On Generalized Reversed Aging Intensity Functions
OOn Generalized Reversed Aging
Intensity Functions
Francesco Buono
Universit`a di Napoli Federico IIItaly
Maria Longobardi
Universit`a di Napoli Federico IIItaly
Magdalena Szymkowiak
Poznan University of TechnologyPolandSeptember 11, 2020
Abstract
The reversed aging intensity function is defined as the ratio of the instantaneous re-versed hazard rate to the baseline value of the reversed hazard rate. It analyzes the agingproperty quantitatively, the higher the reversed aging intensity, the weaker the tendency ofaging. In this paper, a family of generalized reversed aging intensity functions is introducedand studied. Those functions depend on a real parameter. If the parameter is positive theycharacterize uniquely the distribution functions of univariate positive absolutely continuousrandom variables, in the opposite case they characterize families of distributions. Further-more, the generalized reversed aging intensity orders are defined and studied. Finally,several numerical examples are given.
Keywords:
Generalized reversed aging intensity, Reversed hazard rate, Generalized Paretodistribution, Generalized reversed aging intensity order.
AMS Subject Classification : 60E15, 60E20, 62N05
Let X be a non–negative and absolutely continuous random variable with cumulative distri-bution function (cdf) F , probability density function (pdf) f and survival function (sf) F . Inreliability theory F is also known as unreliability function whereas F as reliability function.1 a r X i v : . [ m a t h . S T ] S e p n this context a great importance has the hazard rate function r of X , also known as theforce of mortality or the failure rate, where X is the survival model of a life or a system beingstudied. This definition will cover discrete survival models as well as mixed survival models.In the same way we define the reversed hazard rate ˘ r of X , that has attracted the attentionof researchers. In a certain sense it is the dual function of the hazard rate and it bears someinteresting features useful in reliability analysis (see also Block and Savits 1998; Finkelstein2002).Let X be a random variable with sf F and cdf F . We define, for x such that F ( x ) >
0, thehazard rate function of X at x , r ( x ), in the following way: r ( x ) = lim ∆ x → + P ( x < X ≤ x + ∆ x | X > x )∆ x = 1 F ( x ) lim ∆ x → + P ( x < X ≤ x + ∆ x )∆ x . Moreover we define, for x such that F ( x ) >
0, the reversed hazard rate function of X at x ,˘ r ( x ) (see Bartoszewicz 2009 for the notation), in the following way:˘ r ( x ) = lim ∆ x → + P ( x − ∆ x < X ≤ x | X ≤ x )∆ x = 1 F ( x ) lim ∆ x → + P ( x − ∆ x < X ≤ x )∆ x . The reversed hazard rate ˘ r ( x ) can be treated as the instantaneous failure rate occurring im-mediately before the time point x (the failure occurs just before the time point x , given thatthe unit has not survived longer than time x ).So, if X is an absolutely continuous random variable with density f , for x such that F ( x ) >
0, the hazard rate function is r ( x ) = 1 F ( x ) lim ∆ x → + P ( x < X ≤ x + ∆ x )∆ x = f ( x ) F ( x ) , (1)while, for x such that F ( x ) >
0, the reversed hazard rate function is˘ r ( x ) = 1 F ( x ) lim ∆ x → + P ( x − ∆ x < X ≤ x )∆ x = f ( x ) F ( x ) . (2)By the hazard rate function we introduce the aging intensity function L that is defined for x > L ( x ) = − xf ( x ) F ( x ) log F ( x ) = − xr ( x )log F ( x ) , (3)where log denotes the natural logarithm. It can be showed that the survival function ofan absolutely continuous random variable and its aging intensity function are related by a2elationship and that under some conditions a function determines a family of survival functionsand it is their aging intensity function, for more details see Szymkowiak (2018a).The reversed aging intensity function ˘ L ( x ) is defined, for x >
0, as follows (see also Rezaeiand Khalef 2014) ˘ L ( x ) = − xf ( x ) F ( x ) log F ( x ) = − x ˘ r ( x )log F ( x ) . (4)The reversed aging intensity function can be expressed also in a different way by observingthat the cumulative reversed hazard rate function defined as˘ R ( x ) = (cid:90) + ∞ x ˘ r ( t ) d t = log F ( t ) (cid:12)(cid:12) t → + ∞ t = x = − log F ( x ) , can be treated as the total amount of failures accumulated after the time point x . So ˘ H ( x ) = x ˘ R ( x ), being the proportion between the total amount of failures accumulated after the timepoint x and the time x for which the unit is still survived, can be considered as the baselinevalue of the reversed hazard rate. Then, (4) can be written as˘ L ( x ) = x ˘ r ( x )˘ R ( x ) = ˘ r ( x )˘ H ( x ) , and so the reversed aging intensity function, defined as the ratio of the instantaneous reversedhazard rate ˘ r to the baseline value of the reversed hazard rate ˘ H , expresses the units averageaging behavior: the higher the reversed aging intensity (it means the higher the instantaneousreversed hazard rate, and the smaller the total amount of failures accumulated after the timepoint x , and the higher the the time x for which the unit is still survived), the weaker thetendency of aging.It is the analogous for the future of the aging intensity function, introduced and studiedby Bhattacharjee, Nanda and Misra (2013), Jiang, Ji and Xiao (2003), Nanda, Bhattachar-jee and Alam (2007) and Szymkowiak (2018a). The concept of aging intensity function wasgeneralized by Szymkowiak (2018b). In Section 2, we define the generalized reversed agingintensity functions and in the particular case in which the random variable has a generalizedPareto distribution function we generalize our results and study monotonicity properties. InSection 3, we give some characterizations with use of our new aging intensities. Some examplesof characterization are given in Section 4. In Section 5, we study the family of new stochas-tic orders called α –generalized reversed aging intensity orders. Then, in Section 6 we presentexamples of analysis of α –generalized reversed aging intensity through generated and real data.3 Generalized reversed aging intensity functions
Let, for x > W ( x ) = 1 − exp( − x ), i.e., W is the distribution function of an exponentialvariable with parameter 1, so ˘ R ( x ) = W − (1 − F ( x )). In fact, W − ( x ) = − log(1 − x ) and so W − (1 − F ( x )) = − log F ( x ) = ˘ R ( x ) . Replacing W with a strictly increasing distribution function G with density g , it is possibleto generalize the concepts of reversed hazard rate function, cumulative reversed hazard ratefunction and reversed aging intensity function. The generalization of the hazard rate functionwas introduced by Barlow and Zwet (1969a, 1969b). Definition 1.
Let X be a non–negative and absolutely continuous random variable with cdf F . Let G be a strictly increasing distribution function with density g . We define the G –generalized cumulative reversed hazard rate function, ˘ R G , the G –generalized reversed hazardrate function, ˘ r G , the G –generalized reversed aging intensity function, ˘ L G , of X as˘ R G ( x ) = G − (1 − F ( x )) , (5)˘ r G ( x ) = − d ˘ R G ( x )d x = f ( x ) g ( G − (1 − F ( x ))) , (6)˘ L G ( x ) = x ˘ r G ( x )˘ R G ( x ) = xf ( x ) g ( G − (1 − F ( x ))) G − (1 − F ( x )) . (7)A very interesting case, because it provides intuitive results, is the one in which the distri-bution function G is the distribution function of a generalized Pareto distribution. Definition 2.
A random variable X α follows a generalized Pareto distribution with parameter α ∈ R if the distribution function W α is expressed as (see Pickands 1975): W α ( x ) = − (1 − αx ) α , for x > , if α < < x < α , if α > − exp( − x ) , for x > α = 0 Remark 1.
For α = 0 we have the distribution function of an exponential variable withparameter 1.From the distribution function it is possible to obtain the quantile and the density function.In particular we have W − α ( x ) = α [1 − (1 − x ) α ] , for 0 < x < , if α (cid:54) = 0 − log(1 − x ) , for 0 < x < , if α = 04 α ( x ) = (1 − αx ) − αα , for x > , if α < < x < α , if α > − x ) , for x > , if α = 0Let X be a non–negative and absolutely continuous random variable with cdf F and pdf f .Then it is possible to determine the W α - generalized cumulative reversed hazard rate functionand the W α - generalized reversed hazard rate function in the following way:˘ R W α ( x ) = W − α (1 − F ( x )) = α [1 − F α ( x )] , for x > , if α (cid:54) = 0 − log F ( x ) , for x > , if α = 0˘ r W α ( x ) = − d ˘ R α ( x )d x = F α − ( x ) f ( x ) , for x > . For the sake of simplicity, those functions can be, respectively, indicated by ˘ R α , ˘ r α and wecan refer to them as the α –generalized cumulative reversed hazard rate function and the α –generalized reversed hazard rate function. Remark 2.
The 1–generalized reversed hazard rate function is equal to the density function.In fact the density function gives a first rough illustration of the aging tendency of the randomvariable by its monotonicity. The 0–generalized reversed hazard rate function is equal to theusual reversed hazard rate function.From these functions, it is possible to introduce the α –generalized reversed aging intensityfunction ˘ L α ( x ) = ˘ r α ( x ) x ˘ R α ( x ) = αxF α − ( x ) f ( x )1 − F α ( x ) , for x > , if α (cid:54) = 0 − xf ( x ) F ( x ) log F ( x ) , for x > , if α = 0 (8)The α –generalized reversed aging intensity function describes the relationship between theinstantaneous value of the α –generalized reversed hazard rate function ˘ r α ( x ) and the baselinevalue of the α –generalized reversed hazard rate function x ˘ R α ( x ). The higher the α –generalizedreversed aging intensity function (it means the higher the actual value of the α –generalizedreversed hazard rate function respect to its baseline value), the weaker the tendency of aging.Moreover, the α –generalized reversed aging intensity function can be treated as the elasticity(see Sydsaeter and Hammond 2012), except for the sign, of the α –generalized cumulativereversed hazard rate function, i.e., it indicates how much the function ˘ R α changes if x changesby a small amount. 5e recall the definition of α –generalized aging intensity functions, L α . These functions aredefined by Szymkowiak (2018b) in the following way L α ( x ) = αx (1 − F ( x )) α − f ( x )1 − (1 − F ( x )) α , for x > , if α (cid:54) = 0 − xf ( x )(1 − F ( x )) log(1 − F ( x )) , for x > , if α = 0 (9) Remark 3.
The 0–generalized reversed aging intensity function is equal to the usual reversedaging intensity function. If α = 1 we have˘ L ( x ) = xf ( x ) F ( x ) , i.e., it is the negative of the elasticity of the survival function F , they are equal in modulus.If α = n ∈ N we have ˘ L n ( x ) = nxF n − ( x ) f ( x )1 − F n ( x ) , where the denominator is the survival function of the largest order statistic for a sample of n i.i.d. variables, while the numerator is composed by x multiplied for the density of this orderstatistic. So ˘ L n can be considered as the negative of the elasticity for the survival function ofthe largest order statistic.If α = − L − ( x ) = − x ( F ( x )) − f ( x )1 − ( F ( x )) − = − xf ( x ) F ( x ) F ( x ) − F ( x ) = xf ( x ) F ( x )(1 − F ( x )) , and so ˘ L − ( x ) = xLOR X ( x ) = L − ( x ) , where LOR X is the log-odds rate of X (see Zimmer, Wang and Pathak 1998).The next proposition analyzes the monotonicity of α –generalized reversed aging intensityfunctions respect to the parameter α . This result could be important if we introduce stochas-tic orders based on α –generalized reversed aging intensity functions, i.e., αRAI orders, andcompare these orders as α varies (see Section 5). Proposition 2.1.
Let X be a non–negative and absolutely continuous random variable withcdf F and pdf f . Then the α –generalized reversed aging intensity function is decreasing respectto α ∈ R , ∀ x ∈ (0 , + ∞ ) .Proof. For some c ∈ (0 ,
1) we consider the function h c ( α ) = αc α − c α , for α (cid:54) = 0. Thend h c ( α )d α = c α (1 − c α + log c α )(1 − c α ) . c α ∈ (0 , + ∞ ) and the function k ( t ) = 1 − t +log t is negativefor t > k (1) = 0 and 1 is maximum point for this function. So h c is decreasing in ( −∞ , ∪ (0 , + ∞ ). Defining the extension for continuity in 0 of h c , h c (0) = lim α → h c ( α ) = lim α → αc α − c α = lim α → c α + αc α log c − c α log c = − c , it is possible to say that h c is decreasing in R .Fixing c = F ( x ), with x >
0, and multiplying h F ( x ) ( α ) for the positive factor xf ( x ) F ( x ) we getthat the following function xf ( x ) F ( x ) h F ( x ) ( α ) = αxF α − ( x ) f ( x )1 − F α ( x ) , if α (cid:54) = 0 − xf ( x ) F ( x ) log F ( x ) , if α = 0 = ˘ L α ( x )is decreasing in α as x is fixed. α –generalized reversed agingintensity In reliability theory some functions characterize the associated distribution function. Forexample it was showed in Barlow and Proschan (1996) that the hazard rate of an absolutelycontinuous random variable uniquely determines its distribution function.In the following theorem we show that, for α <
0, the distribution function of a non–negative and absolutely continuous random variable is defined by the α –generalized reversedaging intensity function and that, under some conditions, a function can be considerated asthe α –generalized reversed aging intensity function for a family of random variables. Theorem 3.1.
Let X be a non–negative and absolutely continuous random variable with cdf F and let ˘ L α be its α –generalized reversed aging intensity function with α < . Then F and ˘ L α are related, for all a ∈ (0 , + ∞ ) , by the relationship F ( x ) = (cid:34) − (1 − F α ( a )) exp (cid:32) − (cid:90) xa ˘ L α ( t ) t d t (cid:33)(cid:35) α , x ∈ (0 , + ∞ ) . (10) Moreover, a function ˘ L defined on (0 , + ∞ ) and satisfying, for a ∈ (0 , + ∞ ) , the followingconditions:(1) ≤ ˘ L ( x ) < + ∞ , for all x ∈ (0 , + ∞ ) ;(2) lim x → + (cid:82) ax ˘ L ( t ) t d t = + ∞ ; lim x → + ∞ (cid:82) xa ˘ L ( t ) t d t = + ∞ ;determines, for α < , a family of absolutely continuous distribution functions F k by therelationship F k ( x ) = 1 − W α (cid:32) k exp (cid:32) − (cid:90) xa ˘ L ( t ) t d t (cid:33)(cid:33) = (cid:34) − kα exp (cid:32) − (cid:90) xa ˘ L ( t ) t d t (cid:33)(cid:35) α , x ∈ (0 , + ∞ ) , (11) for varying the parameter k ∈ (0 , + ∞ ) and it is the α –generalized reversed aging intensityfunction for those distribution functions.Proof. Fix distribution function F with respective density function f , and put α <
0. Fromthe definition of ˘ L α it is possible to obtain˘ L α ( t ) t = αF α − ( t ) f ( t )1 − F α ( t ) , t ∈ (0 , + ∞ ) . By integrating both members between a and x , for an arbitrary a ∈ (0 , + ∞ ), we get (cid:90) xa ˘ L α ( t ) t d t = (cid:90) xa αF α − ( t ) f ( t )1 − F α ( t ) d t = − log 1 − F α ( x )1 − F α ( a ) , therefore 1 − F α ( x ) = (1 − F α ( a )) exp (cid:32) − (cid:90) xa ˘ L α ( t ) t d t (cid:33) , and so we get (10).Let ˘ L be a function defined on (0 , + ∞ ) and satisfying, for a ∈ (0 , + ∞ ), the conditions (1) , (2) , (3) . We show that 1 − W α (cid:16) k exp (cid:16) − (cid:82) xa ˘ L ( t ) t d t (cid:17)(cid:17) = F k ( x ) defines a distribution functionof a non–negative and absolutely continuous random variable, with k ∈ (0 , + ∞ ).In fact, from (2) it follows that lim x → + (cid:82) xa ˘ L ( t ) t d t = −∞ and so we conclude thatlim x → + F k ( x ) = 0, while from (3) we have lim x → + ∞ F k ( x ) = 1.Since W α is increasing, k > − (cid:82) xa ˘ L ( t ) t d t is a decreasing function in x i.e., (cid:82) xa ˘ L ( t ) t d t is increasing in x . From (1) and from the assumptions about the interval inwhich we have a and x it follows that the integrand is non negative. Let x , x be such that0 < x < x < + ∞ . If x ≥ a then we have two non negative quantities and (cid:82) x a ˘ L ( t ) t d t ≤ (cid:82) x a ˘ L ( t ) t d t because ( a, x ) ⊂ ( a, x ). If x ≤ a < x , then (cid:82) x a ˘ L ( t ) t d t ≤ ≤ (cid:82) x a ˘ L ( t ) t d t . If,8nally, x < x ≤ a , we have two non positive quantities and, for a reasoning similar to thefirst case, (cid:82) ax ˘ L ( t ) t d t ≥ (cid:82) ax ˘ L ( t ) t d t and so (cid:82) x a ˘ L ( t ) t d t ≤ (cid:82) x a ˘ L ( t ) t d t .Since W α , the exponential function, the multiplication for a scalar and the indefinite integral x (cid:55)→ (cid:82) xa ˘ L ( t ) t d t with respect to the Lebesgue measure are continuous functions, we have acontinuous function. In order to obtain the absolute continuity of F k , it suffices to observethat the derivative F (cid:48) k ( x ) = (cid:34) − kα exp (cid:32) − (cid:90) xa ˘ L ( t ) t d t (cid:33)(cid:35) α − ( − k ) exp (cid:32) − (cid:90) xa ˘ L ( t ) t d t (cid:33) (cid:32) − ˘ L ( x ) x (cid:33) is non–negative in x > L is α –generalized reversed aging intensity function related to those distri-bution functions we have to observe that F k ( a ) = [1 − kα ] α , and so kα = 1 − F αk ( a ) i.e., F k and ˘ L are related by the relationship expressed in the first part of the theorem. Remark 4.
The expression W α (cid:16) k exp (cid:16) − (cid:82) xa ˘ L ( t ) t d t (cid:17)(cid:17) depends only on the parameter k ∈ (0 , + ∞ ) because the dependence from a ∈ (0 , + ∞ ) is fictitious. In fact, replacing a with b ∈ (0 , + ∞ ) we get W α (cid:32) k exp (cid:32) − (cid:90) xb ˘ L ( t ) t d t (cid:33)(cid:33) = W α (cid:32) k exp (cid:32) − (cid:90) ab ˘ L ( t ) t d t (cid:33) exp (cid:32) − (cid:90) xa ˘ L ( t ) t d t (cid:33)(cid:33) = W α (cid:32) k exp (cid:32) − (cid:90) xa ˘ L ( t ) t d t (cid:33)(cid:33) , where k is such that k = k exp (cid:16) − (cid:82) ab ˘ L ( t ) t d t (cid:17) > Remark 5.
If ˘ L is the α –generalized reversed aging intensity function, with α <
0, of a non–negative and absolutely continuous random variable X , it satisfies conditions (1) , (2) , (3) ofTheorem 3.1. In fact, from (8) we observe that ˘ L is non–negative for x ∈ (0 , + ∞ ). Moreover,lim x → + (cid:90) ax ˘ L ( t ) t d t = lim x → + (cid:90) ax αF α − ( t ) f ( t )1 − F α ( t ) d t = lim x → + − log 1 − F α ( a )1 − F α ( x ) = + ∞ , lim x → + ∞ (cid:90) xa ˘ L ( t ) t d t = lim x → + ∞ (cid:90) xa αF α − ( t ) f ( t )1 − F α ( t ) d t = lim x → + ∞ − log 1 − F α ( x )1 − F α ( a ) = + ∞ . emark 6. If ˘ L is a function that satisfies conditions (1) , (2) , (3) of Theorem 3.1 then itdetermines, for α = 0, a family of absolutely continuous distribution functions F k by therelationship F k ( x ) = 1 − W (cid:32) k exp (cid:32) − (cid:90) xa ˘ L ( t ) t d t (cid:33)(cid:33) = exp − k exp − x (cid:90) a ˘ L ( t ) t d t , x ∈ (0 , + ∞ ) , (12)for varying the parameter k ∈ (0 , + ∞ ) and it is the 0–generalized reversed aging intensityfunction (i.e., the reversed aging intensity function) for those distribution functions. Thisfollows from corollary 4 of Szymkowiak (2018a) noting that L X (cid:0) x (cid:1) = ˘ L X ( x ).In the following theorem we show that, for α >
0, the distribution function of a non–negative and absolutely continuous random variable is defined by the α –generalized reversedaging intensity function and that, under some conditions, a function can be considerated asthe α –generalized reversed aging intensity function for a unique random variable. Theorem 3.2.
Let X be a non–negative and absolutely continuous random variable with cdf F and let ˘ L α be its α –generalized reversed aging intensity function with α > . Then F and ˘ L α are related, for all a ∈ (0 , + ∞ ) , by the relationship F ( x ) = (cid:34) − exp (cid:32) − (cid:90) x ˘ L α ( t ) t d t (cid:33)(cid:35) α , x ∈ (0 , + ∞ ) . (13) Moreover, a function ˘ L defined on (0 , + ∞ ) and satisfying, for a ∈ (0 , + ∞ ) , the followingconditions:(1) ≤ ˘ L ( x ) + ∞ , for all x ∈ (0 , + ∞ ) ;(2) lim x → + (cid:82) ax ˘ L ( t ) t d t < + ∞ ;(3) lim x → + ∞ (cid:82) xa ˘ L ( t ) t d t = + ∞ ;determines, for α > , a unique absolutely continuous distribution function F by the relation-ship F ( x ) = 1 − W α (cid:32) α exp (cid:32) − (cid:90) x ˘ L ( t ) t d t (cid:33)(cid:33) = (cid:34) − exp (cid:32) − (cid:90) x ˘ L ( t ) t d t (cid:33)(cid:35) α , x ∈ (0 , + ∞ ) , (14) and it is α –generalized reversed aging intensity function for that distribution function. roof. Fix distribution function F with respective density function f , and put α >
0. Fromthe definition of ˘ L α it is possible to obtain˘ L α ( t ) t = αF α − ( t ) f ( t )1 − F α ( t ) , t ∈ (0 , + ∞ ) . By integrating both members between 0 and x , we get (cid:90) x ˘ L α ( t ) t d t = (cid:90) x αF α − ( t ) f ( t )1 − F α ( t ) d t = − log(1 − F α ( x )) , therefore 1 − F α ( x ) = exp (cid:32) − (cid:90) x ˘ L α ( t ) t d t (cid:33) , and so we get (13).Let ˘ L be a function defined on (0 , + ∞ ) and satisfying, for a ∈ (0 , + ∞ ), the conditions (1) , (2) , (3) . We show that 1 − W α (cid:16) α exp (cid:16) − (cid:82) x L ( t ) t d t (cid:17)(cid:17) = F ( x ) defines a distribution functionof a non–negative and absolutely continuous random variable.In fact, from (2) it follows that lim x → + F ( x ) = 0, whereas from (3) we obtain thatlim x → + ∞ F ( x ) = 1 . Since W α is increasing, α > − (cid:82) x L ( t ) t d t is a decreasing function in x i.e., (cid:82) x L ( t ) t d t is increasing in x , but this is immediate because the integrand is non negativeand as x increases, the integration interval widens.Since W α , the exponential function, the multiplication for a scalar and the indefinite integral x (cid:55)→ (cid:82) x L ( t ) t d t are continuous functions, we have a continuous function. In order to obtain theabsolute continuity of F , it suffices to observe that the derivative F (cid:48) ( x ) = − α (cid:34) − exp (cid:32) − (cid:90) x ˘ L ( t ) t d t (cid:33)(cid:35) α − exp (cid:32) − (cid:90) x ˘ L ( t ) t d t (cid:33) (cid:32) − ˘ L ( x ) x (cid:33) is non–negative in x >
0. Finally, F and ˘ L are related by the same relationship found in thefirst part of the theorem and so ˘ L is α –generalized reversed aging intensity function for thatdistribution function. Remark 7.
If ˘ L is the α –generalized reversed aging intensity function, with α >
0, of a non–negative and absolutely continuous random variable X , it satisfies conditions (1) , (2) , (3) ofTheorem 3.2. In fact, from (8) we observe that ˘ L is non–negative for x ∈ (0 , + ∞ ). Moreover,lim x → + (cid:90) ax ˘ L ( t ) t d t = lim x → + (cid:90) ax αF α − ( t ) f ( t )1 − F α ( t ) d t = lim x → + − log 1 − F α ( a )1 − F α ( x ) < + ∞ , x → + ∞ (cid:90) xa ˘ L ( t ) t d t = lim x → + ∞ (cid:90) xa αF α − ( t ) f ( t )1 − F α ( t ) d t = lim x → + ∞ − log 1 − F α ( x )1 − F α ( a ) = + ∞ . In a concrete situation, if we have data it is possible to obtain an estimation of bothdistribution function and α –generalized reversed aging intensity functions. So it could happenthat the shape of an α –generalized reversed aging intensity function is easier to recognize thanthat of the distribution function. Definition 3.
We say that a random variable X follows an inverse two-parameter Weibulldistribution (see Murthy, Xie and Jiang 2004) if for x ∈ (0 , + ∞ ) and β, λ > F ( x ) = exp (cid:18) − λx β (cid:19) . (15)In that case we write X ∼ invW β, λ ).From the cdf (15) it is possible to obtain other characteristics of the distribution. Inparticular, for x ∈ (0 , + ∞ ), the pdf is f ( x ) = λβx β +1 exp (cid:18) − λx β (cid:19) , the reversed hazard rate function is ˘ r ( x ) = f ( x ) F ( x ) = λβx β +1 , and the α –generalized reversed aging intensity function, for α (cid:54) = 0, is˘ L α ( x ) = αx λβx β +1 exp (cid:0) − λx β (cid:1) exp (cid:16) − λ ( α − x β (cid:17) − exp (cid:0) − λαx β (cid:1) = αβλx β exp (cid:0) − λαx β (cid:1) − exp (cid:0) − λαx β (cid:1) . (16)Let α <
0. By remark 5 we know that (16) satisfies the hypothesis (1) , (2) , (3) of Theo-rem 3.1. So we can apply the theorem by determining the quantity F k ( x ) = (cid:34) − kα exp (cid:32) − (cid:90) xa αβλt β +1 exp (cid:0) − λαt β (cid:1) − exp (cid:0) − λαt β (cid:1) d t (cid:33)(cid:35) α = (cid:34) − kα exp (cid:0) − λαx β (cid:1) − (cid:0) − λαa β (cid:1) − (cid:35) α . orollary 4.1. If a random variable X has α –generalized reversed aging intensity function, α < , ˘ L α ( x ) = αβλx β exp (cid:16) − λαxβ (cid:17) − exp (cid:16) − λαxβ (cid:17) , a.e. x ∈ (0 , + ∞ ) , with β, λ > , then the distribution functionof X is expressed as F ( x ) = (cid:20) − γα (cid:18) exp (cid:18) − λαx β (cid:19) − (cid:19)(cid:21) α , x ∈ (0 , + ∞ ) (17) for γ ∈ (0 , + ∞ ) . Remark 8. If γ = − α , the distribution function of Corollary 4.1 is the distribution functionof an inverse two-parameter Weibull distribution, invW β, λ ).Let α >
0. By remark 7 we know that (16) satisfies the hypothesis (1) , (2) , (3) of Theo-rem 3.2. So we can apply the theorem by determining the quantity F ( x ) = (cid:34) − exp (cid:32) − (cid:90) x αβλt β +1 exp (cid:0) − λαt β (cid:1) − exp (cid:0) − λαt β (cid:1) d t (cid:33)(cid:35) α = (cid:20) − (cid:18) − exp (cid:18) − λαx β (cid:19)(cid:19)(cid:21) α = exp (cid:18) − λx β (cid:19) . Corollary 4.2.
If a random variable X has α –generalized reversed aging intensity function, α > , ˘ L α ( x ) = αβλx β exp (cid:16) − λαxβ (cid:17) − exp (cid:16) − λαxβ (cid:17) , a.e. x ∈ (0 , + ∞ ) , with β, λ > , then X follows an inversetwo-parameter Weibull distribution, X ∼ invW β, λ ) . Let us consider some examples of polynomial α –generalized reversed aging intensity func-tions. Example 1.
Let us consider ˘ L α ( x ) = A >
0, for x >
0. It could be a constant α –generalizedreversed aging intensity function for α ≤
0, in fact for α > α = 0 it determines a family of inverse two-parameter Weibull distributions by therelationship (see Szymkowiak (2018a)) F k ( x ) = exp (cid:34) − k (cid:18) x (cid:19) A (cid:35) , x ∈ (0 , + ∞ ) , (18)where k is a non–negative parameter.For α <
0, it determines a family of continuous distributions by the relationship F k ( x ) = (cid:34) k (cid:18) x (cid:19) A (cid:35) α , x ∈ (0 , + ∞ ) (19)where k is a non–negative parameter. 13 xample 2. Let us consider ˘ L α ( x ) = A + Bx , for x > A, B >
0. It could be a linear α –generalized reversed aging intensity function for α ≤
0, in fact for α > α = 0, it determines a family of continuous distributions by the relationship F k ( x ) = exp (cid:34) − k (cid:18) x (cid:19) A exp( − B x ) (cid:35) , x ∈ (0 , + ∞ ) (20)where k is a non–negative parameter.For α <
0, it determines a family of continuous distributions by the relationship F k ( x ) = (cid:34) k (cid:18) x (cid:19) A exp( − B x ) (cid:35) α , x ∈ (0 , + ∞ ) (21)where k is a non–negative parameter. Example 3.
Let us consider ˘ L α ( x ) = Bx , for x >
0, where
B >
0. It could be a linear α –generalized reversed aging intensity function for α >
0, in fact for α > F ( x ) = [1 − exp( − Bx )] α , x ∈ (0 , + ∞ ) , (22)i.e., an exponentiated exponential distribution (see Gupta and Kundu, 2001). We note thatfor α = 1 this is the distribution function of an exponential random variable with parameter B . So if X has 1–generalized reversed aging intensity function ˘ L ( x ) = Bx , for x > B > X ∼ Exp ( B ). α –generalized reversed aging intensity orders In this section we introduce and study the family of the α –generalized reversed aging intensityorders. In the following, we use the notation L α,X to indicate the α –generalized aging intensityfunction of the random variable X and ˘ L α,X to indicate the α –generalized reversed agingintensity function of the random variable X .In the next proposition we show a useful relationship between L α,X and ˘ L α, X . Proposition 5.1.
Let X be a non–negative and absolutely continuous random variable and let X be its inverse. Then the following equality holds L α,X (cid:18) x (cid:19) = ˘ L α, X ( x ) , x ∈ (0 , + ∞ ) . (23)14 roof. We obtain an expression for the distribution function and the density function of therandom variable X through X , for x > F X ( x ) = P (cid:18) X ≤ x (cid:19) = P (cid:18) X ≥ x (cid:19) = 1 − F X (cid:18) x (cid:19) ,f X ( x ) = 1 x f X (cid:18) x (cid:19) . If α = 0 we have, for x > L , X ( x ) = ˘ L X ( x ) = − xf X ( x ) F X ( x ) log F X ( x )= − x f X (cid:0) x (cid:1) (1 − F X (cid:0) x (cid:1) ) log(1 − F X (cid:0) x (cid:1) ) = L X (cid:18) x (cid:19) = L ,X (cid:18) x (cid:19) . If α (cid:54) = 0 we have, for x > L α, X ( x ) = αx ( F X ( x )) α − f X ( x )1 − ( F X ( x )) α = α x (cid:0) − F X (cid:0) x (cid:1)(cid:1) α − f X (cid:0) x (cid:1) − (1 − F X (cid:0) x (cid:1) ) α = L α,X (cid:18) x (cid:19) . Definition 4.
Let X and Y be non–negative and absolutely continuous random variables andlet α be a real number. We say that X is smaller than Y in the α –generalized reversed agingintensity order, X ≤ αRAI Y , if and only if ˘ L α,X ( x ) ≤ ˘ L α,Y ( x ), ∀ x ∈ (0 , + ∞ ).In the next lemma we show a relationship between the αRAI order and the αAI order.We recall that X ≤ αAI Y if and only if L α,X ( x ) ≥ L α,Y ( x ), ∀ x ∈ (0 , + ∞ ). Lemma 5.1.
Let X and Y be non–negative and absolutely continuous random variables andlet α be a real number. We have X ≤ αRAI Y if and only if X ≥ αAI Y .Proof. We have X ≤ αRAI Y if and only if ˘ L α,X ( x ) ≤ ˘ L α,Y ( x ), ∀ x ∈ (0 , + ∞ ). By proposi-tion 5.1 this is equivalent to L α, X (cid:0) x (cid:1) ≤ L α, Y (cid:0) x (cid:1) , ∀ x ∈ (0 , + ∞ ), i.e. X ≥ αAI Y . Remark 9.
For particular choices of the real number α we find some relationship with otherstochastic orders. Obviously, the reversed aging intensity order coincides with the 0–generalizedreversed aging intensity order. For α = 1 we have showed in remark 3 that ˘ L ,X ( x ) = xr X ( x ) , so we get a relationship with the hazard rate order. In fact, X ≤ hr Y ⇔ r X ( x ) ≥ r Y ( x ) , ∀ x > ⇔ xr X ( x ) ≥ xr Y ( x ) , ∀ x > ⇔ ˘ L ,X ( x ) ≥ ˘ L ,Y ( x ) , ∀ x > ⇔ X ≥ RAI Y. (24)15or α = − L − ,X ( x ) = xLOR X ( x ) = L − ,X ( x ) , so we geta relationship with the log-odds rate order. In fact, X ≤ LOR Y ⇔ LOR X ( x ) ≥ LOR Y ( x ) , ∀ x > ⇔ xLOR X ( x ) ≥ xLOR Y ( x ) , ∀ x > ⇔ ˘ L − ,X ( x ) ≥ ˘ L − ,Y ( x ) , ∀ x > ⇔ X ≥ − RAI Y. (25)Moreover we have X ≥ − RAI Y ⇔ X ≤ − AI Y so they are dual relations. For α = n ∈ N wehave showed in remark 3 that˘ L n,X ( x ) = nx ( F X ( x )) n − f X ( x )1 − ( F X ( x )) n = xr X ( n ) ( x ) , so there is a connection with the largest order statistic and the hazard rate order. In fact X ( n ) ≤ hr Y ( n ) ⇔ r X ( n ) ( x ) ≥ r Y ( n ) ( x ) , ∀ x > ⇔ xr X ( n ) ( x ) ≥ xr Y ( n ) ( x ) , ∀ x > ⇔ ˘ L n,X ( x ) ≥ ˘ L n,Y ( x ) , ∀ x > ⇔ X ≥ nRAI Y. (26) Remark 10.
Lemma 5.1 and Remark 9 provide the following series of relations X ≤ − RAI Y ⇔ X ≤ − RAI Y ⇔ X ≥ − AI Y ⇔ X ≥ − AI Y .
Proposition 5.2.
Let X and Y be non–negative and absolutely continuous random variablessuch that X ≤ st Y , i.e., F X ( x ) ≥ F Y ( x ) for all x > .(1) If exists β ∈ R such that X ≤ βRAI Y then for all α < β we have X ≤ αRAI Y ;(2) If exists β ∈ R such that X ≥ βRAI Y then for all α > β we have X ≥ αRAI Y .Proof. (1). From X ≤ βRAI Y and lemma 5.1 we have X ≥ βAI Y . Moreover from X ≤ st Y weget X ≥ st Y so by proposition 4 of Szymkowiak (2018b) we obtain that ∀ α < β X ≥ αAI Y ,i.e., X ≤ αRAI Y .The proof of part (2) is analogous. Proposition 5.3.
Let X and Y be non–negative and absolutely continuous random variables.(1) If exists β ∈ R such that for all α < β we have X ≥ αRAI Y then X ≥ rh Y , i.e., ˘ r X ( x ) ≥ ˘ r Y ( x ) for all x > ;(2) If exists β ∈ R such that for all α > β we have X ≤ αRAI Y then X ≥ st Y .Proof. (1). From X ≥ αRAI Y and lemma 5.1 we have X ≤ αAI Y , ∀ α < β . So with use ofproposition 5 of Szymkowiak (2018b) we obtain X ≤ hr Y , i.e. X ≥ rh Y .The proof of part (2) is analogous. 16 orollary 5.1. Let X and Y be non–negative and absolutely continuous random variables.(1) X ≤ st Y and X ≥ LOR Y ⇒ X ≤ αRAI Y for all α ∈ ( −∞ , − ;(2) X ≤ st Y and X ≤ LOR Y ⇒ X ≥ αRAI Y for all α ∈ ( − , + ∞ ) .Proof. (1). We have X ≥ LOR Y ⇔ X ≤ − RAI Y so the proof is completed with use ofproposition 5.2.The proof of part (2) is analogous. α –generalized reversed aging intensity func-tion in data analysis Very often it is really a difficult task to recognize the lifetime data distribution analyzingonly the shapes of their density and distribution function estimators. But sometimes, thecorresponding α –generalized reversed aging intensity function for a properly chosen α can havea relatively simple form, and it can be easily recognized with use of the respective reversedaging intensity estimate.For some distribution F with support x ∈ (0 , + ∞ ), we obtain a natural estimator of the α –generalized reversed aging intensity function (cid:98) ˘ L α ( x ) = α x (cid:98) f ( x )[ (cid:98) F ( x )] α − − [ (cid:98) F ( x )] α for x > , α (cid:54) = 0 − x (cid:98) f ( x ) (cid:98) F ( x ) ln[ (cid:98) F ( x )] for x > , α = 0 , (27)where (cid:98) f denotes a nonparametric estimate of the unknown density function f and (cid:98) F ( x ) = (cid:82) x (cid:98) f ( t )d t represents the corresponding distribution function estimate. The proposed estimationof the aging intensity function is possible if we assume that data follow an absolutely continuousdistribution with support (0 , + ∞ ) and if the nonparametric estimate of its density functionexists. Moreover, larger sample sizes generally lead to increased precision of estimation. Weperform our study for both the generated and the real data. α –generalized reversed aging intensity function through gen-erated data In the following example we consider an application of the estimator (27) for α = − xample 4. Our goal is to check if a member of the inverse log-logistic distributions invLLog ( γ, λ )with the distribution function given by F γ,λ ( x ) = (cid:20) (cid:18) λx (cid:19) γ (cid:21) − , x ∈ (0 , + ∞ ) , (28)for some unknown positive parameters of the shape γ and the scale λ , is the parent distributionof a random sample X , . . . , X N .From presented in Section 4, Example 1 we know that for distribution function (28), its − L − ( x ) = γ . So, we checkif the respective reversed aging intensity estimator (27) is indeed an accurate approximationof a constant function.Therefore, we use the following procedure to obtain N independent random variables X , . . . , X N with invLLog ( γ, λ ) lifetime distribution. First, we generate standard uniformrandom variables U , . . . , U N using function random of MATLAB . Then, applying the inversetransform technique with F γ,λ ( x ) = (cid:104) (cid:0) λx (cid:1) γ (cid:105) − , we get Y i = F − γ,λ (1 − U i ) = λ (cid:16) − U i − (cid:17) − γ , i = 1 , . . . , N , with the inverse log-logistic distribution invLLog ( γ, λ ). In this way, applying thefunction random with the seed = 88, we generate N = 1000 independent inverse log-logisticrandom variables with the shape parameter γ = 4, and the scale parameter λ = 0 . MATLAB ksdensity function, (cid:98) f ( x ) = 1 N h N (cid:88) j =1 K (cid:18) x − X j h (cid:19) , (29)with a chosen normal kernel smoothing function and a selected bandwidth h = 0 .
05. Then,the kernel estimator of the distribution function is equal to (cid:98) F ( x ) = 1 N N (cid:88) j =1 I (cid:18) x − X j h (cid:19) , where I ( x ) = (cid:82) x −∞ K ( t )d t . The obtained − (cid:98) ˘ L − ( x ) = x (cid:98) f ( x ) (cid:98) F ( x ) (cid:104) − (cid:98) F ( x ) (cid:105) = x N h (cid:80) Nj =1 K (cid:16) x − X j h (cid:17) N (cid:80) Nj =1 I (cid:16) x − X j h (cid:17) (cid:104) − N (cid:80) Nj =1 I (cid:16) x − X j h (cid:17)(cid:105) . (30)For our simulation data, the plot of the density estimator (29) is presented in Figure 1.Analyzing the plot, it is not easy to decide if the density function belongs to the inverse log-logistic family. But, we can notice that the plot of respective estimator (30) of − (cid:98) ˘ L − ( x ) (see Figure 2), oscillates around a constant function,18igure 1: Density estimator (cid:98) f ( x ) for the data from Example 4especially after removing few outlying values at the right-end. This gives us the motivation toaccept our hypothesis that an inverse log-logistic distribution is the parent distribution of thegenerated sample.Figure 2: (cid:98) ˘ L ( x ) and adjusted regression line for the data from Example 4To justify our intuitive decision, we propose to carry out the following more formal statis-tical procedure. First, we calculate the least squares estimate of the intercept which for ourdata equals to (cid:98) γ = 3 . λ maximizing it. The problem resolves19nto finding the solution to the equation N (cid:88) i =1 (cid:0) x i λ (cid:1) (cid:98) γ + 1 = N . As the result we obtain (cid:98) λ = 0 . (cid:98) γ and (cid:98) λ based on the empirical − invLLog ( γ, λ ) γ λ Theoretical parameters 4 0.5Estimators 3.7990 0.4957Finally, by the chi-square goodness-of-fit test we check if the data really fit the inverse log-logistic distribution. For this purpose, we apply function histogram , available in
MATLAB andgroup the data into k = 20 classes of observations lying into intervals [ x j , x j +1 ) = [ x j , x j +∆ x ), j = 1 , . . . , k , of length ∆ x = 0 .
21. The classes, together with their empirical frequencies N j = N j ( X , . . . , X N ) and theoretical frequencies based on the inverse log-logistic distributionwith parameters replaced by the estimators n j = N (cid:104) F (cid:98) γ, (cid:98) λ ( x j +1 ) − F (cid:98) γ, (cid:98) λ ( x j ) (cid:105) , are presented inTable 2.Furthermore, available in MATLAB function chi2gof determines the value of chi-squarestatistics χ = 9 . ν = 7 degrees of freedom (automatically joining together the lasttwelve classes with low frequencies) and determines the respective p -value, p = 0 . . α –generalized reversed aging intensity through real data Next, we present an example of real data. Analyzing its estimated α –generalized reversedaging intensity we could assume that the data follow the adequate distribution. Example 5.
The real data (see Data Set 6.2 in Murthy et al. 2004) concern failure times of20 components: 0.067 0.068 0.076 0.081 0.084 0.085 0.085 0.086 0.089 0.098 0.098 0.114 0.1140.115 0.121 0.125 0.131 0.149 0.160 0.485.For the given data, the plot of the normal kernel density estimator (see Bowman andAzzalini 1997), obtained by
MATLAB function ksdensity with a returned bandwidth h =0 . x j , x j +1 ) N j n j (cid:98) ˘ L ( x ) = − x (cid:98) f ( x ) (cid:98) F ( x ) ln[ (cid:98) F ( x )] , x ∈ (0 , + ∞ ) . The plot of the estimator (cid:98) ˘ L ( x ) (see Figure 4) can be treated as oscillating around a linearfunction, especially after removing one outlying value at the right-end. This motivates us tostate the hypothesis that data follow an inverse modified Weibull distribution (see Section 4,Example 2) with distribution function F γ,λ,δ ( x ) = exp (cid:20) − (cid:18) λx (cid:19) γ exp( − δ x ) (cid:21) , x ∈ (0 , + ∞ ) , (31)and 0–generalized reversed aging intensity function˘ L ( x ) = δ x + γ, x ∈ (0 , + ∞ ) . (cid:98) f for the data from Example 5‘ Figure 4: (cid:98) ˘ L ( x ), and adjusted regression line for the data from Example 5Moreover, we provide the following procedure. First, we determine the least squares estimates (cid:98) γ = 0 . (cid:98) δ = 31 . L , respectively. Then wedetermine MLE of parameter λ (cid:98) λ = N (cid:80) Ni =1 exp( − (cid:98) δ x i )( x i ) (cid:98) γ (cid:98) γ which maximizes the likelihood function. Here we obtain (cid:98) λ = 549 . MATLAB function kstest ), we determine statistics K = 0 . p -value of the test equal to p = 0 . . In this paper, a family of generalized reversed aging intensity functions was introduced andstudied. In particular, it was showed that, using the generalized Pareto distribution to general-ize the concept of reversed aging intensity function, for α >
0, the α –generalized reversed agingintensity function characterizes a unique distribution function, while for α ≤
0, it determines afamily of distribution functions. Moreover, α –generalized reversed aging intensity orders wereintroduced and some relations with other stochastic orders were studied. Finally, analysis of α –generalized reversed aging intensity through generated data and real one are given. Acknowledgement
Francesco Buono and Maria Longobardi are partially supported by the GNAMPA researchgroup of INdAM (Istituto Nazionale di Alta Matematica) and MIUR-PRIN 2017, Project”Stochastic Models for Complex Systems” (No. 2017 JFFHSH).Magdalena Szymkowiak is partially supported by PUT under grant 0211/SBAD/0911.
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