1. INTRODUCTION
Sequential order statistics have been introduced by Kamps [13,14] as an extension of (ordinary) order statistics in order to model sequential k-out-of-n systems, where the failures of components possibly affect remaining ones. A k-out-of-n system is a system with n components that functions if and only if at least k of the components function. The lifetime of a k-out-of-n system is the same as that of the (n − k + 1)st ordinary order statistic of a set of n independent random variables X1,X2,…, Xn, where Xi denotes the lifetime of the ith component. In the k-out-of-n system, the failure of any component does not affect the remaining ones. Thus, as a more flexible model, sequential k-out-of-n systems are more applicable to practical situations. A formal definition of sequential order statistics is given in Section 2.
The concept of generalized order statistics was also introduced by Kamps [13,14] as a unified approach to a variety of models of ordered random variables. Choosing the parameters appropriately, several other models of ordered random variables are seen to be particular cases. One may refer to Kamps [14] for ordinary order statistics, record values, k-record values, and Pfeifers records, to Balakrishnan, Cramer, and Kamps [2] for progressive type II censored order statistics, and to Belzunce, Mercader, and Ruiz [6] and references therein for order statistics under multivariate imperfect repair.
In the past 10 years, a great number of articles have dealt with stochastic comparisons of order statistics and their spacings. It is natural to obtain stochastic comparisons of generalized order statistics and their spacings by analogy with ordinary order statistics. Some recent articles on the subject are by Franco, Ruiz, and Ruiz [9], Korwar [17], Belzunce, Mercader, and Ruiz [5,6], Hu and Zhuang [10,11], Khaledi and Kochar [16], Khaledi [15], and Hu and Zhuang [12].
Sequential order statistics contain generalized order statistics as their special model. It is interesting to study stochastic properties of sequential order statistics. The purpose of this article is to present some results on multivariate stochastic comparisons of sequential order statistics. In Section 2 we recall the definitions of sequential order statistics and of some multivariate and univariate stochastic orders. The main results are given in Section 3. More precisely, let (X1,n*,…, Xn,n*) and (X1,n+1*,…, Xn+1,n+1*) be sequential order statistics based on {F1,…, Fn} and {F1,…, Fn+1}, respectively. We investigate conditions on {F1,…, Fn+1} under which one of the following relationships holds:
where ≤* is any one of the multivariate likelihood ratio order (≤lr), the multivariate hazard rate order (≤hr), and the usual multivariate stochastic order (≤st). In Section 4 some applications of the main results are given. Some discussions on multivariate comparisons of sequential order statistics from two samples are presented in Section 5.
Throughout this article, the terms “increasing” and “decreasing” mean “nondecreasing” and “nonincreasing,” respectively; a/0 is understood to be ∞ whenever a > 0. For any distribution function F, F = 1 − F denotes its survival function. All random variables are restricted to be nonnegative. Also, we denote by [X|A] any random variable whose distribution is the conditional distribution of X given event A.
2. PRELIMINARIES
2.1. Sequential Order Statistics
Sequential order statistics are defined by means of a triangular scheme of random variables where the rth line contains n − r + 1 random variables with distribution function Fi, i = 1,…, n.
Definition 2.1 (Kamps [13, p.27]): Let F1,…, Fn be continuous distribution functions with F1−1(1) ≤ ··· ≤ Fn−1(1) and let {Yr,n(j),1 ≤ j ≤ n − r + 1} be a sequence of independent and identically distributed random variables each distributed according to Fr, where r = 1,…, n.
Let X1,n(j) = Y1,n(j), 1 ≤ j ≤ n, and denote
For r = 2,…, n, define Xr,n(j) = Fr−1{Fr(Yr,n(j))[1 − Fr(Xr−1,n*)] + Fr(Xr−1,n*)} and denote
Then X1,n*,…, Xn,n* are called sequential order statistics based on {F1,…, Fn}.
If F1,…, Fn are absolutely continuous with densities f1,…, fn, respectively, then the joint density of the first r sequential order statistics (X1,n*,…, Xr,n*) is given by
where x1 < x2 < ··· < xr, 1 ≤ r ≤ n, and
. In particular, the joint density of (X1,n*,…, Xn,n*) is given by
Moreover, sequential order statistics form a Markov chain with transition probabilities
(See Kamps [14, p.29] and Cramer and Kamps [7].)
Cramer and Kamps [8] obtained the following recursion formula for the marginal distribution functions FX1,n*,…, FXn,n* of the sequential order statistics:
where
and r = 2,…, n.
Nonhomogeneous pure birth (NHPB) processes are another useful models of ordered random variables, which arise naturally in many applications of probability (see Belzunce, Lillo, Ruiz, and Shaked [4] and references therein). A counting process {N(t),t ≥ 0} is a NHPB process with intensity functions {rk(·),k ≥ 0} if the following hold:
(a) {N(t),t ≥ 0} has the Markov property.
(b) Pr{N(t + Δt) = k + 1|N(t) = k} = rk(t)Δt + ○ (Δt) for k ≥ 1.
(c) Pr{N(t + Δt) > k + 1|N(t) = k} = ○ (Δt) for k ≥ 1;
the rk′'s are nonnegative functions that satisfy
Condition (2.5) ensures that, with probability 1, the process has a jump after any time point t. In a distributional theoretical sense, there is one-to-one correspondence between sequential order statistics and the first n epoch times of a NHPB process, which is stated in the following proposition.
Proposition 2.1: Let λ1(·),…, λn(·) be the hazard rate functions of distribution functions F1,…, Fn, respectively, and let (X1,n*,…, Xn,n*) be the sequential order statistics based on distributions {F1,…, Fn}. Define
(rk(·) can be chosen arbitrarily for k > n such that (2.5) is satisfied) and denote by {Tk,k ≥ 1} the epoch times of a NHPB process with intensity functions {rk(·),k ≥ 0}. Then
where
means equality in distribution.
Generalized order statistics are contained in the model of sequential order statistics. Let (
) be generalized order statistics based on the distribution function F with parameters
and γi,n > 0 for i = 1,…, n. Then (
) can be regarded as the sequential order statistics (X1,n*,…, Xn,n*) based on distributions {F1,…, Fn}, where
that is, F1,…, Fn satisfy the proportional hazard model. This relationship will be used later. For a formal definition of generalized order statistics and their properties, one refer to Kamps [13,14].
2.2. Some Stochastic Orders
Some multivariate stochastic orders that will be used in this article are recalled in the sequel (see Shaked and Shanthikumar [18]).
First, we recall the definition of the usual multivariate stochastic order. Let X and Y be two n-dimensional random vectors. We say that X is less than Y in the usual multivariate stochastic order, denote by X ≤st Y, if
for all increasing function φ such that the expectations exist. If X and Y are univariate random variables with distribution functions F and G, respectively, then X ≤st Y if and only if F(t) ≤ G(t) for all t.
Next, for the definition of the multivariate hazard rate order, let X and Y be two n-dimensional nonnegative random vectors with multivariate conditional hazard rate functions η·|·(·|·) and λ·|·(·|·) as defined in Shaked and Shanthikumar [18, Sect.4.C.1]. For any vector
and any subset I = {i1,…, ik} ⊆ {1,…, n}, let xI = (xi1,…, xik) and I be the complement of I in {1,…, n}. For a random vector X, the interpretation of XI is similar. Finally, e and 0 denote the vector of 1's and 0's; the dimensions of e and 0 can be determined from the context. Then X is said to be less than Y in the multivariate hazard rate order, denoted by X ≤hr Y, if
whenever I ∩ J = Ø, 0 ≤ sI ≤ tI ≤ ue, and 0 ≤ sJ ≤ ue. In the univariate case, X ≤hr Y if and only if G(x)/F(x) is increasing in t.
Finally, we recall the definition of the multivariate likelihood ratio order. Let X and Y be two n-dimensional random vectors with density functions f and g, respectively. We say that X is less than Y in the multivariate likelihood ratio order, denoted by X ≤lr Y, if
for all (x1,…, xn) and (y1,…, yn) in
, where
In the univariate case, X ≤lr Y if and only if g(x)/f (x) is increasing in x. In the slightly more general case, when X and Y are nonnegative, some of the Xi's might be identically zero and the joint distribution of the rest is absolutely continuous or discrete. Suppose that X1,…, Xm are those that are identically zero for some 0 < m < n. Also, let f denote the joint density of (Xm+1,…, Xn). In that case, we denote X ≤lr Y, if
for all (xm+1,…, xn) and (y1,…, yn).
It is well known that the multivariate likelihood ratio order implies the multivariate hazard rate order, which, in turn, implies the usual multivariate stochastic order; that is,
The multivariate orders ≤lr and ≤st are closed under marginalization. However, the multivariate order ≤hr is not closed under marginalization. Such a closure property is very useful for us to establish the univariate comparison result. Sometimes it is difficult for us to obtain an explicit formula of the density functions of Xi and Yi, but it is easy to get the joint density functions of X and Y. To obtain Xi ≤lr Yi for each i, we can establish X ≤lr Y first. This idea is illustrated in Subsection 3.1.
If the distribution functions of X and Y are F and G, respectively, then X ≤* Y is sometimes denoted by F ≤* G, where ≤* is any one of the above orders.
3. MAIN RESULTS
In this section, we investigate conditions on the underlying distribution functions on which the sequential order statistics are based, to obtain stochastic comparisons of sequential order statistics in the multivariate likelihood ratio, the multivariate hazard rate, and the usual multivariate stochastic orders.
3.1. Multivariate Likelihood Ratio Ordering
Theorem 3.1: Let F1,…, Fn+1 be absolutely continuous distribution functions and let (X1,n*,…, Xn,n*) and (X1,n+1*,…, Xn+1,n+1*) be sequential order statistics based on {F1,…, Fn} and {F1, …,Fn+1}, respectively. If F1 ≤hr F2 ≤hr ··· ≤hr Fn, then
Proof: Let f1,…, fn+1 denote the density functions of F1,…, Fn+1, respectively. By the definition of the order ≤lr, (2.1), and (2.2), it suffices to verify that, for x1 ≤ ··· ≤ xn and y1 ≤ ··· ≤ yn,
Let E1 = {1 ≤ i ≤ n − 1 : xi ≥ yi}. Then, (3.2) reduces to
which follows from F1 ≤hr ··· ≤hr Fn. This completes the proof. █
Theorem 3.2: Let F1,…, Fn be absolutely continuous distribution functions of nonnegative random variables with hazard rates γ1(·),…, γn(·), respectively. Let (X1,n*,…, Xn,n*) be sequential order statistics based on {F1,…, Fn}. If F1 ≤lr F2 ≤lr ··· ≤lr Fn and if
then
Proof: Let f1,…, fn denote the density functions of F1,…, Fn, respectively. Since the multivariate likelihood ratio order is closed under marginalization, it is sufficient to verify that
By (2.1), (2.2), and (2.8), we have to prove that, for all x2 ≤ ··· ≤ xn and y1 ≤ ··· ≤ yn,
Note that condition (3.3) can be written as
Let E2 = {i : xi+1 ≥ yi+1,i = 1,…, n − 2}. Then, (3.5) reduces to
which follows from (3.6), (2.9), and F1 ≤lr F2 ≤lr ··· ≤lr Fn. This completes the proof. █
Theorem 3.3: Let F1,…, Fn+1 be absolutely continuous distribution functions of nonnegative random variables with hazard rates γ1(·),…, γn+1(·), respectively, and let (X1,n*,…, Xn,n*) and (X1,n+1*,…, Xn+1,n+1*) be sequential order statistics based on {F1,…, Fn} and {F1,…, Fn+1}, respectively. If F1 ≤lr F2 ≤lr ··· ≤lr Fn+1 and if
then
Proof: Let f1,…, fn+1 denote the density functions of F1,…, Fn+1, respectively. Since the multivariate likelihood ratio order is closed under marginalization, it is sufficient to verify that
Let E3 = {i : xi+1 ≥ yi+1,i = 1,…, n − 1}. We have to prove that, for all x2 ≤ ··· ≤ xn+1 and y1 ≤ ··· ≤ yn+1,
which follows from F1 ≤lr F2 ≤lr ··· ≤lr Fn+1 and condition (3.7). This completes the proof. █
Since the multivariate likelihood ratio order is closed under marginalization, we get the following result as a corollary of Theorems 3.1–3.3.
Corollary 3.1: Let (X1,n*,…, Xn,n*) and (X1,n+1*,…, Xn+1,n+1*) be the same as in Theorem 3.3.
(1) If F1 ≤hr F2 ≤hr ··· ≤hr Fn, then Xi,n+1* ≤lr Xi,n* for i = 1,…, n.
(2) If F1 ≤lr F2 ≤lr ··· ≤lr Fn and if (3.3) holds, then Xi,n* ≤lr Xi+1,n* for i = 1,…, n − 1.
(3) If F1 ≤lr F2 ≤lr ··· ≤lr Fn+1 and if (3.7) holds, then Xi,n* ≤lr Xi+1,n+1* for i = 1,…, n.
Let (
) and (
) be generalized order statistics based on the distribution function F, where
. Because of the relationship between sequential and generalized order statistics (see (2.6)), (
) and (
) are the sequential order statistics based on {F1,…, Fn} and {G1,…,Gn+1}, respectively, where
and
It is seen that Fi and Gi are not, in general, the same. Thus, parts b and c of Theorem 3.1 in Hu and Zhuang [10] cannot be deduced from parts b and c of Corollary 3.1. Furthermore, part a of Theorem 3.1 in Hu and Zhuang [10] cannot be deduced from part a of Corollary 3.1 either.
3.2. Multivariate Hazard Rate Ordering
We now proceed to stochastic comparisons of sequential order statistics in the multivariate hazard rate order.
Theorem 3.4: Let F1,…, Fn+1 be absolutely continuous distribution functions of nonnegative random variables and let (X1,n*,…, Xn,n*) and (X1,n+1*,…, Xn+1,n+1*) be sequential order statistics based on {F1,…, Fn} and {F1,…, Fn+1}, respectively. Then
Proof: The proof is similar to that of Theorem 3.3 in Belzunce et al. [4]. Let λ1,…, λn+1 denote the hazard rate functions of F1,…, Fn+1, respectively. Denote by η·|·(·|·) and λ·|·(·|·) the multivariate condition hazard rate functions associated with (X1,n+1*,…, Xn,n+1*) and (X1,n*,…, Xn,n*), respectively. We have to verify (2.7).
Observe that X1,n* ≤ ··· ≤ Xn,n* together with (2.3). Then the explicit expression of λk|I(u|tI) is given by
where I must be of the form I = {1,…, r} for some r. Similarly, in ηk|I∪J(u|sI∪J) of (2.7), we must have I = {1,…, r} and J = {r + 1,…, m} for some m ≥ r, or J = Ø (i.e., m = r). Thus,
where I = {1,…, r} and J = {r + 1,…, m} for 0 ≤ r ≤ n − 1.
If m > r, then
thus, (2.7) holds. If m = r (i.e., J = Ø), then
thus, (2.7) also holds. This completes the proof. █
Theorem 3.5: Let F1,…, Fn be absolutely continuous distribution functions of nonnegative random variables and let (X1,n*,…, Xn,n*) be sequential order statistics based on {F1,…, Fn}. If F1 ≤hr F2 ≤hr ··· ≤hr Fn, then
Proof: Let λ1,…, λn denote the hazard rate functions of F1,…, Fn, respectively, and denote by η·|·(·|·) and λ·|·(·|·) the multivariate condition hazard rate functions associated with (X1,n*,…, Xn−1,n*) and (X2,n*,…, Xn,n*), respectively. Suppose that F1 ≤hr ··· ≤hr Fn. It suffices to prove that
where I = {1,…, r} and J = {r + 1,…, m}, 0 ≤ r ≤ n − 2 and sI ≤ tI ≤ ue, sJ ≤ ue.
If m > r ≥ 1, then
thus, (3.13) holds. If m = r ≥ 1 (i.e., J = Ø), then
thus, (3.13) holds in this case too.
If r = 0, (3.14) can be rewritten as ηk|J(u|sJ) ≥ λk|Ø(u|tØ), which is equivalent to
since λk|Ø(u|tØ) = 0 for k ≥ 2. Thus, it remains to prove that X1,n* ≤hr X2,n*. In fact, it follows from (2.4) that
Since F1 ≤hr F2, we get that
is increasing in t. Thus, X1,n* ≤hr X2,n*. This completes the proof. █
Remark 3.1: The assumption that F1 ≤hr F2 ≤hr ··· ≤hr Fn in Theorem 3.5 can be weakened to be
as can be seen from its proof.
Theorem 3.6: Let (X1,n*,…, Xn,n*) and (X1,n+1*,…, Xn+1,n+1*) be the same as in Theorem 3.4. If F1 ≤hr F2 ≤hr ··· ≤hr Fn+1, then
Proof: Let λ1,…, λn+1 be the hazard rate functions of F1,…, Fn+1, respectively, and denote by η·|·(·|·) and λ·|·(·|·) the multivariate condition hazard rate functions associated with (X1,n*,…, Xn,n*) and (X2,n+1*,…, Xn+1,n+1*), respectively. Suppose that F1 ≤hr ··· ≤hr Fn+1. We have to prove that (3.13) and (3.14) hold whenever I = {1,…, r} and J = {r + 1,…, m}, 0 ≤ r ≤ n − 1, sI ≤ tI ≤ ue, sJ ≤ ue.
If m > r ≥ 1, then
If m = r ≥ 1 (i.e., J = Ø), then
Thus, (3.13) holds.
For r = 0, to prove (3.14), we need to prove that X1,n* ≤hr X2,n+1*. In fact, it follows from (2.4) that FX1,n*(t) = [F1(t)]n and
Then F1 ≤hr F2 implies that
is increasing in t. Thus, X1,n* ≤hr X2,n+1*. This completes the proof. █
The multivariate hazard rate order is not closed under marginalization. Thus, we could not establish univariate hazard rate ordering between sequential order statistics from Theorems 3.4–3.6 under the conditions stated there.
3.3. Multivariate Usual Stochastic Ordering
Note that the usual multivariate stochastic order is closed under weak convergence. By (2.9), we get the following result as a corollary of Theorems 3.4 without the assumption that F1,…, Fn+1 are absolutely continuous.
Corollary 3.2: Let (X1,n*,…, Xn,n*) and (X1,n+1*,…, Xn+1,n+1*) be sequential order statistics based on {F1,…, Fn} and {F1,…, Fn+1}, respectively. Then
To prove the next two theorems, we need one useful lemma, which gives a sufficient condition of the usual multivariate stochastic ordering between two random vectors. First, we recall one notion of positive dependence from Barlow and Proschan [3]. A random vector X = (X1,…, Xn) is said to be conditionally increasing in sequence (CIS) if for i = 2,…, n,
whenever xj ≤ xj* for j = 1,…, i.
Lemma 3.1 (Shaked & Shanthikumar [18, Thm.4.B.4]): Let X and Y be two n-dimensional random vectors. If X1 ≤st Y1 and, for i = 2,…, n,
and if either X or Y is CIS, then X ≤st Y.
Theorem 3.7: Let (X1,n*,…, Xn,n*) be sequential order statistics based on {F1,…, Fn}. If F2 ≤hr ··· ≤hr Fn, then
Proof: Note that sequential order statistics have the Markov property. It follows from (2.3) that, for i = 1,…, n − 1,
is increasing with respect to (x1,…, xi) in the sense of the usual stochastic order. Then (X1,n*,…, Xn,n*) is CIS. Since X1,n* ≤ X2,n*, by Lemma 3.1, we have to prove that
that is,
which follows trivially from the assumption F2 ≤hr ··· ≤hr Fn. This completes the proof. █
Remark 3.2: The result of Theorem 3.7 is also true if the assumption F2 ≤hr ··· ≤hr Fn is replaced by
Theorem 3.8: Let (X1,n*,…, Xn,n*) and (X1,n+1*,…, Xn+1,n+1*) be the same as in Corollary 3.2. If F1 ≤st F2 and F2 ≤hr ··· ≤hr Fn+1, then
Proof: By a similar argument to that of Theorem 3.7, we have to prove that X1,n* ≤st X2,n+1* and
Now, (3.23) is equivalent to
which follows trivially from the assumption F2 ≤hr ··· ≤hr Fn. On the other hand, it follows from (3.18) and F1 ≤st F2 that
that is, X1,n* ≤st X2,n+1*. This completes the proof. █
Since the usual multivariate stochastic order is closed under marginalization, we can get univariate comparison results of sequential order statistics from (3.19)–(3.22).
4. SOME APPLICATIONS
In this section, some applications of the main results in Section 3 are presented.
4.1. Ordinary Order Statistics
Ordinary order statistics based on distribution F correspond to the generalized order statistics with γi,n = n − i + 1 for i = 1,…, n. Then the Fi's and Gi's defined in (3.9) and (3.10) are all F. Applying Theorems 3.1–3.3, we get the following result.
Corollary 4.1: Let (X1:n,…, Xn:n) and (X1:n+1,…, Xn+1:n+1) be ordinary order statistics based on an absolutely continuous distribution F. Then
4.2. Sequential Order Statistics [Model (2.6)]
Let {F1,F2,…, Fn+1} satisfy the proportional hazard model (2.6); that is,
If F is absolutely continuous, then Fi ≤lr Fj if and only if αi ≥ αj. On the other hand, conditions (3.3) and (3.7) hold if and only if
Thus, we have the following result as a corollary of Theorems 3.1–3.3.
Corollary 4.2: Let {F1,…, Fn+1} be as defined above with F absolutely continuous and let (X1,n*,…, Xn,n*) and (X1,n+1*,…, Xn+1,n+1*) be sequential order statistics based on {F1,…, Fn} and {F1,…, Fn+1}, respectively. If α1 ≥ α2 ≥ ··· ≥ αn+1 and (4.1) holds, then (3.1), (3.4), and (3.8) hold.
4.3. Progressive Type II Censored Order Statistics
In a progressive type II censoring scheme, N units are placed on a lifetime test. The failure times are described by independent and identically distributed random variables with a common distribution F. A number n (n ≤ N) of units are observed to fail. A predetermined number Ri of surviving units at the time of the ith failure are randomly selected and removed from further testing. Thus,
units are progressively censored; hence,
. The n observed failure times are called progressive type II censored order statistics based on F, denoted by X1:n,NR ≤ X2:n,NR ≤ ··· ≤ Xn:n,NR, where R = (R1,…, Rn). For details on the model of progressive type II censoring, we refer to Korwar [17] and Balakrishnan and Aggarwala [1].
Progressive type II censored order statistics based on F correspond to the generalized order statistics based on distributions {F1,…, Fn} defined in (2.6). The parameter αi,n ≡ αi,R is given by
For censoring policy R with Ri = τ − ic for some τ > 0 and c ≥ 0, we have the following result.
Corollary 4.3: Let R = (R1,…, Rn) and R′ = (R1′,…, Rn′) be two censoring policies with
where τ and τ′ are positive and c and c′ are nonnegative. If F is absolutely continuous, then the following hold:
(a) (X1:n,NR,X2:n,NR,…, Xn−1:n,NR) ≤lr (X2:n,NR,X3:n,NR,…, Xn:n,NR).
(b) For τ < τ′,
Proof: Let (X1:n,NR,X2:n,NR,…, Xn:n,NR) and (X1:n,NR′,X2:n,NR′,…, Xn:n,NR′) be the sequential order statistics based on {F1,…, Fn} and {G1,…,Gn}, respectively, where
(a) First,
It is clear that 2αi,R = αi+1,R + αi−1,R for i = 2,…, n − 1 and that αi,R is decreasing in i. Then the conditions in Theorem 3.2 are fulfilled. Thus, the desired result of part a now follows.
(b) Similarly,
Since N = n(τ + 1) − ½n(n + 1)c = n(τ′ + 1) − ½n(n + 1)c′, it follows that
and, hence, c < c′ when τ < τ′. Then
which implies Fi ≤lr Gi. Thus, part b follows from Theorem 5.2. █
4.4. NHPB Processes
Observing the connection between sequential order statistics and epoch times of NHPB processes, we obtain conditions under which epoch times of a NHPB process can be compared in the sense of the multivariate hazard rate order. This result is a corollary of Theorem 3.5 and Proposition 2.1.
Corollary 4.4: Let {Tn,n ≥ 1} denote the epoch times of a NHPB process {N(t),t ≥ 0} with intensity functions {ri(t),i ≥ 0}. If ri(t) is decreasing in i for each t, then
Proof: Define
Since
, λi(t) can be regarded as the hazard rate function of some distribution function Fi. Let (X1,n*,…, Xn,n*) be the sequential order statistics based on {F1,…, Fn}. Since ri(t) is decreasing in i for each t, condition (3.16) is fulfilled. By Proposition 2.1, we get
Thus, the desired result now follows from Theorem 3.5 and Remark 3.1. █
5. DISCUSSION
Belzunce et al. [4] described various conditions on the parameters of pairs of NHPB processes under which the corresponding epoch times or interepoch intervals are ordered in various senses. Applying Proposition 2.1, we can identify conditions that enable one to compare sequential order statistics from two samples.
Let (X1,n*,…, Xn,n*) and (Y1,n*,…,Yn,n*) respectively be sequential order statistics based on {F1,…, Fn} and {G1,…,Gn} with
Denote by λ1(t),…, λn(t) and μ1(t),…, μn(t) the hazard rates of F1,…, Fn and G1,…,Gn, respectively.
From Theorems 3.11–3.13 and 4.10 of Belzunce et al. [4], we have the following results.
Theorem 5.1: If F1 ≤st G1 and Fi ≤hr Gi for i = 2,…, n, then
In particular, Xi,n* ≤st Yi,n* for i = 1,…, n.
Theorem 5.2: If Fi ≤hr Gi for i = 1,…, n, then
Theorem 5.3: If Fi ≤lr Gi for i = 1,…, n and
then
In particular, Xi,n* ≤lr Yi,n* for i = 1,…, n.
Theorem 5.4: If F1 ≤st G1 and
then
It is worth mentioning that in Theorem 5.3 we weaken the conditions of Theorem 3.13 in Belzunce et al. [4]. Their proof is also valid with minor modification. Because of limited page space, we do not list the comparison results on spacing vectors of sequential order statistics from two samples.
Finally, we give a example, which satisfies all conditions in Theorems 5.1–5.4.
Example 5.1: Let the hazard rate functions of F1,…, Fn and G1,…,Gn respectively are given by
where c and η are positive constants, λ(u) ≥ η for all u, and λ(u) is decreasing. Obviously, λi(t) ≥ μi(t), and μi(t)/λi(t) is increasing in t. By Lemma 3.5 of Belzunce et al. [4], we get that Fi ≤lr Gi for i = 1,…, n. Thus, all conditions in Theorems 5.1–5.4 are satisfied.
Acknowledgments
This work was supported by Program for New Century Excellent Talents in University (NCET-04-0569) and two grants from USTC and Chinese Academy of Sciences.
