The Transmuted Weibull-Pareto Distribution

Pak. J. Statist. 2016 Vol. 32(3), 183-206 THE TRANSMUTED WEIBULL-PARETO DISTRIBUTION Ahmed Z. Afify1, Haitham M. Yousof1, Nadeem Shafique Butt2 and G.G. Hamedani3 1 Department of Statistics, Mathematics and Insurance Benha University, Egypt. Email: ahmed.afify@fcom.bu.edu.eg haitham.yousof@fcom.bu.edu.eg 2 Department of Family and Community Medicine Faculty of Medicine in Rabigh, King Abdulaziz University Jeddah, Saudi Arabia. Email: nshafique@kau.edu.sa 3 Department of Mathematics, Statistics and Computer Science Marquette University, Milwaukee, USA. Email: gholamhoss.hamedani@marquette.edu ABSTRACT A new generalization of the Weibull-Pareto distribution called the transmuted Weibull-Pareto distribution is proposed and studied. Various mathematical properties of this distribution including ordinary and incomplete moments, quantile and generating functions, Bonferroni and Lorenz curves and order statistics are derived. The method of maximum likelihood is used for estimating the model parameters. The flexibility of the new lifetime model is illustrated by means of an application to a real data set. KEY WORDS Transmuted family, Weibull-Pareto Distribution, Bonferroni and Lorenz curves, Order Statistics, Likelihood Estimation. 1. INTRODUCTION Recently, several families of distributions have been proposed via extending common families, by adding one or more parameters to the baseline model, of continuous distributions. These new families provide more flexibility in modeling and analyzing real life data in many applied areas. For that reason the statistical literature contains a good number of new families. For example, the generalized transmuted-G family proposed by Nofal et al. (2015), the transmuted exponentiated generalized-G class defined by Yousof et al. (2015), the transmuted geometric-G family introduced by Afify et al. (2016a) and the Kumaraswamy transmuted-G family introduced by Afify et al. (2016b). Another example is the Weibull-G (W-G for short) class defined by Bourguignon et al. (2014). Using the W-G family, Tahir et al. (2015) defined and studied the Weibull-Pareto (WPa) distribution extending the Pareto (Pa) distribution. The cumulative distribution function (cdf) of the WPa distribution is given (for x >  > 0 ) by © 2016 Pakistan Journal of Statistics 183 184 The Transmuted Weibull-Pareto Distribution b       x  G ( x) = 1  exp     1  ,         (1.1) where  is a positive scale parameter and  and b are positive shape parameters. The corresponding probability density function (pdf) is given by  x   g ( x) =  x 1    1      b b 1 b       x  exp     1  .         (1.2) In this paper, we define and study a new distribution by adding one extra shape parameter in equation (1.2) to provide more flexibility to the generated model. In fact, based on the transmuted-G (TG) family of distributions proposed by Shaw and Buckley (2007), we construct a new model called the transmuted Weibull-Pareto (TWPa) distribution and give a comprehensive descriptions of some of its mathematical properties. We hope that the new model will attract wider applications in reliability, engineering and other areas of applications. Recently, many authors used the TG family to construct new distributions. For example, Afify et al. (2015a) introduced the transmuted Marshall-Olkin Fréchet, Afify et al. (2015b) proposed the transmuted Weibull Lomax, Afify et al. (2014) defined the transmuted complementary Weibull geometric, Khan and King (2013) introduced the transmuted modified Weibull and Aryal and Tsokos (2011) proposed the transmuted Weibull distributions. For an arbitrary baseline cdf G  x  , Shaw and Buckley (2007) defined the TG family F  x  = G  x  1    G  x  with cdf and pdf given by and f  x  = g  x  1    2G  x  , (1.3) (1.4) respectively, where   1 . The TG density is a mixture of the baseline density and the exponentiated-G (Exp-G) density with power parameter two. For  = 0 , (1.3) gives the baseline distribution. The rest of the paper is outlined as follows. In Section 2, we define the TWPa distribution and provide the graphical presentation of its pdf and hazard rate function (hrf). A useful mixture representation for its pdf and cdf is provided in Section 3. Section 4 provides the mathematical properties including ordinary and incomplete moments, quantile and generating functions, Bonferroni, Lorenz and Zenga curves, moments of residual life and moments of the reversed residual life are derived. In Section 5, the order statistics and their moments are discussed. The probability weighted Afify, Yousof, Butt and Hamedani 185 moments (PWMs) are discussed in Section 6. Certain characterizations are presented in Section 7. The maximum likelihood estimates (MLEs) for the model parameters are demonstrated in Section 8. In Section 9, simulation results to assess the performance of the proposed maximum likelihood estimation method are discussed. The TWPa distribution is applied to a real data set to illustrate its usefulness in Section 10. Finally, some concluding remarks are given in Section 11. 2. THE TWPa DISTRIBUTION By inserting Equation (1.1) into Equation (1.3), the cdf of the TWPa model is given (for x >  ) by b   b               x    x  F ( x) = 1  exp     1  1   exp     1   .                       (2.1) The corresponding pdf is obtained by f  x = b  x 1  x      1     b 1 b  b            x    x  exp    1  1    2 exp    1   ,                   (2.2) where  is a scale parameter and  , b and  are positive shape parameters. The random variable X is said to have a TWPa distribution, denoted by X TWPa (, , b, ) , if its cdf is given by Equation (2.1). It is clear that Equation (2.2) reduced to the WPa model for  = 0 . The TWPa distribution due to its flexibility in accommodating all forms of the hrf as shown from Figure 2 seems to be an important model that can be used. A physical interpretation of the cdf of TWPa model is possible if we take a system consisting of two independent components functioning independently at a given time. So, if the two components are connected in parallel, the overall system will have the TWPa cdf with  = 1 . Another motivation for the TWPa distribution follows by taking two iid random variables, say Z1 and Z 2 , with cdf b       x  G  x  = 1  exp     1  .         Let Z1:2 = min(Z1 , Z2 ) and Z2:2 = max(Z1 , Z2 ) . Now, consider the random variable X defined by 186 The Transmuted Weibull-Pareto Distribution 1   ; Z1:2 , with probability   2 X =  Z , with probability 1   . 2:2  2  Then, the cdf of X is given by (2.1). Figure 1 provide some plots of the TWPa density curves for different values of the parameters  ,  ,  and b . Plots of the hrf of TWPa for selected parameter values are given in Figure 2. Some possible shapes for the TWPa cdf are displayed in Figure 3. Figure 1: Plots of the TWPa Density Function for some Parameter Values Afify, Yousof, Butt and Hamedani Figure 2: Plots of the hrf of the TWPa for some Parameter Values 187 188 The Transmuted Weibull-Pareto Distribution Figure 3: Plots of the cdf of the TWPa for some Parameter Values Afify, Yousof, Butt and Hamedani 189 3. MIXTURE REPRESENTATION The TWPa density function given in Equation (2.2) can be expressed as b    G ( x)   f ( x) = 1    bg ( x) exp      G ( x)b 1   G ( x)   b b     G ( x)b 1 G ( x)b 1 G ( x)    G ( x)     2bg ( x)     exp 2 bg ( x ) exp 2     .     G ( x)b 1 G ( x)b 1   G ( x)     G ( x)   G ( x)b 1  1k    k  1 b  j  1   x   1      k!j!   k  1 b  1      k , j =0 Using Equations (1.1) and (1.2) and after a power series, the we have  x f  x  = 1    b   x   x  2b   x   x  2b   x       1k    k  1 b  j  1   x   1      k!j!   k  1 b  1      k , j =0  bk  b  j 1  2 k    k  1 b  j  1   x   1      k!j!   k  1 b  1 k , j =0       bk  b  j 1 bk  b  j 1 . (3.1) Using the generalized binomial series expansion and after some algebra, the TWPa density can be expressed as  1k b   k  1 b  j  1 1    2  2k 1   f  x =  k!j!   k  1 b  1  k  1 b  j  k , j =0  x   k  1 b  j    x  or equivalently    x   1           k 1b  j 1  f ( x) =  k , j h,, k 1b  j ( x), (3.2) k , j =0 where and  1k b   k  1 b  j  1 1    2k 1  1 k , j = . k!j!  k  1 b  j     k  1 b  1  x h,,  k 1b  j  ( x) =  k  1 b  j    x  ,    x   1           k 1b  j 1 190 The Transmuted Weibull-Pareto Distribution is the exponentiated Pareto (EPa) density with parameters ,  and  k  1 b  j . This means that the TWPa density can be expressed as a mixture of EPa densities. So, several of its properties can be derived form those of the EPa model. Integrating (3.2), the cdf of TWPa can be expressed as  F ( x) =  k , j H ,, k 1b  j ( x), where H ,, k 1b j ( x) is the cdf of EPa with parameters  ,  and  k  1 b  j . k , j =0 4. PROPERTIES Established algebraic expansions to determine some structural properties of the TWPa distribution can be more efficient than computing those directly by numerical integration of its density function. The mathematical properties of the TWPa distribution including ordinary and incomplete moments, factorial moments, quantile and generating functions, Bonferroni, Lorenz and Zenga curves, residual life function and reversed residual life function are provided in this section. 4.1 Moments Using (2.2), the r th moment of X , denoted by r , is given by r =  k , j E Yr,, k 1b  j  ,    k , j =0 where E Yr,, k 1b  j  = 0 xr h,, k 1b  j ( x)dx . Then, we obtain (for r   )     r   r =   k  1 b  j  r k , j B 1  ,  k  1 b  j  , k , j =0    where B  m, n  = 0t m1 1  t  1 n 1 (4.1) dt is the beta function. The skewness and kurtosis measures can be calculated from the ordinary moments using well-known relationships. Corollary 1 Using the relation between the central moments and non-centeral moments, we can obtain the n th central moment of a TWPa random variable, denoted by M n , as follows   n n n n r M n = E  X    =     1  E X r , r =0  r  then, Afify, Yousof, Butt and Hamedani 191 n n nr nr M n =     1  1  r r =0  r  and cumulants (  n ) of X are obtained from (4.1) as n 1  n  1 n = n      r   n r  , r =0  r  1  where 1 = 1 hence 2 = 2  12 , 3 = 3  321  13 etc. The skewness and kurtosis measures can be calculated from the ordinary moments using well-known relationships. TWPa (, , , b) , is obtained by 4.2 Quantile and Generating Functions The quantile function (qf) of X , where X inverting (2.1) to obtain xu = F 1  u  as      1   xu =  1    log         1    2 2 1/  1/ b    4u            . Simulating the TWPa random variable is straightforward. If U is a uniform variate on the unit interval (0,1) , then the random variable X = xU follows (2.2).   Here, we will provide two formulas for the moment generating function (mgf) of the TWPa distribution. The mgf is defined by M X  t  =  r =0 M X t  =  t  r tr E X r . Then r!  k , j B 1  ,  k  1 b    k  1 b  j  r! k , j , r =0    r  j .  The second can be computed using Maple. For t  0 , p > 0 and q > 0 Let J  q, p, t  = q x  p etx dx . Using this software, we can obtain    csc  p  p   p  p   p, tq   etq p     J  q, p, t  =  1 q   qt tq  tq  p    tq  p 1   and M X  t  =  dk , j J ,  m  1   1, t  ,  m =0 where ( z, s) = s y z 1 e y dy is the complementary incomplete gamma function and  d k , j =  m 1   k  1 b  j  1 k , j  k  1 b  j   . m k , j =0    1m   192 The Transmuted Weibull-Pareto Distribution 4.3 Incomplete Moments The s th incomplete moments, say s  t  , is given by s  t  = 0x s f  x  dx. t Using equation (3.2) and the lower incomplete beta function, we obtain (for s   )  s s  t  = s  k , j  k  1 b  j  Bt  ,  k  1 b  k , j =0   where Bz  a, b  = 0 wa 1 1  w b 1 z  j ,  (4.2) dw is the incomplete beta function. The first incomplete moment of the TWPa distribution can be obtained by setting s = 1 in (4.2). The main application of the first incomplete moment refers to Bonferroni and Lorenz curves. These curves are very useful in economics, reliability, demography, insurance and medicine. The answers to many important questions in economics require more than just knowing the mean of the distribution, but its shape as well. This is obvious not only in the study of econometrics but in other areas as well. The important application of the first incomplete moment is related to the Lorenz and Bonferroni curves. These curves are very useful in economics, reliability, demography, insurance and medicine. The Lorenz curve, say LF  x  , and Bonferroni curve, say B  F  x  , are defined, respectively, by LF  x  = LF  x  1 1 x x . 0 tf  t  dt and B  F  x  = 0 tf  t  dt = EX  E  X  F  x F  x Then,  1  ,  k  1 b  j  k , j =0    LF  x  =  r    k , j B 1  ,  k  1 b  j  k , j =0    and 1  ,  k  1 b  j   k , j =0  B  F  x   = .  r   F  x   k , j B 1  ,  k  1 b  j  k , j =0      k , j Bt 1      k , j Bt 1  Another application of the first incomplete moment is related to mean residual life and mean waiting time given by m1  t  = 1  1  t  / R(t )  t and M1  t  = t  1  t  / F  t  , respectively. Afify, Yousof, Butt and Hamedani 193 4.4 Residual Life Function Several functions are defined related to the residual life. The failure rate function, mean residual life function and the left censored mean function, also called vitality function. It is well known that these three functions uniquely determine F ( x) . Moreover, the moments of the residual life, mn (t ) = E[( X  t )n | X > t ] , n = 1 , 2 ,..., uniquely determine F ( x) . The n th moment of the residual life of X is given by mn (t ) = 1  n t ( x  t ) dF ( x). 1  F (t ) Then, we can write (for r <  ) n n 1 nr  r mn (t ) =     t  t x dF ( x), 1  F (t ) r =0  r  mn (t ) = n 1 n  r n    k  1 b  j   t  r k , j   R(t ) r =0 k , j =0 r  r  B  ,  k  1 b      r j   Bt  ,  k  1 b      j .  Another interesting function is the mean residual life (MRL) function or the life expectation at age x defined by m1 ( x) = E ( X  x) | X > x  , which represents the expected additional life length for a unit which is alive at age x . The MRL of the TWPa distribution can be obtained by setting n = 1 in the last equation. The moments of the reversed residual life, M n (t ) = E (t  X )n | X  t  for t > 0 ,   n = 1 , 2 ,... uniquely determine F ( x) . We obtain 4.5 Reversed Residual Life Function M n (t ) = 0(t  x)n dF ( x). t Therefore, the n th moment of the reversed residual life of X given that r <  becomes 1 n n r nr t r M n (t ) =     1 t 0x dF ( x), F (t ) r =0  r  1 n   n     r   M n (t ) =     r n  k  1 b  j  k , j Bt 1  ,  k  1 b  j  . F (t ) r =0k , j =0  r  t    r The mean inactivity time (MIT) or mean waiting time (MWT) also called the mean reversed residual life function is defined by M1 (t ) = E[(t  X ) | X  t ] , and it represents the waiting time elapsed since the failure of an item on condition that this failure had occurred in (0, x) . The MIT of X can be obtained by setting n = 1 in the last equation. 194 The Transmuted Weibull-Pareto Distribution 5.ORDER STATISTICS If X1 , X 2 ,..., X n is a random sample of size n from the TWPa distribution and X 1 , X  2 ,..., X  n  are the corresponding order statistics, then the pdf of i th order statistic denoted by fi:n  x  is given by fi:n  x  = n 1 f ( x) j  n  1 i  j 1 ( x).   1  F B  i, n  i  1 j =0  j  (5.1) Using Equations (2.1) (2.2) and (5.1), we can write f  x  F ( x)i  j 1 =  tk ,l h,, k 1b l ( x).  k ,l =0 where t k ,l  1k  m w b m 1   i  j m1    k  1 b  l  1 =  k m, w=0 k!l!  k  1 b  l   w  1    k  1 b  1   i  j  1 i  j  m  1  k k    1    w  1  2  w  2   . w  m   The q th moment of X i:n is    1 j  n  1  q    tk ,l E Y,, k 1b l  . k ,l =0 j =0 B  i, n  i  1  j   n 1 E X iq:n =   Using (4.1), we can write   (5.2)  n  1   j   1 j   q   k  1 b  l  q B 1  ,  k  1 b  l  . k ,l =0 j =0 B  i, n  i  1     n 1 E X iq:n =   Based upon the moments in Equation (5.2), we can derive explicit expressions for the L-moments of X as infinite weighted linear combinations of the means of suitable TWPa distribution. They are linear functions of expected order statistics defined by r =  r  1 1 r 1  1k   E  X r k:r  , r  1. r k =0  k  The first four L-moments are given by: 1 = E  X1:1  ,  2 = 1 E  X 2:2  X1:2  , 2 1 1 E  X 3:3  2 X 2:3  X1:3  and  4 = E  X 4:4  3 X 3:4  3 X 2:4  X1:4  . One simply 4 3 can obtain the  's for X from equation (5.2) with q = 1 . 3 = Afify, Yousof, Butt and Hamedani 195 6. PROBABILITY WEIGHTED MOMENTS The PWMs are expectations of certain functions of a random variable. They can be derived for any random variable whose ordinary moments exist. The PWM approach can be used for estimating parameters of any distribution whose inverse form cannot be expressed explicitly. The ( s, r ) th PWM of X , say s , r , is defined by   s,r = E X s F ( X )r = x s F ( x)r f  x dx.   w  1k    k  1 b  j  1 m r m  r   r  m   1         k!j!   k  1 b  1  k  1 b  j  m w  Using Equations (2.1) and (2.2), we can write f  x  F ( x) r =   b  1 m, w, k , j =0 k  m w x k k  1    w  1  2  w  2    k  1 b  j      x  Then, we have    x   1           k 1b  j 1 . f  x  F ( x)r =  d k , j h,, k 1b  j ( x),  k , j =0 where dk , j  1k  m w b m 1   r m    k  1 b  j  1 =  k m, w=0 k!j!  k  1 b  j   w  1    k  1 b  1     r r  m  k k    1    w  1  2  w  2   . m w  Then, the ( s, r ) th PWM of X can be expressed as s ,r =  dk , j E Ys,, k 1b  j  .    k , j =0 Using (4.1), we can write  s   s ,r =  dk , j  k  1 b  j  s B 1  ,  k  1 b  j  .  k , j =0   7. CHARACTERIZATIONS The problem of characterizing a distribution is an important problem in various fields which has recently attracted the attention of many researchers. These characterizations have been established in many different directions. This section deals with two characterizations of TWPa distribution. These characterizations are based on (i) a simple relationship between two truncated moments and (ii) on conditional expectation of a 196 The Transmuted Weibull-Pareto Distribution function of the random variable. It should be mentioned that for our characterization (i), the cdf need not have a closed form. We believe, due to the nature of the cdf of TWPa, there may not be other possible characterizations of this distribution than the ones presented here. 7.1 Characterizations Based on Two Truncated Moments In this subsection we present characterizations of TWPa distribution in terms of a simple relationship between two truncated moments. Our first characterization result borrows from a theorem due to Glänzel (1987), see Theorem A below. We refer the interested reader to Glänzel (1990) for a proof of Theorem A. Note that the result holds also when the interval H is not closed. Moreover, as mentioned above, it could be also 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. Theorem 7.1. Let  , , P  be a given probability space and let H =  d , e be an interval for some d < e ( d = , e =  might as well be allowed). Let X :   H be a continuous random variable with the distribution function F and let g and h be two real functions defined on H such that E  g  X  | X  x  = E h  X  | X  x    x  , x  H , is defined with some real function  . Assume that g , h  C1  H  ,  C 2  H  and F is twice continuously differentiable and strictly monotone function on the set H . Finally, assume that the equation h = g has no real solution in the interior of H . Then F is uniquely determined by the functions g , h and  , particularly F  x  = aC x   u  u  h u   g u  exp  s  u   du, where the function s is a solution of the differential equation s' = ' h /  h  g  and C is the normalization constant, such that HdF = 1 . Proposition 7.1. Let X :    ,   be a continuous random variable and let b        x   h  x  = 1    2 exp     1           and 1 Afify, Yousof, Butt and Hamedani 197 b       x  g  x  = h  x  exp     1  for x > .         The random variable X belongs to TWPa family (2.2) if and only if the function  defined in Theorem 7.1 has the form b      1  x    x  = exp     1  , x > .  2        (7.1) Proof. Let X be a random variable with density (2.2), then b       x 1  F  x   E h  x  | X  x  = exp      1  , x > ,        and b     1   x  1  F  x   E  g  x  | X  x  = 2 exp 2     1  , x > ,        and finally b      1  x    x  h  x   g  x  =  h  x  exp     1  < 0, for x > .  2        Conversely, if  is given as above, then s  x  =   x  h  x   x h  x  g  x  = b x  1  x       1     b 1 , x > , and hence  x    s  x  =    1 , x > .     b Now, in view of Theorem 7.1, X has density (2.2). Corollary 7.1. Let X :    ,   be a continuous random variable and let h  x  be as in Proposition 7.1. The pdf of X is (6) if and only if there exist functions g and  defined in Theorem 7.1 satisfying the differential equation 198   x  h  x   x h  x  g  x The Transmuted Weibull-Pareto Distribution = b x  1  x       1     b 1 , x > . (7.2) The general solution of the differential equation in Corollary 7.1 is b 1       b x1  x   1   b              x     x  = exp     1   , b               1 x        exp       1   h  x   g  x  dx  D              where D is a constant. Note that a set of functions satisfying the differential equation (7.2) is given in Proposition 7.1 with D = 0. However, it should be also noted that there are other triplets  h, g ,  satisfying the conditions of Theorem 7.1. 7.2 Characterizations Based on Conditional Expectation of a Function of the Random Variable In this subsection we present a characterization result in terms of a function of the random variable X . The following Proposition has appeared in our previous work which will be used to characterize TWPa distribution Proposition 7.2. Let X :    a, b  be a continuous random variable with cdf F and corresponding Let   x  be a differentiable function greater than 1 on that lim xa  x  = 1 and lim xb  x  = 1  c . Then, for 0 < c < 1, pdf f .  a, b  E   X  | X  x  = c  1  c    x  , if and only if    x 1  F  x =    c   such (7.3) 1 c c . (7.4) Remark 7.1. Taking, e.g.,  a, b  =  ,   and b   b   1 c             x    x      x  = 1  c 1  exp     1  1   exp     1    .                       c Proposition 7.3 gives a characterization of TWPa distribution. Afify, Yousof, Butt and Hamedani 199 8. ESTIMATION The maximum likelihood estimators (MLEs) for the parameters of the TWPa distribution are discussed in this section. Let x1 ,..., xn be a random sample of of size n from the TWPa( x;  ) distribution, where  is the unknown parameter vector  =  , , b,   . T Then, the log-likelihood function for the parameters vector  , say expressed as = , can be n ln b  n ln   n ln      1  in=1 ln xi   in=1sib   b  1  in=1 ln si   in=1 ln 1    2pi  ,   sb x  where si =  i   1 and pi = e i .   Assuming U   = known, therefore      are given by = , ,   b    the score vector components, T z z p sb 1  n =  n ln    in=1 ln xi   b  1  in=1 i  b in=1zi sib 1  2b in=1 i i i ,   si 1    2pi p sb ln si  n =   in=1 ln si   in=1sib ln si   in=1 i i b b 1    2pi and 2 pi  1  =  in=1 ,  1    2pi  x  x where zi =  i  ln  i    .  We can find the estimates of the unknown parameters by setting the score vector to ˆ  = 0, and solving them simultaneously to obtain the ML estimators ˆ , bˆ and ̂ . zero, U   These equations cannot be solved analytically and statistical software can be used to solve them numerically by means of iterative techniques such as the Newton-Raphson algorithm. For the four-parameter TWPa distribution all the second order derivatives exist. For interval estimation of the model parameters, we require the 3  3 observed information matrix, J   = J rs   for r, s = , b,   , whose elements are given by 200 The Transmuted Weibull-Pareto Distribution J  = n  2    xi b 1 b n    1  b i =1  b  1 zi si  si ln         x  pi  b  1 zi sib 1  sib ln  i    bzi pi si2b 1     , 1    2pi 2   b  1  in=1  2b in=1 J b =  in=1 J  = 2b in=1 J bb = b 1  b ln s  bs 1    2pi i b i ln si    , 2zi pi2 sib 1  1    2pi  zi pi sib 1  n sb 2  i =1 i J b =  in=1   xi ln    zi bz p 2 s 2b 1 ln si 2   in=1zi sib 1 1  b ln si    2   in=1 i i i si 1    2pi 2 zi pi sib 1 n  2  i =1 n zi si 1    2pi 2 2 pi sib  ln si  1  sib  2 n  ln si   i =1   2 pi  1 2 1    2pi 2 1    2pi and J  =  in=1 ,  2  in=1   2 pi  1  pi2 si2b  ln si  1  sib 2 1    2pi  2 , 2 1    2pi 2 . Under standard regularity conditions, the multivariate normal  N3 0, J  ˆ  1  distribution can be used to construct approximate confidence intervals for the model parameters. Here, J  ˆ  is the total observed information matrix evaluated at ̂ . Therefore, approximate 100(1  )% confidence intervals for ,b and  can be determined as: ˆ  Z  Jˆ , ˆ  Z  Jˆ , and bˆ  Z  Jˆbb , where Z  is the upper  th percentile of the standard normal distribution. 2 2 2 2 9. SIMULATION STUDY Here, we assess the performance of the maximum likelihood estimation procedure for estimating the TWPa parameters using Monte Carlo simulation. An ideal technique for simulating from (5) is the inversion method. For different combinations of  ,  , b and  samples of sizes n = 100, 200, 500 and 1000 are generated from the TWPa model. We repeated the simulation k = 100 times and calculated the MLEs and the standard deviations of the parameter estimates. We use three combinations for the parameter values (I:  =  = b = 1.5 and  = 1 , II:  = 1.5 ,  = b = 2 and  = 0 and III:  =  = 1 ,  = 2 and b = 3 ). The empirical results are given in Table 1. It is evident Afify, Yousof, Butt and Hamedani 201 that the estimates are quite stable and are close to the true value of the parameters for these sample sizes. Additionally, as the sample size increases, the biases and the standard deviations of the MLEs decrease as expected. Table 1 MLEs and Standard Deviations for various Parameter Values Estimated Values (Standard Deviations) Sample size ( n ) ̂ b̂ ̂ ̂ 1.498091143 1.502948857 1.501953638 0.997782415 I (0.11582947) (0.172767349) (0.243121613) (0.144500353) 100 1.491903012 1.48957008 1.527788397 0.999536076 (0.199914811) (0.301097609) (0.381782447) (0.199173712) 200 1.491958427 1.501891389 1.509888485 0.992963765 (0.149228829) (0.198243384) (0.309084222) (0.20302651) 500 1.502175325 1.497094523 1.499390957 0.994431539 (0.119792874) (0.179465332) (0.278695995) (0.159880471) 1000 1.497894409 1.507425395 1.499064534 1.000246218 (0.098759361) (0.151490124) (0.198275817) (0.119637727) 1.500885689 2.00032535 1.997983274 0.003727487 II (0.121687918) (0.228695465) (0.139224288) (0.145980986) 100 1.50426267 1.974097335 1.97696601 0.014608287 (0.288560488) (0.519928598) (0.413326812) (0.521138173) 200 1.495429289 2.00483308 1.983424014 0.017654112 (0.017654112) (0.290681421) (0.305144854) (0.300017145) 500 1.502350282 1.98916705 2.00784619 0.004494032 (0.10061869) (0.204113398) (0.097860177) (0.1008336) 1000 1.500906974 2.007625756 1.998065395 0.000529191 (0.100826821) (0.199465995) (0.099311977) (0.100231728) 1.002653476 2.004604317 2.997469654 1.009583677 III (0.270238689) (0.175851467) (0.300490017) (0.182163635) 100 0.972887849 1.990449446 2.98928733 0.967372304 (0.490687527) (0.300195793) (0.960050826) (0.487313015) 200 0.998436465 2.058020528 2.977709622 1.112873677 (0.421358094) (0.187659576) (0.499413199) (0.403941241) 500 1.021321137 1.999576707 3.011564071 0.990865319 (0.299676141) (0.100350456) (0.301259786) (0.199220749) 1000 0.997139611 1.997850366 2.995192685 1.002505993 (0.203251199) (0.198805918) (0.194364415) (0.09876462) The bold values are combined results of (100 + 200 + 500 + 1000 = 1800). 202 The Transmuted Weibull-Pareto Distribution 10. APPLICATION In this section, we provide an application of the TWPa distribution to show its flexibility and importance. We shall compare the TWPa model with other related models, namely Weibull Pareto (WPa), McDonald Lomax (McL) (Lemonte and Cordeiro 2013), transmuted Weibull Lomax (TWL) (Afify et al. 2015) and transmuted complementary Weibull geometric (Afify et al. 2014) distributions. The density functions (for x > 0 ) associated to these models are given by • The McL pdf given by   x f ( x) = 1    1  B a  , b    • The TWL pdf given by ab  x  1 f ( x) =     b1 f  x  =  x  1 x  e   x   1  1         a 1      x     1 1 1             b 1      . b       x  exp a 1    1            x    1  1         • The TCWG pdf given by  1 b 1  b         x  1    2 exp  a 1    1   .               x     1    e    3   x    1  b         1 e .   The parameters of the above densities are all positive real numbers except for the TWL and TCWG distributions for which |  | 1 . We make use of the data set of gauge lengths of 10 mm from Kundu and Raqab (2009). This data set consists of 63 observations: 1.901, 2.132, 2.203, 2.228, 2.257, 2.350, 2.361, 2.396, 2.397, 2.445, 2.454, 2.474, 2.518, 2.522, 2.525, 2.532, 2.575, 2.614, 2.616, 2.618, 2.624, 2.659, 2.675, 2.738, 2.740, 2.856, 2.917, 2.928, 2.937, 2.937, 2.977, 2.996, 3.030, 3.125, 3.139, 3.145, 3.220, 3.223, 3.235, 3.243, 3.264, 3.272, 3.294, 3.332, 3.346, 3.377, 3.408, 3.435, 3.493, 3.501, 3.537, 3.554, 3.562, 3.628, 3.852, 3.871, 3.886, 3.971, 4.024, 4.027, 4.225, 4.395, 5.020. These data have been used by Afify et al. (2015) to fit the exponentiated transmuted generalized Rayleigh distribution. We consider some criteria like 2 (where is the maximized log-likelihood), AIC (Akaike information criterion) and CAIC (the consistent Akaike information criterion), HQIC (Hannan-Quinn information criterion) and BIC (Bayesian information criterion) in order to compare the distributions. In general, the smaller the values of these statistics, the better the fit to the data. Afify, Yousof, Butt and Hamedani 203 Table 2 provides the MLEs and their corresponding standard errors (SEs) of the model parameters and the numerical values of the 2 , AIC , CAIC , HQIC and BIC . These numerical results are obtained using the Mathcad program. It is shown from Table 2 that the TWPa model has the lowest values for the 2 , AIC , CAIC , HQIC and BIC statistics among all fitted models (except for HQIC and BIC of the WPa distribution). So, the TWPa model could be chosen as the best model. Figure 4 displays the fitted pdf and cdf for the TWPa model and Figure 5 displays the QQ-plot and estimated survival function of the TWPa distribution. It is clear from these plots that the TWPa provides good fit to this data set. Table 2 MLEs, their Standard Errors (SEs) and Goodness-of-Fit Statistics HQIC SEs CAIC AIC BIC Model Estimates 2 TWPa WPa TWL ˆ = 0.1885 ˆ = 0.0909 bˆ = 14.4535 ˆ = 0.7288 ˆ = 0.1834 ˆ = 0.0755 bˆ = 13.9522 ˆ = 0.3922 ˆ = 0.6603 ˆ = 0.5287 bˆ = 8.4451 ˆ = 0.7364 McL 0.065 119.282 127.282 127.972 130.654 135.855 0.115 5.252 0.284 0.102 121.790 127.790 128.197 130.319 134.219 0.159 7.705 0.339 119.688 129.688 130.741 133.903 140.404 1.174 3.32 4.397 0.286 ˆ = 45.9249 59.312 130.597 140.597 141.65 144.812 151.313 ˆ = 48.3024 63.047 8.855 ˆ = 18.1192 bˆ = 195.4633 123.217 ˆ = 353.1435 375.678 TCWG ˆ = 0.2022 ˆ = 3.3482 ˆ = 0.3876 ˆ = -0.0001 0.217 0.783 0.069 0.496 126.895 134.895 135.585 138.267 143.468 204 The Transmuted Weibull-Pareto Distribution 11. CONCLUSIONS In this paper, We propose a new four-parameter model, called the transmuted Weibull-Pareto (TWPa) distribution, which extends the Weibull-Pareto (WPa) distribution introduced by Tahir et al. (2015). We provide some of its mathematical properties. The TWPa density function can be expressed as a mixture of exponentiated Pareto densities. We derive explicit expressions for the ordinary and incomplete moments, quantile and generating functions, Lorenz, Bonferroni and Zenga curves, moments of residual life and moments of the reversed residual life. We also obtain the density function of order statistics and their moments. Further, the Probability weighted moments are investigated and certain characterization results are provided. We discuss the maximum likelihood estimation of the model parameters. The proposed distribution is applied to a real data set and provides a better fit than several nested and non-nested models. 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