2. PRELIMINARIES

2.1. Generalized Order Statistics

Uniform generalized order statistics are defined via some joint density function on a cone of the

. Generalized order statistics based on an arbitrary distribution function F are defined by means of the inverse function of F.

Definition 2.1 (see Kamps [17]): Let

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

. If the rv's

, possess a joint density of the form

on the cone 0 ≤ u₁ ≤ u₂ ≤ ··· ≤ u_n < 1 of

, then they are called uniform generalized order statistics. Now, let F be an arbitrary distribution function. The rv's,

are called the generalized order statistics (GOSs, for short) 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 = ··· = m_n−1 = m, the above rv's are denoted by U(r,n,m,k) and X(r,n,m,k), r = 1,…,n, respectively.

Ordinary order statistics of a random sample from a distribution F are a particular case of GOSs when k = 1 and m_r = 0 for all r = 1,…,n − 1. When k = 1 and m_r = −1 for all r = 1,…,n − 1, then we get the first n record values from a sequence of rv's with distribution F. Choosing the parameters appropriately, several other models of ordered rv's are seen to be particular cases.

It is well known that GOSs from a continuous distribution form a Markov chain with transition probabilities

Throughout this article, we consider the special case of GOSs (m₁ = ··· = m_n−1 = m) in which the marginal distribution and density functions of the rth GOS have closed forms. Stochastic properties of p-spacings of general GOSs are still under our investigation. If F is absolutely continuous with density function f, Lemma 3.3 of Kamps [18] states 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 the marginal density function of

, and M_n = 0. Here, the function

, is defined by

It is easy to see that

and that g_m(x) is nonnegative and increasing in x ∈ [0,1) for each

.

Let F be a distribution function of some nonnegative rv. For a given positive integer p, p ≤ n, let

denote the p-spacings of the GOSs X(1,n,m,k) ≤ X(2,n,m,k) ≤ ··· ≤ X(n,n,m,k). Here, X(0,n,m,k) ≡ 0. For p = 1, 1-spacings are simple spacings in the literature. Let f_r,n^(p)(x), F_r,n^(p)(x), and F_r,n^(p)(x) denote the respective density, distribution, and survival functions of D_r,n^(p), r = 1,…,n − p + 1. Clearly, f_1,n^(p)(x) = f_X(p,n,m,k)(x) given by (2.2). From Lemma 3.5 of Kamps [18], it follows that

and, hence,

for r = 2,…,n − p + 1 and x ≥ 0.

The next proposition states that under suitable restrictions on the parameters of GOSs, the conditional distribution of one GOS given another lower-indexed one based on a continuous distribution has the same distribution as some GOS based on the truncated parent distribution. We denote by [Y |A] any rv whose distribution is the conditional distribution of Y given event A.

Proposition 2.1: Let X(r,n,m,k), r = 1,…,n, be GOSs based on a continuous distribution function F. For each u ∈ Supp(F), the support of F, denote F_u(x) = F(u + x)/F(u) for x ≥ 0. Then

where p ≥ 1 and r = 2,…,n − p + 1, and X^u(p,n − r + 1,m,k) is a GOS based on F_u.

Proof: The proof of the case m ≠ −1 is the immediate consequence of Theorem 3.2 in Keseling [19]. A limiting argument can establish the case of m = −1. █

For the sake of brevity, the constant n in γ_r,n and c_r,n is suppressed when there is no confusion in the following context.

2.2. 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 respectively.

Definition 2.2: Let X and Y be two rv's with respective survival functions F and G. We say that X is smaller than Y if the following hold:

• In the usual stochastic order, denoted by X ≤_st Y or F ≤_st G, if F(t) ≤ G(t) for all t or, equivalently,
  
  for all increasing functions φ
• In the hazard rate order, denoted by X ≤_hr Y or F ≤_hr G, if G(t)/F(t) is increasing in t for which the ratio is well defined
• In the likelihood ratio order, denoted by X ≤_lr Y or F ≤_lr G, if X and Y have respective density functions (or mass functions) f and g and if g(t)/f (t) is increasing in t for which the ratio is well defined.

The relationships among these orders are shown in the following diagram (see Shaked and Shanthikumar [27], and Müller and Stoyan [23]):

Definition 2.3: Let X be a nonnegative rv 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
  
• DRHR (decreasing reversed hazard rate) [IRHR (increasing reversed hazard rate)] if F(x) is logconcave [logconvex] in x ∈
  
  .

If f is logconcave, then F and F are also logconcave (see, e.g., Chandra and Roy [8], Barlow and Proschan [4, p.77]). If f is logconvex, then F is also logconvex while F is logconcave (see Sengupta and Nanda [26]). Furthermore, Block, Savits, and Singh [7] proved that if F is logconvex then F is logconcave. Therefore,

2.3. Some Useful Lemmas

The following lemmas are useful in deriving the main results of this article. Lemma 2.1 is due to Misra and van der Meulen [22], hereafter referred to as MM. Lemma 2.2 is the extension of Lemma 2.1 in MM.

Lemma 2.1: Let Θ be a subset of the real line

and let X be a nonnegative rv having a distribution function belonging to the family

, which satisfies that

Let Ψ(x,θ) be a real-valued function defined on

. Then the following hold:

Lemma 2.2: Let X be a nonnegative rv with distribution function F. If X is DLR [ILR] and if m ≥ 0, then the following hold:

Proof: Denote L_m(x) = g_m(F(x)). Observe that

holds for each

, where f is the density of F. Since the logconvexity [logconcavity] of f (x) implies that F(x) is logconvex [logconcave] in x, it follows that

is also logconvex [logconcave] in x for m ≥ 0. The rest of the proof is the same as that of Lemma 2.1 in MM. █

Remark 2.1: For m = 0, Lemma 2.2 reduces to Lemma 2.1 in MM. It is worthwhile pointing out that Lemma 2.2 is, in general, not true when m < 0, as illustrated by the following counterexample: Let X be uniformly distributed on the interval (0,1). Then X is ILR. For each fixed u ∈ (0,1) and x ∈ (0,1 − u), we have

Observe that

for m ≠ −1 and m < 0, where

means equality in sign, and

Therefore, ψ_u(x) is not decreasing in x; that is, Lemma 2.2(i) is not true in this example. Similarly, it can be checked that Lemma 2.2(ii) does not hold in this example.

Remark 2.2: Observe that (2.7) holds for each m, and that (2.7) can be written as

where λ(t) = f (t)/F(t) is the failure rate function of F. Since the logconcavity of λ(t) implies that f (t) and −log F(t) are both logconcave (see Pellerey, Shaked, and Zinn [25, Appendix]), it follows that the integrand in (2.8) is logconcave when m ∈ [−1,0). Therefore, if λ(t) is logconcave and m ∈ [−1,0), then ψ_u(x) is decreasing in

for each fixed

is decreasing in

for fixed x₂ > x₁ > 0.

To state Lemma 2.4, we first recall the following Prekopa's theorem.

Lemma 2.3 (see Eaton [11, Thm. 5.1]): Suppose that

is a logconcave function and

is finite for each

. Then η(x) is logconcave in

.

Lemma 2.4: Let X be a nonnegative rv with distribution function F. If X is ILR and m ≥ 0, or if the failure rate function of F is logconcave and m ∈ [−1,0), then the following hold:

(i) g_m(F(x)) is logconcave in

₊.

(ii) g_m(F(x + u)) − g_m(F(u)) is logconcave in

.

Proof: The logconcavity of g_m(F(x)) follows from the proof of Lemma 2.2 and Remark 2.2. The same argument as that of the proof of Lemma 3.1 in MM yields part (ii) by applying Lemma 2.3. █

The next lemma gives conditions on the parameters to compare GOSs based on two different distributions in the hazard rate order, which follows from Theorem 3.6 in Franco et al. [12].

Lemma 2.5: Let X(r,n,m,k) and Y(r′,n′,m,k) be two GOSs based on distribution functions F and G, respectively. If F ≤_hr G and m ≥ −1, then

3. PRESERVATION OF LOGCONVEXITY AND LOGCONCAVITY OF p-SPACINGS

For ordinary order statistics, Barlow and Proschan [3] established that if the parent distribution F is DFR, then the corresponding simple spacings D_k,n⁽¹⁾, k = 1,…,n, are also DFR, and MM [12] proved that if F is DLR, then the D_k,n⁽¹⁾ are also DLR. For record values, Gupta and Kirmani [14] noticed that if F is DFR, then the D_k,n⁽¹⁾ are also DFR. In the following theorem, we extend these results to the simple spacings of GOSs.

Theorem 3.1: Let X(r,n,m,k), r = 1,…,n, be GOSs based on distribution function F.

Proof:

where Φ_r−1,n,m,k(F(·)) is the distribution function of X(r − 1,n,m,k). The DFR property of F implies that [F(x + y)/F(y)]^γ_r is logconvex in x for each

. Since the logconvexity is closed under mixture (see Barlow and Proschan [4, p.103]), it follows from (3.1) that F_r,n⁽¹⁾(x) is logconvex in

. Hence, D_r,n⁽¹⁾ is DFR.

is increasing in

by using the DLR property of F, and the nonnegative rv U₁ has a distribution function belonging to the family

with corresponding densities given by

is increasing in

. Therefore, Δ₁(θ) is increasing in

by using Lemma 2.1. This completes the proof. █

From the proof of Theorem 3.1(1), we know that the conclusion is also true for the simple spacings of the GOSs without restriction m₁ = ··· = m_n−1 = m. For p ≥ 2, the p-spacings do not preserve the DFR or DLR property of the parent distribution (see MM [22, Remark 3.1]).

MM also established that the general p-spacings (1 ≤ p ≤ n) of ordinary order statistics preserve the ILR property of the parent distribution. This result is generalized from ordinary order statistics to GOSs under some restriction on the parameters in the following theorem.

Theorem 3.2: Let X(r,n,m,k), r = 1,…,n, be GOSs based on a distribution function F. If F is ILR and m ≥ 0, then D_r,n^(p), r = 1,…,n − p + 1, are also ILR for p = 1,…,n.

Proof: For r = 1, the result follows from (2.2) and (2.3) by using Lemma 2.4(i). For r ≥ 2, a similar argument to that in the proof of Theorem 3.3 in MM yields the desired result by applying Lemmas 2.3 and 2.4 in (2.5). █

Notice that the epoch times of a nonhomogeneous Poisson process with intensity function λ(t) are the record values of a sequence of independent and identically distributed (i.i.d.) nonnegative rv's with the failure rate being λ(t), where

for all

. From Theorem 3.1 in Pellerey et al. [25], we know that if the hazard rate function λ(t) of the underlying distribution is logconcave, then the simple spacings of record values are ILR. This result can be extended to the general spacings of GOSs with m ∈ [−1,0) under the same condition.

Theorem 3.3: Let X(r,n,m,k), r = 1,…,n, be GOSs based on a distribution function F. If the hazard rate function λ(t) of F is logconcave and m ∈ [−1,0), then D_r,n^(p), r = 1,…,n − p + 1, are ILR for p = 1,…,n.

Proof: For r ≥ 2, rewrite (2.5) as

The desired result now follows from Remark 2.2 and Lemmas 2.3 and 2.4.

The proof of the case r = 1 is trivial. █

Remark 3.1: Choosing r = 1 in Theorem 3.3, we obtain that if the hazard rate function λ(t) of F is logconcave, then X(r,n,m,k), r = 1,…,n, are ILR for m ∈ [−1,0). Pellerey et al. [25] considered the special case m = −1 in their Corollary 2.2.

4. STOCHASTIC COMPARISONS BETWEEN p-SPACINGS

For ordinary order statistics, the following are comparison results for general p-spacings:

In this section, we investigate conditions on the parameters to extend the above comparison results (P₁)–(P₇) from ordinary order statistics to GOSs. Theorems 4.1 and 4.4 deal with the likelihood ratio ordering, whereas Theorems 4.2 and 4.3 deal with the hazard rate ordering.

Theorem 4.1: Let X(r,n,m,k), r = 1,…,n, be GOSs based on a distribution function F, where m ≥ 0. Then the following hold:

Proof:

is increasing in

. We consider two cases.

and the nonnegative rv U₂ has a distribution function belonging to the family

with corresponding densities given by

is increasing in

by using Lemma 2.2 and the fact that the logconvexity of f implies the logconvexity of F.

Then, by Lemma 2.1, we conclude that Δ₂(θ) is increasing in

.

is increasing in

by using Lemma 2.2. This completes the proof of part (a).

is decreasing in

. For r = 1, from (2.2), we get that

is decreasing in

. For r ≥ 2, from (2.5), we get that

is also decreasing in

by Lemma 2.1, where the nonnegative rv U₂ has a distribution function belonging to the family

. This completes the proof of part (b).

is increasing [decreasing] in

. For r = 1, from (2.2) and (2.5), we get that

is increasing [decreasing] in

by Lemma 2.2. For r ≥ 2, from (2.5), we get that

does not depend on θ and is increasing in

since γ_p+r,n+1 = k + (n + 1 − p − r)(m + 1) = γ_p+r−1,n, and U₂ has the distribution belonging to the family

. It can be checked that H₂(·|θ₁) ≤_lr [≥_lr] H₂(·|θ₂) for θ₂ ≥ θ₁ ≥ 0 if F is DLR [ILR]. Therefore, applying Lemma 2.1 yields that Δ₄(θ) is increasing [decreasing] in

. This completes the proof. █

Remark 4.1: In views of Remark 2.2, the proof of Theorem 4.1(c) is still valid for m ∈ [−1,0) under the stronger condition that the failure rate of F is logconcave. Therefore, for GOSs, if m ∈ [−1,0) and the failure rate of F is logconcave, then D_r,n^(p) ≥_lr D_r+1,n+1^(p) for r = 1,…,n − p + 1.

An immediate consequence of Theorem 4.1 is the following corollary.

Corollary 4.1: Let X(r,n,m,k), r = 1,…,n, be GOSs based on a distribution function F. If F is DLR and m ≥ 0, then

Choosing m = −1 in Remark 4.1, we have the following corollary.

Corollary 4.2: Let X_L(1),X_L(2),… be record values based on a sequence of i.i.d. rv's with distribution function F. If the failure rate λ(t) of F is logconcave, then

In Theorem 4.1, if, instead, F is assumed to be DFR [IFR], then the results can be weakened from the likelihood ratio order to the hazard rate order (see Theorems 4.2 and 4.3).

Theorem 4.2: Let X(r,n,m,k), r = 1,…,n, be GOSs based on a distribution function F. If F is DFR and m ≥ −1, then the following hold:

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. Let λ_r,n^(p)(t) denote the hazard rate function of D_r,n^(p), and set F_u(x) = F(u + x)/F(u) for x ≥ 0 and u ∈ Supp(F).

(a) First consider r ∈ Θ₁ ≡ {2,3,…,n − p + 1}. From (2.5) and (2.6), we get that

where

and the nonnegative rv U₃ has a distribution function belonging to the family

with corresponding densities given by

here, d₃(θ) is the normalizing constant. Observe the following:

where the nonnegative rv Z₁ has a distribution function belonging to the family

with corresponding densities given by

Again, applying Lemma 2.1 in (4.1) yields that λ_θ,n^(p)(x) is decreasing in θ ∈ Θ₁ for each fixed

. This means that D_r,n^(p) ≤_hr D_r+1,n^(p) for r ∈ Θ₁.

It remains to show that D_1,n^(p) ≤_hr D_2,n^(p). For this, consider

where the last equality follows from Proposition 2.1, and X^u(p,n − 1,m,k) is a GOS based on F_u. Since the DFR property of F, F ≤_hr F_u. By Lemma 2.5, we get that X(p,n,m,k) ≤_hr X^u(p,n − 1,m,k) for

, which, in turn, implies that the ratio in the integrand in (4.3) is increasing in

. Therefore, D_1,n^(p) ≤_hr D_2,n^(p). This completes the proof of part (a).

(b) For r = 1, the desired result D_1,n+1^(p) = X(p,n + 1,m,k) ≤_hr X(p,n,m,k) = D_1,n^(p) follows from Lemma 2.5. Now, consider r ≥ 2, and let p and r be fixed. From (2.5) and (2.6), the failure rate function of D_r,θ^(p) is given by

where

and the nonnegative rv U₅ has a distribution function belonging to the family

with corresponding densities given by

here, d₅(θ) is the normalizing constant. Observe that the following hold:

where Ψ₇(z,u) = z^m+1 is increasing in z ∈ (0,1), not depending on u, and the nonnegative rv Z₂ has a distribution function belonging to the family

with corresponding densities given by

Again, applying Lemma 2.1 in (4.4) yields that λ_r,θ^(p)(x) is increasing in θ ∈ Θ₂ for each fixed

. Therefore, D_r,n+1^(p) ≤_hr D_r,n^(p) for r ≥ 2. This completes the proof of the theorem. █

Remark 4.2: The reversed inequalities in Theorem 4.2 do not, in general, hold when F is IFR, as shown by a counterexample in Hu and Zhuang [16]. Hence, the inequalities in parts (a) and (b) of Theorem 4.1 cannot be reversed when F is ILR.

Remark 4.3: Theorem 4.2 is not true for m < −1, as illustrated by the following counterexample. Let X(r,n,m,k), r = 1,2,…,n, be GOSs based on the exponential distribution F(x) = 1 − e^−x, x ≥ 0, where m < −1. Then γ_r,n < γ_r+1,n for r = 1,…,n − p, and γ_r,n+1 < γ_r,n for r = 1,…,n − p + 1. Denote Y_r,n = γ_r,n D_r,n⁽¹⁾ for r = 1,…,n. From Theorem 3.10 in Kamps [18], we know that F is also the distribution function of Y_r,n. Hence,

Therefore, Theorem 4.2 cannot be true in this example (even in the usual stochastic order).

In the next theorem, there is no further restriction (like m ≥ −1) on the parameter m.

Theorem 4.3: Let X(r,n,m,k), r = 1,…,n, be GOSs based on a distribution function F. If F is DFR [IFR], then D_r,n^(p) ≤_hr [≥_hr] D_r+1,n+1^(p) for r = 1,…,n − p + 1.

Proof: We give the proof of the case m > −1; the proof of the case m < −1 is similar and the proof of the case m = −1 follows by a limiting argument. From (2.5) and (2.6), the failure rate function of D_r+θ,n+θ^(p), θ ∈ Θ₃ ≡ {0,1}, r = 1,…,n − p + 1, is given by

where

and the nonnegative rv U₆ has a distribution function belonging to the family

with corresponding densities given by

here, d₇(θ) is the normalizing constant, and we use the identity that γ_{r+p−1+θ,n+θ} = γ_r+p−1,n for θ ∈ Θ₃. Observe the following:

By Lemma 2.1, we obtain that λ_r+θ,n+θ^(p)(x) is decreasing [increasing] in θ ∈ Θ₃. So we prove the desired result for r ≥ 2.

It remains to verify that D_1,n^(p) ≤_hr D_2,n+1^(p). For this, consider

where the last equality follows from Proposition 2.1, and X^u(p,n,m,k) is a GOS based on F_u. Note that the DFR [IFR] property of F implies that F ≤_hr [≥_hr] F_u. By Lemma 2.5, we get that X(p,n,m,k) ≤_hr [≥_hr] X^u(p,n,m,k) for

, which, in turn, implies that the ratio in the integrand in (4.6) is increasing [decreasing] in

. Therefore, D_1,n^(p) ≤_hr [≥_hr]D_2,n+1^(p). This completes the proof of the theorem. █

In views of Theorems 4.2 and 4.3, we have the following two corollaries.

Corollary 4.3: Let X(r,n,m,k), r = 1,…,n, be GOSs based on a distribution function F. If F is DFR and m ≥ −1, then

Corollary 4.4: Let X_L(1),X_L(2),… be the same as in Corollary 4.2. If the failure rate λ(t) of F is decreasing [increasing], then

Finally, the following theorem extends (P7) from ordinary order statistics to GOSs.

Theorem 4.4: Let X(r,n,m,k), r = 1,…,n, be GOSs based on a distribution function F with failure rate function λ(t). If m ≥ 0 and F is ILR, or if m ∈ [−1,0) and λ(t) is logconcave, then

or, equivalently,

Proof: The equivalence of (4.7) and (4.8) is trivial. We give the proof of (4.7) only. It suffices to prove that for r = 2,…,n − p + 1,

is increasing in

and decreasing in

for m ≥ −1 by Lemma 2.4, and the rv U₂ is defined as in Theorem 4.2. Since H₂(·|θ₁) ≥_lr H₂(·|θ₁) for θ₂ > θ₁ ≥ 0, applying Lemma 2.1 in (4.9) yields that Λ_r(θ) is increasing in

.

which is decreasing in

by using Lemma 2.2 and Remark 2.2. This thus completes the proof. █

Theorem 4.4 cannot be, in general, true when F is DLR, as shown by Example 3.1 in Hu and Zhuang [16].

Choosing m = −1 in Theorem 4.4, we have the following corollary.

Corollary 4.5: Let X_L(1),X_L(2),… be the same as in Corollary 4.2. If the failure rate λ(t) of F is logconcave, then

In Corollaries 4.2, 4.4, and 4.5, {X_L(1),X_L(2),…} can be interpreted as the epoch times {T₁,T₂,…} of a nonhomogeneous Poisson process with intensity function λ(t) (see the paragraph before Theorem 3.3). For various comparison results of nonhomogeneous Poisson processes, one can refer to Belzunce, Lillo, Ruiz, and Shaked [5] and references therein.

It is still an open problem whether (P₈) holds for GOSs.