Stochastic orderings between two finite mixture models with inverted-Kumaraswamy distributed components

In this paper, we consider two finite mixture models (FMMs), with inverted-Kumaraswamy distributed components' lifetimes. Several stochastic ordering results between the FMMs have been obtained. Mainly, we focus on three different cases in terms of the heterogeneity of parameters. The usual stochastic order between the FMMs have been established when heterogeneity presents in one parameter as well as two parameters. In addition, we have also studied ageing faster order in terms of the reversed hazard rate between two FMMs when heterogeneity is in two parameters. For the case of heterogeneity in three parameters, we obtain the comparison results based on reversed hazard rate and likelihood ratio orders. The theoretical developments have been illustrated using several examples and counterexamples.


Introduction
The FMMs have been widely used in many study areas, including biology, reliability, and survival analysis.As a result, both theorists and practitioners have shown a great deal of interest in these models.Due to its unique ability to model heterogeneous data, whose pattern cannot be produced by a single parametric distribution, the mixture model (MM) has acquired a lot of appeal.An unknown model is developed by mixing a collection of homogeneous subpopulations (infinite) in order to capture this heterogeneity.Keep in mind that the mixing is carried out over a latent parameter, which is regarded as a random variable (RV) and chosen from an unknown mixing distribution.Here, we refer to it as mixing proportions or weights throughout this paper.There are many circumstances in which FMMs spontaneously occur.For more information on the different applications of FMMs, see Lindsay (1995), McLachlan and Peel (2000), Amini-Seresht and Zhang (2017), and Wu (2001).For instance, • A FMM can be used in reliability theory to model the "failure time" of a system.The model assumes that the "failure time" is a mixture of two or more distributions, as usually there is more than one reason causing the failures of a component or system (Amini-Seresht and Zhang, 2017).
• To study the distribution of time to death after a major cardiovascular surgery, FMM is useful.Here, one may consider that the lifetime of such patients after surgery contains three phases of time.In the first phase, that is, immediately after surgery, the death risk is relatively high.In the next phase, the hazard rate remains constant upto some certain time.Then, in the final phase, the risk of death of the patient increases.The convenient way to model this situation is to adopt a MM with three components.In each phase, separate parametric model can be assigned to each (here three) components (McLachlan and Peel, 2000).
• A FMM can be used in biological sciences to model the distribution of gene expression levels across different cell types.The model assumes that the gene expression levels are a mixture of two or more distributions, such as a normal distribution and a gamma distribution (Schork et al., 1996 andMcLachlan andPeel, 2000).
Also, FMMs are used in a variety of real-life applications, such as clustering, image segmentation, anomaly detection, and speech recognition.They are also used in medical diagnosis, market segmentation, and customer segmentation, etc.In this paper, we have considered two FMMs for inverted-Kumaraswamy (IK) distributed components.
The topic of stochastic comparisons between two FMMs has been extensively studied.For more details, see Shaked and Shanthikumar (2007), Navarro (2008), Navarro (2016), Amini-Seresht and Zhang (2017), and so on.Hazra and Finkelstein (2018) studied stochastic comparisons of finite mixtures (FMs) where the subpopulations are from semiparametric models, that is, the scale model, proportional hazard rate model, and proportional reversed hazard rate model.Barmalzan et al. (2021) focused on two finite α-MMs and established sufficient conditions for comparing two α-MRVs.Please see, Asadi et al. (2018) for some properties of the α-MM.Sattari et al. (2021) investigated MMs with generalized Lehmann distributed components and presented several ordering outcomes.Barmalzan et al. (2022) studied two FMMs with locationscale family distributed components and established some stochastic comparison results between them in terms of the usual stochastic order and the reversed hazard rate order.By utilizing the majorization idea, Nadeb and Torabi (2022) have been studied a stochastic comparison for two FMs in terms of usual stochastic order, hazard rate order, and reversed hazard rate order.Panja et al. (2022) established ordering results between two finite mixture random variables (FMRVs), where the mixing components are based on proportional odds, proportional hazards, and proportional reversed hazards models.Kayal et al. (2023) obtained some ordering results between two FMMs considering general parametric families of distributions.Mainly, the authors established sufficient conditions for usual stochastic order based on p-larger order and reciprocally majorization order.Very recently, Bhakta et al. (2023) considered similar general parametric families of distributions as in Kayal et al. (2023), and then examined various ordering results with respect to usual stochastic order, hazard rate order, and reversed hazard rate order between two FMMs.
Inverted distributions have several applications in various fields, including econometrics, life testing, biology, engineering sciences, and medicine.Additionally, it is used in reliability theory, survival analysis, financial literature, and environmental research.For more details on inverted distributions and its applications, see Abd EL-Kader (2013).Abd AL-Fattah et al. (2017) developed the IK distribution by using the transformation x = t −1 − 1 from the Kumaraswamy (K) distribution, that is, T ∼ K(α, β), where α and β are the shape parameters.Then X has a IK distribution with cdf and pdf as and respectively, where α and β both are shape parameters.We note that the curves of the pdf and hazard function show that the IK distribution exhibits a long right tail, compared with other commonly used distributions.As a result, it affects long term reliability predictions, producing optimistic predictions of rare events occurring in the right tail of the distribution compared with other well-known distributions.Here, we use the notation X ∼ IK(α, β) for convenience.Many well-known distributions fall under the IK distribution as special cases, e.g., Lomax distribution (for β = 1), Beta Type II distribution (for α = 1) and the log-logistic distribution (for α = β = 1) (Abd AL-Fattah et al., 2017).Also, using appropriate transformations, the IK distribution can be transformed to many well-known distributions such as exponentiated exponential and Weibull, generalized uniform, generalized Lomax, beta Type II and F-distribution, Burr Type III and log logistic distributions (Abd AL-Fattah et al., 2017).Abd AL-Fattah et al. (2017) showed with many real date sets how well the IK distribution fits those real data.
This paper focuses on the stochastic comparison results between two finite mixture models follow IK distributed components.The goal of this paper is to obtain sufficient conditions, for which two finite mixture random variables with IK distributed component lifetimes are comparable in the sense of the usual stochastic order, reversed hazard rate order, likelihood ratio order, and ageing faster order in terms of reversed hazard rate order.
The main contributions and organization of the paper are as follows.In the next section, we present several definitions and lemmas which are essential to obtain our main results.Section 3 contains three subsections, with a description on the proposed model.In Subsection 3.1, we establish usual stochastic order between two FMMs based on the concepts of the weak supermajorization and weak submajorization orders.Subsection 3.2 deals with the ordering results when there is heterogeneity in two parameters.Here, we obtain comparison results with respect to the usual stochastic order and ageing faster order in terms of the reversed hazard rate.In Subsection 3.3, we examine ordering results between the FMMs with respect to the reversed hazard rate and likelihood ratio orders.Besides the theoretical contributions, we present many examples and counterexamples for the validation and justification.

The basic definitions and some prerequisites
This section presents some preliminary definitions and results, which are essential to establish our main results in subsequent section.Let U and V be two continuous and non-negative independent random variables with pdfs f U (•) and g Definition 2.1.The random variable U is smaller than V in the sense of (i) usual stochastic order (abbreviated as (ii) reversed hazard rate order (abbreviated as (iii) likelihood ratio order (abbreviated as (iv) ageing faster order with respect to reversed hazard rate order (abbreviated as Shaked and Shanthikumar (2007) provide a comprehensive overview of stochastic orders and their applications.The idea of majorization and related orders are then discussed, which are highly helpful in establishing the main results in the subsequent sections.Let R n be an n-dimensional Euclidean space.Let ς = (ς 1 , . . ., ς n ) and ε = (ε 1 , . . ., ε n ) be two real vectors.Further, let ς (1) ≤ . . .≤ ς (n) and ε (1) ≤ . . .≤ ε (n) denote the increasing arrangements of the components of the vectors ς and ε, respectively.
Definition 2.2.(Marshall et al., 2011) The vector ς is said to be and n j=1 ς (j) = n j=1 ε (j) ; • weakly supermajorize the vector ε (abbreviated as It is obvious that the majorization order implies both weak supermajorization and weak submajorization orders.In the following, we will present a definition that demonstrates how the Schur-convex function is able to maintain the ordering of the majorization.Definition 2.3.(Marshall et al., 2011) A real valued function Ξ defined on a set S ⊆ R n is said to be Schur-convex (Schur-concave) on S, if ς m ≽ ε implies Ξ(ς) ≥ (≤) Ξ(ε) for any ς, ε ∈ S.
Definition 2.4.Let P = {p ij } and Q = {q ij } be two m×n matrices.Further, let p R 1 , . . ., p R m and q R 1 , . . ., q R m be two rows of P and Q, respectively in such a way that each of these quantities is a row vector of length n.Then, P is said to be chain majorized by Q (abbreviated as A T -transform matrix has the form T = ϖI + (1 − ϖ)Π, where 0 ≤ ϖ ≤ 1 and Π is a permutation matrix that just interchanges two coordinates that is, row and column.Define a matrix Lemma 2.1.A differentiable function Υ : R + 4 → R + satisfies Υ(P ) ≥ Υ(Q) for all P , Q such that P ∈ L 2 , and P ≫ Q if and only if (i) Υ(P ) = Υ(P Π) for all permutation matrices Π, and for all P ∈ L 2 and; and for all P ∈ L 2 , where Υ ij (P ) = ∂Υ(P ) ∂p ij .

Model description and results
Consider n number of homogeneous independent subpopulations of items, denoted by π 1 , . . ., π n , which are infinite in nature.Let X i be the lifetime of an item of π i , i = 1, . . ., n. Further, let R(X, p) denote the RV, representing the mixture of items taken from π 1 , . . ., π n , where X = (X 1 , . . ., X n ) and p = (p 1 , . . ., p n ).Here, p i 's (> 0) are known as the mixing proportions with n i=1 p i = 1.Thus, the survival function of the mixture random variable (MRV) R(X, p) is given by (see McLachlan and Peel, 2000) FR(X,p) where FX i (x) is the survival function of the items of π i .For more details on formulation of mixture distribution, readers are referred to Chen (2017), McLachlan et al. (2019), andMcLachlan andPeel (2000).Because failure rate is a conditional characteristic, the equivalent mixture failure rate is defined using modified conditional weights (on the condition of survival function in x ≥ 0).For detailed discussion on mixture failure rate, one may refer to the references Navarro and Hernandez (2004), Finkelstein (2008), Cha and Finkelstein (2013), and so on.In this paper, we consider that the lifetime of the unit of i-th subpopulation, denoted by where α = (α 1 , . . ., α n ) and β = (β 1 , . . ., β n ).
In this section, we obtain ordering results between two FMMs under three different scenarios.In particular, we consider heterogeneity in one parameter, two parameters, and three parameters.

Ordering results for MMs when heterogeneity presents in one parameter
In this subsection, we obtain stochastic ordering results, by considering heterogeneity in one parameter.The first result studies usual stochastic ordering between two FMMs, when heterogeneity is present in mixing proportions.Furthermore, in this model, we consider α as a common shape parameter vector, and β as a fixed shape parameter.The following lemma is useful in proving the upcoming theorem.
Proof.The proof is straightforward, and hence it is omitted.
The following corollary is an immediate consequence of Theorem 3.1 using the well-known result m ≼⇒≼ w .
Corollary 3.1.Based on the assumptions and conditions as in Theorem 3.1, we have To validate Theorem 3.1, we next provide an example.
Proof.See Appendix.
The following example illustrates Theorem 3.2, for n = 3.
In the upcoming theorem, we consider heterogeneity in the shape parameter α, while β is fixed.In addition, we assume the mixing proportion vector p to be common.Denote X α * ,β = (X α * 1 ,β , . . ., X α * n ,β ).Theorem 3.3.Under the setup as in Assumption 3.1, with Proof.See Appendix.
The following example illustrates Theorem 3.3.

Ordering results for MMs when heterogeneity presents in two parameters
In the previous subsection, we have assumed heterogeneity in one parameter.There are various situations, where more than one parameter is heterogeneous.In this subsection, we consider heterogeneity in two parameters and obtain some ordering results.The results have been established using the concept of chain majorization between the parameter-matrices of two MMs.The following lemma is useful in proving the upcoming theorem.
Proof.The proof is straightforward, and thus it is omitted.
To validate Theorem 3.4, we consider the following example.
It is well-known fact that a finite product of T -transform matrices with a common structure yields a T -transform matrix.Using this, the following corollary is immediate consequence of Theorem 3.5.Corollary 3.2.Consider k number of T -transformed matrices with a common structure, denoted by T ω 1 , . . ., T ωn .Then, under the setup in Theorem 3.5, with (p, α) ∈ L n and β ∈ (0, 1), From the result in corollary 3.2, it is an obvious question that "does the result in corollary 3.2 hold if the T -transform matrices have different structures?"In the next theorem we discuss this issue.We observe that the similar result to corollary 3.2 holds with an additional assumption.
Preceding results of this subsection deal with the stochastic comparison of two MRVs when the matrix (p, α) changes to another matrix (p * , α * ) with fixed β.In the upcoming results, we assume that the matrix (p, β) changes to (p * , β * ) with fixed α.First, we consider two subpopulations.The following lemma is useful to establish the next theorem.
Proof.The proof is straightforward, and thus it is omitted.
In order to justify Theorem 3.7, an example is provided.
Next, we present a result, dealing with n number of subpopulations.
be the sfs of the MRVs R n (X α,β ; p) and R n (X α,β * ; p * ), respectively.For (p, β) ∈ L n and fixed α > 0, Proof.The proof is similar to that of Theorem 3.5, and thus it is omitted.
The following corollary can be established from Theorem 3.8 using the arguments similar to corollary 3.2.

Corollary 3.3. Consider k number of T -transform matrices, denoted by
T ω 1 , . . ., T ω k having a common structure.Then, under the setup in Theorem 3.8, for (p, β) ∈ L n and fixed α > 0, The next theorem proves a result associated with k (≥ 2) number of T -transformed matrices having different structures.
We end this subsection with ageing faster order between two MRVs.Here, we assume that mixing proportions and one of the shape parameter vectors are varying.We recall that using ageing faster order one is capable to compare the relative ageings of two engineering systems.In the following theorem, we study the ageing faster order in terms of the reversed hazard rate function.
The following example illustrates Theorem 3.10, for n = 3.

Ordering results for MMs when heterogeneity presents in three parameters
In the previous subsection, we assume that two parameters are heterogeneous.In this subsection, we present the ordering results considering heterogeneity in three parameters.We mainly obtain the stochastic comparison results between two MRVs with respect to reversed hazard rate and likelihood ratio orders.First, we provide reversed hazard rate order between R n (X α,β ; p) and R n (X α * ,β * ; p * ).
In the next theorem, under some certain parameter restrictions, likelihood ratio ordering between two MRVs R n (X α,β ; p) and R n (X α * ,β * ; p * ) is presented.Proof.See Appendix.
The following example illustrates Theorem 3.12, for n = 3.

Concluding remarks
In this paper, MMs are considered as suitable tools for analyzing population heterogeneity.We are interested in heterogeneous populations with distinct components such as lifetime.We have derived some sufficient conditions for the comparison between two FMMs of IK distributed components with respect to usual stochastic order, reversed hazard rate order, likelihood ratio order, ageing faster order in terms of reversed hazard rate order corresponding to the heterogeneity in the model parameters in the sense of some majorization orders, namely, weakly supermajorized and weakly submajorized order.Here, we have considered heterogeneity in one parameter, two parameters, and three parameters, respectively.We have presented some numerical examples and counterexamples to illustrate the established results in this paper.
The presented results of this paper are mostly theoretical.However, one may find some applications of the established results.Below, we consider an example.
Assume two engineering systems, with components produced by different companies.It is further reasonable to assume that the components have different reliability characteristics.Each of the components can operate in n operational regimes with corresponding different probabilities p i and p * i .Let the lifetimes of the ith regime have different distributions, say IK(α i , β) and IK(α * i , β), respectively.Then, the important question is: which of these two systems perform better in some stochastic sense?By using Theorem 3.1, we conclude that under the condition p * ≼ w p, the first system performs better than the second system.Similar applications can be found for the other established results.
It is naturally be of interest that one can extend this work with respect to some stronger stochastic orders such as hazard rate order, or with respect to the variability orders, like star order, dispersive order, Lorenz order, right-spread order, convex transform order, increasing convex order etc.Future research on the generalizations of these findings may be considered.
Proof of Theorem 3.3.
Proof.Without loss of generality, we assume p 1 ≥ . . .≥ p n > 0. Thus, from the assumptions made, we have α From (A7), we obtain the partial derivative of ψ(α) with respect to α i as which is clearly non-positive.Thus, ψ(α) is decreasing with respect to α i .Further, using Lemma 3.1, for 1 ≤ i ≤ j ≤ n, we obtain after some calculations as Now, from Lemma 2.4 of Bhakta et al. (2023) it can be shown that ψ(α) is Schur-convex with respect to α.Thus, the remaining proof of the theorem follows from Theorem 3.A.8 of Marshall et al. (2011), p. 87.This completes the proof of the Theorem.
Proof of Theorem 3.4.
Proof.The theorem will be proved if the conditions of Lemma 2.1 are satisfied.Here, FR 2 (X α,β ;p) (x) is clearly permutation invariant on L 2 .Further, On differentiation (A14) with respect to x, we obtain Our goal is to show that ξ(x) is non-positive.Now, using the model assumptions, from (A15), we obtain Consider 1 ≤ i, j ≤ n.Under the assumptions made, from (A16), we obtain ∆ i,j,k,l (x) ≤ 0, and hence ξ(x) ≤ 0. This completes the proof of the theorem.
Proof of Theorem 3.11.
Proof.The proof will be completed if we show that is non-increasing with respect to x > 0. The partial derivative of (A17) with respect to x is ∂ ∂x F Rn(X α,β ;p) (x) F Rn(X α * ,β * ;p * ) (x) where For 1 ≤ i, j ≤ n, under the assumptions made, it can be shown that ξ(x) ≤ 0, as desired.
Proof of Theorem 3.12.
Proof.To prove the result, it is required to establish that is non-decreasing with respect to x > 0. The derivative of ξ(x) with respect to x is obtained as Consider 1 ≤ i, j ≤ n.Under the assumptions made, from (A21), we obtain ξ ′ (x) ≥ 0, implying that ξ(x) is non-decreasing with respect to x > 0. Hence the result follows.