On Generalized Reversed Aging Intensity Functions

The reversed aging intensity function is defined as the ratio of the instantaneous reversed hazard rate to the baseline value of the reversed hazard rate. It analyzes the aging property quantitatively, the higher the reversed aging intensity, the weaker the tendency of aging. In this paper, a family of generalized reversed aging intensity functions is introduced and studied. Those functions depend on a real parameter. If the parameter is positive they characterize uniquely the distribution functions of univariate positive absolutely continuous random variables, in the opposite case they characterize families of distributions. Furthermore, the generalized reversed aging intensity orders are defined and studied. Finally, several numerical examples are given.


Introduction
Let X be a non-negative and absolutely continuous random variable with cumulative distribution function (cdf) F , probability density function (pdf) f and survival function (sf) F . In reliability theory F is also known as unreliability function whereas F as reliability function.
In this context a great importance has the hazard rate function r of X, also known as the force of mortality or the failure rate, where X is the survival model of a life or a system being studied. This definition will cover discrete survival models as well as mixed survival models.
In the same way we define the reversed hazard rateȓ of X, that has attracted the attention of researchers. In a certain sense it is the dual function of the hazard rate and it bears some interesting features useful in reliability analysis (see also Block and Savits 1998;Finkelstein 2002).
Let X be a random variable with sf F and cdf F . We define, for x such that F (x) > 0, the hazard rate function of X at x, r(x), in the following way: 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: The reversed hazard rateȓ(x) can be treated as the instantaneous failure rate occurring immediately before the time point x (the failure occurs just before the time point x, given that the 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 while, for x such that F (x) > 0, the reversed hazard rate function is By the hazard rate function we introduce the aging intensity function L that is defined for x > 0 as where log denotes the natural logarithm. It can be showed that the survival function of an absolutely continuous random variable and its aging intensity function are related by a relationship and that under some conditions a function determines a family of survival functions and it is their aging intensity function, for more details see Szymkowiak (2018a).
The reversed aging intensity functionL(x) is defined, for x > 0, as follows (see also Rezaei and Khalef 2014)L The reversed aging intensity function can be expressed also in a different way by observing that the cumulative reversed hazard rate function defined as can be treated as the total amount of failures accumulated after the time point x. SoH(x) = 1 xȒ (x), being the proportion between the total amount of failures accumulated after the time point x and the time x for which the unit is still survived, can be considered as the baseline value of the reversed hazard rate. Then, (4) can be written as and so the reversed aging intensity function, defined as the ratio of the instantaneous reversed hazard rateȓ to the baseline value of the reversed hazard rateH, expresses the units average aging behavior: the higher the reversed aging intensity (it means the higher the instantaneous reversed hazard rate, and the smaller the total amount of failures accumulated after the time point x, and the higher the the time x for which the unit is still survived), the weaker the tendency of aging.
It is the analogous for the future of the aging intensity function, introduced and studied Let, for x > 0, W 0 (x) = 1 − exp(−x), i.e., W 0 is the distribution function of an exponential variable with parameter 1, soȒ( Replacing W 0 with a strictly increasing distribution function G with density g, it is possible to generalize the concepts of reversed hazard rate function, cumulative reversed hazard rate function and reversed aging intensity function. The generalization of the hazard rate function was introduced by 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 Ggeneralized cumulative reversed hazard rate function,Ȓ G , the G-generalized reversed hazard rate function,ȓ G , the G-generalized reversed aging intensity function,L G , of X as A very interesting case, because it provides intuitive results, is the one in which the distribution 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): Remark 1. For α = 0 we have the distribution function of an exponential variable with parameter 1.
From the distribution function it is possible to obtain the quantile and the density function.
In particular we have Let 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 function and the W α -generalized reversed hazard rate function in the following way: For the sake of simplicity, those functions can be, respectively, indicated byȒ α ,ȓ α and we can refer to them as the α-generalized cumulative reversed hazard rate function and the αgeneralized 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 random variable by its monotonicity. The 0-generalized reversed hazard rate function is equal to the usual reversed hazard rate function.
From these functions, it is possible to introduce the α-generalized reversed aging intensity The α-generalized reversed aging intensity function describes the relationship between the instantaneous value of the α-generalized reversed hazard rate functionȓ α (x) and the baseline value of the α-generalized reversed hazard rate function 1 xȒ α (x). The higher the α-generalized reversed aging intensity function (it means the higher the actual value of the α-generalized reversed 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 We recall the definition of α-generalized aging intensity functions, L α . These functions are defined by Szymkowiak (2018b) in the following way Remark 3. The 0-generalized reversed aging intensity function is equal to the usual reversed aging intensity function. If α = 1 we havȇ i.e., it is the negative of the elasticity of the survival function F , they are equal in modulus.
If α = n ∈ N we haveL 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 order statistic. SoL n can be considered as the negative of the elasticity for the survival function of the largest order statistic.
If α = −1 we havȇ , 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 intensity functions respect to the parameter α. This result could be important if we introduce stochastic orders based on α-generalized reversed aging intensity functions, i.e., αRAI orders, and compare these orders as α varies (see Section 5).
Proposition 2.1. Let X be a non-negative and absolutely continuous random variable with cdf F and pdf f . Then the α-generalized reversed aging intensity function is decreasing respect to α ∈ R, ∀x ∈ (0, +∞).
Proof. For some c ∈ (0, 1) we consider the function h c (α) = αc α 1−c α , for α = 0. Then That derivative is negative because c α ∈ (0, +∞) and the function k(t) = 1−t+log t is negative for t > 0 and different from 1. In fact, k(1) = 0 and 1 is maximum point for this function. So h c is decreasing in (−∞, 0) ∪ (0, +∞). Defining the extension for continuity in 0 of h 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 get that the following function is decreasing in α as x is fixed.
3 Characterizations with use of α-generalized reversed aging intensity In reliability theory some functions characterize the associated distribution function. For example it was showed in Barlow and Proschan (1996) that the hazard rate of an absolutely continuous random variable uniquely determines its distribution function.
In the following theorem we show that, for α < 0, the distribution function of a nonnegative and absolutely continuous random variable is defined by the α-generalized reversed aging intensity function and that, under some conditions, a function can be considerated as the 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 letL α be its α-generalized reversed aging intensity function with α < 0. Then F and L α are related, for all a ∈ (0, +∞), by the relationship Moreover, a functionL defined on (0, +∞) and satisfying, for a ∈ (0, +∞), the following conditions: (1) 0 ≤L(x) < +∞, for all x ∈ (0, +∞); determines, for α < 0, a family of absolutely continuous distribution functions F k by the for varying the parameter k ∈ (0, +∞) and it is the α-generalized reversed aging intensity function for those distribution functions.
Proof. Fix distribution function F with respective density function f , and put α < 0. From the definition ofL α it is possible to obtain By integrating both members between a and x, for an arbitrary a ∈ (0, +∞), we get and so we get (10).
Since W α is increasing, k > 0 and the exponential function is increasing, in order to show that it is an increasing function we have to prove that − t dt is increasing in x. From (1) and from the assumptions about the interval in which we have a and x it follows that the integrand is non negative. Let x 1 , x 2 be such that 0 < x 1 < x 2 < +∞. If x 1 ≥ a then we have two non negative quantities and finally, x 1 < x 2 ≤ a, we have two non positive quantities and, for a reasoning similar to the first case, (t) t dt and so Since W α , the exponential function, the multiplication for a scalar and the indefinite integral dt with respect to the Lebesgue measure are continuous functions, we have a continuous function. In order to obtain the absolute continuity of F k , it suffices to observe that the derivative To show thatL is α-generalized reversed aging intensity function related to those distribution functions we have to observe that F k (a) = [1 − kα] 1 α , and so kα = 1 − F α k (a) i.e., F k andL are related by the relationship expressed in the first part of the theorem.
depends only on the parameter k ∈ (0, +∞) because the dependence from a ∈ (0, +∞) is fictitious. In fact, replacing a with b ∈ (0, +∞) we get where k 1 is such that Remark 5. IfL is the α-generalized reversed aging intensity function, with α < 0, of a nonnegative and absolutely continuous random variable X, it satisfies conditions (1), (2), (3) of Theorem 3.1. In fact, from (8) we observe thatL is non-negative for x ∈ (0, +∞). Moreover, for varying the parameter k ∈ (0, +∞) and it is the 0-generalized reversed aging intensity function (i.e., the reversed aging intensity function) for those distribution functions. This follows from corollary 4 of Szymkowiak (2018a) noting that L X 1 In the following theorem we show that, for α > 0, the distribution function of a nonnegative and absolutely continuous random variable is defined by the α-generalized reversed aging intensity function and that, under some conditions, a function can be considerated as the 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 letL α be its α-generalized reversed aging intensity function with α > 0. Then F and L α are related, for all a ∈ (0, +∞), by the relationship Moreover, a functionL defined on (0, +∞) and satisfying, for a ∈ (0, +∞), the following conditions: (1) 0 ≤L(x) + ∞, for all x ∈ (0, +∞); determines, for α > 0, a unique absolutely continuous distribution function F by the relation- and it is α-generalized reversed aging intensity function for that distribution function.
Proof. Fix distribution function F with respective density function f , and put α > 0. From the definition ofL α it is possible to obtain , t ∈ (0, +∞).
By integrating both members between 0 and x, we get and so we get (13).
LetL be a function defined on (0, +∞) and satisfying, for a ∈ (0, +∞), the conditions (1), (t) t dt = F (x) defines a distribution function of a non-negative and absolutely continuous random variable.
Since W α is increasing, α > 0 and the exponential function is increasing, in order to show that it is an increasing function we have to prove that − x 0L (t) t dt is a decreasing function in x i.e., x 0L (t) t dt is increasing in x, but this is immediate because the integrand is non negative and as x increases, the integration interval widens.
Since W α , the exponential function, the multiplication for a scalar and the indefinite integral t dt are continuous functions, we have a continuous function. In order to obtain the absolute continuity of F , it suffices to observe that the derivative is non-negative in x > 0. Finally, F andL are related by the same relationship found in the first part of the theorem and soL is α-generalized reversed aging intensity function for that distribution function.

Remark 7.
IfL is the α-generalized reversed aging intensity function, with α > 0, of a nonnegative and absolutely continuous random variable X, it satisfies conditions (1), (2), (3) of Theorem 3.2. In fact, from (8) we observe thatL is non-negative for x ∈ (0, +∞). Moreover, In a concrete situation, if we have data it is possible to obtain an estimation of both distribution function and α-generalized reversed aging intensity functions. So it could happen that the shape of an α-generalized reversed aging intensity function is easier to recognize than that of the distribution function.

Examples of characterization
Definition 3. We say that a random variable X follows an inverse two-parameter Weibull distribution (see Murthy, Xie and Jiang 2004) if for x ∈ (0, +∞) and β, λ > 0 the distribution function is expressed as In that case we write X ∼ invW 2(β, λ).
From the cdf (15) it is possible to obtain other characteristics of the distribution. In particular, for x ∈ (0, +∞), the pdf is the reversed hazard rate function isȓ and the α-generalized reversed aging intensity function, for α = 0, is Let α < 0. By remark 5 we know that (16) satisfies the hypothesis (1), (2), (3) of Theorem 3.1. So we can apply the theorem by determining the quantity Corollary 4.1. If a random variable X has α-generalized reversed aging intensity function, , a.e. x ∈ (0, +∞), with β, λ > 0, then the distribution function of X is expressed as for γ ∈ (0, +∞).
Let α > 0. By remark 7 we know that (16) satisfies the hypothesis (1), (2), (3) of Theorem 3.2. So we can apply the theorem by determining the quantity where k is a non-negative parameter.
For α < 0, it determines a family of continuous distributions by the relationship where k is a non-negative parameter. For α = 0, it determines a family of continuous distributions by the relationship where k is a non-negative parameter.
For α < 0, it determines a family of continuous distributions by the relationship where k is a non-negative parameter.
i.e., an exponentiated exponential distribution (see Gupta and Kundu, 2001). We note that for α = 1 this is the distribution function of an exponential random variable with parameter B. So if X has 1-generalized reversed aging intensity functionL 1 (x) = Bx, for x > 0 and B > 0 then X ∼ Exp(B).

α-generalized reversed aging intensity orders
In this section we introduce and study the family of the α-generalized reversed aging intensity orders. In the following, we use the notation L α,X to indicate the α-generalized aging intensity function of the random variable X andL α,X to indicate the α-generalized reversed aging intensity function of the random variable X.
In the next proposition we show a useful relationship between L α,X andL α, 1 X .
Proposition 5.1. Let X be a non-negative and absolutely continuous random variable and let 1 X be its inverse. Then the following equality holds Proof. We obtain an expression for the distribution function and the density function of the random variable 1 X through X, for x > 0 we have x .
If α = 0 we have, for x > 0, If α = 0 we have, for x > 0, Definition 4. Let X and Y be non-negative and absolutely continuous random variables and let α be a real number. We say that X is smaller than Y in the α-generalized reversed aging intensity order, X ≤ αRAI Y , if and only ifL 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.
Lemma 5.1. Let X and Y be non-negative and absolutely continuous random variables and let α be a real number. We have X ≤ αRAI Y if and only if 1 Proof. We have X ≤ αRAI Y if and only ifL α,X (x) ≤L α,Y (x), ∀x ∈ (0, +∞). By proposition 5.1 this is equivalent to L α, 1 Remark 9. For particular choices of the real number α we find some relationship with other stochastic orders. Obviously, the reversed aging intensity order coincides with the 0-generalized reversed aging intensity order. For α = 1 we have showed in remark 3 thatL 1,X (x) = xr X (x), so we get a relationship with the hazard rate order. In fact, For α = −1 we have showed in remark 3 thatL −1,X (x) = xLOR X (x) = L −1,X (x), so we get a relationship with the log-odds rate order. In fact, Moreover we have X ≥ −1RAI Y ⇔ X ≤ −1AI Y so they are dual relations. For α = n ∈ N we have showed in remark 3 that so there is a connection with the largest order statistic and the hazard rate order. In fact Remark 10. Lemma 5.1 and Remark 9 provide the following series of relations Proposition 5.2. Let X and Y be non-negative and absolutely continuous random variables (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 1 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) ≥ȓ Y (x) for all x > 0; (2) If exists β ∈ R such that for all α > β we have X ≤ αRAI Y then X ≥ st Y .
Corollary 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 ∈ (−∞, −1); For some distribution F with support x ∈ (0, +∞), we obtain a natural estimator of the α-generalized reversed aging intensity function where f denotes a nonparametric estimate of the unknown density function f and F (x) = x 0 f (t)dt represents the corresponding distribution function estimate. The proposed estimation of the aging intensity function is possible if we assume that data follow an absolutely continuous distribution with support (0, +∞) and if the nonparametric estimate of its density function exists. Moreover, larger sample sizes generally lead to increased precision of estimation. We perform our study for both the generated and the real data.
6.1 Analysis of α-generalized reversed aging intensity function through generated data In the following example we consider an application of the estimator (27) for α = −1 to verify the hypothesis that some simulated data come from the family of inverse log-logistic distributions.

Example 4. Our goal is to check if a member of the inverse log-logistic distributions invLLog(γ, λ)
with the distribution function given by for some unknown positive parameters of the shape γ and the scale λ, is the parent distribution of a random sample X 1 , . . . , X N .
From presented in Section 4, Example 1 we know that for distribution function (28), its −1- Therefore, we use the following procedure to obtain N independent random variables X 1 , . . . , X N with invLLog(γ, λ) lifetime distribution. First, we generate standard uniform random variables U 1 , . . . , U N using function random of MATLAB. Then, applying the inverse . . , N , with the inverse log-logistic distribution invLLog(γ, λ). In this way, applying the function random with the seed= 88, we generate N = 1000 independent inverse log-logistic random variables with the shape parameter γ = 4, and the scale parameter λ = 0.5.
To calculate the reversed aging intensity estimator (27), we apply a kernel density estimator (see Bowman and Azzalini 1997), i.e., given in MATLAB ksdensity function, 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 where I(x) = 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 loglogistic family. But, we can notice that the plot of respective estimator (30) of −1-generalized reversed aging intensity function L −1 (x) (see Figure 2), oscillates around a constant function,  To justify our intuitive decision, we propose to carry out the following more formal statistical procedure. First, we calculate the least squares estimate of the intercept which for our data equals to γ = 3.7990. Next, we put it into the log-likelihood function, and determine maximum likelihood estimator (MLE) of parameter λ maximizing it. The problem resolves into finding the solution to the equation As the result we obtain λ = 0.4957. Note that the estimators γ and λ based on the empirical −1-generalized reversed aging intensity are quite precise (cf. Table 1).
. , X N ) and theoretical frequencies based on the inverse log-logistic distribution with parameters replaced by the estimators n j = N F γ, λ (x j+1 ) − F γ, λ (x j ) , are presented in Table 2.
Furthermore, available in MATLAB function chi2gof determines the value of chi-square statistics χ 2 = 9.3209 with ν = 7 degrees of freedom (automatically joining together the last twelve classes with low frequencies) and determines the respective p-value, p = 0.2304. It means that for a given significance level less than 0.2304 we do not reject the hypothesis that the considered data follow the inverse log-logistic distribution.

Analysis of α-generalized reversed aging intensity through real data
Next, we present an example of real data. Analyzing its estimated α-generalized reversed aging intensity we could assume that the data follow the adequate distribution. For the given data, the plot of the normal kernel density estimator (see Bowman and Azzalini 1997), obtained by MATLAB function ksdensity with a returned bandwidth h = 0.0147, is presented in Figure 3. An analysis of the graph does not enable us to recognize the data distribution. To identify the data distribution we propose to estimate 0-generalized , x ∈ (0, +∞).
The plot of the estimator L 0 (x) (see Figure 4) can be treated as oscillating around a linear function, especially after removing one outlying value at the right-end. This motivates us to state the hypothesis that data follow an inverse modified Weibull distribution (see Section 4, Example 2) with distribution function F γ,λ,δ (x) = exp − λ x γ exp(−δ x) , x ∈ (0, +∞), and 0-generalized reversed aging intensity function L 0 (x) = δ x + γ, x ∈ (0, +∞). Then, to check if the data fit the inverse modified Weibull distribution we use adequate for small data the Kolmogorov-Smirnov goodness-of-fit test (avaliable in MATLAB function kstest), we determine statistics K = 0.1496 and p-value of the test equal to p = 0.7072. It means that for a given significance level less than 0.7072 we do not reject the hypothesis that the considered data follow the inverse modified Weibull distribution.

Conclusion
In this paper, a family of generalized reversed aging intensity functions was introduced and studied. In particular, it was showed that, using the generalized Pareto distribution to generalize the concept of reversed aging intensity function, for α > 0, the α-generalized reversed aging intensity function characterizes a unique distribution function, while for α ≤ 0, it determines a family of distribution functions. Moreover, α-generalized reversed aging intensity orders were introduced 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.