2. DEFINITIONS AND BASIC PROPERTIES

In this section, we first recall some definitions that will be used in the sequel; then we discuss some basic properties about IMIT class and MIT order.

Definition 2.1:

For a more comprehensive discussion of the properties as well as other details of those stochastic orderings, readers is referred to Shaked and Shanthikumar [20], Müller and Stoyan [17], Ahmad et al. [1], and Kayid and Ahmad [9].

The following aging properties are closely related to our main theme. For further details on them, refer to Barlow and Proschan [2], Müller and Stoyan [17], and Nanda, Singh, Misra, and Paul [18].

Definition 2.2:

Now let us turn to the following basic properties.

Proposition 2.3: X is IMIT if and only if X ≤_MIT X + Y for any Y independent of X.

Proof: Necessity: If m_X(t) increases in t > 0, then, by Fubini's theorem, we have, for any t > 0,

Thus, X ≤_MIT X + Y.

Sufficiency: In view of the fact that

the desired result follows immediately by putting Y = s ≥ 0. █

According to Theorem 1.D.8 of Shaked and Shanthikumar [20], X is of decreasing mean residual life (DMRL) if and only if X_t ≤_mrl X for any t ≥ 0. One may wonder whether IMIT property of X is also equivalent to X_t ≤_MIT X for any t ≥ 0. Proposition 2.4 shows that X_t ≤_MIT X for any t ≥ 0 implies that X is of IMIT; However, Example 2.5 states that the inverse is not valid.

Proposition 2.4: If X_t ≤_MIT X for any t ≥ 0, then X is of IMIT.

Proof: For any t ≥ 0 and s > x ≥ 0,

For any t ≥ 0 and s > 0, X_t ≤_MIT X is equivalent to

Then for any t ≥ 0 and s > 0,

Also,

Equivalently, for any t ≥ 0 and s > 0,

which asserts that m_X(s) is increasing in s > 0; that is, X is of IMIT. █

Example 2.5: For a random life X with distribution function

Thus, X is of IMIT.

Put s = t = ½; it can be easily found that

The inequality

tells that X_t ≤_MIT X is not true.

3. PRESERVATION PROPERTIES

This section will develop some preservation properties as well as reversed preservation properties of MIT order and IMIT class under both monotonic transformations and the taking of maximum, respectively.

Theorem 3.1: Assume that φ is strictly increasing and concave; φ(0) = 0. If X ≤_MIT Y, then φ(X) ≤_MIT φ(Y).

Proof: Without loss of generality, assume that φ is differentiable. X ≤_MIT Y implies that for any t > 0,

Since φ′(t) is nonnegative and decreasing, by Lemma 7.1(b) of Barlow and Proschan [2] it holds that

Equivalently,

that is, for any t > 0,

Note that

we have for any t > 0 that

which tells that φ(X) ≤_MIT φ(Y). █

Theorem 3.2: Let X₁,…, X_n and Y₁,…,Y_n be independent and identically distributed (i.i.d.) copies of X and Y, respectively. If max{X₁,…, X_n} ≤_MIT max{Y₁,…,Y_n}, then X ≤_MIT Y.

Proof: max{X₁,…, X_n} ≤_MIT max{Y₁,…,Y_n} implies that

that is,

Since, for any t > 0,

is nonnegative and decreasing in x ≥ 0, by Lemma 7.1(b) of Barlow and Proschan [2] we have

which states that X ≤_MIT Y. █

Remark: Theorem 3.2 in fact also holds for general variables. Since X ≤_MIT Y if and only if −X ≥_mrl − Y of Ahmad et al. [1] and

by Theorem 3.2, we immediately reach the following corollary.

Corollary 3.3: Let X₁,…, X_n and Y₁,…,Y_n be i.i.d. copies of X and Y, respectively. If min{X₁,…, X_n} ≥_mrl min{Y₁,…,Y_n}, then X ≥_mrl Y.

Based on Corollary 3.3 and Lemma 3.4, Theorem 3.5 presents the reversed preservation property of IMIT under the taking of maximum.

Lemma 3.4: X is of IMIT if and only if X_(s) ≤_mrl X_(t) for all 0 < s ≤ t.

Proof: It can be easily established; hence, the proof is omitted. █

Theorem 3.5: Let X,X₁,…, X_n be i.i.d. random variables. If max{X₁,…, X_n} is of IMIT, then X is also of IMIT.

Proof: For any s > x ≥ 0,

that is, for all s ≥ 0,

max{X₁,…, X_n} is of IMIT and by Lemma 3.4, it holds that

Thus,

Now it follows from Corollary 3.3 that

By Lemma 3.4 again, X is of IMIT. █

4. MEAN RESIDUAL LIFE AT A RANDOM TIME

Stochastic comparison under certain conditions on the concerned total life and random time as well as preservation properties of decreasing reversed hazard rate (DRHR) and IMIT have been conducted by Yue and Cao [23] and Li and Zuo [15]. This section reports some results from further comparisons of RLRT.

Yue and Cao [24, Thm.3(a)] proved that X is NBU_Lt (new better than used in Laplace transform order) if and only if X_Y is smaller than X in Laplace transform order for all Y independent of X. Recently, Li [11, Thm.4.1(i)] showed that X is NBU_Mg (new better than used in moment generating function order) if and only if X_Y is smaller than X in the moment generating function order for all Y independent of X. The following theorem gives a parallel result about the MIT order.

Theorem 4.1: X_Y ≤_MIT X for any Y that is independent of X if and only if X_t ≤_MIT X for all t ≥ 0.

Proof: Sufficiency: Since X_t ≤_MIT X for all t ≥ 0, it follows from (2) that

In view of (1), this shows X_Y ≤_MIT X.

Necessity: Suppose X_Y ≤_MIT X holds for any nonnegative random variable Y. Then X_t ≤_MIT X for all t ≥ 0 follows by taking Y as a degenerate variable. █

Recall that for two subsets Θ and Γ of the real line, a nonnegative function h defined on Θ × Γ is said to be totally positive of order 2 (TP2) on Θ × Γ if

whenever x ≤ x′ and y ≤ y′, and x,x′ ∈ Θ and y, y′ ∈ Γ. The following lemma, which is due to Joag-Dev, Kochar, and Proschan [8], will be utilized to derive Theorem 4.3.

Lemma 4.2: Let ψ(x,y) be any TP2 function (not necessarily a reliability function) in x ∈ Θ and y ∈ Γ and F_i(x) be a distribution function for each i. Denote

If F_i(x) is TP2 in i ∈ {1,2} and x ∈ Θ and if ψ(x,y) is increasing in x for every y, then H_i(y) is TP2 in y ∈ Γ and i ∈ {1,2}.

Proof of the next result is based on the idea of Theorem 4.2 of Gao, Belzunce, Hu, and Pellerey [7].

Theorem 4.3: Assume that Z is independent of X and Y. If X ≤_hr Y and Z is of IMIT, then X_Z ≤_mrl Y_Z.

Proof: Denote by F₁, F₂, and G distribution functions of X, Y, and Z, respectively. Since Z is of IMIT, we have, for all y ≥ 0 and Δ > 0,

that is,

is decreasing in y ≥ 0. Then

for all 0 < t₁ ≤ t₂ < y₁ ≤ y₂.

Denote

The last inequality states that

holds for all (t₁,t₂,y₁,y₂) ∈ S = {(t₁,t₂,y₁,y₂) : 0 < t₁ ≤ t₂ < y₁ ≤ y₂}.

It can be easily verified that (3) is also valid for those (t₁,t₂,y₁,y₂) ∈ {(t₁,t₂,y₁,y₂) : 0 < t₁ ≤ t₂, 0 < y₁ ≤ y₂} − S. Thus, ψ(y,t) is TP2 in (y,t) ∈ (0,∞) × (0,∞).

For i = 1,2, let

X ≤_hr Y implies that F_i(x) is TP2 in (i,x) ∈ {1,2} × (0,∞) and ψ(y,t) is increasing in y for each fixed t. From Lemma 4.2 it follows that H_i(t) is TP2 in (i,t) ∈ {1,2} × (0,∞); that is,

is increasing in t ≥ 0, which is equivalent to

Hence, X_Z ≤_mrl Y_Z. █

Remark: Assume X, Y, and Z are three independent nonnegative random variables. Li and Zuo [15, Thm.5] showed that if X ≤_hr Y and Z is of IMIT, then Z_(X) ≤_icx Z_(Y). In view of the fact that X_Z = Z_(X), it is also valid, under the condition of Theorem 4.3, that Z_(X) ≤_mrl Z_(Y). According to Theorem 3.A.13 of Shaked and Shanthikumar [20], X ≤_mrl Y implies X ≤_icx Y and we in fact develop a stronger conclusion here.

As a direct application of Theorem 4.3, we can get a simple and brief proof of Theorem 6 about DMRL in Li and Zuo [15].

Corollary 4.4: Assume X and Z are independent. If X is of IFR and Z is of IMIT, then X_Z is of DMRL.

Proof: According to Theorem 1.B.19 of Shaked and Shanthikumar [20], X is of IFR if and only if X_t ≤_hr X for all t ≥ 0. By Theorem 4.3, (X_t)_Z ≤_mrl X_Z for all t ≥ 0. Note that

for all t ≥ 0; it holds that (X_Z)_t ≤_mrl X_Z for all t ≥ 0. Now, from Theorem 1.D.8 of Shaked and Shanthikumar [20], it follows immediately that X_Z is of DMRL. █

5. STOCHASTIC COMPARISON OF RENEWAL EXCESS LIFETIME

Consider a renewal process with i.i.d. interarrival times X_i with common distribution F. For k = 1,2,…, let

be the time of the kth arrival and S₀ = 0. The renewal counting process N(t) = sup{n : S_n ≤ t} gives the number of arrivals until time t ≥ 0, and the excess lifetime γ(t) = S_N(t)+1 − t at time t ≥ 0 is the time elapsed from the time t to the first arrival after time t. Of course, it is obvious that

. M(t) = EN(t) is called as renewal function, which satisfies the well-known fundamental renewal equation

Brown [5] is among the first to survey the stochastic monotonicity of the excess lifetime; afterward, Shaked and Zhu [21] had some further discussions on this line of research. Subsequently, many authors devoted themselves to investigate the behavior of the renewal excess lifetime of a renewal process with interarrivals having certain aging properties, such as NBU, NBUC, NBU(2), NBUE, NBU_Lt and NBU_Mg, and so forth. Their results in fact assert that various NBU interarrivals can reduce to the corresponding NBU properties of the excess lifetime. For more details, readers are referred to Li [11], Li and Kochar [12], Belzunce, Ortega, and Ruiz [3], Li, Li, and Jing [13], Chen [6], and Barlow and Proschan [2]. In this section, we will investigate the behavior of the excess lifetime of a renewal process with IMIT interarrivals.

Theorem 5.1: If γ(t) is decreasing in the MIT order in t ≥ 0, then X is of IMIT.

Proof: According to Karlin and Taylor [10, p.193], for any t ≥ 0 and x ≥ 0,

Then

for any t ≥ 0 and x ≥ 0.

Since γ(t) is decreasing in t in the sense of MIT order, we have, for all s > 0 and t ≥ 0,

where the last equality is due to (5).

Note that

it holds that

On the other hand,

it holds that for all t ≥ 0 and s > 0,

which is equivalent to (2); that is, X_t ≤_MIT X for all t ≥ 0. From Proposition 2.4, the desired result follows immediately. █

Theorem 5.2: If X_t ≤_MIT X for all t ≥ 0, then γ(t) ≤_MIT γ(0) for all t ≥ 0.

Proof: According to Barlow and Proschan [2], it holds that

Thus,

X_t ≤_MIT X, by (2), for any t ≥ 0 and s > 0,

Now based on (4) and inequality (6), we have, for any t ≥ 0 and s > 0,

Hence, it holds that for all t ≥ 0 and s > 0,

That is, γ(t) ≤_MIT γ(0), for all t ≥ 0. █