3.1. Epoch Times of Discrete-Time Minimal Repair Processes

Let us consider two discrete-time minimal repair processes with underlying distributions P and Q, and we denote by (S₁,…, S_n) and (T₁,…,T_n) the random vector of the first n discrete epoch times. The purpose is to obtain conditions under which it is possible to compare (S₁,…, S_n) and (T₁,…,T_n), as well as the marginal distributions.

We begin by considering the usual multivariate stochastic order. Given two random vectors X and Y, we say that X is less than Y in the multivariate stochastic order (denoted by X ≤_st Y) if E [φ(X)] ≤ E [φ(Y)] for all increasing functions φ, for which the previous expectations exist. In the univariate case, let two random variables X and Y with distribution functions F and G be given. Then X ≤_st Y if F(x) ≥ G(x) for all

, and it can be denoted by F ≤_st G.

It is possible to prove, by construction on the same probability space, that (S₁,…, S_n) ≤_st (T₁,…,T_n), under the assumption that P and Q are ordered in the hazard rate order. However, under this assumption, it is possible to prove a stronger result for the multivariate hazard rate order. First, we recall the definition of the (univariate and multivariate) hazard rate order in the discrete case.

In order to define the multivariate hazard rate order, we need to introduce the concept of history of a random vector. Let us consider a random vector X of dimension n that takes values on

. Given

, let h_t be the event

where X_I denotes the list of components of X with index in I, for some I ⊆ {1,2,…, n} and x_I < te, where e is a vector of 1's and the dimension is determined from the context in which it appears. If the components of X denote the discrete times in which the events A₁,A₂,…, A_n occur and given that the set of indexes of events A_i that have occurred at time t is I, then h_t denotes the list of times, X_I, in which the events with index in I occur, whether for the remaining components X_I ≥ te. The event h_t usually is called a history at time t of X.

Given a history h_t, we can define the discrete multivariate conditional hazard rate at time t for a component with index in I as (see Shaked et al. [9,10,11])

Also, given a discrete random variable X, the previous concept corresponds to the discrete hazard rate at time t defined by (see Salvia and Bollinger [7])

Now we define the discrete multivariate hazard rate order. We will see later a result for the multivariate hazard rate order of (S₁,…, S_n) and (T₁,…,T_n); in this case, the components are strictly ordered in increasing order. If, for two random vectors X = (X₁,X₂,…, X_n) and Y = (Y₁,Y₂,…,Y_n), there cannot be ties for the components of X and for the components of Y, then the definition of the multivariate hazard rate order reduces to the following (see Shaked et al. [10]).

Definition 3.1: Let X and Y be two discrete random vectors of dimension n taking values on

. Let λ_·|·(·|·) and η_·|·(·|·) be the multivariate conditional hazard rates of X and Y, respectively. We say that X is less than Y in the multivariate hazard rate order (denoted by X ≤_hr Y) if

where x_I ≤ y_I < te, x_J < te, I ∩ J = Ø, and i ∈ I ∪ J.

In the univariate case, given two discrete random variables X and Y taking values on

with discrete hazard rates r and s, then the multivariate hazard rate order reduces to

In the univariate case, the previous condition is equivalent to the monotonicity of the ratio of the survival functions; that is, X ≤_hr Y if and only if

Again, if X and Y have distribution functions F and G, respectively, then the hazard rate order will be denoted by F ≤_hr G. It is well known that the hazard rate order is stronger than the stochastic order; that is, ≤_hr ⇒ ≤_st.

Theorem 3.2: For two discrete-time minimal repair processes, as above, with underlying distributions P and Q, then P ≤_hr Q if and only if

Proof: First we prove that P ≤_hr Q is a sufficient condition for Eq. (4). Given that T₁ < ··· < T_n, a.s., then, for a history at time l of (T₁,…,T_n), the set of indexes I of the epoch times that have occurred until the time point l must be of the form I = {1,2,…, m}, for some m ≥ 1, or I = Ø.

If we denote by η_·|·(l|·) the multivariate conditional hazard rate of (T₁,…,T_n) at time l with I = {1,2,…, m} and by s the discrete hazard rate for the distribution function Q, then

where the value for i = m + 1 follows from Lemma 2.1 and for i > m + 1 given that T₁ < ··· < T_n, a.s.

In a similar way, if we denote by λ_·|·(l|·) the multivariate conditional hazard rate of (S₁,…, S_n) at time l for a history in the set of components I ∪ J = {1,2,…, m,m + 1,…, k} and by r the discrete hazard rate for the distribution function P, then

Given i ∈ I ∪ J (i.e., i > k) if m < k, then

If m = k, since F ≤_hr G, then

So, Eq. (4) holds.

On the other hand, P ≤_hr Q is a necessary condition taking n = 1 in Eq. (4). █

Given that the hazard rate order is stronger than the stochastic order and from the fact that the stochastic order is preserved under marginalization (see Shaked and Shanthikumar [8]), we have the following corollary.

Corollary 3.3: For two discrete-time minimal repair processes, as above, with underlying distributions P and Q, if P ≤_hr Q, then

In particular, we can get bounds for the distribution function of the epoch time of a discrete-time minimal repair process.

Next we get a stronger result related to the multivariate likelihood ratio order of two discrete-time minimal repair processes. In particular, we get the multivariate likelihood rate order between (S₁,…, S_n) and (T₁,…,T_n) with stronger conditions on the underlying distributions P and Q.

First, we recall the definition of the multivariate likelihood rate order in the discrete case.

Definition 3.4: Let X and Y be two discrete random vectors of dimension n taking values on

. Let f and g be the joint probability density function of X and Y, respectively. We say that X is less than Y in the multivariate likelihood rate order (denoted by X ≤_lr Y) if

for all (x₁,x₂,…, x_n) and (y₁,y₂,…, y_n) on

, where ∧ and ∨ denote the minimum and the maximum, respectively.

If X and Y are discrete random variables with distribution functions F and G and probability mass functions f and g, respectively, then X ≤_lr Y if and only if g/f is increasing, and in this case, it will be denoted by F ≤_lr G.

Finally, we observe that the likelihood rate order and the stochastic order are preserved under marginalization, but for the hazard rate order, this property does not hold.

Next we fix the following notation. Given two distribution functions P and Q, with probability mass functions p and q, respectively, we will denote

and

Also, we will consider the values

The following lemma will be used to provide some sufficient conditions for the likelihood rate order of the underlying distributions of two discrete-time minimal repair processes. Its proof is immediate from Eqs. (5) and (6).

Lemma 3.5: For two discrete distribution functions P and Q, if P ≤_hr Q and m_i /l_i is increasing in i ≥ 1, then P ≤_lr Q.

Theorem 3.6: For two discrete-time minimal repair processes, as above, with underlying distributions P and Q, if P ≤_hr Q and m_i /l_i is increasing in i ≥ 1, then

Proof: For n = 1, the result is trivial from Lemma 3.5 and by the fact that P and Q have the same distribution as S₁ and T₁, respectively.

Let us assume that n ≥ 2. From the definition of the multivariate likelihood rate order and Eq. (3) we have to prove that

for all x₁ < ··· < x_n and y₁ < ··· < y_n.

If we denote E = {i ≤ n − 1 : x_i ≥ y_i}, then the last inequality becomes

From Lemma 3.5, we have that P ≤_lr Q and, therefore,

On the other hand, from the condition that m_i /l_i is increasing in i ≥ 1, for all j ∈ E we have that

Now, from Eqs. (8) and (9), Eq. (7) holds. █

A consequence of the preservation property under marginalization of the likelihood rate order (see Shaked and Shanthikumar [8, Thm.4.E.3(b)]) and the last theorem is the following corollary in the univariate case.

Corollary 3.7: For two discrete-time minimal repair processes, as above, with underlying distributions P and Q, if P ≤_hr Q and m_i /l_i is increasing in i ≥ 1, then S_n ≤_lr T_n for all n ≥ 1.

However, it is possible to give a stronger result with an independent proof.

Theorem 3.8: For two discrete-time minimal repair processes, as above, with underlying distributions P and Q, if P ≤_lr Q and M_n(i)/L_n(i) is increasing in i ≥ n, then S_n ≤_lr T_n for all n ≥ 1.

Proof: For n = 1, the result is trivial. Let us assume that n ≥ 2 and let us denote by p_n and q_n the probability function of S_n and T_n, respectively. In order to prove that q_n(i)/p_n(i) is increasing in i ≥ n, we observe that

where q_i /p_i is increasing by the hypothesis P ≤_lr Q and M_n−1(i − 1)/L_n−1(i − 1) is increasing in i ≥ n by hypothesis. So, the result holds. █

At this point, the next step is to find out if the conditions of Theorem 3.8 are weaker than those of Theorem 3.6. In order to see this and from Lemma 3.5, it would be sufficient to prove that the condition m_i /l_i is increasing in i ≥ 1 implies that M_n(i)/L_n(i) is increasing in i ≥ n. The next lemma states this conclusion.

Lemma 3.9: For two discrete-time minimal repair processes with underlying distributions P and Q, if m_i /l_i is increasing in i ≥ 1, then M_n(i)/L_n(i) is increasing in i ≥ n.

Proof: We proceed by induction in n. For n = 1, we have to prove that

From the definitions of M₁ and L₁ we have that

where =_sgn denotes the equality in sign.

Therefore, Eq. (10) is equivalent to the condition

that is,

Now from the hypothesis we have that

and then Eq. (11) holds.

Next, let us assume that the result holds for n and we will prove it for n + 1. We have to prove that

From the definitions of M_n and L_n we have that

Therefore, Eq. (13) is equivalent to the condition

Now from the induction hypothesis we have that

Therefore, from Eqs. (12) and (15), Eq. (14) holds. █

As we have observed in Section 2, these results provide stochastic comparisons for discrete record values; next we will provide results for discrete “weak” record values.

3.2. Discrete Weak Record Values

In this subsection we consider two sequences of discrete weak record values and provide conditions under which the first n weak record values of the two sequences can be ordered in the stochastic and likelihood ratio orders.

Let {X_i}_i=1^∞ and {Y_i}_i=1^∞ be two sequences of i.i.d. random variables taking on values 1,2,…, with distribution functions P and Q, respectively, and let us denote by {X_w(i)}_i=1^∞ and {Y_w(i)}_i=1^∞ the corresponding sequences of discrete weak record values.

First, we observe that we have not been able to provide a result for the hazard rate as in Theorem 3.2. However, it is possible to provide the following result for the stochastic order.

Now we recall the CIS notion, which will be used in the next theorem. Given a random vector X = (X₁,…, X_n), X is said to be conditionally increasing in sequence (CIS) (see Lehman [3]) if, for i = 2,3,…, n,

whenever x_j ≤ x_j′, j = 1,2,…, i − 1.

Theorem 3.10: For two sequences {X_w(i)}_i=1^∞ and {Y_w(i)}_i=1^∞ of discrete weak record values, as above, from distribution functions P and Q, if P ≤_hr Q, then

Proof: The proof follows from Theorem 4.B.4 in Shaked and Shanthikumar [8]. Provided that (X_w(1),…, X_w(n)) and (Y_w(1),…,Y_w(n)) are CIS, we just need to prove that

and for i = 2,3,…, n,

The CIS property of (X_w(1),…, X_w(n)) and (Y_w(1),…,Y_w(n)) follows from the Markovian property of the sequence of weak record values (see Section 2), where the transition probabilities P(X_w(n + 1) = j|X_w(n) = i) are clearly increasing in i.

Equation (16) follows from the fact that X_w(1) and Y_w(1) have distribution functions P and Q, respectively, and the fact that the hazard rate order is stronger than the stochastic order.

Finally, Eq. (17) follows from the previously mentioned Markovian property and, again, the fact that the hazard rate order is stronger than the stochastic order. █

Next we consider the likelihood hazard rate order. Under arguments similar to those in Lemma 3.5 and Theorem 3.6, we can prove the following results.

Lemma 3.11: For two discrete distribution functions P and Q, with hazard rates r and s, respectively, if P ≤_hr Q and s_i /r_i is increasing in i ≥ 1, then P ≤_lr Q.

Theorem 3.12: For two sequences of discrete weak record values, as above, from distribution functions P and Q, if P ≤_hr Q and s_i /r_i is increasing in i ≥ 1, then

4.1. Interepoch Times of Discrete-Time Minimal Repair Processes

Again consider two discrete-time minimal repair processes with underlying distributions P and Q and denote by (S₁,…, S_n) and (T₁,…,T_n) the random vector of the first n discrete epoch times. Let (U₁,…,U_n) and (V₁,…,V_n) be the corresponding random vectors of interepoch times; that is, U_i ≡ S_i − S_i−1 and V_i ≡ T_i − T_i−1, for i = 1,2,…, n, where U₀ ≡ V₀ ≡ 0. The purpose is to study whether we can compare in the stochastic and likelihood ratio orders the random vectors (U₁,…,U_n) and (V₁,…,V_n).

Next we give conditions under which (U₁,…,U_n) and (V₁,…,V_n) are ordered in the stochastic order. In order to give the result first, we need to give the decreasing failure rate (DFR) notion.

Let X be a discrete random variable taking values in

, with distribution function P and hazard rate r; we say that X is DFR if r is decreasing. It is well known that X is DFR if and only if P(X > t + x|X > x) is increasing in

for all

.

Theorem 4.1: For two discrete-time minimal repair processes, as above, with underlying distributions P and Q, if P ≤_hr Q and either P or Q are DFR, then

Proof: From Theorem 4.B.4 in Shaked and Shanthikumar [8], the proof follows if (U₁,…,U_n) or (V₁,…,V_n) are CIS, and

and for i = 2,3,…, n,

We observe that, for i = 2,…, n,

and, similarly,

Then, from previous equalities, the CIS property of (U₁,…,U_n) (or (V₁,…,V_n)) follows from the DFR property of P (or Q) and Eqs. (18) and (19) follows from the hypothesis P ≤_hr Q. █

Again, given that the stochastic order is preserved under marginalization, we can get the comparison in the stochastic order of U_n and V_n.

Corollary 4.2: For two discrete-time minimal repair processes as above, with underlying distributions P and Q, if P ≤_hr Q and P or Q or both are DFR, then

Next we consider the likelihood rate order between (U₁,…,U_n) and (V₁,…,V_n).

Theorem 4.3: For two discrete-time minimal repair processes with underlying distributions P and Q, if P ≤_hr Q and m_i /l_i is increasing in i ≥ 1, P or Q have logconvex probability mass functions, and l_i or m_i is logconvex in i, then

Proof: Let us assume that the probability mass function p of P and l are logconvex functions. The proof in other cases is similar.

Let us consider u_k, v_k > 0, k = 1,…, n, and fix the following notation for j = 1,…, n :

where the last equality is easy to prove and also a_j ≥ b_j, b_j′ ≥ a_j′, and c_j ≥ 0.

The logconvex property of l implies for j = 1,…, n − 1 that

and the logconvex property of p implies

So, we have to prove that

where q is the probability function of Q. From Eqs. (21) and (22), we have that

On the other hand, from the hypothesis and Lemma 3.5,

Therefore, from Eq. (24) and the last inequality, Eq. (23) is true and the result holds. █

4.2. Weak Interrecord Values

By following the same notation as in Section 3.2, we consider the random vectors (U_w(1),…,U_w(n)) and (V_w(1),…,V_w(n)) of interrecord values for discrete weak record values, where U_w(i) ≡ X_w(i) − X_w(i − 1) and V_w(i) ≡ Y_w(i) − Y_w(i − 1), for i = 1,2,…, n, where U_w(0) ≡ V_w(0) ≡ 0. In this subsection we get similar results to those of Section 4.1; the proofs are similar and are therefore omitted.

Theorem 4.4: For two sequences of discrete weak record values, as above, with underlying distributions P and Q, if P ≤_hr Q and P or Q is DFR, then

Theorem 4.5: For two sequences of discrete weak record values, as above, with underlying distributions P and Q and failure rates r and s, respectively, if P ≤_hr Q and s_i /r_i is increasing in i ≥ 1, F or G have logconvex probability mass functions, and r_i or s_i is logconvex in i, then

Remark 4.6: Some of the results stated in this article can be used to also provide results of positive association for random vectors of record values or interrecord values. For example, it is well known that given a random vector X, X is said to be MTP2 (multivariate totally positive of order 2; see Karlin and Rinot [2]) if and only if X ≤_lr X. Now, from Theorem 4.3, given a random vector of interepoch times (U₁,…,U_n) from a discrete-time minimal repair process with underlying distribution P and values {l_i}_i=1^∞, defined as in Section 2, if P has a logconvex density function and l_i is logconvex, then (U₁,…,U_n) is MTP2. A similar result can be obtained from Theorem 4.5. By the same arguments and from Theorems 3.6 and 3.12, a random vector of epoch times of a discrete-time minimal repair process and a random vector of discrete weak record values, respectively, is always MTP2.