On the Generalized Decreasing Mean Time to Failure or Replaced Ordering

In this paper, we establish two new stochastic orders, DMTFR (decreasing mean time to failure or replaced) and GDMTFR (generalized decreasing mean time to failure or replaced), and mainly investigate properties of the GDMTFR order. Some characterizations of the GDMTFR order are given. The implication relationships between the DMTFR and the GDMTFR orders are considered. Also, closure and reversed closure properties of the new order GDMTFR are investigated. Meanwhile, several illustrative examples that meet the GDMTFR order are shown as well.


Introduction and Preliminaries
In risk investment fields, we often need to compare or select two risk assets [1]. Similarly, two lifetimes of two systems or units also need to be compared in engineering technologies [2]. To do these things, some refined stochastic orders were defined in statistics. For more details on stochastic orders, one may refer to the studies by Maria Fernandez-Ponce et al. [3], Belzunce [4], Müller and Stoyan [5], Li and Yam [6], Shaked and Shanthikumar [7,8], Zhao and Balakrishnan [9], Sunoj et al. [10], Kang [11,12], Yan [13], Kang and Yan [14], Vineshkumar [15], and the references therein. However, sometimes, we need to compare two risk assets by the aid of the third referred system, such as when we compare two risk assets [16], their values are changeable with the settlement time or with the kind of valuation currency [17]. To solve this problem, we establish a new stochastic order by introducing a common measure factor, namely, referred function.
Let X be a nonnegative and absolutely continuous random variable. X has distribution function F X with 0 being the left endpoint of its support, survival function F X � 1 − F X , and density function f X . Let again F − 1 X be the right-continuous inverse function of F X defined by In many important real areas, such as reliability, economics, management sciences, information sciences, and other related fields, stochastic comparisons are paid much attention and have been used as sharp tools in dealing with some random problems in recent years. Based on comparisons of residual life, mean residual life, mean inactivity time, failure rate, and many aging concepts are presented earlier or later. In these aspects, interested readers can refer to the studies by Barlow and Proschan [18], Shaked et al. [7], Shaked and Shanthikumar [8], Müller and Stoyan [5], Kochar et al. [19], Ahmad et al. [20], Knopik [21], Kochar et al. [19], and Li and Shaked [22] for some existing results.
In reality, a nonnegative random variable X is often called a life representing the lifetimes of a device. e life distribution classes IFR (increasing failure rate), IFRA (increasing failure rate average), DMRL (decreasing mean residual life), NBUE (new better than used in expectation), and MTFR (mean time to failure or replaced) classes of life distributions are commonly used to describe the ageing features of units or systems, for example, see Barlow and Proschan [18].
Consider an age replacement police as the one in which a unit is replaced by t time units after installation or at failure, whichever occurs first, and then the expected value for the first time to an in-service is (see Barlow and Proschan [18]) When M X (t) is monotonic, the case was considered by Barlow and Campo [23], Marschall and Proschan [27], and Klefsjö [24]. In Knopik [21], and the ageing class MTFR (mean time to failure or replaced) of lifetime distribution is introduced, and it is proved that the MTFR class is closed under the operation of maximum for independence. In Kochar et al. [19], it is showed that the class MTFR is closed under weak convergence of distributions and convolution, and the dual family MTFR D is closed under noncrossing mixtures.

Definition 1.
e random variable X belongs to the mean time to failure with replacement (MTFR) class if the function X is decreasing for t ∈ x|F X (x) > 0 .
Assume that X has finite mean E(X) � μ X . e residual life of X at time t > 0 is defined as X t � [X − t | X > t]. en, the mean residual life of X at time t > 0 is , for t ≥ 0.
If μ X (t) is decreasing, then we say that X (or F X ) is in the decreasing mean residual (DMRL) life distribution class and denoted by X(or F X ) ∈ DMRL.
Emad-Edlin [28] proposed and studied the decreasing mean residual life (DMRL) ordering. Maria Fernandez-Ponce et al. [3] and Belzunce [4] proposed and studied the right spread order. Subsequently, Kochar et al. [19] proposed and studied the total time on test transform order and excess wealth order which is equivalent to the right spread order. e total time on test (TTT) transform functions of X and Y are defined, respectively, as e right spread (RS, for short) functions of X and Y are defined as, for all p ∈ (0, 1), respectively, where F X � 1 − F X and G Y � 1 − G Y are the survival functions of F X and G Y , respectively, and (x) + � max x, 0 { }. Now, we recall several stochastic orders from Shaked and Shanthikumar [8].
(2) X is said to be smaller than Y in the right spread (3) X is said to be smaller than Y in the NBUE (new better than used in expectation) order (denoted by (4) X is said to be smaller than Y in the location independent riskier order (denoted by X ≤ lir Y) if (5) X is said to be smaller than Y in the DMRL order Computational Intelligence and Neuroscience is increasing in p ∈ (0, 1).
From Definition 1, we easily have the following result without proof. Lemma 1. Let X and Y be two continuous and nonnegative random variables with respective distribution functions F X and G Y , density functions f X and g Y , and the right-continuous inverse functions F − 1 X (p) and G − 1 Y (p) of F X and G Y , respectively. en, the following statements are equivalent: is valid.
is valid.
Definition 3. Let X and Y be two nonnegative random variables with distribution functions F X , G Y such that Let X and X be two absolutely continuous nonnegative random variables with respective distribution functions F X and G Y such that F X (0) � G Y (0) � 0, and density functions f X and g Y , and right-continuous inverse functions of F − 1 X and G − 1 Y , respectively. We say that X is smaller than Y in the increasing concave order (denoted by Here, Hence, the order ≤ icv is a direct comparison for the mean service times of X and Y. In insurance theory, is the average losses undertaken by the insured when the deductible excess is d. Hence, that is, T X (p) and T Y (p) are the mean service times of X and Y before the respective equal probability time points. We wish to compare T X (p) and T Y (p) by means of a ratio T Y (p)/T X (p) Based on such an idea, we give Definition 4. Let X and Z be two nonnegative continuous random variables with respective distribution functions F X and H Z such that F X (0) � H Z (0) � 0, right-continuous inverse functions F − 1 X and H − 1 Z , survival functions F X and H Z , and density functions f X and h Z , respectively. Definition 4. X is said to be smaller than Z in the decreasing mean time to failure (DMTFR) order (denoted by is increasing in p ∈ (0, 1). From Definition 3, the following lemma is obvious, the proof is omitted here.

Lemma 2.
e following statements are equivalent: Computational Intelligence and Neuroscience 3 is valid.
is valid.
Shaked and Shanthikumar [8] studied the generalized TTT transform and proposed the generalized TTT transform order. Bartoszewicz and Benduch [29] further studied some properties of the generalized TTT transform by iteration. Motivated by their excellent works, we establish the following two new stochastic orders, the RDMTFR and the GDMTFR orders, see Definitions 5 and 6, respectively.
Definition 5. Let X, Z, and Y be three random variables. X is said to be smaller than Z relative to Y in the relative DMTFR order (denoted by is increasing in p ∈ (0, 1). From Definition 5, the following lemma is obvious, and the proof is omitted here.

Lemma 3.
e following statements are equivalent: is valid.
is valid.
It can be easily seen that when Y is an exponential random variable with a rate λ > 0, that is, Y ∼ E(λ), there exists a simple relationship between the relative DMTFR order and the DMTFR order: Assume that ψ is a real function defined on (0, 1). Denote by T X (p; ψ) and T Z (p; ψ) are called the generalized TTT transforms of X and Z relative to ψ, respectively. en, we give the following definition. Definition 6. Let X and Z be two absolutely continuous nonnegative random variables; assume that ψ is a real function defined on (0, 1). X is said to be smaller than Z with respect to ψ in the generalized DMTFR order (denoted by is increasing in p ∈ (0, 1).

Remark 1.
One can verify that the GDMTFR order is a partial order relation. e reflexive and transitive are evident; the antisymmetric is as follows (see eorem 1 (30) as follows). X ≤ gdmtfr Z w.r.t. ψ and Z ≤ gdmtfr X w.r.t. ψ hold simultaneously, if and only if Y � kX + b almost surely, where k and b are any real numbers such that Y is nonnegative and k ≠ 0.
, u ∈ (0, 1), there exists a simple relationship between the GDMTFR order and the RDMTFR order: Now, we consider a system composed of same components. We say that the system preserves some properties if the components of this system possess some properties, and we conclude from the structure of the system that the system also possesses the same property. And conversely, we say that this system has the reversed preservation for some property if the system has some property. According to the structure of the system, we conclude that the components of this system also have the same property.
For two systems, if their two components satisfy some stochastic order relation, by the structure of these two systems, we conclude that the two systems also satisfy the same stochastic order relation; then, we say that these systems of the structure preserve this stochastic order relation, or equivalently, we say that this stochastic order relation possesses closure property under the structure. Conversely, if two systems satisfy some stochastic order relation, by the structure of these two systems, we conclude that the two components of the two systems also satisfy the same stochastic order relation, and then we say that these systems of the structure reversely preserve this stochastic order relation, or equivalently, we say that this stochastic order relation possesses reversed closure property under the structure.
In reliability theory, such two problems are often of interest: one is to investigate the closure properties of a stochastic order, and the other is to examine the reversed closure properties of a stochastic order under several reliability operations, such as increasing convex transforms and taking of maxima and minima.
To prove our main results, we first introduce the following lemma from Barlow and Proschan [18], which plays a key role in the proofs of this paper and are repeatedly used in the sequel.
In the next, we assume that all the random variables, under consideration, are nonnegative and continuous. roughout this paper, the term increasing stands for monotone nondecreasing and decreasing stands for monotone nonincreasing. Assume that all the random variables under considerations are nonnegative and absolutely continuous with 0 as the left endpoint of their supports, and that all the integrals and expectations involved are always finite. All the encountered ratios are always supposed to be well defined.
In this paper, we devote our interest to the closure properties of the GDMTFR order. e paper is organized as follows. First, in Section 2, we consider characterizations of GDMTFR order. We investigate the implication relationships between DMTFR and GDMTFR orders in Section 3. e closure and reversed closure properties of the GDMTFR order are studied in Section 4. Finally, in Section 5, we give two examples which meet the GDMTFR order.
In the following, we always assume that X and Z are two absolutely continuous and nonnegative random variables with distribution functions F X and H Z such that F X (0) � H Z (0) � 0, survival functions F X ≡ 1 − F and H Z ≡ 1 − H Z , density functions f X and h Z , and rightcontinuous inverse functions F − 1 X and H − 1 Z of F X and H Z , respectively.

Characterizations of the GDMTFR Order
Now, we explore some characterizations of the GDMTFR order. First, we give a result by Definition 1.6, which will be useful in the proofs of upcoming theorems.

Theorem 1.
e following statements are equivalent: Computational Intelligence and Neuroscience 5 is valid.
is valid.
Proof. We only give proof for the case of (1)⟺ (2). Suppose that the function is increasing in p ∈ (0, 1). By differentiating, we have that the numerator of the derivative of this ratio is Since the function is increasing in p ∈ (0, 1), we have Letting H Z (x) � F X (y) in the second integral of the lefthand side of inequality (35) and then letting F − 1 X (p) � t, we obtain that And, the above deduction is reversible. erefore, the proof of the theorem is complete. □ Definition 7. Let X and Z be nonnegative absolutely continuous random variables with distribution functions F X , H Z such that F X (0) � H Z (0) � 0, density functions f X , h Z , and right-continuous inverse functions F − 1 X , H − 1 Z , respectively. (1) X is said to be smaller than Z in the starshaped order (denoted by X ≤ * Z), if the function H − 1 Z (F X (x))/x is increasing in x > 0.
(2) X is said to be smaller than Z in the convex order (3) X is said to be smaller than Z in the dispersive order ] for all p ∈ (0, 1). (0, 1). en, for any θ > 0, X ≤ gdmtfr θX w.r.t. ψ. 6 Computational Intelligence and Neuroscience

Remark 4.
eorem 4 states that the ≤ gdmtfr order is scale invariant with respect to the referred function.

Remark 6.
eorem 6 states that the order ≤ gdmtfr is scale invariant with respect to the compared random variables and the referred random variable.
at is, Differentiating both sides of above equality, we obtain where c is any real number. By the assumption of H Z (0) � F X (0) � 0, c � 0 and then H − 1 Z (p) � F − 1 θX (p). Hence, Z � d θX, and this is the stated result.
e following theorem gives a sufficient condition for the GDMTFR order.

Theorem 8. Let ψ be a nonnegative function defined on
, for all 0 ≤ x ≤ t.

Implication Relationships between DMTFR and GDMTFR Orders
Definition 8. For a positive integer number n, we say that (1) h is star-shaped (anti-star-shaped) with respect to the Let X and Z be two nonnegative continuous random variables with respective distribution functions F X and H Z having 0 as the common left endpoint of their supports. Let again ψ be a nonnegative function defined on (0, 1). Assume that X 1 , . . . , X n and Z 1 , . . . , Z n are the i.i.d. copies of X and Z, respectively, and that X k:n and Z k:n , k � 1, 2, . . . , n, are the order statistics of X and Z, respectively. Let X k:n have distribution function F X k:n (x), survival function F X k:n (x), and density function f X k:n (x), then for all x ≥ 0, where B k,n− k+1 (·) and β k,n− k+1 (·) are the distribution function and density function of a beta distribution with parameters k and n − k + 1, respectively. e following eorem 9 gives some conditions under which the DMTFR order implies the GDMTFR order.
(3) Suppose that X 1:n ≤ dmtfr Z 1:n . en, for all t > 0, Since ψ is star-shaped of order n with respect to the point (1, 0), the function is decreasing in x, which leads to the function being nonnegative decreasing. Moreover, From Lemma 4, (57), and (60), we have Computational Intelligence and Neuroscience 9 which is equivalent to (4) Suppose that X n: n ≤ dmtfr Z n: n . en, for all t > 0, t 0 F X n: n (x) f X n: n (t) h Z n:n H − 1 Z n: n F X n: n (t) − f X n: n (x) h Z n: n H − 1 Z n: n F X n: n (x) Since ψ is dual anti-star-shaped of order n, the function ψ(x)/(1 − x n ) is nonnegative decreasing in x ∈ (0, 1), which leads to the function being nonnegative decreasing. Moreover, From Lemma 4, (63)-(65), we have and this asserts that (5) Let the function ψ be anti-star-shaped with respect to the point (1, 0). Suppose that, for some 1 ≤ k ≤ n, X k:n ≤ dmtfr Z k:n , then for all t > 0, Since the function ψ is star-shaped with respect to the point (1, 0), the function ψ(x)/(1 − x) is nonnegatively decreasing in x ∈ (0, 1). us, φ(x) � ψ[F X k:n (x)]/F X k:n (x) is nonnegatively decreasing. From Lemma 4 and (68), we have which asserts that

Proof
(1) Suppose that, for some 1 ≤ k ≤ n, X k:n ≤ gdmtfr Z k:n w.r.t. ψ. en, for all t > 0, Since the function ψ is anti-star-shaped with respect to the point (1, 0), the function is nonnegatively decreasing. From Lemma 4 and (76), we have which asserts that X k:n ≤ dmtfr Z k:n , for all k � 1, 2, . . . , n. (78) Since the function ψ is anti-star-shaped with respect to the point (1, 0), the function ] is nonnegatively decreasing. From Lemma 4 and (79), we have which states that Since ψ is dual star-shaped of order n, the function (1 − u n )/ψ(u) is nonnegative decreasing in u ∈ (0, 1), which leads to the function and this asserts that X n: n ≤ dmtfr Z n: n . (86) Since ψ is anti-star-shaped of order n with respect to the point (1, 0), the function (1 − u) n /ψ(u) is decreasing in u ∈ (0, 1), and this leads to the function being nonnegatively decreasing. Moreover, From Lemma 4, (87)-(89), we have which is equivalent to (5) Suppose that X ≤ dmtfr Z w.r.t. ψ. en, we have, for all t > 0, Since ψ is increasing in u ∈ (0, 1), then  erefore, the proof of the theorem is complete. Denote by X 1: N � min(X 1 , X 2 , . . . , X N ) and X N: N � max(X 1 , X 2 , . . . , X N ). Z 1: N and Z N: N are similar.