Ordering results for extreme claim amounts based on random number of claims

Ordering results for extreme c laim amoun ts based on ra ndom n u m b er of claims Sangita Das a ∗ a The o r e tic a l Statistics a nd Math ematics Unit, Indian Statistic al Institute, Bangalor e-560059, Indi a T o app e ar in Ric er che di Matematic a. Abstract Consider tw o seq uences of heterogeneous and indep endent p ortfolio s of risks T 1 , T 2 , . . . and T ∗ 1 , T ∗ 2 , . . . and, let N 1 and N 2 be t w o po sitive integer-v a lued r a ndom v ariables, indep endent o f T ′ i and T ∗ i , res pec tiv ely . In this article, we in vestigate differen t stochastic inequalities involving min { T 1 , . . . , T N 1 } and min { T ∗ 1 , . . . , T ∗ N 2 } , and max { T 1 , . . . , T N 1 } and max { T ∗ 1 , . . . , T ∗ N 2 } in the sense o f usual sto chastic orde r and reversed hazard rate order concerning maltiv ariate chain ma jor ization order. These new r esults strengthen and gener alize some of the w ell known results in the literature, including Ba rmalzan et al. (20 17), Ba lakrishnan et a l. (2018) and Kundu and Chowdh ur y (20 21) for the ca se o f random claim sizes. Different numerical examples are provided to highlight the applicability of this w ork. Finally , some in teresting applications of our results in reliability theory and auction theory ar e presented. Keyw ords: Row weak ma jor ization order, matrix chain ma jor ization order, usual stochastic o rder, reversed hazard r ate o rder. Mathematics Sub ject Cl assification: 6 0E15 ; 62 G30; 60 K10; 90B2 5. 1. In tro duction A se mipa rametric family of life distributions is a family that cor resp onds to a cla s s of statistical mo dels which pro vides a fle x ible alter native framew ork compared to fully par ametric approa ches. These mo dels ar e extensively applied in insurance and actuaria l science for modeling mortality , surviv al, and time-to-dec r ement pro cesses (see Marshall and Olkin (2007 ) and Marshall and Olkin (1997)). A ra ndom v ariable X is said to hav e a semiparametr ic family of distr ibution if its s urviv al function can be written as ¯ F X ( x ) = ¯ F ( · ; α ) , α > 0 . (1.1) Denote ¯ F ( · ; α ), f ( · ; α ), r ( · ; α ) and ˜ r ( . ; α ) are the surviv al, dens it y , hazar d rate and r eversed hazard r ate functions, r esp ectively . Among all semipara metric mo dels, the sca le and prop ortiona l hazard models are one of the mo st widely recog nized semipara metric mo dels commonly used in reliability , sur viv al s tudies and nu merous o ther disciplines (see Finkelstein (200 8); Finkelstein and Cha (2013); Kumar and Klefsj¨ o (1994); Cox (199 2)). Particularly in insurance analysis, scale mo dels play a crucial role in asses sing risks, predicting claims, and estimating policy holder b ehavior. Depending on differen t factors, such as cov erage, regio n, o r inflation, the size of cla ims can v ary . T o handle these situations, scale mo dels are used to adjust these fa c to rs without c ha nging the shap e of the baseline dis tr ibution. Similar ly , prop ortiona l hazard rate mo dels ha v e also bee n widely used to determine the timing o f events such as mortality , illness , p olicy lapses, disability , and claims w hile accounting for multiple risk factors. In pa r ticular, it is useful in life insurance, health insura nce, ∗ Email address: sangita das118@gmail.com, 1 disability insur ance, and p olicy reten tion studies b ecause it allows for the inclusion o f v ar ious cov ariates (e.g ., age, gender, lifestyle, economic conditions) without making strict assumptions ab out the underlying bas e line hazard function. In an insur ance pe rio d, for i = 1 , . . . , n , let U i be a p ositive random v ariable repr e sent the claim se verit y (or loss a mount) for the i th risk, a ssuming that a c laim o ccurs. Associa ted with U i , let J i be a B ernoulli random v a riable with E ( J i ) = p i , such that J i =  1 , if the i th insured p erso n makes ra ndom claim U i , 0 , otherwise , where i = 1 , . . . , n . Then, the quantit y T i = J i U i represent the individua l cla im amount corresp o nd- ing to the i th insure d p erson in a por tfolio o f risks { T 1 , . . . , T n } . Denote T 1: n = min { T 1 , . . . , T n } and T n : n = max { T 1 , . . . , T n } are the smallest and lar gest claim a mo unt s, resp ectively , cor resp onding to the po rtfolio of risks { T 1 , . . . , T n } . Let us denote ψ : (0 , 1 ) → (0 , ∞ ) b e a differentiable function. Also denote ψ ( p ) = ( ψ ( p 1 ) , . . . , ψ ( p n )) . In the context of risk ana lysis, the function ψ ( · ) r epresents the transfor med vector of occurr ence proba bilities, will be used to mo dify probability distributions corresp onding to risk assessment, premium c alculation, r einsurance pricing, a nd ca pital allo ca tion. Insurers use these transfor mations to change actual probabilities to displa y v a rious r is k attitudes, regulato r y requir ement s, or pricing str a tegies. In partic- ular, decre asing tra nsformation will help the higher probability even t to get less w eight, and by the help of conv e x transformation, a risk-av erse insur er increases the probability of lar ge claims. In insurance analysis, it is of great imp orta nce to compare the amount of extre me claims to analyze the risk measur e , as they provide imp o rtant information to determine the annual lose or gain. Therefor e, studying sto chastic pr op erties of extreme claim a mounts are imp ortant b oth in practical and theoretical p oint of view. In this direction, a large amount of research hav e been done by many authors by considering hetero geneous independent/interdependent po r tfolios of r isk having different general families o f risk severities for the case of fixed claim size. Among them, Barmalza n et a l. (2017 ) hav e co nsidered indepe ndent hetero g eneous general scale severities and es tablished different ordering proper ties b etw een smallest claim amoun ts with r esp ect to usual s to chastic order when the matrix o f parameters ( ψ ( p ) , α ; n ) changes to ( ψ ( p ∗ ) , β ; n ) in the sense of m ultiv ariate c hain ma jor ization o rder and hazard rate order when the scale parameters are connected with w eakly sup erma jorization order as well as weakly subma joriza tio n order. Consequently , Balak r ishnan et al. (2018) hav e inv estigated the ordering pr op erties of the larg est claim amounts arising from tw o s e ts of heterogeneous p ortfolio s for independent observ ation according to the usual sto chastic o r der when the matrix o f parameter s ( ψ ( p ) , α ; n ) changes to ( ψ ( p ∗ ) , β ; n ) in the sense of row weakly ma jo rization o rder. Subsequently , Zhang e t al. (20 19) hav e established sufficien t conditions for compar ing extreme cla im amounts arising from t wo sets of hetero geneous insurance por tfolios accor ding to v ario us sto chastic orders. Recently Kundu et al. (2 023) hav e in v estigated if the model par ameters and the transfor m v ectors a re connected with weakly subma jo r ization o rder then the la rgest c la im amounts a re comparable with re s pec t to us ual sto chastic and rev ersed hazard rate o rders for both indepe nden t/interdependent set-up. Different comparison results o f extreme claim amounts ha ving indep endent/in terdep e ndent general claim severities hav e developed by s everal resear chers, like lo cation-sca le sev erities (see , Barmalzan et al. (2020 ), Sa meen et al. (202 0) Das et al. (2021) and Das et al. (202 2)), transmuted-G model (see Nadeb et al. (202 0a)), prop ortiona l o dds mo del (see Panja et al. (2024)). Randomness and v ar iability are fundamental a sp e ct of real-life data due to occurr ence of the ev ent , mea- surement errors, and differ e nces among samples. In many biological and agr icultural studies, o btaining a fix e d sample size can b e challenging, as observ a tions are often los t due to v ar ious facto rs, a s a result the sample size bec omes random. In insurance analysis, the n um ber o f claims received b y an insurer from all its custo mer s ov er the y ear is random. In a c tuarial science, randomness a rises fr om differ ent sources, clos e ly related to uncertaint y in future ev ents. In finance a nd risk management, it is often in ter esting to get information ab out the minimum a nd max im um loss from a por tfolio of loans or securities when the num ber of assets o r liabilities is random (see Es c udero and Or teg a (2 008)). In transp orta tion theor y , to measure the acciden t-free distance of a shipmen t, such a s explosives, the num b er of defective items may b e random which will make a p o tent ial accident (see Shak ed and W ong (19 97)). I n biostatistics, for the case of a cance r patient treatment the n um ber of carcinoge nic cells that are left to b e activ a ted after the first shot of the trea tmen t is giv en will be random (see Co oner et al. (2 0 07)). In h ydrology , the study inv o lving annual maximum rainfall or flo o ds , it is very 2 common that the num b er of storms or floo d per y ear is r andom ra ther than fixed (se e Koutsoyiannis (200 4)). Let X 1 , X 2 , . . . b e a sequence of independent random sample a nd N be a p ositive int eger-v alued random v ariable, independent of X i . Denote X 1: N = min { X 1 , . . . , X N } and X N : N = max { X 1 , . . . , X N } are the random minima and random ma xima, res pec tively . The random v ariable X 1: N naturally emer ges in transp o r tation theory to r epresent the accident-free distance of a shipment, where N defectiv e items may detonate after trav eling X 1 , . . . , X N miles, resp ectively , le a ding to a p otential a ccident. In a c tua rial s c ience, X N : N and X 1: N represent the larges t and smallest a mount of cla ims in a particular time p erio d. Th us, comparing tw o random maxima or minima sto chastically is of significant interest to rese archers o wing to its imp o rtant applications across v ar ious area s of statistics . In this context, relatively little work has b een done b y res e archers on general families o f distributions. Among them, in Sha ked and W ong (1997) the authors hav e considered t wo different independent and identically distributed samples X 1 , X 2 , . . . and Y 1 , Y 2 , . . . having the same (random) sample size ( N ) and established that if X i ≤ st Y i , for i = 1 , . . . , N then X 1: N ≤ st Y 1: N and X N : N ≤ st Y N : N . Also, they hav e proved that if N 1 and N 2 , a re connected with Laplac e transfor m and Lapla c e tr ansform ratio order s then the usual sto chastic, hazar d rate, reversed hazard rate and likelihoo d ratio o rders are hold b etw een X 1: N 1 , X 1: N 2 and X N 1 : N 1 , X N 2 : N 2 , where N 1 and N 2 are tw o p ositive in teger-v alued random v ar iables, independent of X ′ i s. F ollowing this, Bartos z ewicz (200 1) hav e show ed that if conv ex, star and s uper -additive orders hold betw een X 1 and Y 1 , then X 1: N and X N : N are sma ller tha n Y 1: N and Y N : N , res pe c tively . After that, Li and Zuo (2004) hav e established some certain conditions such that the right s pread and the incr easing conv ex order ings hold b etw een X N : N and Y N : N . F urther more, they hav e show e d that tota l time on test tra nsform and increa sing concav e o rderings hold betw een X 1: N and Y 1: N . Subsequently , Ahmad and Kayid (2007 ) ha ve r e po rted that reversed preserv ation pro per ty o f righ t sprea d and total time on test tr ansform orders under random minima and maxima. V ery recently , in Kundu et a l. (2024) the authors hav e proved s everal compar ison r esults o f random minima and maxima from a random num b er of non- identical rando m v ariables. T o get an overview of the r esults on sto chastic compariso ns of random maxima and minima, the in terested reader may refer to Nanda a nd Shaked (200 8), Mane s h et al. (20 23), Das and Ba lakrishnan (2026). Motiv ated from these work in literature, let us c onsider { U 1 , . . . , U N 1 } and { V 1 , . . . , V N 2 } b e tw o sets of independent and hetero geneous random v ariables such that i = 1 , . . . , N 1 , U i ∼ ¯ F ( x ; α i ) a nd for i = 1 , . . . , N 2 , V i ∼ ¯ F ( x ; β i ) . Also, let T i = J i U i and T ∗ i = J ∗ i V i , where J i and J ∗ i be indep endent Bernoulli random v ariables , indep endent of U ′ i s and V ′ i s such that E ( J i ) = p i and E ( J ∗ i ) = p ∗ i and, N 1 and N 2 be another t wo positive integer-v alued rando m v ariables independently o f T ′ i s a nd T ∗ i ′ s, resp ectively . Under the ab ov e set-up, the smallest and largest claim amount aris ing from T 1 , . . . , T N 1 and T ∗ 1 , . . . , T ∗ N 2 are, r esp ectively , denoted by T 1: N 1 = min { T 1 , . . . , T N 1 } , T ∗ 1: N 2 = min { T ∗ 1 , . . . , T ∗ N 2 } , T N 1 : N 1 = ma x { T 1 , . . . , T N 1 } and T ∗ N 2 : N 2 = max { T ∗ 1 , . . . , T ∗ N 2 } , r esp ectively . T o b e st of o ur knowledge, the prop os ed mo del presented in this s tudy has not b een inv estig ated in the exist- ing literature. In this article, our aim is to study the s to chastic pr o p e rties of r andom cla ims ( T 1: N 1 and T ∗ 1: N 2 ) and ( T N 1 : N 1 and T ∗ N 2 : N 2 ) , r e sp e ctively , w he n the matrix of par ameters ( ψ ( p ) , α ; n ) changes to ( ψ ( p ∗ ) , β ; n ) in the sense of the row w eak ma joriza tion orde r in the spac e M n based on the usual sto chastic and reversed haz- ard rate order s under the co ndition that “ ψ is stric tly decreasing conv ex function”. T o pr ov e our main result based on usual sto chastic order pres ent ed in Theorem 3.3, we fir st es tablish the results (see Theorems 3.1 a nd 3.2) based on vector c ha in mejorization order for fix sample size and then using the condition N 1 ≤ st N 2 we prov e T 1: N 1 is sto chastically less than T 1: N 2 . Similar idea is used for Theor em 3.5, whic h compar es T N 1 : N 1 and T N 2 : N 2 sto chastically . T o prove the results based on reversed haz a rd r ate order for same and random sample size N (see Theorems 3.13 a nd 3.17), we apply some Theorems from K undu e t al. (2 024), and using the idea of Theorem 3.3. As a consequence, our results extend the results in Barma lzan e t al. (201 7 ), Balakr ishnan et al. (201 8) and K undu a nd Cho wdh ury (202 1) to a genera l set-up. Moreov er, these results established here provide impor tant insig ht into determining the final price in auction theory and the b est reliabilit y system in reliability theory . The remaining part o f the article is orga nized as follows. In Section 2, s ome imp or ta nt definitions a nd preliminary results ar e pr esented. The ordering results p er taining to differe nt sto chastic o rders, such as the usual s to chastic and reversed haza rd rate o rders, a re developed in Section 3 using the concepts o f matrix row ma joriza tio n and row ma joriz a tion orders. In Section 4, we consider some applica tio ns of our es ta blished theoretical r esults for the purp oses of illustratio n in r eliability theory and auction theory . Finally , Section 5 provides a co nclusion o f this w ork. 3 2. Preliminary In this s ection, we review some imp orta n t definitions and well-known concepts inv olving the no tion o f ma joriza- tion and sto chastic orders . The rea der ma y consult to Shaked and Shanthikumar (2007) and Mar shall et al. (2011) for further details on these notions. Let V 1 and V 2 be tw o univ a riate random v ariables ha ving densit y functions (PDFs) f V 1 ( · ) a nd f V 2 ( · ), dis tr ibution functions (CDFs) F V 1 ( · ) a nd F V 2 ( · ), s ur viv al functions ¯ F V 1 ( · ) and ¯ F V 2 ( · ), and reversed hazar d rate functions ˜ r V 1 ( · ) = f V 1 ( · ) /F V 1 ( · ), ˜ r V 2 ( · ) = f V 2 ( · ) /F V 2 ( · ), r esp ectively . Definition 2.1. A r andom variable V 1 is said to b e smal ler than V 2 in the • r everse d hazar d r ate or der (denote d by V 1 ≤ r h V 2 ) if ˜ r V 1 ( x ) ≤ ˜ r V 2 ( x ) , for al l x ; • usual sto chastic or der (denote d by V 1 ≤ st V 2 ) if ¯ F V 1 ( x ) ≤ ¯ F V 2 ( x ) , for al l x. Let a = ( a 1 , · · · , a n ) and b = ( b 1 , . . . , b n ) be t w o n - dimensional vectors, wher e a , b ∈ A . Here, A ⊂ R n and R n is a n -dimensiona l Euclidea n spac e . F urthermore, let the order coordina tes of the vectors a and b b e a 1: n ≤ · · · ≤ a n : n and b 1: n ≤ · · · ≤ b n : n , res pec tively . Deno te J n = { 1 , 2 , . . . , n } . Definition 2.2. A ve ctor b is said to • majorize t he ve ctor a , (denote d by a  m b ), for l ∈ J n − 1 , P l i =1 a i : n ≥ P l i =1 b i : n and P n i =1 a i : n = P n i =1 b i : n ; • we akly su p ermajorize ve ctor b , denote d by a  w b , if P l i =1 a i : n ≥ P l i =1 b i : n for l ∈ J n . In the following, we pr esent the definition of Schu r-conv ex and Sch ur-concave functions and, an impo r tant lemma which will b e useful in the conseque nce section. Definition 2.3. A function h : R n → R is said to b e Schur-c onvex (Schur- c onc ave) on R n if x m  y ⇒ h ( x ) ≥ ( ≤ ) h ( y ) , for al l x , y ∈ R n . Lemma 2.1. ( Marshal l et al. (2011)) Conside r the re al-value d c ontinuously differ entiable fun ction φ on J n , wher e J ⊆ R is an op en interval. Then, φ is Schur-c onvex ( Schur-c onc ave ) on J n if and only if φ is symmetric on J n , and for al l i 6 = j and al l u ∈ J n , ( u i − u j )  ∂ φ ( u ) ∂ u i − ∂ φ ( u ) ∂ u j  ≥ ( ≤ )0 , wher e ∂ φ ( u ) ∂ u i denotes the p artial derivative of φ ( u ) with r esp e ct to its i -th ar gument. Next, w e discuss the concept of ma jorization on matrices. A squa r e matrix Π is said to be a per mut ation matrix if eac h row and column hav e exactly one unity and zer os elsewher e. Such ma tr ices can be constructed by interc hanging r ows (or columns) of the n × n identit y matrix I n . Therefor e, a T -trans form matrix is o f the form T w = w I n + (1 − w )Π , 0 ≤ w ≤ 1 . (2.2) Consider tw o T -transfor m matric e s T w 1 = w 1 I n + (1 − w 1 )Π 1 and T w 2 = w 2 I n + (1 − w 2 )Π 2 , where Π 1 and Π 2 are tw o p er mutation matrices a nd 0 ≤ w 1 , w 2 ≤ 1. If Π 1 = Π 2 , then the matric e s T w 1 and T w 2 hav e the sa me structure, otherwise they hav e different structures. In the following, we describ e the concept of m ultiv ariate ma jorization. Definition 2.4. Consider two m × n m atric es A = { a ij } and B = { b ij } , wher e i = 1 , . . . , m and j = 1 , . . . , n . Then, • A is said to majorize B , (denote d by A >> B ), if ther e exists a fi nite set of n × n T -tr ansform matric es T w 1 , . . . , T w k such that B = AT w 1 . . . T w k ; 4 • A is said to chain majorize B , (denote d by A > B ), if B = AP , wher e the n × n matrix P is doubly sto chastic. S inc e a pr o duct of T - t r ansform is double sto chastic, it fol lows that A > > B ⇒ A > B ; • A is said to r ow m ajorize B , ( denote d by A r ow > B ), if a R i m  b R i for i = 1 , . . . , m, wher e a R 1 , . . . , a R m and b R 1 , . . . , b R m ar e the ro ws of A and B , so that these quantities ar e r ow ve ctors of length n. It is known that A > B ⇒ A r ow > B ; • A is said t o r ow we akly maj orize B , (denote d by A w > B ), if a R i w  b R i for i = 1 , . . . , m, wher e a R 1 , . . . , a R m and b R 1 , . . . , b R m b e the r ows of A and B , so that these quantities ar e r ow ve ctors of length n. It is known that A r ow > B ⇒ A w > B . F o r detailed discussio n on this topic, we ma y refer to Marsha ll et al. (2011). F or simplicity , from now on we denote the matrix of order m × n as ( r 1 , r 2 , . . . , r m ; n ) , where the vectors with real v alue r 1 , r 2 , . . . , r m are the first, second, . . . , m -th rows, resp ectively , each having n elements. Le t us denote M n = { ( x , y ; n ) : x i > 0 , y j > 0 and ( x i − x j )( y i − y j ) ≥ 0 , i, j = 1 , . . . , n } and h ′ ( z ) = dh ( z ) dz . 3. Ordering results In this section, we consider t wo sets of independent and hetero geneous insurance p o rtfolios of risks { T 1 , . . . , T N 1 } and { T ∗ 1 , . . . , T ∗ N 2 } with T i = J i U i and T ∗ i = J ∗ i V i where for i = 1 , . . . , N 1 , U i ∼ ¯ F ( · ; α i ) and J i ∼ B e rnou l i ( p i ) independent of U ′ i s a nd for i = 1 , . . . , N 2 , V i ∼ ¯ F ( · ; β i ) and J ∗ i ∼ B er noul i ( p ∗ i ) indep endent of V ′ i s. Here, N 1 and N 2 are tw o discrete random v a riables with p ositive integer v alues and have s upp or t as { 1 , 2 , . . . , } , independent of T ′ i s and T ∗ i ′ s, r esp ectively . Under this se t-up, here we establish order ing results b etw een t w o extreme claim amo un ts, by using the concept of vector and matrix ma jorizatio n in the sense of usual sto chastic and reversed haza rd rate or ders. Here, the num ber o f claims N 1 and N 2 are rando m and sto chastically compa- rable, indep endent of T i ’s and T ∗ i ’s, resp ectively . F urthermore, let T n : n and T 1: n , a nd T ∗ n : n and T ∗ 1: n denote the largest and sma llest c la im amounts c o rresp onding to the p o rtfolios of risks { T 1 , . . . , T n } and { T ∗ 1 , . . . , T ∗ n } , re- sp ectively . W e start this section with the result ba sed on usua l sto chastic or de r which states tha t if the matrix of parameter s ( ψ ( p ) , α ; n ) changes to ( ψ ( p ∗ ) , β ; n ) in the sense of the row w eak ma jorizatio n or der in the space M n then T 1: N 1 is smaller than T ∗ 1: N 2 in terms of usual sto chastic or der. T o establis h this theor em, we first pr ov e t wo compar ison results based on vector ma jorization wher eas the first one shows if the mo deled par ameter are connected with weakly sup er ma jorizatio n order then the usual stochastic o rder ho lds b etw een tw o smalle s t claim amounts for the c a se of common p and the second one pro vides that the s ame inequality holds b etw een the smalles t claim a mount s whenever the occur rence probabilities a re ass o ciated with weakly sup er ma jo r iza- tion order with common α . Thro ughout this section, w e denote ψ ( p ) = ( ψ ( p 1 ) , . . . , ψ ( p n )) = ( v 1 , . . . , v n ) and ψ ( p ∗ ) = ( ψ ( p ∗ 1 ) , . . . , ψ ( p ∗ n )) = ( u 1 , . . . , u n ) . Theorem 3.1. L et { U 1 , . . . , U n } and { V 1 , . . . , V n } b e two set s of indep endent r andom variables with U i ∼ ¯ F ( x ; α i ) and V i ∼ ¯ F ( x ; β i ) , r esp e ctively. A lso, let { J 1 , . . . , J n } and { J ∗ 1 , . . . , J ∗ n } b e another two sets of inde- p endent Bernoul li r andom variables, indep endently of U ′ i s and V ′ i s with E ( J i ) = p i and E ( J ∗ i ) = p i , r esp e ctively. Assume that ¯ F ( x ; α i ) is de cr e asing and lo g- c onvex in α i for any x . Then, for ( ψ ( p ) , α ; n ) , ( ψ ( p ) , β ; n ) ∈ M n , we have α w  β ⇒ T 1: n ≥ st T ∗ 1: n . Pr o of. In or der to complete the theorem, w e only need to show whether α w  β ⇒ ¯ F T 1: n ( x ) ≥ ¯ F T ∗ 1: n ( x ) , which is equiv a le n t to establishing that ¯ F T 1: n ( x ) is decreasing and Sc h ur-conv ex in α a ccording to Theorem A. 8 of Marshall et a l. (20 11). The relia bilit y function of ¯ F T 1: n ( x ) c a n b e w r itten a s ¯ F T 1: n ( x ) = Q n i =1 ψ − 1 ( v i ) ¯ F ( x ; α i ) . After taking partial deriv ative o f ¯ F T 1: n ( x ) with resp ect to α i , we have ∂ ¯ F T 1: n ( x ) ∂ α i = d ¯ F ( x ; α i ) dα i ¯ F ( x ; α i ) ¯ F T 1: n ( x ) ≤ 0 , (3.3) 5 due to the decreasing prop erty of ¯ F ( x ; α i ) in α i . Co nsider the case α i ≤ α j and v i ≤ v j for any pair i, j such that 1 < i ≤ j < n. Ba sed on the condition that ¯ F ( x ; α i ) is log-convex in α i , w e hav e the following inequality ( α i − α j )  ∂ ¯ F T 1: n ( x ) ∂ α i − ∂ ¯ F T 1: n ( x ) ∂ α j  ≥ 0 , which y ie lds ¯ F T 1: n ( x ) is Sch ur- conv e x in α . Hence, the pro of. Theorem 3.2. L et { U 1 , . . . , U n } and { V 1 , . . . , V n } b e two set s of indep endent r andom variables with U i ∼ ¯ F ( x ; α i ) and V i ∼ ¯ F ( x ; α i ) , r esp e ctively. A lso, let { J 1 , . . . , J n } and { J ∗ 1 , . . . , J ∗ n } b e another two sets of in- dep endent Bernoul li ra ndom variables, indep endently of U ′ i s and V ′ i s with E ( J i ) = p i and E ( J ∗ i ) = p ∗ i , r e- sp e ct ively. Assume that ψ : (0 , 1 ) → (0 , ∞ ) is a differ entiable, de cr e asing, and lo g-c onvex function. Then, for ( ψ ( p ) , α ; n ) , ( ψ ( p ∗ ) , α ; n ) ∈ M n , we have ψ ( p ) w  ψ ( p ∗ ) ⇒ T 1: n ≥ st T ∗ 1: n . Pr o of. T o obtain the required result, we ha ve to prov e that ψ ( p ) w  ψ ( p ∗ ) ⇒ ¯ F T 1: n ( x ) ≥ ¯ F T ∗ 1: n ( x ) . In other word, w e have to show that ¯ F T 1: n ( x ) is decreas ing and Sc h ur-conv ex in α by Theorem A. 8 of Marshall et al. (2011). The reliability function of ¯ F T 1: n ( x ) can b e wr itten a s ¯ F T 1: n ( x ) = Q n i =1 ψ − 1 ( v i ) ¯ F ( x ; α i ) . The partial deriv ative of ¯ F T 1: n ( x ) with resp ect to v i , is given by ∂ ¯ F T 1: n ( x ) ∂ v i = dψ − 1 ( v i ) dv i ψ − 1 ( v i ) ¯ F T 1: n ( x ) ≤ 0 , (3.4) due to the decreasing prop er t y of ψ − 1 ( v i ) in v i for i = 1 , . . . , n . Co nsider the case v i ≤ v j for an y pair i, j such that 1 < i ≤ j < n. Using the log-conv exit y pr o p e rty of ψ − 1 ( v i ) in v i , w e hav e ( v i − v j )  ∂ ¯ F T 1: n ( x ) ∂ v i − ∂ ¯ F T 1: n ( x ) ∂ v j  ≥ 0 , which co ncludes that ¯ F T 1: n ( x ) is Sch ur- conv e x in v . Hence, the pro o f. Next, we prove that if N 1 ≤ st N 2 then the usua l sto chastic o rder also holds b etw een tw o smallest claim amounts T 1: N 1 and T ∗ 1: N 2 from tw o se ts of independent and heterogeneous portfolios of risk s when the matrix of par ameters ( ψ ( p ) , α ; n ) changes to ( ψ ( p ∗ ) , β ; n ) in the sense of the row w eak ma jo r ization or der in the space M n . Theorem 3.3. L et { U 1 , . . . , U n } and { V 1 , . . . , V n } b e two set s of indep endent r andom variables with U i ∼ ¯ F ( x ; α i ) and V i ∼ ¯ F ( x ; β i ) , r esp e ctively. A lso, let { J 1 , . . . , J n } and { J ∗ 1 , . . . , J ∗ n } b e another two sets of inde- p endent Bernoul li r andom variables, indep endently of U ′ i s and V ′ i s with E ( J i ) = p i and E ( J ∗ i ) = p ∗ i , r esp e c- tively. F urther, let N 1 and N 2 b e two p ositive inte ger-value d r andom vari ables indep endently of T ′ i s and T ∗ i ′ s satisfying N 1 ≤ st N 2 , r esp e ctively. Assume that t he fol lowing c onditions hold: (i) ¯ F ( x ; α i ) is de cr e asing and lo g- c onvex in α i for any x ; (ii) ψ : (0 , 1) → (0 , ∞ ) is a differ ent iable, strictly de cr e asing and lo g-c onvex funct ion. Then, for ( ψ ( p ) , α ; n ) , ( ψ ( p ∗ ) , β ; n ) ∈ M n , we have ( ψ ( p ) , α ; n ) w > ( ψ ( p ∗ ) , β ; n ) ⇒ T 1: N 1 ≥ st T ∗ 1: N 2 . Pr o of. The main step in proving the theorem is to establis h that the following tw o inequalities hold: (i) ( ψ ( p ) , α ; n ) w > ( ψ ( p ∗ ) , β ; n ) ⇒ ¯ F T 1: n ( x ) ≥ ¯ F T ∗ 1: n ( x ); (ii) ¯ F T 1: N 1 ( x ) ≥ ¯ F T ∗ 1: N 2 ( x ) . 6 T o co mplete the first part, let us supp ose W 1: n , Z 1: n and M 1: n be the smallest claim amounts from sample J p 1: n U α 1: n , . . . , J p n : n U α n : n , J p 1: n V β 1: n , . . . , J p n : n V β n : n and J ∗ p ∗ 1: n V β 1: n , . . . , J ∗ p ∗ n : n V β n : n , resp ectively . It is e asy to observe that T 1: n st = W 1: n and T ∗ 1: n st = M 1: n . How ever, accor ding to Theorem 3.1 we have W 1: n ≥ st Z 1: n and Theorem 3.2, we hav e Z 1: n ≥ st M 1: n , whic h concludes ¯ F T 1: n ( x ) ≥ ¯ F T ∗ 1: n ( x ) . T o prov e the se c ond part, first we need to show ¯ F T ∗ 1: m ( x ) is increas ing in m, which ca n b e done by Theorem 1 .B . 28 of Shaked and Sha n thikumar (2007). Now ¯ F T 1: N 1 ( x ) = n X m =1 P ( T 1: N 1 > x | N 1 = m ) P ( N 1 = m ) = n X m =1 P ( T 1: m > x ) P ( N 1 = m ) ≥ n X m =1 P ( T ∗ 1: m > x ) P ( N 1 = m ) [as T 1: m ≥ st T ∗ 1: m ] ≥ n X m =1 P ( N 2 = m ) ¯ F T ∗ 1: m ( x ) [as N 1 ≤ st N 2 and ¯ F T ∗ 1: m ( x ) is monoto ne in m ] = ¯ F T ∗ 1: N 2 ( x ) , (3.5) which co mpletes the pro o f of the theorem. The co ndition “ N 1 ≤ st N 2 ” pr ovided in Theor e m 3.3 plays an crucial role to establis h the result. In the following co un terexample, we s e e that if the condition “ N 1 ≤ st N 2 ” is violated then Theorem 3.3 do es not hold. Coun terexample 3.1. Consider the density fun ction of a Gamma distribution (denote d by Gamma ( θ , α ) ), f ( x, θ , α ) = 1 Γ( α ) θ α x α − 1 e − x θ , x , θ , α > 0 . L et ψ ( p ) = − ln( p ) . Her e b oth ψ ( p ) and the s u rvival function of Gamma distribution ( ¯ F ( x ; α ) ), ar e de cr e asing and lo g-c onvex in p and α for fix θ , r esp e ctively. Supp ose, U i ∼ Gamma (10 . 09 , α i ) and V i ∼ Gamma (10 . 09 , β i ) , for i = 1 , 2 , 3 , 4 , 5 . F urther, set ( α 1 , α 2 , α 3 , α 4 , α 5 ) = (1 . 9 , 2 , 3 , 5 , 6) and ( β 1 , β 2 , β 3 , β 4 , β 5 ) = (4 . 9 , 6 . 5 , 7 . 6 , 8 . 2 , 10 . 9) , ( p 1 , p 2 , p 3 , p 4 , p 5 ) = ( e − 0 . 7 , e − 2 . 1 , e − 3 . 2 , e − 4 . 9 , e − 6 . 9 ) and ( q 1 , q 2 , q 3 , q 4 , q 5 ) = ( e − 1 . 5 , e − 1 . 6 , e − 2 . 6 , e − 3 . 9 , e − 4 . 2 ) . Cle arly, ( ψ ( p ) , α ; n ) w > ( ψ ( p ∗ ) , β ; n ) . L et N 1 ∼ Poisson ( λ 1 ) and N 2 ∼ Poisson ( λ 2 ) , wher e λ 1 = 10 . 9 and λ 2 = 2 . Also, c onsider ( U 1 , U 2 , U 3 ) ar e sele ct e d with pr ob ability P ( N 1 = 3) = e − λ 1 λ 3 1 3! , ( U 1 , U 2 , U 3 , U 4 ) ar e sele cte d with pr ob ability P ( N 1 = 4) = e − λ 1 λ 4 1 4! and ( U 1 , U 2 , U 3 , U 4 , U 5 ) ar e sele cte d with pr ob ability P ( N 1 = 5 ) = e − λ 1 λ 5 1 5! , and ( V 1 , V 2 , V 3 ) ar e sele cte d with pr ob ability P ( N 2 = 3) = e − λ 2 λ 3 2 3! , ( V 1 , V 2 , V 3 , V 4 ) ar e sele cte d with pr ob abili ty P ( N 2 = 4) = e − λ 2 λ 4 2 4! and ( V 1 , V 2 , V 3 , V 4 , V 5 ) ar e sele cte d with pr ob ability P ( N 2 = 5) = e − λ 2 λ 5 2 5! . Her e al l the c onditions ar e satisfie d of The or em 3.3 exc ept N 1 ≤ st N 2 . Figur e 2( b ) pr esents t hat the differ enc e b etwe en ¯ F T 1: N 1 ( x ) and ¯ F T ∗ 1: N 2 ( x ) cr osses x axis for some x ≥ 0 which violate d t he s t atement of the The or em 3.3. In Balakrishnan et a l. (20 18), the author s ha v e pr oved the following theorem wher e the us ual s to chastic order ho lds b etw een larg est claim amounts T n : n and T ∗ n : n when the parameter ma trix ( ψ ( p ) , α ; n ) c ha ng es to another matr ix ( ψ ( p ∗ ) , β ; n ) . Theorem 3.4. L et { U 1 , . . . , U n } and { V 1 , . . . , V n } b e two set s of indep endent r andom variables with U i ∼ ¯ F ( x ; α i ) and V i ∼ ¯ F ( x ; β i ) , r esp e ctively. Also, let { J 1 , . . . , J n } and { J ∗ 1 , . . . , J ∗ n } b e another two set s of indep en- dent Bernoul li r andom variables, indep endently of U ′ i s and V ′ i s with E ( J i ) = p i and E ( J ∗ i ) = p ∗ i , r esp e ctively. Assume that the fol lowing c onditions hold: (i) ¯ F ( x ; α i ) is de cr e asing and c onvex in α i for al l x ; (ii) ψ : (0 , 1) → (0 , ∞ ) is a differ ent iable, strictly de cr e asing c onvex function. 7 Then, for ( ψ ( p ) , α ; n ) , ( ψ ( p ∗ ) , β ; n ) ∈ M n , we have ( ψ ( p ) , α ; n ) w > ( ψ ( p ∗ ) , β ; n ) ⇒ T n : n ≥ st T ∗ n : n . Now, w e generalize Theore m 3.4 from the c ase of fixed claim size to the case of random claim sizes. Theorem 3.5. L et { U 1 , . . . , U n } and { V 1 , . . . , V n } b e two set s of indep endent r andom variables with U i ∼ ¯ F ( x ; α i ) and V i ∼ ¯ F ( x ; β i ) , r esp e ctively. A lso, let { J 1 , . . . , J n } and { J ∗ 1 , . . . , J ∗ n } b e another two sets of inde- p endent Bernoul li r andom variables, indep endently of U ′ i s and V ′ i s with E ( J i ) = p i and E ( J ∗ i ) = p ∗ i , r esp e c- tively. F urther, let N 1 and N 2 b e two p ositive inte ger-value d r andom vari ables indep endently of T ′ i s and T ∗ i ′ s satisfying N 1 ≤ st N 2 , r esp e ctively. Assume that t he fol lowing c onditions hold: (i) ¯ F ( x ; α i ) is de cr e asing and c onvex in α i for al l x ; (ii) ψ : (0 , 1) → (0 , ∞ ) is a differ ent iable, strictly de cr e asing c onvex function. Then, for ( ψ ( p ) , α ; n ) , ( ψ ( p ∗ ) , β ; n ) ∈ M n , we have ( ψ ( p ) , α ; n ) w > ( ψ ( p ∗ ) , β ; n ) ⇒ T N 1 : N 1 ≥ st T ∗ N 2 : N 2 . Pr o of. There ar e tw o steps to complete the proof of the theorem: (i) ( ψ ( p ) , α ; n ) w > ( ψ ( p ∗ ) , β ; n ) ⇒ F T n : n ( x ) ≤ F T ∗ n : n ( x ); (ii) F T N 1 : N 1 ( x ) ≤ F T ∗ N 2 : N 2 ( x ) . Using Theorem 3.4 , we can write the first inequa lit y . T o establish the second inequa lity , first we need to sho w that F T ∗ m : m ( x ) is increa sing in m, whic h c a n be prov e easily as F T ∗ m : m ( x ) < F T ∗ m +1: m +1 ( x ) for a ll m = 1 , . . . , n . Now F T N 1 : N 1 ( x ) = n X m =1 P ( T N 1 : N 1 < x | N 1 = m ) P ( N 1 = m ) = n X m =1 P ( T m : m < x ) P ( N 1 = m ) ≤ n X m =1 P ( T m : m < x ) P ( N 2 = m ) [ as N 1 ≤ st N 2 ] ≤ n X m =1 P ( T ∗ m : m < x ) P ( N 2 = m ) , [ a s T m : m ≥ st T ∗ m : m ] ≤ F T ∗ N 2 : N 2 ( x ) , (3.6) which co mpletes the pro o f of the theorem. Now w e consider the follo wing exa mple to illustrate Theorem 3.5. Example 3.1. Supp ose, U i ∼ Gamma ( k , α i ) and V i ∼ Gamma ( k , β i ) , with k = 1 . 5 for i = 1 , 2 , 3 , 4 , 5 . F ur- ther, set ( α 1 , α 2 , α 3 , α 4 , α 5 ) = (1 . 9 , 2 , 3 , 5 , 6 ) and ( β 1 , β 2 , β 3 , β 4 , β 5 ) = (4 . 9 , 6 . 5 , 7 . 6 , 8 . 2 , 10 . 9 ) , ψ ( p ) = − ln( p ) , ( p 1 , p 2 , p 3 , p 4 , p 5 ) = ( e − 0 . 7 , e − 0 . 9 , e − 3 , e − 4 . 9 , e 3 . 9 ) and ( q 1 , q 2 , q 3 , q 4 , q 5 ) = ( e − 1 . 2 , e − 1 . 5 , e − 1 . 6 , e − 2 . 6 , e 3 . 9 ) . Cle arly, ( ψ ( p ) , α ; n ) w > ( ψ ( p ∗ ) , β ; n ) . L et N 1 ∼ Poisson ( λ 1 ) and N 2 ∼ Poisson ( λ 2 ) , wher e λ 1 = 0 . 9 and λ 2 = 1 . 9 . Also , c onsider ( U 1 , U 2 , U 3 ) ar e sele cte d with pr ob abili ty P ( N 1 = 3) = e − λ 1 λ 3 1 3! , ( U 1 , U 2 , U 3 , U 4 ) ar e sele cte d with pr ob ability P ( N 1 = 4 ) = e − λ 1 λ 4 1 4! and ( U 1 , U 2 , U 3 , U 4 , U 5 ) ar e sele cte d with pr ob ability P ( N 1 = 5) = e − λ 1 λ 5 1 5! , and ( V 1 , V 2 , V 3 ) ar e sele cte d with pr ob ability P ( N 2 = 3) = e − λ 2 λ 3 2 3! , ( V 1 , V 2 , V 3 , V 4 ) ar e sele cte d with pr ob ability P ( N 2 = 4) = e − λ 2 λ 4 2 4! and ( V 1 , V 2 , V 3 , V 4 , V 5 ) ar e sele cte d with pr ob ability P ( N 2 = 5) = e − λ 2 λ 5 2 5! . It is cle ar that N 1 ≤ st N 2 . Her e, al l the c onditio ns ar e satisfie d of The or em 3.5. Figur e 1( a ) pr esents that the differ enc e b etwe en F T N 1 : N 1 ( x ) and F T ∗ N 2 : N 2 ( x ) takes ne gative values for al l x ≥ 0 . 8 Next counterexample shows that the result pres ent ed in Theorem 3.5 may not b e true if w e remov e the condition ( ψ ( p ) , α ; n ) , ( ψ ( p ∗ ) , β ; n ) ∈ M n . Coun terexample 3.2. Supp ose, U i ∼ Gamma ( k , α i ) and V i ∼ Gamma ( k , β i ) , with k = 1 . 5 for i = 1 , 2 , 3 , 4 , 5 . F urther, set ( α 1 , α 2 , α 3 , α 4 , α 5 ) = (1 . 9 , 5 , 5 . 5 , 6 , 10) and ( β 1 , β 2 , β 3 , β 4 , β 5 ) = (0 . 7 , 0 . 9 , 3 , 4 . 9 , 3 . 9) , ψ ( p ) = − ln( p ) , ( p 1 , p 2 , p 3 , p 4 , p 5 ) = ( e − 0 . 7 , e − 0 . 9 , e − 3 , e − 4 . 9 , e 3 . 9 ) and ( q 1 , q 2 , q 3 , q 4 , q 5 ) = ( e − 3 . 9 , e − 1 . 5 , e − 1 . 6 , e − 4 . 2 , e 2 . 6 ) . Cle arly, ( ψ ( p ) , α ; n ) w > ( ψ ( p ∗ ) , β ; n ) . L et N 1 ∼ Poisson ( λ 1 ) and N 2 ∼ Poisson ( λ 2 ) , wher e λ 1 = 0 . 9 and λ 2 = 1 . 9 . Also, c onsider ( U 1 , U 2 , U 3 ) ar e sele cte d with pr ob ability P ( N 1 = 3) = e − λ 1 λ 3 1 3! , ( U 1 , U 2 , U 3 , U 4 ) ar e sele cte d with pr ob abili ty P ( N 1 = 4 ) = e − λ 1 λ 4 1 4! and ( U 1 , U 2 , U 3 , U 4 , U 5 ) ar e sele cte d with pr ob ability P ( N 1 = 5) = e − λ 1 λ 5 1 5! , and ( V 1 , V 2 , V 3 ) ar e sele cte d with pr ob ability P ( N 2 = 3) = e − λ 2 λ 3 2 3! , ( V 1 , V 2 , V 3 , V 4 ) ar e sele cte d with pr ob ability P ( N 2 = 4) = e − λ 2 λ 4 2 4! and ( V 1 , V 2 , V 3 , V 4 , V 5 ) ar e sele cte d with pr ob ability P ( N 2 = 5) = e − λ 2 λ 5 2 5! . Her e it is cle ar that N 1 ≤ st N 2 . Her e al l the c onditions ar e satisfie d of The or em 3.5 exc ept ( ψ ( p ) , α ; n ) , ( ψ ( p ∗ ) , β ; n ) ∈ M n . Figur e 1( b ) pr esents that the differ enc e b etwe en F T N 1 : N 1 ( x ) and F T ∗ N 2 : N 2 ( x ) cr osses x ax is for some x ≥ 0 , which violates the statement of the The or em 3.5. x axis 0 10 20 30 40 50 60 70 80 90 100 y axis -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 (a) x axis 10 20 30 40 50 60 70 80 90 y axis -0.3 -0.2 -0.1 0 0.1 0.2 0.3 (b) Figure 1: (a) Plot of ¯ F T N 1 : N 1 ( x ) − ¯ F T ∗ N 2 : N 2 ( x ) as in Example 3.1. (b) Plo t of F X N 1 : N 1 ( x ) − F Y N 2 : N 2 ( x ) as in Counterexample 3.2. Remark 3.1. The c ondition “ ψ : (0 , 1) → (0 , ∞ ) is a differ entiable, strictly de cr e asing c onvex function” pr esente d in The or em 3.5, is qu ite e asy t o verify for differ ent tr ansform funct ions. Her e, we c onsider some widely use d de cr e asing and c onvex tr ansformation functions. L et X i b e the i th claim amount asso ciate d with o c cu rrenc e pr ob ability p i , for i ∈ J n . Consider (i) the Exp onen t ial tr ansformation with ψ ( p i ) = e − αp i , wher e α > 0 . (ii) the p ower (inverse) tr ansformation with ψ ( p i ) = p β i , wher e β > 0 a risk aversion c o efficient, wil l b e use d in distorte d risk me asur es, hig hlighting ex tr eme t ails. (iii) the Ne gative lo garithm with ψ ( p i ) = − ln( p i ) . In insurance analy s is, it is quite co mmon that the b ehavior of insurance claims data will b e a symmetry and heavy-tailed. Therefore, to mo de l this type da ta, an insur er alwa ys try to find o ut some distributions 9 having such pr op erties. In this con text, Kuma raswam y-G (Kw-G) distribution is a generalized fa mily of distributions, whic h is a more flexible model having g reater con trol over the shap e, sk ewness, and k urtosis of the distr ibution. Due to the flexibility prop erty of Kw-G distr ibutions, insurer frequently use this distribution to mo del b etter tail ris ks. It is interesting to no te that the condition ( i ) presented in Theorem 3 .5 contains the Kw-G family , as a sp ecial ca s e. Therefor e, the following r esult is a direct a pplication of Theorem 3.5 for the case o f indep endent and hetero geneous Kw-G ris ks. The rando m v aria ble { U 1 , . . . , U n } is s aid to b elong to Kw - G family if U i ∼ (1 − G γ ( x )) α i , where γ , α i > 0 for i = 1 , . . . , n. Theorem 3.6. L et ¯ F ( x ; α i ) = (1 − G γ ( x )) α i and ¯ F ( x ; β i ) = (1 − G γ ( x )) β i for i = 1 , . . . , n. Under the set-up of The or em 3.5 , supp ose ψ : (0 , 1) → (0 , ∞ ) is a differ entiable, strictly de cr e asing c onvex function. Then, for ( ψ ( p ) , α ; n ) , ( ψ ( p ∗ ) , β ; n ) ∈ M n , we have ( ψ ( p ) , α ; n ) w > ( ψ ( p ∗ ) , β ; n ) ⇒ T N 1 : N 1 ≥ st T ∗ N 2 : N 2 , Pr o of. Using The o rem 3.5, w e only need to c heck that ‘ ¯ F ( x ; α i ) is decreas ing and convex in α i for a ll x ”. F o r the given model ¯ F ( x ; α i ) = (1 − G γ ( x )) α i for i = 1 , . . . , n. T aking fir st or der and se c ond or der partial deriv atives of ¯ F ( x ; α i ) with resp ect to α i for i = 1 , . . . , n, w e get ∂ ¯ F ( x ; α i ) ∂ α i = l og (1 − G γ ( x ))(1 − G γ ( x )) α i < 0 (3.7) and ∂ 2 ¯ F ( x ; α i ) ∂ α 2 i = [ l og (1 − G γ ( x ))] 2 (1 − G γ ( x )) α i > 0 , (3.8) which co mpletes the pro o f of the theorem. Similar to Theorem 3 .6, it is imp ortant to note that the well-known scale family a nd the prop or tio nal haza r d rate family also satisfy the condition ( i ) pr esented in The o rem 3 .5, thus using Theorem 3.5, w e can wr ite the following tw o theorems which ar e extensions of Theorems 3 . 5 and 3 . 6 in Ba lakrishnan et al. (2018) fr o m fixed claim size to random claim sizes for the case of heterogeneous a nd independent scale a nd prop ortio na l ha zard rate risks. The random v ariable { U 1 , . . . , U n } is s aid to b elong to the scale family if U i ∼ ¯ F ( xα i ) , where α i > 0 for i = 1 , . . . , n. Here, f ( x ) is the densit y function corresp onding to the distribution F ( x ) . Theorem 3 .7. L et ¯ F ( x ; α i ) = ¯ F ( xα i ) and ¯ F ( x ; β i ) = ¯ F ( xβ i ) for i = 1 , . . . , n. Under the set-up of The or em 3.5, supp ose t he fol lowing two c onditions hold: (i) ψ : (0 , 1 ) → (0 , ∞ ) is a differ entiable, strictly de cr e asing c onvex function; (ii) f ( x ) is de cr e asing in x. Then, for ( ψ ( p ) , α ; n ) , ( ψ ( p ∗ ) , β ; n ) ∈ M n , we have ( ψ ( p ) , α ; n ) w > ( ψ ( p ∗ ) , β ; n ) ⇒ T N 1 : N 1 ≥ st T ∗ N 2 : N 2 . Pr o of. F or the given model ¯ F ( x ; α i ) = ¯ F ( xα i ) for i = 1 , . . . , n. Ev aluating the first or der and second order partial deriv atives o f ¯ F ( x ; α i ) with resp ect to α i for i = 1 , . . . , n, we obtain ∂ ¯ F ( x ; α i ) ∂ α i = − xf ( α i x ) < 0 and ∂ 2 ¯ F ( x ; α i ) ∂ α 2 i = − x 2 ∂ f ( α i x ) ∂ x > 0 , as f ( x ) is decr easing. Therefore, ¯ F ( x ; α i ) is decreas ing and conv ex in α i for all x . Finally , using Theorem 3.5, we e stablish the required result. In the following, w e w ill apply T heo rem 3.5 for the prop ortio nal haza rd rate mo del. A rando m v a riable U i is sa id to be longs to the propor tional haza rd rate family if U i ∼ ¯ F α i ( x ) , wher e α i > 0 for i = 1 , . . . , n. 10 Theorem 3. 8 . L et ¯ F ( x ; α i ) = ¯ F α i ( x ) and ¯ F ( x ; β i ) = ¯ F β i ( x ) for i = 1 , . . . , n. Under the set-up of The- or em 3.5, su pp ose ψ : (0 , 1 ) → (0 , ∞ ) is a differ entiable, strictly de cr e asing c onvex function. Then, fo r ( ψ ( p ) , α ; n ) , ( ψ ( p ∗ ) , β ; n ) ∈ M n , we have ( ψ ( p ) , α ; n ) w > ( ψ ( p ∗ ) , β ; n ) ⇒ T N 1 : N 1 ≥ st T ∗ N 2 : N 2 . Pr o of. According to The o rem 3.5, it is r equired to sho w that ¯ F ( x ; α i ) is decreas ing and co nv ex in α i for all x . F o r the given model ¯ F ( x ; α i ) = ¯ F α i ( x ) for i = 1 , . . . , n. Differentiating ¯ F ( x ; α i ) par tially with resp ect to α i up to second order for i = 1 , . . . , n, we get ∂ ¯ F ( x ; α i ) ∂ α i = log ¯ F ( x ) ¯ F α i ( x ) < 0 and ∂ 2 ¯ F ( x ; α i ) ∂ α 2 i = [log ¯ F ( x )] 2 ¯ F α i ( x ) > 0 , which co mpletes the pro o f of the theorem. As w e kno w that A >> B ⇒ A > B ⇒ A r ow > B ⇒ A w > B . Thu s using Theo rem 3.5, w e can write the following r esult based on m ultiv ariate chain ma jorization o rder which is a stro nger co ndition. Theorem 3.9. L et { U 1 , . . . , U n } and { V 1 , . . . , V n } b e two set s of indep endent r andom variables with U i ∼ ¯ F ( x ; α i ) and V i ∼ ¯ F ( x ; β i ) , r esp e ctively. A lso, let { J 1 , . . . , J n } and { J ∗ 1 , . . . , J ∗ n } b e another two sets of inde- p endent Bernoul li r andom variables, indep endently of U ′ i s and V ′ i s with E ( J i ) = p i and E ( J ∗ i ) = p ∗ i , r esp e c- tively. F urther, let N 1 and N 2 b e two p ositive inte ger-value d r andom vari ables indep endently of T ′ i s and T ∗ i ′ s satisfying N 1 ≤ st N 2 , r esp e ctively. Assume that t he fol lowing c onditions hold: (i) ¯ F ( x ; α i ) is de cr e asing and c onvex in α i for al l x ; (ii) ψ : (0 , 1) → (0 , ∞ ) is a differ ent iable, strictly de cr e asing c onvex function. Then, for ( ψ ( p ) , α ; n ) , ( ψ ( p ∗ ) , β ; n ) ∈ M n , we have ( ψ ( p ∗ ) , β ; n ) = ( ψ ( p ) , α ; n ) T ⇒ T N 1 : N 1 ≥ st T ∗ N 2 : N 2 . Similar to Theor em 3.7, using the re sult presented in Theo rem 3.9, we can wr ite the following theorem which is a generalization o f Theorem 2 of Barmalzan et al. (201 7) for the cas e of ra ndom claim sizes. Theorem 3.10. L et ¯ F ( x ; α i ) = ¯ F ( xα i ) and ¯ F ( x ; β i ) = ¯ F ( xβ i ) for i = 1 , . . . , n. Under the set-up of The or em 3.9, supp ose t he fol lowing two c onditions hold: (i) ψ : (0 , 1 ) → (0 , ∞ ) is a differ entiable, strictly de cr e asing c onvex function; (ii) f ( x ) is de cr e asing in x. Then, for ( ψ ( p ) , α ; n ) , ( ψ ( p ∗ ) , β ; n ) ∈ M n , we have ( ψ ( p ∗ ) , β ; n ) = ( ψ ( p ) , α ; n ) T ⇒ T N 1 : N 1 ≥ st T ∗ N 2 : N 2 . Still no w, our presented compariso n results a re base d on usual sto chastic order under the condition that N 1 ≤ st N 2 . Therefore, it will be a common question whether the usua l sto chastic o rder can b e extended to other stro nger or derings like ha zard ra te or rev ersed hazard rate orders. The following counterexample s hows that w e can not generalize d Theorem 3.5 from usual sto chastic order to r eversed hazard ra te or der under the same c onditions. Coun terexample 3.3. Under the same set-up as of Example 3.1, Figur e 2 ( a ) pr esents that the r atio b etwe en F T N 1 : N 1 ( x ) and F T ∗ N 2 : N 2 ( x ) is not an incr e asing function of x for al l x ≥ 0 , which states that The or em 3.5 c an not b e ext en de d to r everse d hazar d r ate or der. 11 x axis 0 10 20 30 40 50 60 70 80 90 100 y axis 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8 0.82 (a) x axis 10 20 30 40 50 60 -7 -6 -5 -4 -3 -2 -1 0 1 y axis × 10 -4 (b) Figure 2: (a) Plot o f F T N 1 : N 1 ( x ) /F T ∗ N 2 : N 2 ( x ) as in Co un terexample 3.3. (b) Plot of ¯ F X 1: N 1 ( x ) − ¯ F Y 1: N 2 ( x ) as in Counterexample 3.1. Next, we establish so me cer tain conditions such that the reversed hazard r ate o r der holds b etw een tw o largest claim amounts T N : N and T ∗ N : N having r andom claim s ize N when the matrix o f parameter s ( ψ ( p ) , α ; n ) changes to ( ψ ( p ∗ ) , β ; n ) in the sense of the row w eak ma jor ization order in the space M n . T o co mplete the theorem, w e need to prov e the next t w o res ults which are related to weakly supe r ma jorization o rder. Theorem 3.11. L et { U 1 , . . . , U n } and { V 1 , . . . , V n } b e two sets of indep endent r andom variables with U i ∼ ¯ F ( x ; α i ) and V i ∼ ¯ F ( x ; β i ) , r esp e ctively. Also, let { J 1 , . . . , J n } and { J ∗ 1 , . . . , J ∗ n } b e another two set s of indep en- dent Bernoul li ra ndom variables, indep endently of U ′ i s and V ′ i s with E ( J i ) = p i and E ( J ∗ i ) = p i , r esp e ctively. Assume that the fol lowing c onditions hold: (i) ψ : (0 , 1 ) → (0 , ∞ ) is a differ entiable and de cr e asing fun ction; (ii) ¯ F ( x ; α i ) is de cr e asing and lo g- c onvex in α i for al l x ; (iii) r ( x ; α i ) is de cr e asing and c onvex in α i for al l x. Then, for ( ψ ( p ) , α ; n ) , ( ψ ( p ) , β ; n ) ∈ M n , we have α w  β ⇒ T n : n ≥ r h T ∗ n : n . Pr o of. T o complete the result, it is enough to show that ˜ r T n : n ( x ) , where ˜ r T n : n ( x ) = f T n : n ( x ) F T n : n ( x ) is decr easing a nd Sch ur-c o nv ex in α b y using Theorem A. 8 of Marshall et a l. (201 1). Here, F T n : n ( x ) re pr esents the distr ibution function of T n : n corres p o nding to the densit y function f T n : n ( x ). Then, for x ≥ 0 , f T n : n ( x ) = n Y i =1 (1 − ψ − 1 ( v i ) ¯ F ( x ; α i )) n X i =1 ψ − 1 ( v i ) ¯ F ( x ; α i ) r ( x ; α i ) 1 − ψ − 1 ( v i ) ¯ F ( x ; α i ) . Therefore, the reversed hazar d rate function of T n : n is giv en by ˜ r T n : n ( x ) = n X i =1 ψ − 1 ( v i ) ¯ F ( x ; α i ) r ( x ; α i ) 1 − ψ − 1 ( v i ) ¯ F ( x ; α i ) . (3.9) T a king pa rtial deriv ativ e of ˜ r T n : n ( x ) with resp ect to α i , we g et ∂ ˜ r T n : n ( x ) ∂ α i = d ¯ F ( x ; α i ) dα i ¯ F ( x ; α i ) r ( x ; α i ) ¯ F ( x ; α i ) ψ − 1 ( v i ) (1 − ψ − 1 ( v i ) ¯ F ( x ; α i )) 2 + dr ( x ; α i ) dα i ¯ F ( x ; α i ) ψ − 1 ( v i ) (1 − ψ − 1 ( v i ) ¯ F ( x ; α i )) 2 ≤ 0 , 12 due to the decreasing prop er ties of ¯ F ( x ; α i ) and r ( x ; α i ) in α i . Consider the case α i ≤ α j and v i ≤ v j for a ny pair i, j such that 1 ≤ i < j ≤ n . Accor ding to the conditions that ¯ F ( x ; α i ) is decreasing in α i and ψ − 1 ( v i ) is decreasing in v i , w e hav e r ( x ; α i ) ¯ F ( x ; α i ) ψ − 1 ( v i ) (1 − ψ − 1 ( v i ) ¯ F ( x ; α i )) 2 ≥ r ( x ; α j ) ¯ F ( x ; α j ) ψ − 1 ( v j ) (1 − ψ − 1 ( v j ) ¯ F ( x ; α j )) 2 . (3.10) Now, the decr easing a nd log- conv exit y pro p e rties o f ¯ F ( x ; α i ) in α i provides d ¯ F ( x ; α i ) dα i ¯ F ( x ; α i ) ≤ d ¯ F ( x ; α j ) dα j ¯ F ( x ; α j ) ≤ 0 . (3.11) As r ( x, α i ) is decreasing and conv ex in α i , th us we hav e dr ( x ; α j ) dα j ¯ F ( x ; α j ) ψ − 1 ( v j ) (1 − ψ − 1 ( v j ) ¯ F ( x ; α j )) 2 ≤ dr ( x ; α i ) dα i ¯ F ( x ; α i ) ψ − 1 ( v i ) (1 − ψ − 1 ( v i ) ¯ F ( x ; α i )) 2 ≤ 0 . (3.12) Finally , combining equations (3.10)-(3.12), we hav e the following ineq ua lity ( α i − α j )  ∂ ˜ r T n : n ( x ) ∂ α i − ∂ ˜ r T n : n ( x ) ∂ α j  ≥ 0 , which implies that ˜ r T n : n ( x ) is Sch ur- conv e x in α using Lemmas 2.1. Hence, the pro o f. Theorem 3.12. L et { U 1 , . . . , U n } and { V 1 , . . . , V n } b e two sets of indep endent r andom variables with U i ∼ ¯ F ( x ; α i ) and V i ∼ ¯ F ( x ; α i ) , r esp e ctively. Also , let { J 1 , . . . , J n } and { J ∗ 1 , . . . , J ∗ n } b e another two s et s of indep en- dent Bernoul li r andom variables, indep endently of U ′ i s and V ′ i s with E ( J i ) = p i and E ( J ∗ i ) = p ∗ i , r esp e ctively. Assume that the fol lowing c onditions hold: (i) ψ : (0 , 1 ) → (0 , ∞ ) is a differ entiable de cr e asing and lo g-c onvex function; (ii) ¯ F ( x ; α i ) and r ( x ; α i ) ar e de cr e asing in α i for al l x. Then, for ( ψ ( p ) , α ; n ) , ( ψ ( p ∗ ) , α ; n ) ∈ M n , we have ψ ( p ) w  ψ ( p ∗ ) ⇒ T n : n ≥ r h T ∗ n : n . Pr o of. In order to complete the theorem, it remains to s how whether ˜ r T n : n ( x ) is decreasing and Sc hur-conv ex in v , where v = ( ψ ( p 1 ) , . . . , ψ ( p n )) . T aking partial deriv ative o f ˜ r T n : n ( x ) with resp ect to v i , we hav e ∂ ˜ r T n : n ( x ) ∂ v i = dψ − 1 ( v i ) dv i ψ − 1 ( v i ) ¯ F 2 ( x ; α i ) r ( x ; α i ) (1 − ψ − 1 ( v i ) ¯ F ( x ; α i )) 2 ≤ 0 by decreas ing prop erty of ψ − 1 . Consider the ca se α i ≤ α j and v i ≤ v j for any pair i, j such that 1 ≤ i < j ≤ n. Now, decreasing pr op erties of ψ − 1 ( v i ) in v i , r ( x ; α i ) in α i and ¯ F ( x ; α i ) in α i provide ¯ F 2 ( x ; α i ) r ( x ; α i ) (1 − ψ − 1 ( v i ) ¯ F ( x ; α i )) 2 ≥ ¯ F 2 ( x ; α j ) r ( x ; α j ) (1 − ψ − 1 ( v j ) ¯ F ( x ; α j )) 2 . (3.13) Using decr easing a nd log-conv ex ity prop er ties of ψ − 1 , we g et dψ − 1 ( v i ) dv i ψ − 1 ( v i ) ≤ dψ − 1 ( v j ) dv j ψ − 1 ( v j ) ≤ 0 . (3.14) Finally , combining equations (3.13) and (3.14), we have the following ine q uality ( v i − v j )  ∂ ˜ r T n : n ( x ) ∂ v i − ∂ ˜ r T n : n ( x ) ∂ v j  ≥ 0 , (3.15) which concludes that ˜ r T n : n ( x ) is decreasing and Sch ur -conv ex in v according to Theorem A. 8 of Marshall et al. (2011). Hence, the pr o of. 13 Theorem 3.13. L et { U 1 , . . . , U n } and { V 1 , . . . , V n } b e two sets of indep endent r andom variables with U i ∼ ¯ F ( x ; α i ) and V i ∼ ¯ F ( x ; β i ) , r esp e ctively with α n ≥ β n and p n ≤ p ∗ n . A lso, let { J 1 , . . . , J n } and { J ∗ 1 , . . . , J ∗ n } b e another two sets of indep endent Bernoul li r andom variables, indep endently of U ′ i s and V ′ i s with E ( J i ) = p i and E ( J ∗ i ) = p ∗ i , r esp e ct ively. F urther, let N 1 and N 2 b e two p ositive inte ger-value d r andom variables indep endently of T ′ i s and T ∗ i ′ s satisfying N 1 st = N 2 st = N , r esp e ctively. Assu me that the fol lowing c onditions hold: (i) ψ : (0 , 1 ) → (0 , ∞ ) is a differ entiable, strictly de cr e asing and lo g-c onvex funct ion. (ii) ¯ F ( x ; α i ) is de cr e asing and lo g- c onvex in α i for al l x ; (iii) r ( x ; α i ) is de cr e asing and c onvex in α i for al l x . Then, for ( ψ ( p ) , α ; n ) , ( ψ ( p ∗ ) , β ; n ) ∈ M n , we have ( ψ ( p ) , α ; n ) w > ( ψ ( p ∗ ) , β ; n ) ⇒ T N : N ≥ r h T ∗ N : N . Pr o of. Applying Theo rem 3 . 4 of K undu et al. (2024), our aim is to chec k (i) F T n : n ( x ) F T ∗ n : n ( x ) is increasing in n ; (ii) ˜ r T ∗ n : n ( x ) is increasing in n ; (iii) T n : n ≥ r h T ∗ n : n . Denote A ( n ) = F T n : n ( x ) F T ∗ n : n ( x ) = Q n i =1 (1 − ψ − 1 ( v i ) ¯ F ( x ; α i )) Q n i =1 (1 − ψ − 1 ( u i ) ¯ F ( x ; β i )) . Using α n ≥ β n and p n ≤ p ∗ n , we ca n easily verify that A ( n ) > A ( n − 1) which implies F T n : n ( x ) F T ∗ n : n ( x ) is incr easing in n. Also, we can easily chec k that ˜ r T ∗ n +1: n +1 ( x ) − ˜ r T ∗ n : n ( x ) > 0 , which shows ˜ r X n : n ( x ) is increasing in n. Using the same lines as for Theorem 3.3, we ca n wr ite the following result ( ψ ( p ) , α ; n ) w > ( ψ ( p ∗ ) , β ; n ) ⇒ T n : n ≥ r h T ∗ n : n , with the help of Theorem 3.1 1 and Theorem 3 .12. Finally applying Theor em 3 . 4 of Kundu et al. (202 4), we can o btain the required result. It is noteworth y to mention that the condition “ ¯ F ( x ; α i ) is decreasing and lo g-conv ex in α i for all x ” presented in Theorems 3.3 and 3.13, is very imp or tant for mo deling claims. An insur er usually applies the decreasing surviv al function to mo del the time until a claim o c c urs. In addition, the decreasing log-c onv ex ity prop erty ca n b e suitable in a situation where the risk o f a claim increase s ov er time, yet the rate of increa s e gradually slows down. Thus, in insurance analysis, a decr easing log-co nv ex sur viv al function is appropriate as it allows for a mor e r elev ant model of risk over time, capturing the evolving behavior o f claim probabilities, which will be used for ma k ing appropria te premium and capital allo cation p olicies. In Theorem 3.1 3, the results pres e n ted are for the largest claims amounts in the sense of reversed hazard rate o rder. Now a natural question that arises whether the result is true for the sma llest claim a mounts ha ving random claim size. T o prov e the result, we first need to establish the fo llowing tw o theor e ms based on vector ma jorizatio n, where the first theorem shows under some certain conditions if α m  β then T 1: n is smaller than T ∗ 1: n with resp ect to the reversed hazar d rate or der and the second one provides the same or dering holds betw een T 1: n and T ∗ 1: n whenever ψ ( p ) m  ψ ( p ∗ ) . Theorem 3.14. L et { U 1 , . . . , U n } and { V 1 , . . . , V n } b e two sets of indep endent r andom variables with U i ∼ ¯ F ( x ; α i ) and V i ∼ ¯ F ( x ; β i ) , r esp e ctively. Also, let { J 1 , . . . , J n } and { J ∗ 1 , . . . , J ∗ n } b e another two set s of indep en- dent Bernoul li ra ndom variables, indep endently of U ′ i s and V ′ i s with E ( J i ) = p i and E ( J ∗ i ) = p i , r esp e ctively. Assume that the fol lowing c onditions hold: (i) ψ : (0 , 1 ) → (0 , ∞ ) is a differ entiable function; 14 (ii) F ( x ; α i ) is incr e asing and lo g-c onvex in α i for al l x ; (iii) r ( x ; α i ) is c onvex in α i for al l x . Then, for ( ψ ( p ) , α ; n ) , ( ψ ( p ) , β ; n ) ∈ M n , we have α m  β ⇒ T 1: n ≥ r h T ∗ 1: n . Pr o of. In proving the result, utilizing Theorem A. 8 of Marshall et al. (201 1), w e only need to chec k whether ˜ r T 1: n ( x ) is Sch ur-conv ex in α , where ˜ r T 1: n ( x ) = f T 1: n ( x ) F T 1: n ( x ) and ˜ r T ∗ 1: n ( x ) = f T ∗ 1: n ( x ) F T ∗ 1: n ( x ) . Here, F T 1: n ( x ) represents the distribution function of T 1: n corres p o nding to the density function f T 1: n ( x ). Then, for x ≥ 0 , F T 1: n ( x ) = P ( T 1: n ≤ x ) = 1 − n Y i =1  ψ − 1 ( v i ) ¯ F ( x ; α i )  and f T 1: n ( x ) = n Y i =1  ψ − 1 ( v i ) ¯ F ( x ; α i )  n X i =1 r ( x ; α i ) = ¯ F T 1: n ( x ) n X i =1 r ( x ; α i ) . Therefore, the reversed hazar d rate function of T 1: n is giv en by ˜ r T 1: n ( x ) = Q n i =1  ψ − 1 ( v i ) ¯ F ( x ; α i )  1 − Q n i =1  ψ − 1 ( v i ) ¯ F ( x ; α i )  n X i =1 r ( x ; α i ) = ¯ F T 1: n ( x ) F T 1: n ( x ) n X i =1 r ( x ; α i ) . Now, taking pa rtial der iv ative of ˜ r T 1: n ( x ) with resp ect to α i we have ∂ ˜ r T 1: n ( x ) ∂ α i = ∂ ∂ α i  ¯ F T 1: n ( x ) F T 1: n ( x )  n X i =1 r ( x ; α i ) +  ¯ F T 1: n ( x ) F T 1: n ( x )  dr ( x ; α i ) dα i , where ∂ ∂ α i  ¯ F T 1: n ( x ) F T 1: n ( x )  = dF ( x ; α i ) dα i F ( x ; α i )  F T 1: n ( x ) [ F T 1: n ( x )] 2  . No w ( α i − α j )  ∂ ˜ r T 1: n ( x ) ∂ α i − ∂ ˜ r T 1: n ( x ) ∂ α j  ≥ 0 , by the given conditions, whic h co ncludes ˜ r T 1: n ( x ) is Sch ur- conv e x in α . Hence, the pro of. T o prove the next theorem, w e need the fo llowing result. Theorem 3.15. S upp ose X 1: n ∼ ¯ F 1: n ( x ) and X 1: n ∼ ¯ G 1: n ( x ) . Also let the supp ort of a p ositive int e ger-value d r andom variable N having pmf p ( n ) b e N + . Now, for al l x ≥ l , if ˜ r Y 1: n ( x ) , t he r everse d hazar d r ate function of Y 1: n is incr e asing in n ∈ N + and F 1: n ( x ) G 1: n ( x ) is incr e asing in n ∈ N + , then X 1: n ≥ r h Y 1: n implies X 1: N ≥ r h Y 1: N . Pr o of. If ˜ r Y 1: n ( x ) is increasing in n, then for any tw o positive integers n 1 ≤ n 2 and for a ll x ≥ l w e ca n write ˜ r Y 1: n 1 ( x ) ≤ ˜ r Y 1: n 2 ( x ) , which is equiv alent to say that G 1: n 1 ( x ) G 1 : n 2 ( x ) is decr easing in l ≤ x ≤ u. Thus we can prov e the required result using Pr op osition 3 . 4 and Theo rem 3 . 1 o f Kundu et a l. (20 2 3). Theorem 3.16. L et { U 1 , . . . , U n } and { V 1 , . . . , V n } b e two sets of indep endent r andom variables with U i ∼ ¯ F ( x ; α i ) and V i ∼ ¯ F ( x ; α i ) , r esp e ctively. A lso, let { J 1 , . . . , J n } and { J ∗ 1 , . . . , J ∗ n } b e another two sets of in- dep endent Bernoul li ra ndom variables, indep endently of U ′ i s and V ′ i s with E ( J i ) = p i and E ( J ∗ i ) = p ∗ i , r e- sp e ct ively. Assume that ψ : (0 , 1) → (0 , ∞ ) is a differ entiable incr e asing and lo g-c onvex function. Then, for ( ψ ( p ) , α ; n ) , ( ψ ( p ∗ ) , α ; n ) ∈ M n , we have ψ ( p ) m  ψ ( p ∗ ) ⇒ T 1: n ≥ r h T ∗ 1: n . 15 Pr o of. Similar to Theorem 3 .1 4, a fter tak ing partial deriv ative of ˜ r T 1: n ( x ) with resp ect to v i , w e hav e ∂ ˜ r T 1: n ( x ) ∂ v i = ∂ ∂ v i  ¯ F T 1: n ( x ) F T 1: n ( x )  n X i =1 r ( x ; α i ) , where ∂ ∂ v i  ¯ F T 1: n ( x ) F T 1: n ( x )  = dψ − 1 ( v i ) dv i ψ − 1 ( v i )  ¯ F T 1: n ( x ) [ F T 1: n ( x )] 2  . No w ( v i − v j )  ∂ ˜ r T 1: n ( x ) ∂ v i − ∂ ˜ r T 1: n ( x ) ∂ v j  ≥ 0 , by the given conditions whic h yields ˜ r T 1: n ( x ) is Sch ur-conv ex in v . Hence, the pro o f. Theorem 3.17. L et { U 1 , . . . , U n } and { V 1 , . . . , V n } b e two sets of indep endent r andom variables with U i ∼ ¯ F ( x ; α i ) and V i ∼ ¯ F ( x ; β i ) , r esp e ctively. Also, let { J 1 , . . . , J n } and { J ∗ 1 , . . . , J ∗ n } b e another two set s of indep en- dent Bernoul li r andom variables, indep endently of U ′ i s and V ′ i s with E ( J i ) = p i and E ( J ∗ i ) = p ∗ i , r esp e ctively. F urther, let N 1 and N 2 b e t wo p ositive inte ger-value d r andom variables indep endently of T ′ i s and T ∗ i ′ s satisfying N 1 st = N 2 st = N , r esp e ctively. Assume that the fol lowing c onditions hold: (i) ψ : (0 , 1 ) → (0 , ∞ ) is a differ entiable, strictly incr e asing and lo g- c onvex funct ion; (ii) F ( x ; α i ) is lo g-c onvex in α i for al l x ; (iii) r ( x ; α i ) is c onvex in α i for any x . Then, for ( ψ ( p ) , α ; n ) , ( ψ ( p ∗ ) , β ; n ) ∈ M n , we have ( ψ ( p ) , α ; n ) r ow > ( ψ ( p ∗ ) , β ; n ) ⇒ T 1: N ≥ r h T ∗ 1: N . Pr o of. Applying the similar idea as of Theorem 3.3, we can state the following result ( ψ ( p ) , α ; n ) r ow > ( ψ ( p ∗ ) , β ; n ) ⇒ T 1: n ≥ r h T ∗ 1: n , with the help o f Theorem 3 .14 and Theorem 3 .16. Finally , utilizing Theorem 3 . 3 of Kundu et al. (2024), one can ea sily o btain the desired result. Remark 3 . 2. It is quite natur al t o find some distribution fu n ctions having ( i ) ¯ F ( x ; α i ) is de cr e asing and lo g-c onvex in α i for al l x ; ( ii ) r ( x ; α i ) is de cr e asing and c onvex in α i for al l x ; ( ii i ) r ( x ; α i ) is c onvex in α i for al l x ; ( iv ) F ( x ; α i ) is lo g-c onvex in α i . F or example, c onsider (i) exp onent ial distribution with su rvival function ¯ F ( x ; α ) = e − αx , x, α > 0 , which satisfies al l the c onditions pr ovide d in ( i ) - ( iv ); (ii) Weibul l distribution with survival function ¯ F ( x ; α, β ) = e − β x α , x, α, β > 0 , which satisfies al l t he c onditions with r esp e ct to β pr esente d in ( i ) - ( iv ); (iii) Power-gener alize d Weibul l distribut ion with survival fun ction ¯ F ( x ; c, k ) = e 1 − (1+ x c ) 1 k , x, c, k > 0 , which satisfies pr op ert ies ( i ) and ( ii ) with r esp e ct to k ; (iv) Gamma distribution with density function f ( x ; α, β ) = 1 Γ( α ) β α x α − 1 e − x β , α, β > 0 , whi ch satisfies c ondi- tions ( iii ) and ( iv ) with r esp e ct to β . It is imp ortant to note t hat a distribution having c onstant hazar d r ate is very useful in insu r anc e claim m o deling, p articularly in ac cident insu r anc e as the events ar e r andom and age indep endent. As do cumente d in the liter atur e (se e Mer aou et al. (202 2) and Abub akar and Danrimi (202 3)), the exp onent ial distribution, Weibul l distribution play a crucial r ole for mo deling the claim severity in actuarial scienc e which p erform wel l to analyze skewe d data. 16 4. Application Ordering results b etw een tw o random extremes is useful in r eliability and auction theory . In the following, we consider so me applica tions o f o ur established theoretical results for the purpose of illustration. 4.1 Reliability theory Consider a r eliability system having n comp onents in working condition and suppos e each components ar e sub jected to a sho ck that may r esult in failur e . The lifetimes of each co mpo ne nts of the system ar e r epresented by the non-nega tive ra ndom v ariables U 1 , . . . , U n , which ar e exp er ienced with a r andom sho ck at the b eginning. Let J 1 , . . . , J n be another collectio n of indep endent Bernoulli random v a riables, independently of U ′ i s with E ( J i ) = p i , whic h can b e called as sho ck para meter. Also , let with proba bility p i the i th c omp o nent still working after re ceiving the ra ndom sho ck if J i = 1 , otherwis e it fail to work with probability 1 − p i . Then the random v a riable T i = U i J i represent the lifetime of the i th component in a system under sho ck. In re liability analysis, measuring the proba bilit y of remaining health y or the av er age lifetime of a system whic h are under random sho ck plays a v ital role. Sto chastic compar ison of such reliability sy stems has attracted significant resear ch interest. Several co mpa rison results based on fixed sample size for these r eliability mo del have b een developed by differ e n t authors (see F a ng and Balakr is hnan (2018), Kundu and Chowdh ury (20 21), Abdolahi et al. (2023) and Amini-Seresht et al. (2 024)). In practice, it is not alwa y s p ossible to get a system having fixed sample siz e because some observ ations g et lo st for differ e nt reasons a nd sometimes it depends up on the o ccurrence of some even ts, whic h makes the sample size rando m. Therefor e, a nalyzing the reliability of a system ha ving rando m num b er of co mpo ne nts which ar e under random shock is of gre at imp ortance from a practical po in t of view. In this subsection, w e apply our establish results for the ab ove presented model to compare t wo series as well as parallel systems having ra ndo m num b er of comp onents which are under rando m sho ck and g eneralized so me existing results fr om fixed sample size to differen t (r andom) sa mple sizes . Let us consider a sys tem ha ving N 1 nu m be r of comp onents, where U 1 , . . . , U N 1 with U i ∼ ¯ F ( x ; α i ) repre- senting the life-times of the comp onents o f the s ystem which may received a random sho ck at the b eginning. Let { J 1 , . . . , J N 1 } b e another set of indep endent Bernoulli random v ar iables, indep endently of U ′ i s with E ( J i ) = p i such that for a given time p erio d J i = 1 if the i th comp onent is remain op er ating after a n exp erience of rando m sho cks a nd, J i = 0 if the i th component fa il due to the random shock. Let T i = U i J i , for i = 1 , . . . , N 1 . Here, N 1 is a pos itive integer-v a lued random v a riable with supp ort { 1 , 2 , . . . } , indep endently of T ′ i s. Under this set-up, T 1 , . . . , T N 1 represent the life-times of compo ne nts o f the system that are sub ject to random sho cks. Thu s, T N 1 : N 1 = max { T 1 , . . . , T N 1 } a nd T 1: N 1 = min { T 1 , . . . , T N 1 } r epresent the life-times o f the parallel and series systems, resp ectively having random nu m be r of observ ations . Therefor e, using o ur e stablished results presented in Theorem 3.5 (Theor em 3 .3), we can say that under some certain co nditions if N 1 ≤ st N 2 then the surviv a l time of the parallel sys tem T N 1 : N 1 (series system T 1: N 1 ) is sma lle r (greater) than the surviv a l time of the paralle l sy s tem T ∗ N 2 : N 2 (series system T ∗ 1: N 2 ) when the matrix of par ameters ( ψ ( p ) , α , n ) changes to ( ψ ( p ∗ ) , β , n ) in the s ense o f the row weak ma joriza tion order in the spa ce M n . Similar ly , Theorem 3 .13 and Theo rem 3.17 pro vide that based on some cer tain co nditions tw o parallel a s well a s s eries systems that are sub ject to r andom sho cks a re comparable acco rding to the reversed ha z a rd rate or der when the matrix of parameters ( ψ ( p ) , α , n ) changes to ( ψ ( p ∗ ) , β , n ) in the sense o f the r ow weak ma jorization order and row ma jorize o rder, resp ectively in the space M n . Simila rly , Theo rems 3.6, 3.7 and 3 .8 can b e in terpreted as ab ov e when the system co mp onents follow Kw-G family , scale family and pro po rtional hazard rate family , resp ectively . Remark 4.1. It s hould b e mentione d that The or em 3.6 is an extens ion of The or em 3 . 3 of Kundu and Chowd- hury (2021) for the c ase of r andom nu mb er of sho cks. 4.2 Auct ion Theory There ar e v arious kinds o f auctions . Among them, in the first-price reverse auction, the bidders (seller s ) need to pr ovide their sealed bids to the a uctioneer (buyer) who initiates the auction to purchase some sp ecific items. The bidder who submit the low e st price is likely to win the bid and will be received the lowest bid from the auctioneer. At the same time, due to some unant icipated s itua tions, some of the bidders may wis h to opt o ut 17 from the auction b efore it starts. F or this reason, the auction ta kes a discussion tha t the final cost turns o ut to b e the low est bid (the smallest order statistics) that comes from J 1 U 1 . . . , J n U n , where J i denotes whether the bidder i th participates in the auction o r not, and U i is in ter preted a s the bidding price for the i th bidder if he/ she takes a part in the auctio n, i = 1 , . . . , n . In the ab ove situation, the num ber o f bidders may b e random. Therefo re, the re s ults presented in the previous sectio n can b e applicable in quantitativ e analysis on the effects of num ber o f bidders, their attending proba bilities and biding price distributions o n the final auction pr ice (see Zhang et al. (2019)). In this r egard, let the bids U 1 , . . . , U N 1 with U i ∼ ¯ F ( x ; α i ) . F ur ther, let { J 1 , . . . , J N 1 } b e another set of independent Bernoulli r andom v ariables, independently of U ′ i s with E ( J i ) = p i such that a g iven time perio d J i = 1 if the i th bidder participa tes in the a uction and, J i = 0 if the i th bidder opt out from the auction. Suppo se T i = U i J i , for i = 1 , . . . , N 1 . Here, N 1 is the num b er o f bidders which is a p o sitive in teger-v alued random v ar iable with supp or t { 1 , 2 , . . . } , indep endently of T ′ i s. Under this set-up, T 1: N 1 = min { T 1 , . . . , T N 1 } represent the low est bid a mount in the auction where the n um ber o f bidder is ra ndom. T hus, from Theorem 3.3, w e ca n say tha t under some certain conditions if N 1 ≤ st N 2 then the final pric e will b e sto chastically lar ger when the matr ix of para meters ( ψ ( p ) , α , n ) changes to ( ψ ( p ∗ ) , β , n ) in the s ense of the r ow weak ma jorization order in the space M n . Similar interpretation can be found from Theorem 3.17. 5. Conclusion In this article, we hav e co nsidered t w o indep endent heterogeneo us p o rtfolios of risks having different and random nu mber o f claims a nd prov e d several comparis o n results base d on usual sto chastic and re versed ha z ard rate orders when the matr ix of parameters ( ψ ( p ) , α , n ) changes to ( ψ ( p ∗ ) , β , n ) in the sense o f the row weak ma joriza tio n order and row ma jorize in the space M n . F urthermore, utilizing o ur res ults, we ha ve see n that our obtained results genera lize the r esult established in Barmalza n et a l. (2017), Bala krishnan et al. (2018) and K undu and Chowdhury (2021). In this work, our developed r esults are mainly based on independent heteroge ne o us portfolio s of risks for the case of random claim sizes . Howev er , the growing complexity of insurance and reinsur ance pro ducts has sparked considerable interest in modeling dep endent risks. In this regard, there ar e many research a v ailable in litera ture r elated to dep endent p or tfolio s of risk s fo r the ca se of fixed c la im sizes (see Na de b et al. (2020 b); T o rrado and Nav arro (202 1); Das et a l. (2022); Liu a nd Y an (202 2); Panahi et al. (202 3 ); Zhang et a l. (202 3)). Thu s, we can think to ex tend the existing r esults for the cas e of r andom claim sizes, which could be some int eresting future research pr o blems. It is also imp ortant to mention that our es ta blished results are mainly based on row w eak ma jorizatio n or der in a particula r space. Th us, it will also b e interesting to obtain the same res ults based on o ther ma jor ization orders (like m ultiv ariate c hain ma jor ization order) in different spa c e s. Moreov er, our established ordering results a re based on the usual sto chastic and reversed hazar d rate orders. Therefore, we can also extend these results for other sto chastic orders . 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