Orderings of extremes among dependent extended Weibull random variables

In this work, we consider two sets of dependent variables $\{X_{1},\ldots,X_{n}\}$ and $\{Y_{1},\ldots,Y_{n}\}$, where $X_{i}\sim EW(\alpha_{i},\lambda_{i},k_{i})$ and $Y_{i}\sim EW(\beta_{i},\mu_{i},l_{i})$, for $i=1,\ldots, n$, which are coupled by Archimedean copulas having different generators. Also, let $N_{1}$ and $N_{2}$ be two non-negative integer-valued random variables, independent of $X_{i}'$s and $Y_{i}'$s, respectively. We then establish different inequalities between two extremes, namely, $X_{1:n}$ and $Y_{1:n}$ and $X_{n:n}$ and $Y_{n:n}$, in terms of the usual stochastic, star, Lorenz, hazard rate, reversed hazard rate and dispersive orders. We also establish some ordering results between $X_{1:{N_{1}}}$ and $Y_{1:{N_{2}}}$ and $X_{{N_{1}}:{N_{1}}}$ and $Y_{{N_{2}}:{N_{2}}}$ in terms of the usual stochastic order. Several examples and counterexamples are presented for illustrating all the results established here. Some of the results here extend the existing results of Barmalzan et al. (2020).


Introduction
A r-out-of-n system will function if at least r of the n components are functioning.This includes parallel, fail-safe and series systems, corresponding to r = 1, r = n − 1, and r = n, respectively.We denote the lifetimes of the components by X 1 , • • • , X n , and the corresponding order statistics by X 1:n ≤ • • • ≤ X n:n .Then, the lifetime of the r-out-of-n system is given by X n−r+1:n and so, the theory of order statistics has been used extensively to study the properties of (n − r + 1)-out-of-n systems.For detailed information on order statistics and their applications, interested readers may refer to Arnold et al. (1992), Balakrishnan and Rao (1998a) and Balakrishnan and Rao (1998b).
The Weibull distribution has been used in a wide variety of areas, ranging from engineering to finance.Numerous works have been conducted to further explore and analyze properties and different uses of the Weibull distribution.These works further highlight the broad applicability of the Weibull distribution, and its potential for use in many different fields; see, for example, Mudholkar et al. (1996) and Lim et al. (2004).
A flexible family of statistical models is frequently required for data analysis to achieve flexibility while modeling real-life data.Several techniques have been devised to enhance the malleability of a given statistical distribution.One approach is to leverage already wellstudied classic distributions, such as gamma, Weibull, and log-normal.Alternatively, one can increase the flexiblity of a distribution by including an additional shape parameter; for instance, the Weibull distribution is generated by taking powers of exponentially distributed random variables.Another popular strategy for achieving this objective, as proposed by Marshall and Olkin (1997), is to add an extra parameter to any distribution function, resulting in a new family of distributions.To be specific, let G(x) and Ḡ(x) = 1 − G(x) be the distribution and survival functions of a baseline distribution, respectively.We assume that the distributions have non-negative support.Then, it is easy to verify that and x, α ∈ (0, ∞), ᾱ = 1 − α, (1.2) are both valid cumulative distribution functions.Here, the newly added parameter α is referred to as the tilt parameter.When G(x) has probability density and hazard rate functions as g(x) and r G (x), respectively, then the hazard rate function of F (x; α) in (1.1) is seen to be x, α ∈ (0, ∞), ᾱ = 1 − α.
(1.3) Thus, if r G (x) is decreasing (increasing) in x, then for 0 < α ≤ 1 (α ≥ 1), r F (x; α) is also decreasing (increasing) in x.This method has been used by different authors to introduce new extended family of distributions; see, for example, Mudholkar and Srivastava (1993).
Comparison of two order statistics stochastically has been studied rather extensively, and especially the comparison of various characteristics of lifetimes of different systems having Weibull components, based on different stochastic orderings.For example, one may see Khaledi and Kochar (2006), Fang and Zhang (2013), Fang and Tang (2014), Torrado and Kochor (2015), Torrado (2015), Zhao et al. (2016), Fang et al. (2015), and the references therein, for stochastic comparisons of series and parallel systems with heterogeneous components with various lifetime distributions.The majority of existing research on the comparison of series and parallel systems has only considered the case of components that are all independent.However, the operating environment of such technical systems is often subject to a range of factors, such as operating conditions, environmental conditions and the stress factors on the components.For this reason, it would be prudent to take into account the dependence of the lifetimes of components.There are various methods to model this dependence, with the theory of copulas being a popular tool; for example, Nelsen (2006) provides a comprehensive account of copulas.Archimedean copulas are a type of multivariate probability distributions used to model the dependence between random variables.They are frequently used in financial applications, such as insurance, risk modeling and portfolio optimization.Many researchers have given consideration to the Archimedean copula due to its flexibility, as it includes the renowned Clayton copula, Ali-Mikhail-Haq copula, and Gumbel-Hougaard copula.Moreover, it also incorporates the independence copula as a special case.As such, results of comparison established under an Archimedean copula for the joint distribution of components' lifespans in a system are general, and would naturally include the corresponding results for the case of independent components.
In this article, we consider the following family of distributions known as extended Weibull family of distributions, with G(x) = 1 − e −(xλ) k , x, λ, k > 0, as the baseline distribution in (1.1).The distribution function of the extended Weibull family is then given by (1.4) We denote this variable by X ∼ EW (α, λ, k), where α, λ and k are respectively known as tilt, scale and shape parameters.In (1.4), if we take α = 1 and k = 1, then the extended Weibull family of distributions reduces to the Weibull family of distributions and the extended exponential family of distributions (see, Barmalzan et al. (2020)), respectively.Similarly, if we take both α = 1 and k = 1, the extended Weibull family of distributions reduces to the exponential family of distributions.Now, let us consider two sets of dependent variables {X 1 , . . ., X n } and {Y 1 , . . ., Y n }, where for i = 1, . . ., n, X i ∼ EW (α i , λ i , k i ) and Y i ∼ EW (β i , µ i , l i ) are combined with Archimedean (survival) copula having different generators.We then establish here different ordering results between two series and parallel systems, where the systems' components follow extended Weibull family of distributions.The obtained results are based on the usual stochastic, star, Lorenz and dispersive orders.Moreover, considering {X 1 , . . ., X N 1 } and {Y 1 , . . ., Y N 2 }, where X i ∼ EW (α i , λ i , k i ) and Y i ∼ EW (β i , µ i , l i ) and N 1 and N 2 are two random integer-valued random variables independently of X i s and Y i s, respectively, we then compare X 1:N 1 and Y 1:N 2 and X N 1 :N 1 and Y N 2 :N 2 stochastically.The rest of this paper is organized as follows.In Section 2, we recall some basic stochastic orders and some important lemmas.The main results are presented in Section 3.This part is divided into two subsections.The ordering results between two extreme order statistics are established in Subsection 3.1 when the number of variables in the two sets of observations are the same and that the dependent extended Weibull family of distributions have Archimedean (survival) copulas, while in Subsection 3.2, we focus on the case when the two sets have random numbers of variables satisfying the usual stochastic order.In Subsection 3.1, the ordering results are based on the usual stochastic, star, Lorenz, hazard rate, reversed hazard rate and dispersive orders.Finally, Section 4 presents a brief summary of the work.
Here, we focus on random variables which are defined on (0, ∞) respresenting lifetimes.The terms 'increasing' and 'decreasing' are used in the nonstrict sense.Also, ' sign = ' is used to denote that both sides of an equality have the same sign.

Preliminaries
In this section, we review some important definitions and well-known concepts of stochastic order and majorization which are most pertinant to ensuing discussions.Let c = (c 1 , . . ., c n ) and d = (d 1 , . . ., d n ) to be two n dimensional vectors such that c , d ∈ A. Here, A ⊂ R n and R n is an n-dimensional Euclidean space.Also, consider the order of the elements of the vectors c and d to be c 1:n ≤ . . .≤ c n:n and d 1:n ≤ . . .≤ d n:n , respectively.Note that c m d implies both c w d and c w d.But, the converse is not always true.For an introduction to majorization order and their applications, are may refer to Marshall et al. (2011).
Throughout this paper, we are concerned only with non-negative random variables.Now, we discuss some stochastic orderings.For this purpose, let us suppose Y and Z are two nonnegative random variables with probability density functions (PDFs) f Y and f Z , cumulative distribution functions (CDFs) being the corresponding hazard rate and reversed hazard rate functions, respectively.Definition 2.2.A random variable Y is said to be smaller than Z in the • reversed hazard rate order (denoted by Y ≤ rh Z) if rY (x) ≤ rZ (x), for all x; • usual stochastic order (denoted by Y ≤ st Z) if FY (x) ≤ FZ (x), for all x; It is known that the star ordering implies the Lorenz ordering.One may refer to Shaked and Shanthikumar (2007) for an exhaustive discussion on stochastic orderings.Next, we introduce Schur-convex and Schur-concave functions.
Next, we present some important lemmas which are essential for the results developed in the following sections.
Lemma 2.2.(Lemma 7.1 of Li and Fang (2015)).For two n-dimensional Archimedean copulas , for all x and y in the domain of f.Here, φ 2 is the right-continuous inverse of ψ 2 .
Proof.Taking derivative with respect to x, we get Now, as e x ≥ x+1 for x ≥ 0, we have e b k e kx ≥ b k e kx + 1 which implies e b k e kx ≥ a• b k e kx + 1 , for 0 ≤ a ≤ 1.Hence, f (x) ≥ 0 and therefore f (x) is increasing in x ∈ (0, ∞).
Proof.Differentiating m 1 (x) with respect to x, we get ke kx • e e kx − a ≥ 0.
Because at x = 0, we have e e kx − a(e kx + 1) ≥ 0, the required result as follows.
Lemma 2.7.Let m 3 (λ) : (0, ∞) → (0, ∞) be a function given by where 0 ≤ a ≤ 1 and k ≥ 1.Then, it is convex with respect to λ Proof.Taking first and second order partial derivatives of m 3 (λ) with respect to λ, we get To establish the required result, we only need to show that f (λ) ≥ 0. We first set (xλ) k = t and then observe that It is evident that the above polynomial is greater than 0 at t = 0. Now, upon differentiating the above expression with respect to t, we get 3at 2 + 2(a 2 + 1)t + (3a 2 − 3a + 2) ≥ 0, which proves that Hence, we get f 1 (λ) ≥ 0, as required.

Main Results
In this section, we establish different comparison results between two series as well as parallel systems, wherein the systems' components follow extended Weibull distributions with different parameters.The results obtained are in terms of usual stochastic, dispersive and star orders.
The modeled parameters are connected with different majorization orders.The main results established here are presented in two subsections; in the first, we consider the sample sizes for the two sets of variables to be equal while in the second they are taken to be random.

Ordering results based on equal number of variables
Let us consider two sets of (equal size) of dependent variables {X 1 . . ., X n } and {Y 1 . . ., Y n }, where X i and Y i follow dependent extended Weibull distributions having different parameters . ., l n ), respectively.In the following, we present some results for comparing two extreme order statistics according to their survival functions.
have their associated Archimedean survival copulas to be with generators ψ 1 and ψ 2 , respectively.Further, suppose φ 2 • ψ 1 is super-additive and ψ 1 is log-concave.Then, for 0 < α ≤ 1, we have Proof.The distribution functions of X 1:n and Y 1:n can be written as Therefore, to establish the required result, we only need to prove that , where e v = (e v 1 , . . ., e vn ) and (v 1 , . . ., v n ) = (log λ 1 , . . ., log λ n ).Due to Theorem A.8 of Marshall et al. (2011), we just have to -show that δ(e v ) is increasing and Schur-convex in v. Taking partial derivative of δ(e v ) with respect to v i , for i = 1, . . ., n, we have , for i = 1, . . ., n.Therefore, from (3.6), we can see that δ(e v ) is increasing in v i , for i = 1, . . ., n.Now, the derivative of Γ(v i ) with respect to v i , is given by since ψ 1 is decreasing and log-concave.This implies that Γ(v i ) is decreasing and non-positive Hence, δ(e v ) is Schur-convex in v from Lemma 2.1, which completes the proof of the theorem.
From Theorem 3.1, we can say that if α and β, the shape parameters k and l are the same and scalar-valued, then under the stated assumptions, we can say that the lifetime X 1:n is stochastically less than the lifetime Y 1:n .
Note in Theorem 3.1 that we have considered the tilt parameter α to lie in (0, 1].A natural question that arises is whether under the same condition, we can establish the inequality between two largest order statistics with respect to the usual stochastic order.The following counterexample gives the answer to be negative. ) and (µ 1 , µ 2 , µ 3 ) = (0.77, 0.8, 0.8).It is then easy to see that (log λ 1 , log λ 2 , log λ 3 ) w (log µ 1 , log µ 2 , log µ 3 ).Now, suppose we choose the Gumbel-Hougaard copula with parameters θ 1 = 3 and θ 2 = 0.6 and k = 1.63.Then, Figure 1 presents plots of F X 3:3 (x) and F Y 3:3 (x), from which it is evident that when k ≥ 1, the graph of F X 3:3 (x) intersects with that of F Y 3:3 (x) for some x ≥ 0, where the two distribution functions are given by and So, from this counterexample, we show that in order to establish comparisons results between the lifetimes of X n:n and Y n:n , we require some other sufficient conditions.
Remark 3.1.In Theorem 3.2, if we take k = 1, we simply get the result in Theorem 1 of Barmalzan et al. (2020).
A natural question that arises here is whether we can extend Theorem 3.2 for k ≥ 1.The answer to this question is negative as the following counterexample illustrates.
and the associated Archimedean copula be with generators ψ 1 and ψ 2 , respectively.Further, let φ 2 • ψ 1 be super-additive and αtφ 1 (t) + φ 1 (t) ≥ 0.Then, for 0 < α ≤ 1, we have Proof.The distribution functions of X n:n and Y n:n can be expressed as and So, to establish the required result, we only need to prove that For this purpose, let us define where k = (k 1 , . . ., k n ).Upon, differentiating δ 3 (k) with respect to k i , we get Let us now define a function I 3 (k i ) as which, upon differentiating with respect to k i , yields Now, since for 0 ≤ a ≤ 1 and x ≥ 0, as αtφ 1 (t) + φ 1 (t) ≥ 0, and ≤ 0, which implies δ 3 (k) is Schur-concave in k Lemma 2.1.This completes the proof the theorem.
Remark 3.3.It is useful to observe that the condition "φ 2 • ψ 1 is super-additive" in Theorem 3.5 is quite general and is easy to verify for many well-known Archimedean copulas.For example, we consider the copula φ(t) = e θ t − e θ , for t ≥ 0, that satisfies the relation where t ≥ 0 and the inverse of φ(t) is .
Suppose we have two such copulas, but with parameters α and β, and we want to find the condition for φ 2 • ψ 1 to be super-additive.Taking double-derivative of φ 2 • ψ 1 with respect to x, we get that For illustrating the result in Theorem 3.3, let us consider the following example.
It is useful to observe that the condition "φ 2 • ψ 1 is super-additive" provides the copula with generator ψ 2 to be more positively dependent than the copula with generator ψ 1 .In Theorem 3.3, we have considered φ 2 • ψ 1 to be super-additive which is important to establish the inequality between the survival functions of X n:n and Y n:n when the parameters l and k are comparable in terms of majorization order.We now present a counterexample which allows us to show that if the condition is violated, then the theorem does not hold.
In Theorem 3.3, if we replace the condition αtφ 1 (t) + φ 1 (t) ≥ 0 by tφ 1 (t) + φ 1 (t) ≥ 0, then we can also compare the smallest order statistics X 1:n and Y 1:n with respect to the usual stochastic order under the same conditions as in Theorem 3.3.
and the associated Archimedean survival copulas be with generators ψ 1 and ψ 2 , respectively.Also, let φ 2 • ψ 1 be super-additive and tφ 1 (t) + φ 1 (t) ≥ 0.Then, for 0 < α ≤ 1, we have Proof.The distribution functions of X 1:n and Y 1:n can be written as respectively.Now from Lemma 2.2, the super-additivity of φ 2 o ψ 1 provides Hence, to establish the required result, we only need to prove that where k = (k 1 , . . ., k n ).Upon taking partial-derivative of δ 4 (k) with respect to k i , we get Next, let us define a function I 4 (k i ) as which upon taking partial-derivative with respect to k i yields Now, since for 0 ≤ a ≤ 1 and x ≥ 0, (x − 1)e x + ax + a x e −x ≤ 1, we obtain and since tφ 1 (t) + φ 1 (t) ≥ 0, it shows I 4 (k i ) is increasing in k i , for i = 1, . . ., n.Finally, for i = j, ≤ 0, which implies δ 4 (k) is Schur-concave in k by Lemma 2.1.This completes the proof of the theorem.
In all the previous theorems, we have developed results concerning the usual stochastic order between two extremes, where the tilt parameters for both sets of variables are the same and scalar-valued.Next, we prove another result for comparing two parallel systems containing n number of dependent components following extended Weibull distribution wherein the dependency is modelled by Archimedean copulas having different generators and the tilt parameters are connected in weakly sub-majorization order.To establish the following theorem, we need φ 2 • ψ 1 to be super-additive and tφ 1 (t) + 2φ 1 (t) ≥ 0, where φ 1 is the inverse of ψ 1 .
Then, we have α w β ⇒ X n:n st Y n:n .
Proof.The distribution functions of X n:n and Y n:n can be written as respectively.Now from Lemma 2.2, the super-additivity of φ 2 o ψ 1 provides Hence, to establish the required result, we only need to prove that Now, differentiating δ 5 (α) with respect to α i , we get ∂δ 5 (α) Upon partial derivative of Γ 5 (α i ) with respect to α i , we get and so, δ 5 (α) is decreasing and Schur-concave in α, from Lemma 2.1.This completes the proof of the theorem.
Similarly, we can also derive conditions under which two series systems are comparable when the tilt parameter vectors are connected by weakly super-majorization order, as done in the following theorem.
For the purpose of illustrating Theorem 3.5, present the following example.
As in Counterexample 3.3, in the counterexample below, we show that if we violate the condition "φ 2 • ψ 1 is super-additive" in Theorem 3.5, then the distribution functions of X n:n and Y n:n cross each other.
Next, we establish another result with regard to the comparison of X 1:n and Y 1:n in terms of the hazard rate order where the Archimedean survival copula is taken as independence copula with same generators.
Theorem 3.7.Let X i ∼ EW (α, λ i , k) and Y i ∼ EW (α, µ i , k), for i = 1, . . ., n, and the associated Archimedean survival copulas are with generators ψ 1 = ψ 2 = e −x , that is , X i 's and Y i 's are independent random variables.Also, let 0 < α ≤ 1 and k ≥ 1.Then, Proof.Under the independence copula, the hazard rate function of X 1:n is given by (3.11) By applying Lemma 2.7, it is easy to observe that r X 1:n (x) is convex in λ.Now, upon using Proposition C1 of Marshall et al. (2011), we observe that r X 1:n (x) is Schur-convex with respect to λ, which proves the theorem.
We then present two more results concerning the hazard rate order and the reversed hazard rate order between the smallest order statistics when the tilt parameters are connected in weakly super-majorization order and under independence copula with generator ψ 1 (x) = ψ 2 (x) = e −x , x > 0. The proofs of two results can be completed by using Lemma 3.2 of Balakrishnan et al. (2018).
Theorem 3.8.Let X i ∼ EW (α i , λ, k) and Y i ∼ EW (β i , λ, k), for i = 1, . . ., n, and the associated Archimedean survival copulas be with generators ψ 1 = ψ 2 = e −x , respectively.Then, Theorem 3.9.Let X i ∼ EW (α i , λ, k) and Y i ∼ EW (β i , λ, k), for i = 1, . . ., n, and the associated Archimedean survival copulas be with generators ψ 1 = ψ 2 = e −x , respectively.Then, We now derive some conditions on model parameters, for comparing the extremes with respect to the dispersive, star and Lorenz orders, when the variables are dependent and follow extended Weibull distributions structured with Archimedean copula having the same generator.
Before stating our main resluts, we present the following two lemmas which will be used to prove the main results.
Lemma 3.1.(Saunders and Moran (1978)) Let X a be a random variable with distribution function F a , for each a ∈ (0, ∞), such that (i) F a is supported on some interval (x (ii) The derivative of F a with respect to a exists and is denoted by F a .
Then, X a ≥ * X a * , for a, a * ∈ (0, ∞) and a > a * , iff F a (x)/xf a (x) is decreasing in x.
We now establish some sufficient conditions for the comparison of two extremes in the sense of star order, with the first result being for parallel systems and the second one being for series systems.
Proof.Assume that (λ 1 − λ 2 )(µ 1 − µ 2 ) ≥ 0. Now, without loss of generality, let us assume that λ 1 ≤ λ 2 and µ 1 ≤ µ 2 .The distribution functions of X n:n and Y n:n are and where q = n − p.In this case, the proof can be completed by considering the following two cases.
Case (i): Then, under this setting, the distribution functions of X n:n and Y n:n are respectively.Now, to obtain the required result, it is sufficient to show that On the other hand, the density function corresponding to F λ has the form So, we have Thus, it suffices to show that, for λ ∈ (1/2, 1], From the fact that λ ∈ (1/2, 1], we have t 1 < t 2 for all x ∈ (0, ∞), and so from which we get the derivative of ∆(x) with respect to x to be It is easy to show that the derivatives of t 1 and t 2 with respect to x are Hence, we get As t 1 < t 2 , ∆ < 0 if and only if In this case, we can note that where k is a scalar.We then have (kµ 1 , kµ 2 ) m (λ 1 , λ 2 ).Let W 1:n be the lifetime of a series system having n dependent extended exponentially distributed components whose lifetimes have an Archimedean copula with generator ψ, where W i ∼ EW (α, k, µ 1 ) (i = 1, . . ., p) and W j ∼ EW (α, k, µ 2 ) (j = p+1, . . ., n).From the result in Case (i), we then have W n:n ≤ * X n:n .But, since star order is scale invariant, it then follows that Y n:n ≤ * X n:n .
Case (i): For convenience, let us assume that Then, under this setting, the distribution functions of X 1:n and Y 1:n are and respectively.Now, to obtain the required result, it is sufficient to show that On the other hand, the density function corresponding to F λ has the form So, we have We can then conclude that and e (1−λ) k x k = α+ ᾱt 2 t 2 , and so Ω(x) = t 1 (α + ᾱt 1 )φ (t 1 ) t 2 (α + ᾱt 2 )φ (t 2 ) , whose derivative with respect to x is It is easy to show that Hence, we have is decreasing in t ∈ [0, 1], as required.
It is important to mention that X ≤ * Y implies X ≤ Loenz Y. Therefore, from Theorems 3.10 and 3.11, we readily obtain the following two corollaries.
We now present some conditions for comparing the smallest order statistics in terms of dispersive order.In the following theorem, we use the notation and the associated Archimedean survival copula for both be with generator ψ, 0 ≤ k ≤ 1 and 0 Proof.First, let us consider the function Let us define another function h Upon taking derivative of h(e x ) with respect to x, from Lemma 2.5, we get (3.17) From (3.17), we have h(e x ) to be increasing in x and so h(x) is increasing in x.Again, the second-order partial derivative of h(e x ) with respect to x is given by where 0 ≤ k ≤ 1 and 0 ≤ α ≤ 1.Hence, from (3.19), we see that F (x) is log-convex in x ≥ 0. Under the considered set-up, X 1:n and Y 1:n have their distribution functions as ) and H 1 (x) = 1 − nψ(n( F (λx))) for x ≥ 0, and their density functions as Again, concavity property of ψ/ψ yields ) .
As h(x) is increasing and ψ/ψ is decreasing, ln F (e x ) is concave and ln ψ is convex.Now, using the given assumption that λ , and the fact that ln F (x) ≤ 0 is decreasing, we have ln F (λx) ≥ ln F ((Π n k=1 λ k x) Observe that ln F (e x ) is concave, ln ψ is convex, and λ ∈ I + or D + .Hence, from Chebychev's inequality, it follows that ln F (λx So, from (3.22), we have L 1 (x; λ) ≥ λx.Moreover, we have h(x) to be decreasing as h(x) is increasing and so, h(x) is convex.Therefore, using λ ≤ ( n k=1 λ k ) 1 n , we have Once again, by using Chebychev's inequality, increasing property of h, decreasing property of ψ/ψ and λ ∈ I + or D + , we obtain . (3.23) Now, using the inequality in (3.23), (3.20) and (3.21), we obtain, for all x ≥ 0, ), for all x ∈ (0, 1).This completes the proof of the theorem.

Ordering results based on ramdom number of variables
In this subsection, we will consider two sets of dependent N 1 and N 2 variables {X 1 , . . ., X N 1 } and {Y 1 , . . ., Y N 2 }, where X i follows EW (α i , λ i , l i ) and Y i follows EW (β i , λ i , k i ) coupled with Archimedean copulas having different generators.Under this set-up, we develop different ordering results based on the usual stochastic order, where in the model parameters are connected by different majorization orders.Here, the number of observations N 1 and N 2 are stochastically comparable, independent of X i s and Y i s, respectively.In the following theorem, if N 1 ≤ st N 2 and the tilt parameter α ∈ (0, ∞), then under the same conditions as in Theorem 3.1, we can extend the corresponding results as follows, and their associated Archimedean survival copulas be with generators ψ 1 and ψ 2 , respectively.Let N 1 be a non-negative integer-valued random variable independently of X i s and N 2 be a non-negative integer-valued random variable independently of Y i s.Further, let N 1 ≤ st N 2 , φ 2 • ψ 1 be super-additive and ψ 1 be log-concave.Then, for 0 < α ≤ 1, we have (log λ 1 , . . ., log λ n ) w (log µ 1 , . . ., log µ n ) ⇒ Y 1:N 2 st X 1:N 1 .
Proof.The survival functions of X 1:n and Y 1:n are given by We similarly extend Theorem 3.2 by considering N 1 and N 2 to be random.The result provides us sufficient conditions for comparing two parallel systems, wherein the components lifetimes follow dependent extended Weibull family of distributions having different scale parameters.
Theorem 3.14.Let X i ∼ EW (α, λ i , k) (i = 1, . . ., n) and Y i ∼ EW (α, µ i , k) (i = 1, . . ., n), where 0 < k ≤ 1 and their associated Archimedean copulas be with generators ψ 1 and ψ 2 , respectively.Let N 1 be a non-negative integer-valued random variable independently of X i s and N 2 be a non-negative integer-valued random variable independently of Y i s.Also, let N 1 ≤ st N 2 and φ 2 • ψ 1 be super-additive.Then, for 0 < α ≤ 1, we have Proof.The distribution functions of X 1:n and Y 1:n are given by The following theorem is an extension of Theorem 3.3 for the case when N 1 and N 2 are random when the shape parameters l and k are connected in majorization order and the tilt parameters α and β are equal, scalar-valued and lie between 0 and 1.
Theorem 3.15.Let X i ∼ EW (α, λ, k i ) (i = 1, . . ., n) and Y i ∼ EW (α, λ, l i ) (i = 1, . . ., n), and the associated Archimedean copulas be with generators ψ 1 and ψ 2 , respectively.Let N 1 be a non-negative integer-valued random variable independently of X i s and N 2 be a non-negative integer-valued random variable independently of Y i s.Further, let N 1 ≤ st N 2 , φ 2 • ψ 1 be superadditive and αtφ 1 (t) + φ 1 (t) ≥ 0.Then, for 0 < α ≤ 1, we have The above result states that the survival function of X N 1 :N 1 is stochastically less than that of Y N 2 :N 2 , whereas the following theorem presents conditions under which the survival function of X 1:N 1 is stochastically less than that of Y 1:N 2 .These can be proved using the same arguments as in the proofs of Theorems 3.3 and 3.13.Theorem 3.16.Let X i ∼ EW (α, λ, k i ) (i = 1, . . ., n) and Y i ∼ EW (α, µ, l i ) (i = 1, . . ., n) and the associated Archimedean survival copulas be with generators ψ 1 and ψ 2 , respectively.Let N 1 be a non-negative integer-valued random variable independently of X i s and N 2 be a non-negative integer-valued random variable independently of Y i s.Further, let N 1 ≤ st N 2 , φ 2 • ψ 1 be super-additive and tφ 1 (t) + φ 1 (t) ≥ 0.Then, for 0 < α ≤ 1, we have We finally present similar comaparison reults when the tilt parameter vectors are connected in majorization order and the parallel and series systems have extended Weibull components.
Let N 1 be a non-negative integer-valued random variable independently of X i s and N 2 be a non-negative integer-valued random variable independently of Y i s.Further, let N 1 ≤ st N 2 and φ 2 • ψ 1 be super-additive.Then, we have

Concluding remarks
The purpose of this article is to establish ordering results between two given sets of dependent variables {X 1 , . . ., X n } and {Y 1 , . . ., Y n }, wherein the underlying variables are from extended Weibull family of distributions.The random variables are then associated with Archimedean (survival) copulas with different generators.Several conditions are presented for the stochastic comparisons of extremes in the sense of usual stochastic, star, Lorenz, hazard rate, reversed hazard rate and dispersive orders.Further, we have derived inequalities between the extreme order statistics when the number of variables in the two sets are random satisfying the usual stochastic order.We have also presented several examples and counterexamples to illustrate all the established results and their implications.

Disclosure statement
No potential conflict of interest was reported by the authors.
Funding SD thanks the Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore for the financial support and the hospitality during her stay.
Definition 2.1.A vector c is said to be • majorized by another vector d (denoted by c m d) if, for each l = 1, . . ., n − 1, we have l i=1 c i:n ≥ l i=1 d i:n and n i=1 c i:n = n i=1 d i:n ; • weakly submajorized by another vector d (denoted by c w d) if, for each l = 1, . . ., n, we have n i=l c i:n ≤ n i=l d i:n ; • weakly supermajorized by another vector d, denoted by c w d, if for each l = 1, . . ., n, we have l i=1 c i:n ≥ l i=1 d i:n .

Figure 1 :
Figure 1: Plots of F X 3:3 (x) and F Y 3:3 (x) in Counterexample 3.1, where the red line corresponds to F X 3:3 (x) and the blue line corresponds to F Y 3:3 (x).

Figure 4 :F
Figure 4: Plots of F X 3:3 (x) and F Y 3:3 (x) as in Counterexample 3.4.Here, the red line corresponds to F X 3:3 (x) and the blue line corresponds to F Y 3:3 (x).
+ ) ⊂ (0, ∞) and has density f a which does not vanish in any subinterval of (x