1. INTRODUCTION
Generalized order statistics (GOSs) have been of interest during the last few years because they are more flexible in statistical modeling and inference (see Cramer and Kamps [6] and references therein). The concept of GOSs was introduced by Kamps [11,12] as a unified approach to a variety of models of ordered random variables (r.v.s), which was defined through uniform GOSs.
Definition 1.1: (see Kamps [11]): Let
, be parameters such that γr,n = k + n − r + Mr ≥ 1 for all r = 1,…, n − 1, and let
if n ≥ 2 (
arbitrary if n = 1). If the r.v.'s
, possess a joint density of the form
on the cone 0 ≤ u1 ≤ u2 ≤ ··· ≤ un < 1 of
, then they are called uniform GOSs. Now let F be an arbitrary distribution function. The r.v.'s,
are called the GOSs based on F, where F−1 is the inverse of F defined by F−1(u) = sup{x : F(x) ≤ u} for u ∈ [0,1]. In the particular case m1 = ··· = mn−1 = m, the above r.v.'s are denoted by U(r,n,m,k) and X(r,n,m,k), r = 1,…, n, respectively.
Throughout this article, we consider the special case of GOSs: m1 = ··· = mn−1 = m, in which the marginal distribution and density functions of the rth GOS have closed forms, and γr,n = k + (n − r)(m + 1) for each r.
Ordinary order statistics of a random sample from a distribution F are a particular case of GOSs when k = 1 and mr = 0 for all r = 1,…, n − 1. When k = 1 and mr = −1 for all r = 1,…, n − 1, then we get the first n record values from a sequence of independent and identically distributed (i.i.d.) r.v.'s with distribution F. Choosing the parameters appropriately, several other models of ordered r.v.'s are seen to be particular cases (see Kamps [12], Balakrishnan, Cramer, and Kamps [2], and Belzunce, Mercader, and Ruiz [5]).
Let X and Y be two nonnegative r.v.'s with associated distribution functions F and G, and let X(r,n,m,k) and Y(r,n,m,k), r = 1,…, n, be GOSs based on F and G, respectively. For a given positive integer p, p ≤ n, let
denote respectively the p-spacings and the normalizing (simple) spacings of the GOSs {X(i,n,m,k), i = 1,…, n}. Here, X(0,n,m,k) ≡ 0. For p = 1, 1-spacings are simple spacings in the literature. Similarly, define
Stochastic comparisons of ordinary order statistics, record values, and their spacings have been studied extensively by many authors during the last 10 years. Because of the similarity of some structural properties among ordinary order statistics, record values, and GOSs, it is natural and interesting to study stochastic properties of GOSs and their spacings. Some recent works on this subject are by Franco, Ruiz, and Ruiz [7], Khaledi [15], Hu and Zhuang [8,9,10], Khaledi and Kochar [17], and Belzunce et al. [5]. For the comparison of spacings of GOSs from two samples, Belzunce et al. [5] in their Theorem 4.4 proved that if X ≤hr Y and m ≥ −1 and if either X or Y is DFR (decreasing failure rate), then
where ≤hr and ≤st denote the hazard rate and the usual stochastic orders, respectively (the formal definitions of the stochastic orders and of aging notion mentioned in this section can be found in Section 2). Franco et al. [7] proved that if X ≤lr Y and m ≥ 0 and if either X or Y is DFR, then
and
The purpose of this article is to investigate conditions on the distributions and the parameters on which the GOSs are based to establish relationship (5) in the likelihood ratio order and to obtain (6) and (7) for m ≥ −1. We thus extend some recent results on ordinary order statistics and record values. This article is a continuation of Hu and Zhuang [10] which is devoted to stochastic properties of p-spacings of GOSs from one sample. In Section 2, we recall the definitions of some stochastic orders and of some aging notions and we give some useful lemmas, which will be used in Section 3, where the main results are given.
Throughout, the terms “increasing” and “decreasing” mean “nondecreasing” and “nonincreasing,” respectively. All ratios are well defined, and a/0 is understood to be ∞ whenever a > 0. All integrals and expectations are implicitly assumed to exist whenever they are written.
2. PRELIMINARIES
2.1. Stochastic Orders and Aging Notions
Some stochastic orders and aging notions that will be used in this article are recalled in the following two definitions.
Definition 2.1: Let X and Y be two r.v.'s with respective survival functions F and G. We say that X is smaller than Y
- in the usual stochastic order, denoted by X ≤st Y, if F(t) ≤ G(t) for all t
- in the hazard rate order, denoted by X ≤hr Y, if G(t)/F(t) is increasing in t
- in the likelihood ratio order, denoted by X ≤lr Y, if X and Y have respective density functions (or mass functions) f and g and if g(t)/f (t) is increasing in t
- in the dispersive order, denoted by X ≤disp Y, if F−1(β) − F−1(α) ≤ G−1(β) − G−1(α) whenever 0 ≤ α < β ≤ 1.
The relationships among these orders are shown in the following (see Shaked and Shanthikumar [22] and Müller and Stoyan [20]):
Definition 2.2: Let X be a nonnegative r.v. with distribution function F. X or F is said to be
- ILR (increasing likelihood ratio) [DLR (decreasing
likelihood ratio)] if its density function f (x)
exists and is logconcave [logconvex] in
- IFR (increasing failure rate) [DFR (decreasing failure
rate)] if F(x) is logconcave
[logconvex] in
.
Remark 2.1: Bagai and Kochar [1] proved that if X ≤hr Y and either X or Y is DFR, then X ≤disp Y.
Remark 2.2: If f is logconcave, then F and F are also logconcave (see Barlow and Proschan [3, p.77]). If f is logconvex, then F is also logconvex while F is logconcave (see Sengupta and Nanda [21]). Therefore, ILR ⇒ IFR and DLR ⇒ DFR.
2.2. Some Useful Lemmas
The following lemmas are useful in deriving the main results of this article.
Lemma 2.1(Misra and van der Meulen [19]): Let Θ be a subset of the real line
and let X be a nonnegative r.v. having a distribution function belonging to the family
, which satisfies
Let Ψ(x,θ) be a real-valued function defined on
. Then
(i)
is increasing in θ if Ψ(x,θ) is increasing in θ and increasing [decreasing] in x;
(ii)
is decreasing in θ if Ψ(x,θ) is decreasing in θ and decreasing [increasing] in x.
To state the next lemmas, we define the function
, by
Lemma 2.2: Let X be a nonnegative r.v. with distribution function F(·) and hazard rate function λ(·). If X is DLR and m ≥ 0, or if X is DLR, λ(x) is logconvex in
and m ∈ [−1,0), then
is increasing in u
for fixed x2 > x1 ≥ 0.
Proof: Denote by lm(x) the derivative of δm(F(x)). Then
Since the logconvexity of f implies the logconvexity of F, it follows that lm(x) is logconvex in
under the assumption of the lemma. Observe that
where 1A is the indicator function of set A, and that the logconvexity of lm(x) in
is equivalent to lm(u + v) is TP2 in
. For the definition of TP2, one can refer to Karlin [13]. Here it should be mentioned that the logconvexity of a function h does not in general imply that h(x + y) is TP2 in
, where
are some subsets of
, if h has a special support of the form [a,∞) with a ≠ 0. Also, 1{0≤v≤x} is TP2 in
. By applying the basic composition formula to (9) (see Karlin [13, p.17]) or by Lemma 2 in Karlin and Proschan [14], we conclude that δm(F(x + u)) − δm(F(x)) is TP2 in
. This means that ψx1,x2(u) is increasing in
for 0 ≤ x1 < x2. █
Lemma 2.3: Let X and Y be two nonnegative r.v.'s with respective distribution functions F(·) and G(·) and hazard rate functions λ(·) and κ(·). If X ≤lr Y and m ≥ 0, or if X ≤hr Y, κ(x)/λ(x) is increasing in
, and m ∈ [−1,0), then
(i) δm(G(x))/δm(F(x)) is increasing in
(ii)
is increasing in
.
Proof: We only give the proof of part (ii) since part (i) follows from part (ii) directly by noting that δm(G(x))/δm(F(x)) = φ(x,0).
Let h1(x) and h2(x) denote the respective derivatives of δm(F(x)) and δm(G(x)), and define
An argument similar to that in the proof of Lemma 2.2 yields that
It is well known that if X ≤hr Y and κ(x)/λ(x) is increasing, then X ≤lr Y (see Belzunce, Lillo, Ruiz, and Shaked [4, Lemma3.5]). Since the likelihood ratio order implies the hazard rate order, it follows that, under the assumptions of the lemma,
is increasing in
; that is, hi(w) is TP2 in
. It can be checked that 1{u≤w≤x+u} is both TP2 in
for each
, and TP2 in
for each
. Again, by applying the basic composition formula to (10) or by Lemma 2 in Karlin and Proschan [14], we get that Ψ(i,x,u) is TP2 in
for each
, and TP2 in
for each
. Therefore, φ(x,u) is increasing in
. █
At the end of this section, we give two examples of r.v.'s satisfying the assumptions in Lemmas 2.2 and 2.3.
Example 2.1: For α ∈ (0,1], let Xα be a r.v. with the following Weibull distribution function:
The corresponding hazard rate and density functions of Xα are respectively given by
It can be checked that λα(x) and fα(x) are both logconvex in
for α ∈ (0,1].
Example 2.2: Consider a r.v. X with the density function f (x) given by
where θ ∈ (0,1) and β > α > 0. Since the logconvexity is closed under mixtures (see Barlow and Proschan [3, p.102]), it follows that X is DLR, and hence X is DFR. Now let Y be an exponential r.v. with parameter α. Then it is easy to see that X ≤lr Y, κ(t)/λ(t) is increasing in
, and both g and κ are logconvex.
3. MAIN RESULTS
Throughout this section, let X and Y be two nonnegative r.v.'s with associated distribution functions F and G, densities f and g, and hazard rate functions λ and κ, respectively. Let X(r,n,m,k) and Y(r,n,m,k), r = 1,…, n, be GOSs based on F and G, respectively.
From Lemma 3.3 of Kamps [12], it follows that for each r = 1,…, n, the marginal density function of the rth GOS X(r,n,m,k) based on F is given by
where
is defined by (8). Let fr,n(p)(x), Fr,n(p)(x), and Fr,n(p)(x) denote the respective density, distribution, and survival functions of Vr,n(p), r = 1,…, n − p + 1. Clearly, f1,n(p)(x) = fX(p,n,m,k)(x) given by Eq. (11). From Lemma 3.5 of Kamps [12], it follows that
and, hence,
for r = 2,…, n − p + 1 and x ≥ 0.
Similarly, let gr,n(p)(x), Gr,n(p)(x), and Gr,n(p)(x) denote the respective density, distribution, and survival functions of Wr,n(p), r = 1,…, n − p + 1.
3.1. Likelihood Ratio Ordering
Theorem 3.1: Let X and Y be two nonnegative r.v.'s with corresponding distribution functions F and G and hazard rate functions λ and κ, and let X(r,n,m,k) and Y(r,n,m,k), r = 1,…, n, be GOSs based on F and G, respectively. Then
if either of the following assumptions [(A1) or (A2)] holds:
(A1) X ≤lr Y, either X or Y is DLR, and m ≥ 0
(A2) X ≤hr Y, κ(x)/λ(x) is increasing, either X is DLR and λ is logconvex or Y is DLR and κ is logconvex, and m ∈ [−1,0).
Proof: To prove (14), it suffices to prove that for fixed r = 1,…, n − p + 1,
is increasing in
.
For r = 1, it follows from (11) that
is increasing in θ since γ1,n ≥ 1, Assumption (A2) implies X ≤lr Y, and X ≤lr Y implies X ≤hr Y.
For r = 2,…, n − p + 1, it follows from (12) that
where
and the nonnegative r.v. U1 has a distribution function belonging to the family
with corresponding densities given by
here d1(θ) is the normalizing constant. Now consider three cases.
Case 1: Suppose that X ≤lr Y, X is DLR, and m ≥ 0. It is seen that
- Ψ1(u,θ) is increasing in
by using Lemma 2.3 and the fact that X ≤lr Y implies X ≤hr Y
- H1(·|θ1) ≤lr H1(·|θ2) for θ2 ≥ θ1 ≥ 0 by using Lemma 2.2 and the fact that the logconvexity of f implies the logconvexity of F (see the proof of Theorem 4.1 in Hu and Zhuang [10]).
Then, by Lemma 2.1(i), we conclude that Δ1(θ) is increasing in
.
Case 2: Suppose that X ≤lr Y, Y is DLR, and m ≥ 0. An argument similar to the preceding paragraphs can establish that Δ1−1(θ) = fr,n(p)(θ)/gr,n(p)(θ) is decreasing in
by Lemma 2.1(ii).
Case 3: Suppose assumption (A2) holds. Then rewrite Ψ1(u,θ) and h1(u|θ) as follows:
The rest of the proof is similar to those of Cases 1 and 2 and hence omitted. This completes the proof of the theorem. █
Remark 3.1: Belzunce et al. [5] in their Theorem 4.5 established that under the same assumptions as Theorem 3.1,
where ≤lr denotes the multivariate likelihood ratio order (see Shaked and Shanthikumar [22, Sect.4.E]). Since the multivariate likelihood ratio order is closed under marginalization (see [22, Thm.4.E.3(b)]), it follows from (15) that
Franco et al. [7] proved (16) under the assumption (A1). Our Theorem 3.1 generalizes (16) from simple spacings to general p-spacings.
Remark 3.2: Theorem 4.1 in Hu and Zhuang [10] states that if X is DLR and m ≥ 0, then
However, the proofs of Theorem 4.1 in Hu and Zhuang [10] are also valid for the case when X is DLR, its hazard rate function λ(x) is also logconvex in
, and m ∈ [−1,0) by using Lemma 2.2. Therefore, from (17)–(19) we have the following corollary.
Corollary 3.1: Let X be a nonnegative r.v. with distribution function F and hazard rate function λ, and let X(r,n,m,k), r = 1,…, n, be GOSs based on F. If X is DLR and m ≥ 0, or if X is DLR, λ(x) is logconvex in
, and m ∈ [−1,0), then
Combining Corollary 3.1 with Theorem 3.1, we have the following.
Corollary 3.2: Under the same assumptions as those of Theorem 3.1,
By choosing m = −1 and m = 0 in Corollaries 3.1 and 3.2, we get the following interesting results concerning the likelihood ratio orderings of record values and ordinary order statistics.
Corollary 3.3: Let XL(1) and XL(2),… be record values based on a sequence of i.i.d. nonnegative r.v.'s with distribution function F. If X is DLR and its hazard rate λ(t) is logconvex, then
where XL(0) = 0.
It is worthwhile pointing out that if the hazard rate function λ of F is logconcave, then
(see Corollary 4.2 in Hu and Zhuang [10]).
Corollary 3.4: Let {XL(n),n ≥ 1} and {YL(n),n ≥ 1} be record values based on two sequences of i.i.d. nonnegative r.v.'s with respective distribution functions F and G. If X ≤hr Y and κ(x)/λ(x) is increasing in x and if either X is DLR and λ(·) is logconvex or Y is DLR and κ(·) is logconvex, then
where XL(0) = YL(0) = 0.
Note that the epoch times of a nonhomogeneous Poisson process with intensity function λ(t) are the record values of a sequence of i.i.d. nonnegative r.v.'s with the hazard rate being λ(t), where
for all
. Observing this fact, the special case of Corollary 3.4 with p = 1 is Corollary 4.7 in Belzunce et al. [4].
Corollary 3.5: Let X1:n ≤ X2 : n ≤ ··· ≤ Xn:n and Y1:n ≤ Y2:n ≤ ··· ≤ Yn:n denote the corresponding ordinary order statistics of two random samples of size n from distribution functions F and G, respectively. Define
If X ≤lr Y and if either X or Y is DLR, then
where X0:n = Y0,n ≡ 0.
Kochar [18] obtained Corollary 3.5 for the special case p = 1. In fact, he established the multivariate likelihood ratio ordering (15) between the random vectors of simple spacings of ordinary order statistics from two samples under the assumption that X ≤lr Y and either X or Y is DLR. Xu and Li [23] proved (20) under the assumption that X ≤lr Y and X is DLR.
3.2. Hazard Rate Ordering
In Theorem 3.1, if, instead, either X or Y is assumed to be DFR, then the result for simple spacings can be weakened from the likelihood ratio order to the hazard rate order (see Theorem 3.2). It is still an open problem whether the hazard rate orderings between the general p-spacings Vr,n(p) and Wr,n(p) hold under the same assumptions as those of Theorem 3.2.
Theorem 3.2: Let X and Y be two nonnegative r.v.'s with corresponding distribution functions F and G and hazard rate functions λ and κ, and let X(r,n,m,k) and Y(r,n,m,k), r = 1,…, n, be GOSs based on F and G, respectively. Then
if either one of the following assumptions [(A3) and (A4)] holds:
(A3) X ≤lr Y, either X or Y is DFR, and m ≥ 0.
(A4) X ≤hr Y, κ(x)/λ(x) is increasing, either X or Y is DFR and m ∈ [−1,0).
Proof: We give the proof of the case that m > −1; the proof of the case m = −1 follows from the closure property of the hazard rate order under weak convergence. For r = 1, the desired result (21) follows from Theorem 3.3 in Hu and Zhuang [9] or Theorem 3.3 in Franco et al. [7].
Now consider the case r = 2,…, n, and let λr,n(1)(·) and κr,n(1)(·) denote the hazard rate functions of Vr,n(1) and Wr,n(1), respectively. From (12) and (13), we get that
and
where the nonnegative r.v.'s U1 and U3 have distribution functions belonging to the families
, respectively, with corresponding densities given by
here d2(x) and d3(x) are the normalizing constants. It can be checked that under assumption (A3) or (A4),
is increasing in
for each fixed x by Lemma 2.3; that is, U2 ≤lr U3 for each
. Note that X ≤hr Y implies κ(t) ≤ λ(t) for all
. Therefore, if X is DFR, then
and if Y is DFR, then
This completes the proof of the theorem. █
Corollary 4.3 in Hu and Zhuang [10] states that if X is DFR and m ≥ −1, then
Combining this with Theorem 3.2, we have the following.
Corollary 3.6: Under the same assumptions as those in Theorem 3.2,
It is known that simple spacings of GOSs from a DFR distribution are also DFR; see Kamps [11, Thm.1.15] or Hu and Zhuang [10, Thm.3.1(1)]. Combining this result and Remark 2.1 with Corollary 3.6, we get the next corollary.
Corollary 3.7: Under the same assumptions as those in Theorem 3.2,
Remark 3.3: Franco et al. [7] proved Theorem 3.2 and Corollaries 3.6 and 3.7 for GOSs under assumption (A3) by using a different method.
Theorem 2.1 of Kochar [18] on ordinary order statistics is the special case of Theorem 3.2 by choosing k = 1 and m = 0. Choosing k = 1 and m = −1 in Corollaries 3.6 and 3.7, we get the next result concerning stochastic comparisons of simple spacings of record values in the hazard rate and the dispersive orders, which was proved by Belzunce et al. [4, Thm.4.4] under the additional assumption that both X and Y are DFR.
Corollary 3.8: Let {XL(n),n ≥ 1} and {YL(n),n ≥ 1} be as defined in Corollary 3.4. If X ≤hr Y and κ(x)/λ(x) is increasing in x and if either X or Y is DFR, then
where XL(0) = YL(0) = 0.
Franco et al. [7] also proved that
under assumption (A3), which contains Theorem 2.2 and Corollary 2.2 in Khaledi and Kochar [16] on ordinary order statistics as its special case. We close this section with a comparison result of normalizing spacings of GOSs in the hazard rate and dispersive orders under assumption (A4).
Theorem 3.3: Let X(r,n,m,k), Y(r,n,m,k), and assumptions (A3) and (A4) be as defined in Theorem 3.2. If either assumption (A3) or (A4) is satisfied, then (24) holds.
Proof: We only give the proof of the hazard rate order in (24) under assumption (A4). Assume that s ≥ r and s − r ≥ l − n. Then γr,n ≥ γs,l. Let
denote the hazard rate functions of
, respectively.
For r = 2,…, n, it follows from (22) and (23) that
where U2 is defined in the proof of Theorem 3.2 and U4 has a distribution function belonging to the families
with corresponding densities given by
here d4(x) is the normalizing constant. Then
is increasing in u for each fixed
by Lemma 2.3 and the fact that δm(·) is increasing; that is, U2 ≤lr U4 for each x. A similar argument to that in the proof of Theorem 3.2 yields that
for each
.
For r = 1, it follows from (11) that
If X is DFR, then
and if Y is DFR, then
This means that (24) holds for r = 1. Therefore, we complete the proof. █
Acknowledgments
This work was supported by the Program for New Century Excellent Talents in University (No. NCET-04-0569), a PhD Program Foundation of the Ministry of Education of China, and two grants from USTC and the Chinese Academy of Sciences
