2, 3 and 4 present skewness( τ 3 ) and kurtosis( τ 4 ) measures for various parameter ce pt values. We can show that τ 3 lies in (0.25,1) and τ 4 lies in (0.12,0.67) using numerical optimization techniques. Thus the proposed class of distributions (2.3) consists only positively skewed distributions. The curves of τ 3 and τ 4 increase with α for fixed β and p , decrease with β for fixed α and p , and first increase and then decreases with p for fixed α and β . 4.1. Order statistics Ac If X r:n is the r th order statistic in a random sample of size n , then the density function of X r:n can be written as, 1 f r ( x) = f ( x) F r −1 ( x)(1 − F ( x)) n − r . B(r , n − r + 1) From (2.5), we have 1 (1 − F ( x)) n − r (1 − pF ( x))(1 − ( F ( x) 2 )( F ( x)) r −1 . f r ( x) = B(r , n − r + 1) α (1 − p )(1 + F ( x)) + 2(1 − pF ( x)) β Hence, 6 UJSP_A_1448732 n−r ∞ (1 − F ( x )) 1 (1 − pF ( x))(1 − ( F ( x) 2 )( F ( x)) r −1 x dx. α (1 − p )(1 + F ( x)) + 2(1 − pF ( x)) β B (r , n − r + 1) 0 In quantile terms, we have 1 1 (1 − u )n − r (1 − pu )(1 − u 2 )u r −1 E ( X r:n ) = Q u dx. ( ) B(r , n − r + 1) 0 α (1 − p)(1 + u ) + 2(1 − p + u ) β For the class of distributions (2.3), the first order statistic X 1:n has the quantile function, E ( X r:n ) = 1 = α log ( p − ( p − 1)(1 − u ) −1/ n ) + β log ( 2(1 − u ) −1/ n − 1) , and nth order statistic X n:n has the quantile function, an  1 − pu1/ n   u1/ n + 1  + = α log  log . β  1/ n  1/ n   1− u   1− u  us c 1 Q( n ) (u) = Q(u n ) rip t Q(1) (u ) = Q(1 − (1 − u ) n ) 5. Reliability properties M One of the basic concepts employed for modeling and analysis of lifetime data is the hazard rate. In quantile setup, Nair and Sankaran [11] defined hazard quantile function; which is equivalent to the hazard rate. The hazard quantile function H (u) is defined as ed H (u )= h(Q(u )) = [(1 − u )q(u )]−1. (5.1) Thus H (u) can be interpreted as the conditional probability of failure of a unit in the next small interval of time given the survival of the unit until 100(1 − u ) % point of the distribution. Note that H (u) uniquely determines the distribution using the identity u dp (5.2) . (1 − p ) H ( p ) 0 Since the proposed class of distributions is the sum of quantile functions of exponential geometric and half logistic quantile functions, (5.1) and (2.2) give, 1 1 1 (5.3) + = , H (u ) H1 (u ) H 2 (u ) where H (u ), H1 (u ) and H 2 (u ) are the hazard quantile functions of the proposed class of distributions, exponential geometric and half logistic quantile functions respectively. From (5.3), the proposed class of distributions (2.3) has hazard quantile function proportional to the harmonic average of the hazard quantile functions of exponential geometric and half logistic quantile functions. For the class of distributions (2.3), we have (u + 1)( pu − 1) . H (u ) = (5.4) α ( p − 1)(u + 1) + 2β ( pu − 1) The shape of the hazard function is determined by the derivative of H (u ) , which is obtained as, Ac ce pt Q (u ) =  α p( p − 1)(u + 1) 2 + 2 β ( pu − 1) 2 H (u ) = . (α ( p − 1)(u + 1) + 2 β ( pu − 1)) 2 ' 7 (5.5) UJSP_A_1448732 rip t Since (α ( p − 1)(u + 1) + 2β ( pu − 1)) 2 > 0 for all values of the parameters, the sign of H ′(u ) depends only on, K (u ) = α p( p − 1)(u + 1)2 + 2β ( pu − 1)2 . (5.6) The hazard quantile function accommodates increasing, decreasing, linear and upside- down bathtub shapes for different choices of parameters. Plots of hazard quantile function for different values of parameters is given in Figure 5. Now we consider the following cases. Case 1. p = 0, α > 0 and β > 0 . K (u ) = 2β . The first term in K (u ) is zero and the second term is positive, so that K (u ) > 0 for all 0 < u < 1 and the distribution have increasing hazard rate (IHR). Case 2. p = 1, α > 0 and β > 0 . an α us c K (u ) = 2β (u − 1) 2 . The first term in K (u ) is zero and the second term is positive, so that K (u ) > 0 for all 0 < u < 1 and the distribution have increasing hazard rate (IHR). Case 3. p = 0 , β = 0 and α > 0 . 1 H (u ) = , aconstant. M Thus the distribution is exponential. 2α p Case 4. 0 < p < 1, α > 0 and β > . (1 − p ) Now X is IHR if and only if K (u ) > 0 for all u ∈ (0,1) . This holds if and only if, p( p − 1)α (1 + u )2 > −2β ( pu − 1)2 , (5.7) ed which gives Ac ce pt 2β (1 + u ) 2 (5.8) > . α p(1 − p) ( pu − 1) 2 Since (1 + u ) 2 > ( pu − 1) 2 ,forall 0 < u < 1and 0 < p < 1 , we have the right side of (5.8) is increasing in u and attains its maximum when u = 1 . Now for u = 1 , the 2α p , thus it is clear that H (u) is increasing in inequality (5.8) reduces to β > (1 − p ) this case. α p(1 − p) Case 5. 0 < p < 1, α > 0 and 0 < β < . 2 Similar to the Case 4, we can show that H (u) have decreasing hazard rate (DHR) if and only if, (5.9) p( p − 1)α (1 + u )2 < −2 β ( pu − 1) 2 or 2β (1 + u ) 2 (5.10) < . α p(1 − p) ( pu − 1) 2 Since right side of (5.10) is increasing in u and attains its minimum when u = 0 , 8 UJSP_A_1448732 the inequality (5.8) reduces to β < α p(1 − p rip t . Thus the distribution is DHR. 2 α p(1 − p 2α p <β < . Case 6. 0 < p < 1, α > 0 and 2 1− p First term of K (u ) is negative and second term is positive, so that K (u ) attains a zero in this case. This, in turn, gives H ′(u ) = 0 suggesting the possibility for H (u ) to be non-monotone. Let u0 be the solution of the equation K (u ) = 0 . From (5.6), we have u0 is the solution corresponding to the quadratic equation, u 2 (α p( p − 1) + 2β p 2 ) + u (2α p( p − 1) − 4 pβ ) + (α p( p − 1) + 2 β ) = 0, which provides, −α p 2 − 2 −αβ p 4 − αβ p 3 + αβ p 2 + αβ p + α p + 2β p . α p 2 + 2β p 2 − α p For further analysis, we note that the second derivative of H (u ) as us c u0 = 4αβ (1 − p)( p + 1) 2 . (α ( p − 1)(u + 1) + 2β ( pu − 1))3 For the change point u 0 obtained in (5.12), we get, H ′′(u0 ) = − an H ′′(u ) = 2 p2 . (5.11) (5.12) (5.13) (5.14) M αβ (1 − p ) p ( p + 1) 2 Since H ′′(u0 ) < 0 , we have H (u) attains a maximum at u0 . Hence X has an pt ed upside-down bathtub-shaped hazard quantile function (see Nair et al. [12]). The ageing pattern of H (u ) for various parameter values are summarized in Table 1. We can easily show the following lemma, which is useful for finding bounds of H (u) . Lemma 5.1 The limits of HLEG(α , β , p) hazard quantile function is given by, lim H (u ) = (5.15) ce u →0 1 1 and lim H (u ) = , u →1 α (1 − p) + 2β α +β where α > 0, β > 0 and 0 < p < 1 . Proof. It is straight forward to show the results of (45) by taking the limit of HLEG(α , β , p) hazard quantile function (5.4). Ac □ Theorem 5.1 If X  HLEG(α , β ,1) , then the two limits of hazard quantile function are independent of the parameter α as given below; (i) limu →1 H (u ) = 2 limu →0 H (u ) and 1 1 (ii ) < H (u )< , for all 0 < u < 1 and β > 0 . 2β β Proof. (i) The proof is direct once we note that, 9 UJSP_A_1448732 1 1 and limH (u ) = . (5.16) u →0 u →1 β 2β (ii) From Table 1, H (u ) is IHR for p = 1,α > 0 and β > 0 . Thus lower and upper bounds for H (u ) exist when u approaches to 0 and 1 respectively. Now from (5.16), we get 1 1 < H (u )< for all0 < u < 1and β > 0. β 2β This completes the proof. □ Theorem 5.2 If X  HLEG (α , β , 0) . Then the bounds of H (u ) are given by, 1 1 < H (u )< , for all 0 < u < 1and β > 0. α + 2β α +β us c rip t limH (u ) = (5.17) an Proof. The proof is similar to that of Theorem 5.1 once we note that, 1 1 and lim H (u ) = , lim H (u ) = u →0 u →1 α + 2β α +β and, H (u) is increasing for p = 0, α > 0 and β > 0 . □ ed M Theorem 5.3 Let X  HLEG(α , β , p) . Then the hazard quantile function satisfies following; 2α p 1 1 (i ) If β > then < H (u )< α (1 − p ) + 2β (1 − p) α +β α p(1 − p) 1 1 (ii) If 0 < β < then < H (u )< α +β 2 α (1 − p) + 2β Proof. pt 2α p . Now from Lemma 6.1, we get, (1 − p ) 1 1 < H (u )< . (5.18) α (1 − p) + 2β α +β α p(1 − p) . Since H (u ) is decreasing over To prove (ii), note that X is DHR for 0 < β < 2 u , boundary values are reversed. This completes the proof. □ Ac ce From Table 1, note that X is IHR when β > Mean residual function is a well known measure, which has been widely used for modelling lifetime data in reliability and survival analysis. For a non negative random variable X , the mean residual life function is defined as, ∞ 1 (5.19) m( x ) = E ( X − x | X > x ) = (1 − F (t ))dt. 1 − F ( x) x The mean residual quantile function, which is the quantile version of the mean residual 10 UJSP_A_1448732 function (5.19), defined by Nair and Sankaran [11] has the expression, 1 1 M (u ) = (Q( p ) − Q(u ))dp. 1 − u u For the class of distributions (2.3), M (u ) has the form (5.20)  p −1    pu − 1  α ( p − 1) log  − 2β log(u + 1) p M (u )= . (5.21) 1− u It is well known that increasing(decreasing) failure rate implies decreasing(increasing) mean residual life(See Lai and Xie [9]). The ageing behaviour of the class of distributions (2.3) based on mean residual quantile function can be defined from Table1. There exists closed form expressions of the hazard quantile function and mean residual quantile function defined in reverse time (see Nair and Sankaran [11]) for the proposed class of distributions (2.3). The total time on test transform(TTT) is a widely accepted statistical tool, which has many applications in reliability analysis( see Lai and Xie [9]). The quantile based TTT introduced in Nair et al. [14] has the form, us c rip t β log(4) + u 0 an T (u )=  (1 − p ) q ( p)dp. (5.22) pt ed M For the class of distributions (10), we obtain T (u ) as α ( p − 1) log(1 − pu ) T (u )= + 2β log(u + 1). (5.23) p In a fundamental paper on exploratory data analysis using quantile functions, Parzen [16] has introduced the score function defined as, q ' (u ) J (u ) = 2 , (5.24) q (u ) where q ' (u ) is the derivative of q(u ) . Nair et al. [13] studied properties of J (u ) in the context of lifetime data analysis. For the class of distributions (2.3), J (u ) is obtained as, q ' (u ) α ( p − 1)(u + 1) 2 ( p (2u − 1) − 1) + 4β u ( pu − 1) 2 (5.25) = . (α ( p − 1)(u + 1) + 2 β ( pu − 1)) 2 q 2 (u ) It is customary to characterize life distributions by the relationships among reliability concepts. In the same spirit, we prove the following characterization theorem. Theorem 5.4 A non negative continuous random variable X follows; (a) HLEG(u; α , 0, p) if and only if any one of the following properties hold. (i) H (u ) = A1 − A2 u , 0 < A2 < A1 (ii) J (u ) = H (u ) + C (1 − u ), C > 0 Ac ce J (u ) = (iii) T (u ) =  H (u )  −1 log   A2  A1  and (b) HLEG(u;0, β , p) if and only if any one of the following properties hold. (i) H (u ) = K (1 + u ) , K > 0 11 UJSP_A_1448732 rip  A − A2 u  log  1  A1 (1 − u )   , Q (u ) = A1 − A2 t (ii) J (u ) = 2 Ku 1 (iii) T (u ) = log( K K H (u )) K Proof. We prove the result for (a). The proof for (b) is similar. (a) Suppose the identity (i) is true. Then H (u ) = A1 − A2 u , so that the corresponding quantile function is obtained as, (5.26) 1 A > 0 and 0 < p = 2 < 1 . The A1 − A2 A1 converse part is direct from the definition of H (u ) given in Section 6. When (ii) is true, we have from Nair and Sankaran [11], (1 − u ) H ′(u ) = H (u ) − J (u ). (5.27) Thus we obtain, (1 − u ) H ′(u ) = C (u − 1). (5.28) The solution of the ordinary differential equation (5.28) is, H (u ) = D − Cu, C > 0, D − C > 0, (5.29) which satisfies (i), so that proof is completed. Suppose (iii) is true. Differentiating (iii) with respect to u we get, − H ′(u ) (5.30) T ′(u ) = . A2 H (u ) Differentiating (5.22) with respect to u we get 1 T ′(u ) = (1 − u )q(u ) = . (5.31) H (u ) From (5.30) and (5.31), we get H ′(u ) = − A2 , (5.32) which leads to (i). Conversely for the class of distributions HLEG(u; α , 0, p) we obtain, α ( p − 1) log(1 − pu ) T (u ) = , (5.33) p or  H (u )  −1 log  T (u ) = (5.34) , A2  A1  Ac ce pt ed M an us c which is equivalent to HLEG(u; α , 0, p) , with α = 1 p and A2 = . α (1 − p ) α (1 − p) This completes the proof. where A1 = □ 6. Inference and applications There are different methods for the estimation of parameters of the quantile function. The method of percentiles, method of L-moments, method of minimum absolute deviation, method of 12 UJSP_A_1448732 least squares and method of maximum likelihood are commonly used techniques. To estimate the parameters of (2.3), we use the method of L-moments. We equate sample L moments to corresponding population L moments. Let X 1 , X 2 ,..., X n be a random sample of size n from the population with quantile function (2.3), then the sample L-moments are given by 1 n l1 =   x(i )  n  i =1 where x( i )  1  n l3 =      3  3 is the ith order statistic. −1   i − 1  n − i   −   x(i ) i =1   1   1   n t    i − 1  i − 1 n − i   n − i    − 2  +   x(i ) i =1   2   1  1   2   n  us c l2 −1 rip  1  n =     2  2 For estimating the parameters α , β , and σ , we equate first three sample L moments to population L-moments given in Section 4. The parameters are obtained by solving the equations, lr = Lr ; r = 1, 2,3. (6.1) an Since L1 is the mean of the distribution, mean survival time is estimated as l1. Similarly estimate 1 of variance is obtained as Vˆ ( x) =  (Qˆ (u ))2 du − l12 , which can be evaluated with the help of 0 M numerical integration techniques. Hosking [8] has shown that the L-moment estimates are asymptotically normal and consistent. Specifically, Hosking [8] has shown that n (lr − Lr ) , r = 1, 2,...., m , converges to the multivariate normal distribution N (0, Δ) , where the elements Δ r , s of Δ are given by,  {Pr*−1 (u ) Ps*−1 (v) + Ps*−1 (u ) Pr*−1 (v)}u (1 − v)q (u )q(v)dudv, ed Λ r ,s = (6.2) 0< u < v <1 Ac ce pt  r  r + k  k r where Pr* ( x) =  k =0(−1)r − k    x . Since the set of equations (6.1) are non-linear in α ,  k  k  β and p, asymptotic variances of the L-moment estimates are difficult to compute. One can use the bootstrap method to obtain the asymptotic variance of the estimates. To illustrate the application of the proposed class of distributions we consider a real data set reported in Zimmer et al. [25]. The data consist of time to first failure of 20 electric carts. We estimate the parameters using the method of L-moments. The sample L-moments are obtained as, l1 = 12.66 l2 = 5.91 and l3 = 1.57. (6.3) We then equate these values to the corresponding population L-moments given in (4.6), (4.7) and (4.8), so that we have three non-linear equations. The Newton-Raphson method is used to find the solutions of these equations. Least square method of estimation for quantile functions given in Öztürk and Dale [15] east was employed for fixing the initial estimates for the NewtonRaphson iterative procedure. The estimates of the parameters are obtained as, (6.4) αˆ = 8.518 βˆ = 1.209 and pˆ = 0.329. To examine the adequacy of the model, two goodness-of-fit techniques are employed. The first 13 UJSP_A_1448732 one is the Q-Q plot, which is given in Figure 6. The Q-Q plot shows that the proposed model is appropriate for the given data set. We also carry out the chi-square goodness of fit test. The chi-square test statistic value is 0.210, giving p-value 0.647 with one degree of freedom. This also indicates the adequacy of proposed model for the given data set. We compute the estimate of H (u) by substituting the parameter values (6.4) in (5.4), which is given in Figure 7. Note that the estimate Hˆ (u ) is decreasing in u , which is t consistent with our claim in Table 1. rip 7. Conclusion M an us c In this paper, we have introduced a class of distributions (2.3), which is the sum of quantile functions of the half logistic and exponential geometric quantile functions. Various reliability properties are studied. We have identified several well-known distributions which are members of the proposed class of distributions. The estimation of the parameters of the model using Lmoments was studied and discussed the estimation procedure with the aid of a real dataset. The proposed model has several advantages over the existing quantile function models. The analysis of hazard quantile function over the whole parameter space can be done without using numerical methods. The model is useful for fitting different types of lifetime data due to the flexible behaviour of hazard quantile function. Unlike generalized lambda distribution and generalized Tukey lambda distribution, the estimation of parameters does not involve any computational difficulties. There are several properties and extensions of the new family of distributions not considered in this paper, such as stochastic orderings and quantile based cumulative residual entropy. Estimation using maximum likelihood method and Bayes technique need numerical approximations. The study on multivariate generalizations of the HLEG distribution is interesting, which will be addressed later. ed Acknowledgement pt We thank the referee and the editor for their constructive comments. The second author is thankful to Kerala State Council for Science Technology and Environment(KSCSTE) for the financial support. ce [1] Adamidis, K., Loukas, S., 1998. A lifetime distribution with decreasing failure rate. Statistics & Probability Letters 39 (1), 35–42. Ac [2] Balakrishnan, N., 1985. Order statistics from the half logistic distribution. Journal of Statistical Computation and Simulation 20 (4), 287–309. [3] Freimer, M., Kollia, G., Mudholkar, G. S., Lin, C. T., 1988. A study of the generalized Tukey lambda family. Communications in Statistics-Theory and Methods 17 (10), 3547–3567. [4] Gilchrist, W., 2000. Statistical Modelling with Quantile Functions. CRC Press, Abingdon. [5] Govindarajulu, 1977. A class of distributions useful in life testing and reliability. IEEE Transactions on Reliability 26 (1), 67–69. [6] Hankin, R. K., Lee, A., 2006. A new family of non-negative distributions. Australian & New 14 UJSP_A_1448732 Zealand Journal of Statistics 48 (1), 67–78. [7] Hastings, C., Mosteller, F., Tukey, J. W., Winsor, C. P., 1947. Low moments for small samples: a comparative study of order statistics. The Annals of Mathematical Statistics, 413– 426. t [8] Hosking, J. R. M., 1990. L-moments: analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society. Series B, 105–124. rip [9] Lai, C. D., Xie, M., 2006. Stochastic Ageing and Dependence for Reliability. Springer Science & Business Media. us c [10] Midhu, N. N., Sankaran, P. G., Nair, N. U., 2014. A class of distributions with linear hazard quantile function. Communications in Statistics-Theory and Methods 43 (17), 3674–3689. [11] Nair, N. U., Sankaran, P. G., 2009. Quantile-based reliability analysis. Communications in Statistics-Theory and Methods 38 (2), 222–232. an [12] Nair, N. U., Sankaran, P. G., Balakrishnan, N., 2013. Quantile-Based Reliability Analysis. Springer, Birkhauser, New York. M [13] Nair, N. U., Sankaran, P. G., Kumar, B. V., 2012. Modelling lifetimes by quantile functions using parzen’s score function. Statistics 46 (6), 799–811. ed [14] Nair, N. U., Sankaran, P. G., Vineshkumar, B., 2008. Total time on test transforms of order n and their implications in reliability analysis. Journal of Applied Probability 45 (4), 1126–1139. pt [15] Öztürk, A., Dale, R. F., 1985. Least squares estimation of the parameters of the generalized lambda distribution. Technometrics 27 (1), 81–84. ce [16] Parzen, E., 1979. Nonparametric statistical data modeling. Journal of the American Statistical Association 74 (365), 105–121. Ac [17] Ramberg, J. S., 1975. A probability distribution with applications to Monte Carlo simulation studies. In: A Modern Course on Statistical Distributions in Scientific Work. Springer, pp. 51–64. [18] Ramberg, J. S., Dudewicz, E. J., Tadikamalla, P. R., Mykytka, E. F., 1979. A probability distribution and its uses in fitting data. Technometrics 21 (2), 201–214. [19] Ramberg, J. S., Schmeiser, B. W., 1972. An approximate method for generating symmetric random variables. Communications of the ACM 15 (11), 987–990. [20] Sankaran, P. G., Nair, N. U., Midhu, N. N., 2016. A new quantile function with applications to reliability analysis. Communications in Statistics-Simulation and Computation 45 (2), 566–582. 15 UJSP_A_1448732 [21] Sankaran, P. G., Thomas, B., Midhu, N. N., 2015. On bilinear hazard quantile functions. METRON 73 (1), 135–148. [22] Sankarasubramanian, A., Srinivasan, K., 1999. Investigation and comparison of sampling properties of L-moments and conventional moments. Journal of Hydrology 218 (1), 13–34. rip t [23] Tukey, J. W., 1962. The future of data analysis. The Annals of Mathematical Statistics 33 (1), 1–67. us c [24] Van Staden, P. J., Loots, M. T., 2009. Method of L-moment estimation for the generalized lambda distribution. In: Proceedings of the Third Annual ASEARC Conference, New Castle, Australia. Ac ce pt ed M an [25] Zimmer, W. J., Keats, J. B., Wang, F. K., 1998. The burr xii distribution in reliability analysis. Journal of Quality Technology 30 (4), 386–394. 16 UJSP_A_1448732 Ac ce pt ed M an us c rip t Figure 1: Plots of density function for different values of parameters 17 UJSP_A_1448732 Ac ce pt ed M an us c rip t Figure 2: Skewness and kurtosis of the HLEG(α , β , p) distribution for selected values of β and p as a function of the parameter α 18 UJSP_A_1448732 Ac ce pt ed M an us c rip t Figure 3: Skewness and kurtosis of the HLEG(α , β , p) distribution for selected values of α and p as a function of the parameter β 19 UJSP_A_1448732 Ac ce pt ed M an us c rip t Figure 4: Skewness and kurtosis of the HLEG(α , β , p) distribution for selected values of α and β as a function of the parameter p 20 UJSP_A_1448732 Ac ce pt ed M an us c rip t Figure 5: Plots of hazard quantile function for different values of parameters 21 UJSP_A_1448732 Ac ce pt ed M an us c rip t Figure 6: Q-Q plot for the dataset 22 UJSP_A_1448732 Ac ce pt ed M an us c rip t Figure 7: H(u) for the data set 23 UJSP_A_1448732 Ac ce pt ed M an 6 us c 5 rip t Sl.No 1 2 3 4 Table 1: Ageing behaviour of the hazard quantile function for different regions of parameter space Parameter Region Shape of hazard quantile function IHR p = 0, α > 0 and β > 0 IHR p = 1, α > 0 and β > 0 Constant p = 0 , β = 0 and α > 0 0 < p < 1, α > 0 and IHR 2α p β> (1 − p ) 0 < p < 1, α > 0 and DHR α p(1 − p) 0<β < 2 0 < p < 1, α > 0 and upside-down bathtub α p(1 − p ) 2α p <β < 2 1− p 24 UJSP_A_1448732