2. PROBABILISTIC MODELS OF RANDOM VECTORS WITH ORDERED COMPONENTS

Perhaps the most well-known model of random vector with ordered components is the random vector of order statistics. Given a set of n i.i.d. random variables X₁,…,X_n, if we arrange these random variables in ascending order of magnitude, then the random vector of the ordered quantities (X₍₁₎,…,X_(n)) is called the random vector of order statistics.

Other models are given from the epoch times of a counting process. For example, we can consider the epoch times of a nonhomogeneous Poisson counting process. A counting process {N(t),t ≥ 0} is a nonhomogeneous Poisson process with intensity function

if the following hold:

If we denote by T_n = inf{t : N(t) = n} the epoch times of the nonhomogeneous Poisson process, clearly the components of the random vector of the first n epoch times, (T₁,…,T_n), are ordered.

Epoch times of nonhomogeneous Poisson processes can be introduced as record values of a proper sequence of random variables.

Let {X_i}_i=1^∞ be a sequence of i.i.d. random variables and denote by

the record times; then, the record values are defined as

Now, if we consider a sequence {X_i}_i=1^∞ of i.i.d. random variables (r.v.'s) with hazard rate r, then

where the T_n's are the epoch times of a nonhomogeneous Poisson process, with intensity r.

In order to present a unified approach to study order statistics and record values (and a variety of models of ordered random variables), Kamps [19] introduced the concept of generalized order statistics.

Next, we give the definition of generalized order statistics following Kamps [19,20].

Definition 2.1: Let

, be parameters such that γ_r = k + n − r + M_r ≥ 1 for all r ∈ 1,…,n − 1, and let

. We call uniform generalized order statistics to the random vector

with joint density function

on the cone 0 ≤ u₁ ≤···≤ u_n ≤ 1. Now, given a distribution function F, we call generalized order statistics based on F to the random vector

If F is an absolutely continuous distribution with density f, the joint density function of

is given by

on the cone F⁻¹(0) ≤ x₁ ≤···≤ x_n ≤ F⁻¹(1).

Several models of ordered random variables are included in this model. As we have mentioned previously, order statistics and record values are a particular case of this model. Taking m_i = 0 for all i = 1,…,n − 1 and k = 1, we get the order statistics from a distribution F, and taking m_i = −1 for all i = 1,…,n − 1 and k = 1, we get the first n record values from a sequence of random variables with distribution F. Some other models contained in the generalized order statistics (GOS) model will be seen in Section 5.

A property that will be used later is that GOS from a continuous distribution form a Markov chain with transition probabilities

Another random vector with ordered components that has recently received great interest is the model of the first n epoch times of a nonhomogeneous pure birth (NHPB) process. Such processes are called “relevation counting processes” in Pellerey, Shaked, and Zinn [28]. Epoch times of NHPB processes correspond also to Pfeifer's record values. Pfeifer [29] established a model of record values based on a double sequence of nonidentically distributed random variables.

Applications of NHPB processes in insurance, reliability theory, epidemiology, and load-sharing models can be found in Belzunce et al. [4], Hu and Pan [16], Pellerey et al. [28], and Shaked, Spizzichino, and Suter [33].

A NHPB process is a Markovian point process in which the jump intensity at any time t depends not only on t but also on the number n of jumps before time t. Formally, we consider a point process N = {N(t), t ≥ 0} such that the following hold:

The r_n's are nonnegative functions that satisfy

In a distributional theoretical sense GOS are contained in the model of epoch times of a NHPB process. Consider GOS based on F with failure rate r and parameters k, n, and M_r, r = 1,…,n − 1; then,

where the T_i's are the epoch times of a NHPB process with intensities r_i = (k + n − i + M_i)r, for i = 1,…,n. This relationship will be used later to obtain stochastic comparisons of GOS from some previous known results about NHPB processes.

3. STOCHASTIC COMPARISONS OF GENERALIZED ORDER STATISTICS

In this section, we give several results about stochastic comparisons of GOS. As we will see, some of them are given for the whole random vector of generalized order statistics, from which we can obtain stochastic comparisons of the coordinates. These results extend some recent results given by other authors.

We start by considering the usual multivariate stochastic order. Recall that given two n-dimensional random vectors X and Y, then X is said to be less than Y in the multivariate stochastic order (denoted by X ≤_st Y) if E [φ(X)] ≤_st E [φ(Y)] for all increasing functions

, provided the previous expectations exist. In the univariate case given two r.v.'s X and Y with distribution functions F and G, respectively, X ≤_st Y if and only if F ≤ G.

As we have mentioned in the previous section, GOS can be considered as a particular case of epoch times of NHPB. For this last model, Belzunce et al. [4] gave conditions in Theorem 3.11 under which two random vectors of epoch times of NHPB processes are ordered in the multivariate stochastic order. Such conditions can be translated into conditions for the multivariate stochastic order of two random vectors of GOS based on distributions F and G. In fact, under the hazard rate order of F and G (see the definition of the hazard rate order later), we get the multivariate stochastic order of the corresponding random vectors of generalized order statistics. However, as we see next, this result can be given under weaker conditions.

Theorem 3.1: Let X and Y be r.v.'s with distribution functions F and G, respectively, and let

be random vectors of generalized order statistics based on F and G, respectively. If X ≤_st Y, then

Proof: We prove the result by constructing random vectors

on the same probability space with

, such that

, and the result follows from Theorem 4.B.1 in Shaked and Shanthikumar [31]. Let

be random vectors, where

and

and

be the random vector of uniform generalized order statistics. Clearly,

are defined on the same probability space, and by Definition 2.1,

. On the other hand,

where the last equality follows from the assumption X ≤_st Y. The result follows from Theorem 4.B.1 in Shaked and Shanthikumar [31]. █

The multivariate stochastic order is preserved under marginalization. Therefore, if X ≤_st Y, any subset of coordinates of the random vector of GOS based on F is smaller in the stochastic order than the corresponding subset of coordinates based on G. In particular, we can obtain the following result.

Corollary 3.2: Let X and Y be r.v.'s with distribution functions F and G, respectively, and let

be random vectors of generalized order statistics based on F and G, respectively. If X ≤_st Y, then

This corollary extends Theorem 3.2 of Franco et al. [13] to the more general case in which the m_i's are not equal.

Given two nonnegative random variables X and Y with hazard rates r and s, respectively, then X is said to be less than Y in the hazard rate order (denoted by X ≤_hr Y) if r(t) ≥ s(t) for all t ≥ 0. This order can be characterized in terms of the survival functions as follows: X ≤_hr Y if F/G is decreasing, where F and G are the survival functions of X and Y, respectively.

This partial order is stronger than the stochastic order (i.e., ≤_hr ⇒ ≤_st), so the previous theorem is given under weaker conditions than those that follow from Theorem 3.11 in Belzunce et al. [4]. Under the hazard rate order assumption, it is possible to obtain the stronger result of the multivariate hazard rate order of generalized order statistics (we refer the reader to [31] for the definition of the multivariate hazard rate order). The following result follows from Theorem 3.12 by Belzunce et al. [4].

Theorem 3.3: Let X and Y be r.v.'s with distribution functions F and G, respectively, and let

be random vectors of generalized order statistics based on F and G, respectively. If X ≤_hr Y, then

Remark 3.4. The hazard rate order is not known to be preserved under marginalization; therefore, we cannot obtain from the previous result the hazard rate order of the corresponding coordinates. A partial result in this direction is given in Theorem 3.3 by Franco et al. [13] when m_i = m, for all i = 1,…,n − 1.

Next, we provide a result for the general case (i.e., when the m_i′s do not need to be equal). To provide the result, we recall, first, the following equality given recently by Cramer and Kamps [11].

The hazard rate, λ_r, of the rth generalized order statistic

based on an absolutely continuous distribution function F, with hazard rate r(t), is given by

where h_r is defined by

and G is a particular Meijer's G-function. Cramer and Kamps [11] show also (see Remark 4.7) that h_r(z) is decreasing in z.

Next, we give the following result for the hazard rate order.

Theorem 3.5: Let X and Y be absolutely continuous r.v.'s with distribution functions F and G and with hazard rates r and s, respectively. Let

be the rth generalized order statistics based on F and G, respectively. If X ≤_hr Y, then

.

Proof: From (3.1), the hazard rates of

are given by

respectively.

From the condition X ≤_hr Y, we get that r(t) ≥ s(t) and F(t) ≤ G(t). Now, given that h_r(z) is decreasing in z, we get the following inequality:

and, therefore, the result. █

Next, we consider the multivariate likelihood ratio order. Let X and Y be two n-dimensional random vectors with density functions f_X and f_Y, respectively. We say that X is less than Y in the multivariate likelihood ratio order, denoted by X ≤_lr Y, if (∧ and ∨ denote, respectively, the minimum and the maximum operations)

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

(see Shaked and Shanthikumar [31, Chap. 4, and references therein]).

A property that will be used in the proof of the following theorem is that the likelihood ratio order is stronger than the hazard rate order (i.e., ≤_lr ⇒ ≤_hr).

Theorem 3.6: Let X and Y be absolutely continuous r.v.'s with distribution functions F and G and with hazard rates r and s, respectively. Let

be random vectors of generalized order statistics based on F and G, respectively. If (a) m_i ≥ 0 for all i and X ≤_lr Y or (b) m_i ≥ −1 for all i, X ≤_hr Y, and s/r is increasing, then

Proof: By (2.1), we have that the joint density function of

is given by

and the joint density function of

is given by

Therefore, we have to prove that

If we consider the set E = {i ≤ n − 1 : x_i ≥ y_i}, then the previous inequality is equivalent to

From the assumption X ≤_lr Y (⇒ X ≤_hr Y) in parts (a) and (b), we have

Now, under the assumptions in part (a) we have that, for all j ∈ E,

Therefore, under the conditions in part (a), we get the result.

On the other hand, under the assumptions in part (b), we get that for all j ∈ E,

Again, under the assumptions in part (b) we get the result. █

The likelihood ratio order is preserved under marginalization; therefore, we have the following result.

Corollary 3.7: Let X and Y be r.v.'s with distribution functions F and G, respectively, and let

be random vectors of generalized order statistics based on F and G, respectively. If (a) m_i ≥ 0 for all i and X ≤_lr Y or (b) m_i ≥ −1 for all i, X ≤_hr Y and s/r is increasing, then

Part (a) of this corollary extends Theorem 3.4 by Franco et al. [13] to the more general case in which the m_i's are not equal.

Now, we consider a different problem. We consider stochastic comparisons of two random vectors of GOS with different parameters but based on the same distribution F. We start by giving a simple result for random vectors of uniform generalized order statistics.

Theorem 3.8: Let

be random vectors of uniform generalized order statistics with parameters k and m_i, i = 1,…,n − 1, and k′ and m_i′, i = 1,…,n − 1, respectively. Then, U ≤_lr V if and only if k ≥ k′ and m_i ≥ m_i′, for all i = 1,…,n − 1.

Proof: From Definition 2.1 we need to prove that

for all 0 < u₁ ≤···≤ u_n < 1 and 0 < v₁ ≤···≤ v_n < 1.

The previous inequality holds if and only if

and

for all 0 < u₁ ≤···≤ u_n < 1 and 0 < v₁ ≤···≤ v_n < 1.

Now, it is easy to see that the previous inequalities are equivalent to k ≥ k′ and m_j ≥ m_j′, for all j = 1,…,n − 1. █

Now, from the preservation of the multivariate likelihood ratio order under increasing transformations of each individual coordinate (see [31, Thm.4.E.2]), we can obtain the following result.

Theorem 3.9: Let

be random vectors of generalized order statistics based on the same distribution F and with parameters k and m_i, i = 1,…,n − 1, and k′ and m_i′, i = 1,…,n − 1, respectively. Then, X ≤_lr X′ if and only if k ≥ k′ and m_i ≥ m_i′ for all i = 1,…,n − 1.

Now, a combination of Theorem 3.9 and Theorems 3.1, 3.3, and 3.6 leads to the following result.

Theorem 3.10: Let X and Y be r.v.'s with distribution functions F and G, respectively, and let

be random vectors of generalized order statistics based on F and G and with parameters k and m_i, i = 1,…,n − 1, and k′ and m_i′, i = 1,…,n − 1, respectively. Let us suppose that k ≥ k′ and m_i ≥ m_i′ for all i = 1,…,n − 1. Then, we have the following results:

To finish this section, we consider two results under the dispersive order assumption.

Given two random variables X and Y, with distribution functions F and G, respectively, we say that X is less than Y in the dispersive order (denoted by X ≤_disp Y) if (see [24]):

or, equivalently, in the continuous case, if

Shaked and Shanthikumar [32, p.16] consider the following multivariate generalization of (3.2).

Let X = (X₁,…,X_n) be a random vector and consider the transformation Ψ_X* : (0,1)ⁿ → Rⁿ defined by

where the x_i's are defined recursively by

and

where F₁⁻¹ is the inverse of the marginal distribution of X₁ and F_{i|1,…,i−1}⁻¹(u_i|x₁,…,x_i−1) is the inverse of the conditional distribution of X_i given that X₁ = x₁,…,X_i−1 = x_i−1.

Now, given Ψ_X* and Ψ_Y* for two random vectors, X and Y, of dimension n, a multivariate generalization of (3.2) is the condition

Next, we show that under the dispersive order of F and G, then (3.3) holds for the random vectors of GOS based on F and G.

Theorem 3.11: Let X and Y be r.v.'s with distribution functions F and G, respectively, and let

be random vectors of generalized order statistics based on F and G, respectively. If X ≤_disp Y, then

Proof: We observe first that

and the same holds for Y; that is,

Therefore, we just need to prove that

and

for all r = 2,…,n, where x_r = F⁻¹(1 − F(x_r−1)(1 − u_r)^1/γ_r) and y_r = G⁻¹(1 − G(y_r−1)(1 − u_r)^1/γ_r). If we define recursively the values p₁ = 1 − (1 − u₁)^1/n and p_r = 1 − (1 − p_r−1)(1 − u_r)^1/(n−r+1) for r = 2,…,n, it is not difficult to see that (3.4) and (3.5) are equivalent to

for all r = 1,…,n and that p_r is an increasing function of (u₁,…,u_r). Therefore, the result follows from the assumption X ≤_disp Y. █

Again, condition (3.3) does not imply the dispersive order among coordinates. However, it is possible to obtain such result, as we show in Theorem 3.12. The result is easy to obtain observing that the distribution function of

based on an arbitrary distribution function F is given by

where Φ_r,n is the distribution function of the uniform generalized order statistic

.

Now, we have the following result:

Theorem 3.12: Let X and Y be r.v.'s with distribution functions F and G, respectively, and let

be random vectors of generalized order statistics based on F and G, respectively. If X ≤_disp Y, then

Proof: The proof follows by observing that

Similar results hold for the convex, star-shaped, and superadditive orders.

Given two random variables X and Y, with distribution functions F and G, respectively, we say that X is less than Y in the convex (star-shaped, superadditive) order (denoted by X ≤_cx[*,su] Y) if

Under the same arguments as in Theorem 3.12, we get the following result.

Theorem 3.13: Let X and Y be r.v.'s with distribution functions F and G, respectively, and let

be random vectors of generalized order statistics based on F and G, respectively. If X ≤_cx[*,su] Y, then

4. STOCHASTIC COMPARISONS OF RANDOM VECTORS OF SPACINGS OF GENERALIZED ORDER STATISTICS

In this section, we provide conditions for the stochastic comparisons of spacings from random vectors of generalized order statistics.

Given a random variable X with distribution function F and a random vector of

based on F, we will denote the spacings from X by

for r ≥ 2 and

, and the random vector of spacings from X by

We start giving conditions for the multivariate stochastic order of random vectors of spacings from GOS. As we will see later, the following result is given under weaker conditions than those that follow from Theorem 4.10 of [4].

Theorem 4.1: Let X and Y be r.v.'s with distribution functions F and G, respectively, and let

be random vectors of generalized order statistics based on F and G, respectively. If

then

Proof: We proceed, as in Theorem 3.1, to construct two random variables

on the same probability space, such that

.

Let us consider a random vector V = (V₁,V₂,…,V_n) such that V =_st X. Then, we define

and

Clearly,

and

therefore,

.

Now, since F ≤_disp G and V_j−1 ≤ V_j,

and, therefore,

. █

If we denote by r and s the failure rates of X and Y, respectively, then the conditions given in Theorem 4.10 of [4], translated to this context are r(t) ≥ s(t + x), t,x ≥ 0. Now, from Proposition 4.2 of [4], this condition is stronger than the dispersive order of X and Y. By the preservation of the stochastic order under marginalization, we obtain Theorem 2.7 by Kamps [20, p.183].

Next, we proceed to give a result for the stochastic comparison of univariate spacings with gaps and from random vectors of GOS with different dimensions.

In order to state the result, we need first the following two technical results. Let X be a random variable with continuous distribution function F and let

be a random vector of generalized order statistics based on F. Let us denote the spacing between pth and qth coordinates from X by

for p < q. Then, we have the following result.

Lemma 4.2: Let X be a random variable with continuous distribution function F, let

be a random vector of generalized order statistics based on F, and let m_r = m ≥ −1 for all r = 1,…,n − 1. Then, we have the following:

(i) If X has a decreasing failure rate (DFR), then

is stochastically increasing in

.

(ii)

for all y and for all p ≥ i and q − j ≥ p − i ≥ m − n.

Proof: (i) We consider first the case m ≠ − 1. After some computations, we have that the survival function of

at a point x is given by

where

with

Clearly, H_p,q(t) is an increasing function and F(x + y)/F(y) is increasing in y if X is DFR, and, therefore, M_p,q(x|y) is increasing in y.

For m = −1, the proof follows taking limits when m → −1. It is not hard to prove that

converges in distribution to

when m → −1. By the preservation of the stochastic order by the convergence in distribution, we get part (i) when m = −1.

(ii) Let us consider p ≥ i and q − j ≥ p − i ≥ m − n. In this case, it can be easily checked that

, where

denotes the uniform generalized order statistics. The proof is easy and can be obtained by showing, instead of the stochastic order, the likelihood ratio order. Now, by the preservation of the stochastic order by increasing transformations, we get

for all y and for all p ≥ i and q − j ≥ p − i ≥ m − n. █

Now, we recall the following result.

Proposition 4.3: Let

be two families of random variables. Let Θ₁ and Θ₂ be two random variables with supports in

. Let Y₁ and Y₂ be two random variables such that Y_i =_st X_i(Θ_i), i = 1,2. Sufficient conditions for the implication

to be true for every such Θ₁ and Θ₂ are

and E [φ(X₁(θ))] or E [φ(X₂(θ))] or both are increasing in θ for every increasing function φ.

Belzunce and Shaked [7] prove similar results for the (increasing) convex and concave order and mention that these results can be generalized to the (increasing) convex s-orders described in Denuit, Lefèvre, and Shaked [12]. In particular, when s = 1, we get the stochastic order and, therefore, the previous result.

Let X and Y be r.v.'s with continuous distribution functions F and G, respectively, and let

be random vectors of generalized order statistics based on F and G, respectively. Let us denote

for 0 ≤ i < j ≤ n and

for 0 ≤ i < j ≤ l. Then, we have the following result.

Theorem 4.4: Under the above notation, if either X or Y or both are DFR and X ≤_hr Y, then

Proof: We give the proof for the case when F is DFR; in the other case, the proof is similar.

By Theorem 4.1 and the preservation under convolution of the stochastic order, we get that S_q,p:l ≤_st V_q,p:l and, therefore, we just need to prove

Now, consider

. Then S_j,i:n =_st X₁(Θ₁) and S_q,p:l =_st X₂(Θ₂). Now from our Lemma 4.2 and Theorem 2.6 in Kamps [20, p.182], these random variables satisfy the conditions of Proposition 4.3 and, therefore, S_j,i:n ≤_st S_q,p:l whenever p ≥ i and q − j ≥ p − i ≥ l − n. █

Now, we consider the multivariate likelihood ratio order of random vectors of spacings from GOS.

Theorem 4.5: Let X and Y be r.v.'s with distribution functions F and G, density functions f and g, and failure rates r and s, respectively, and let

be random vectors of generalized order statistics based on F and G, respectively. If (a) X ≤_lr Y, m_j ≥ 0 for all j = 1,…,n − 1 and if f or g or both are logconvex, or (b) X ≤_hr Y, m_j ≥ −1 for all j = 1,…,n − 1, r and f or s and g are logconvex, and s/r is increasing, then

Proof: (a) First, we note that the multivariate density functions of S_X and S_Y are given respectively by

and

for x_i ≥ 0, for all i = 1,…,n.

Let us suppose that f is logconvex (and, in fact, F is also logconvex); the proof in the other case is similar. Under this assumption, we have the following inequality:

Now, given that g/f and G/F are increasing functions and m_j ≥ 0 and from the previous inequality, we get the following inequality;

and, therefore, the result.

(b) In this case, we write the multivariate density functions of S_X and S_Y respectively by

for x_i ≥ 0, for all i = 1,…,n. Now, the proof follows as in the previous case. █

To finish this section, we consider again the situation of generalized order statistics based on the same distribution but with different parameters. First, we give a result for the multivariate stochastic order, which follows from Theorem 4.10 in [4].

Theorem 4.6: Let

be random vectors of generalized order statistics based on the same distribution F and with parameters k and m_i, i = 1,…,n − 1, and k′ and m_i′, i = 1,…,n − 1, respectively. If F has a decreasing failure rate and k ≥ k′ and m_i ≥ m_i′ for all i = 1,…,n − 1, then X ≤_st X′.

For the multivariate likelihood ratio order, it is possible to get a result under stronger conditions, which improves the conditions that follow from Theorem 4.11 in [4].

Theorem 4.7: Let

be random vectors of generalized order statistics based on the same distribution F and with parameters k and m_i, i = 1,…,n − 1, and k′ and m_i′, i = 1,…,n − 1, respectively. If X has a decreasing and logconvex failure rate and k ≥ k′ and m_i ≥ m_i′ for all i = 1,…,n − 1, then X ≤_lr X′.

Proof: The proof follows the steps similar to the proof of Theorem 4.5. █

Again, as in Section 3, we can combine Theorems 4.1 and 4.5 with Theorems 4.6 and 4.7 to get stochastic comparisons of random vectors of spacings of GOS based on different distributions and different parameters.

5. APPLICATIONS

In this section, we apply the previous results to several models of random vectors with ordered components. As a particular case, we obtain several results given by other authors and we also give some new results. As we have mentioned in Section 1, most of the results given by other authors for generalized order statistics are given under the assumption that m_i = m for all i = 1,…,n, which is not valid for some models, as we will see in this section.

5.3. k-Record Values

A generalization of the previous model is the case in which

, obtaining what is called k-records (see [20, Sect.1.5] for references).

Let

be a sequence of i.i.d r.v.'s with distribution function F and let

. Denoting the sequence of kth largest order statistics by

the r.v.'s given by

are called the record times and the quantities

are called kth records or k-records.

For this model, there are no results about stochastic comparisons, at least as far as we know; thus, from the results given in the previous sections, we get a great variety of results in this context, which generalize the results given for the usual record values (k = 1).

5.4. Order Statistics Under Multivariate Imperfect Repair

An interesting model contained in the model of generalized order statistics is the model of order statistics under multivariate imperfect repair. We consider the following minimal repair policy introduced by Shaked and Shanthikumar [30]. n items start to function at the same time 0. Upon failure, an item undergoes a repair. If i items (i = 0,1,…,n − 1) have already been scrapped, then, with probability p_i+1, the repair is unsuccessful and the item is scrapped, and with probability 1 − p_i+1, the repair is successful and minimal.

Let us now consider n items with i.i.d. random lifetimes T₁,….,T_n, with the same distribution F and density function f. Let (T₍₁₎,…,T_(n)) be the ordered random lifetimes resulting from T₁,….,T_n under a minimal repair policy as above; the joint multivariate density function of (T₍₁₎,…,T_(n)) is given by

This density function is a particular case of the density of generalized order statistics based on F with parameters k = p_n and m_j = (n − j + 1)p_j − (n − j)p_j+1 − 1. Therefore, we can apply previous results to get stochastic comparisons for this model. For example, from Theorem 3.8, we get as a particular case Theorem 1 given by Hu [15]. Theorems 4.6 and 4.7 can be applied to provide stochastic comparisons of spacings for this model.

If we choose F(x) = 1 − e^−x, x > 0, then the joint density of the corresponding generalized order statistics is the density of Weinman's multivariate exponential distribution. For this model, it is not required that p_i be a probability; it can be any positive real number. Again, we can apply the previous results for this model.

Recall also that Theorem 3.8 and Theorems 4.6 and 4.7 and their consequences can be applied to compare different models (e.g., order statistics from i.i.d. r.v.'s and order statistics under multivariate imperfect repair).