SOME COMMENTS ON METHODOLOGY OF CUBIC RANK TRANSMUTED DISTRIBUTIONS

In this study, at first a new polynomial rank transmutation is proposed. Then, a new cubic rank transmutation is introduced by simplifying the set of transmutation parameters in order to improve its usefulness in statistical modeling. The purpose of this comment is to clarify some issues that exist in the methodology of obtaining the distribution by the cubic transmutation and the stage of proofing it. In this way, both the parameter space is expanded and the process of establishing the cubic transformed distribution family is given.


Introduction
In this study, we inspire the quadratic rank transmutation map (QRTM) proposed by [16]. The mapping is given as where u 2 [0; 1] and 2 [ 1; 1]. Using this transmutation many distributions have been derived and still continue to be derived. Beside this, there are also some studies on the modi…cations of the QRTM. Some of the pioneering works on proposing modi…ed QRTM can be given as follows: [1] proposed a new Weibull distribution by using exponentiated QRTM. [2] generated a new distribution family by considering exponentiated distribution as the baseline distribution. [11] studied a new distribution by taking the baseline distribution as exponentiated exponential distribution. [5] introduced transmuted exponentiated modi…ed Weibull distribution, and [3] introduced transmuted exponentiated Lomax distribution. The last three 1162 M . HAM ELDARBANDI, M . YILM AZ studies can be seen as a special case of [2]. [9] introduced a new transmutation map by adding extra two parameters to get more ‡exible distribution. Then, [10] introduced a new Lindley distribution by using this new transmutation map approach. [4] introduced a kind of generalization of QRTM by considering sum of k-dimensional vector of transmutation parameters. There are two similar studies which are the generalized transmuted G family by [12] and generalized transmuted Weibull distribution by [13]. Also, by taking into account recent works, [15] introduced a new distribution named as transmuted generalized Gamma distribution. They use QRTM to generate this distribution family.
In this study, a new polynomial rank transmutation is proposed additionally to [17]. Since the parameter set is still complex, a new cubic rank transmutation is introduced in the light of the idea behind QRTM. In our study, since an extra transmutation parameter is added, the distribution has become more ‡exible.

2.
Motivation [17] proposed polynomial rank transmutation map to demonstrate Skew-kurtotic transmutations. Figure 3 of [17] indicates that admissible parameter region. However this region is quite complex structure, the points on Figure 5 of them show some special cases related to family of order statistics up to 3-sized sample. Under the leadership of this idea, we propose a new polynomial rank transmutation to get simpler structure of parameter region. Let G(u) stand for the polynomial rank transmutation de…ned on [0; 1]. Then, we have with G(0) = 0 and G(1) = 1. Note that, 1 and 2 are the transmutation parameters. Parameter region will be de…ned with following discussion. Since G should be non-decreasing, non-negativity of the …rst derivative of G with respect to u is examined. Thus, the shape of the parameter region is determined. By calling this derivative with g, we have Non-negativity of g (u; 1 ; 2 ) at the end-points, namely the inequalities g (0; 1 ; 2 ) = 1 + 1 0 and g (1; 1 ; 2 ) = 1 1 2 0 both requires that From these two inequalities, it is clear that 2 2. When the eq. (3) is taken into account, g (u; 1 ; 2 ) is a concave function for 2 2 (0; 2]. As long as the inequality (4) is valid, g (u; 1 ; 2 ) will take non-negative values. For 2 0, we will investigate the su¢ cient conditions on non-negativity of g (u; 1 ; 2 ). In this case, g (u; 1 ; 2 ) has a minimum point since it is a convex function. If this minimum point is within (0; 1), the value at that point of the function g (u; 1 ; 2 ) must be positive. Accordingly, the minimum point is obtained by taking the derivative of the eq. (3) and equating them to zero as follows: Then, the value of g (u; 1 ; 2 ) at u must satisfy Hence, it is necessary to say that the value of the numerator in (6) is non-positive. If this statement given by the numerator is considered as a second order polynomial of 1 , the roots are given by Here, we can say that the condition 4 2 must also occur in order for the roots to be real valued. Thus, under the condition 4 2 < 0, we have bounds for 1 as follows: For these bounds, the numerator in (6) has a negative sign. This leads to the following conclusion: The range of 1 is as in (8)  Thus, combining this results, the parameter region for ( 1 ; 2 ) appears as shown in the Figure 1. By considering this parameter set of ( 1 ; 2 ), many well de…ned distributions are generated from the eq. (2) with the baseline distribution F . Now, let's get a map of the integer values of the pair ( 2 ; 1 ) to see the known distributions tabulated in Table 1. The distributions speci…ed by the star in Table 1 are described below how they correspond to some known failure distributions. Let X r:n be the rth order statistic in a sample of size n. By noting that, for  Some Generated Distributions On the other hand, for 1 = 1; 2 = 0 generated distribution indicates the failure distribution of the lifetime of the two-component parallel system, namely distribution of X 2:2 = max fX 1 ; X 2 g. For 1 = 1; 2 = 0 generated distribution indicates the failure distribution of the lifetime of the two-component series system, namely distribution of X 1:2 = min fX 1 ; X 2 g.
In this case, in addition to the known distributions introduced by the quadratic transmutation, more informative distribution functions occure. However, the set of the transformation parameters of the proposed cubic transmutation is still complicated.
In order to eliminate of this complexity, by referring to the concept of reliability evaluation of coherent system by using signature (see, [6,7] ), we come up with an idea inspired by both works of [16,18] as follows: Pr (X 2:2 t) = Pr (max fX 1 ; X 2 g t) = F 2 (t) and Pr (X 1:2 t) = Pr (min fX 1 ; X 2 g t) = 2F (t) F 2 (t) where F (t) indicates the failure distribution of the component lifetime, namely, Pr (X 1 t) = F (t). Hence there exists a stochastic ordering relation such as X 1:2 st X st X 2:2 . In this case, these three failure distributions can be ordered as From the latter inequality, we can say that F (t) is represented by a convex combination of 2F (t) F 2 (t) and F 2 (t) where the value of the combination parameter is 1 2 . On the other hand, it is possible to obtain many distributions besides F . Let G stand for the distribution obtained by this convex combination. Then, for 2 [0; 1], we have Here, the combination parameter is reparametrized by taking = 1+ 2 to attain quadratic rank transmutation. Now, the new parameter takes the values in [ 1; 1]. As can be seen immediately, = 0 corresponds to = 1 2 . In eq. (9), substituting by , we have The above expression is the quadratic rank trasmutation proposed by [16]. Now, we concentrate on 3-component systems with similar thinking. Let X 1 ; X 2 and X 3 be independent random variables distributed as F . Let X r:3 denote rth order statistic of (X 1 ; X 2 ; X 3 ) with corresponding distribution F r:3 . Then we have F 3:3 (t) = Pr (X 3:3 t) = Pr (max fX 1 ; X 2 ; X 3 g t) = F 3 (t) (11) F 2:3 (t) = Pr (X 2:3 t) = Pr (max fmin fX 1 ; X 2 g ; min fX 1 ; X 3 g ; min fX 2 ; X 3 gg t) According to [18], the properties F 3:3 F 2:3 F 1:3 and F = 1 3 F 3:3 + 1 3 F 2:3 + . Hence, we can suggest a convex combination to cover both ordering situations. Our aim is to determine exactly where F is. In this case, we can write the following convex combination obtained by F 1:3 and F 2:3 , called as G .
where 1 2 [0; 1] . Now, let's write a convex combination between G and F 3:3 . Denoting this convex combination by G, we have where 2 2 [0; 1]. Combining with the equations (14) and (15), we obtain G as If the notation F is used for the representation of F r:3 , and rearranging with respect to polynomial degree of F , the following expression is obtained: Undoubtedly, G is a distribution function. However, reparameterization is made on the model in order to achieve the similar structure of the quadratic rank transmutation. Now, by taking w 1 = 1 2 and w 2 = 2 1 2 , eq. (17) can be rewritten as follows: where w 1 ; w 2 2 [0; 1]. In eq. (18), by the reparametrizating as w 1 = 1+ 1 3 and where 1 ; 2 2 [ 1; 2]. Since 2 = w 1 + w 2 , the parameter set is also constrained by the condition 1 + 2 1. Consequently, the parameter set of 1 and 2 is presented in a simpler form than the parameter region given in Figure 1. This transmutation de…ned in eq. (19) is called as cubic rank transmutation and transformed distribution G is named as CRT-F. As can be seen immediately, CRT-F de…nes a quadratic rank transmuted distribution at 2 = 0, and 1 = 2 = 0 gives the baseline distribution F . For this reason, CRT-F can be seen as a generalized form of QRT. The parameter set of 1 and 2 , which is de…ned as f( 1 ; 2 ) : 1 ; 2 2 [ 1; 2] ; 1 + 2 1g can be …gure out in Figure 2.Now, referring to the integer values of 1 and 2 , we can determine the generated distribution functions by the Table 2. Identi…cations given in Table 2 show that Table 1 of [17] is included by CRT-F according to special choices of transmutation parameters.
Note that, by taking into account the parameter set of (19), the distribution family CRT-F is di¤erent as compared with the families proposed by [8,14].
[8] proposed a cubic rank transmuted distribution family motivated by a study of [17]. The paper contained one theorem (referred to as Theorem 2.1), deriving cubic transmuted distribution. Here, We would like to point out that the result of Theorem 2.1 can be reduced to an explicit and understandable form.

Conclusion
In this article, we propose a new version of polynomial rank transmutation. Since the parameter set is still complex, a new cubic rank transmutation is introduced in the light of the idea behind QRTM technique. Compared to the two techniques in the literature, it is seen that the proposed technique covers them in terms of parameter space.