On stochastic properties of m-spacings

Journal of Statistical Planning andInference 115 (2003) 683– 697

www.elsevier.com/locate/jspi

On stochastic properties of m-spacingsNeeraj Misraa ; ∗, Edward C van der Meulenb

aDepartment of Mathematics, Indian Institute of Technology, Kanpur 208 016, IndiabDepartment of Mathematics, Katholieke University Leuven, B-3001 Heverlee, Belgium

Received 8 March 2001; accepted 16 February 2002

Abstract

Let X be a nonnegative and continuous random variable having the probability density func-tion (pdf) f(:). Let Xk:n (k = 1; 2; : : : ; n) denote the kth order statistic based on n independentobservations on X and, for a given positive integer m (6 n), let D(m)

k;n = Xk+m−1:n − Xk−1:n;k = 1; 2; : : : ; n − m + 1, denote the successive (overlapping) spacings of gap size m (to be re-ferred as m-spacings); here X0:n ≡ 0. It is shown that if f(:) is log convex, then the pdfof corresponding simple (gap size one) spacings D(1)

k;n; k = 1; 2; : : : ; n, are also log convex. Itis also shown that the m-spacings D(m)

k;n ; k = 1; 2; : : : ; n − m + 1, preserve the log concavity ofthe parent pdf f(:). Under the log convexity of the parent pdf f(:), we further show that, fork = 1; 2; : : : ; n − m; D(m)

k;n is smaller than D(m)k+1; n in the likelihood ratio ordering and that, for a

;xed 16 k6 n− m + 1 and n¿ k + m− 1; D(m)k;n+1 is smaller than D(m)

k;n in the likelihood ratioordering. Finally, we show that if X has a decreasing failure rate then, for k=1; 2; : : : ; n−m; D(m)

k;n

is smaller than D(m)k+1; n in the failure rate ordering and that, for a ;xed 16 k6 n − m + 1 and

n¿ k +m− 1; D(m)k;n+1 is smaller than D(m)

k;n in the failure rate ordering. c© 2002 Elsevier ScienceB.V. All rights reserved.

MSC: 62N05

Keywords: DFR; IFR; Likelihood ratio order; Failure (hazard) rate order; Log convex; Log concavedistribution; m-Spacings

1. Introduction

Let X be a nonnegative and continuous random variable (rv) having distributionfunction (df) F(x), probability density function (pdf) f(x) and survival function (sf)

∗ Corresponding author. Department of Statistics, West Virginia University, Hodges Hall, Morgantown,WV 26505, USA. Tel.: +1-304-293-3607/1055; fax: +1-304-293-2272.

E-mail address: [email protected] (N. Misra).

0378-3758/02/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S0 3 7 8 -3 7 5 8 (02)00157 -X

684 N. Misra, E. C van der Meulen / Journal of Statistical Planning and Inference 115 (2003) 683– 697

GF(x) = 1 − F(x). Let Xk:n (k = 1; 2; : : : ; n) denote the kth order statistic based on nindependent observations on X . For a given positive integer m (6 n) and X0:n ≡ 0, letD(m)

k;n = Xk+m−1:n − Xk−1:n; k = 1; 2; : : : n − m + 1, denote the successive (overlapping)sample spacings of gap size m, to be referred as m-spacings hereafter. Throughout, thespacings of gap size one will be referred as simple spacings and the random variablesDk;n = (n− k + 1)D(1)

k;n; k = 1; 2; : : : ; n, will be referred as normalized simple spacings.There is an extensive literature on the properties of sample spacings and their

usage in goodness-of-;t tests. For a basic work on spacings, one may refer to Pyke(1965, 1972). Goodness-of-;t tests based on spacings have been proposed, amongothers, by Kirmani and Alam (1974); Cressie (1976) and Dudewicz and van derMeulen (1981). Barlow and Proschan (1966) studied some reliability-theoretic proper-ties of the normalized simple spacings. Recently, there has been renewed interest in thereliability-theoretic properties of the sample spacings (cf. Pledger and Proschan, 1971;Kochar and Kirmani, 1995; Kirmani, 1996, 1998; Kochar and Korwar, 1996; Kocharand Rojo, 1996; Kochar, 1999; Khaledi and Kochar, 1999). However, all this workis restricted to the reliability-theoretic properties of the simple spacings, i.e. m = 1.For a class of goodness-of-;t tests based symmetrically on spacings, Rao and Kuo(1984) established that a goodness-of-;t test based on m-spacings (m¿ 2) is alwaysasymptotically superior to its analog based on simple spacings. This, along with thefact that an m-spacing is a sum of dependent random variables (simple spacings),motivates us to study the reliability-theoretic properties of m-spacings, for a generalm (16m6 n). Recently, Hu and Wei (2001) derived some stochastic comparisonresults for m-spacings (which they call as generalized spacings). In this paper, wewill derive some additional results on stochastic comparisons of m-spacings. Through-out this paper, the term increasing (decreasing) is used for monotone nondecreasing(nonincreasing).

A nonnegative and continuous rv Y , having df G(x), pdf g(x) and sf GG(x) is said tobe log concave (log convex) if log(g(x)) is concave (convex), or equivalently, for each�¿ 0; g(x + �)=g(x) is a decreasing (increasing) function of x∈ (0;∞). The rv Y issaid to be IFR (increasing failure rate) if its failure rate function rY (x) = g(x)= GG(x) isan increasing function of x∈ (0;∞). Similarly, the rv Y is said to be DFR (decreasingfailure rate) if the failure rate function rY (x) is a decreasing function of x∈ (0;∞).Here it must be noted that the log concavity (log convexity) of g implies that, foreach ;xed �¿ 0; GG(x + �)= GG(x) is a decreasing (increasing) function of x, which isequivalent to saying that GG(x) is log concave (log convex) or that Y is IFR (DFR) (cf.Barlow and Proschan, 1981). Moreover, if g(x) is log concave, then G(x) is also logconcave. If Z is another rv having df H (x), pdf h(x), sf GH (x) and failure rate functionrZ(x)=h(x)= GH (x), then Z is said to be stochastically larger than Y (written as Y 6st Z)if, for each x¿ 0; GH (x)¿ GG(x). The rv Z is said to be larger than Y in likelihood ratioordering (written as Y 6lr Z) if h(x)=g(x) is an increasing function of x. Similarly, therv Z is said to be larger than Y in failure rate ordering (written as Y 6fr Z) if, foreach x¿ 0; rY (x)¿ rZ(x) or equivalently GH (x)= GG(x) is an increasing function of x.It is well known that likelihood ratio ordering implies failure rate ordering, which inturn implies stochastic ordering. For the properties of these and other orderings, seeShaked and Shanthikumar, 1994.

N. Misra, E. C van der Meulen / Journal of Statistical Planning and Inference 115 (2003) 683– 697 685

In Section 2 of this paper, we introduce preliminaries, which are used in subsequentsections. In Section 3, we prove that the log convexity of the pdf f(:) is preservedby the simple spacings D(1)

k;n; k = 1; 2; : : : ; n (cf. Theorem 3.1), which is an extensionof a result by Barlow and Proschan (1966), who proved that simple spacings preservethe log concavity of the parent pdf. In Section 3, we also show that the m-spacingsD(m)

k;n ; k = 1; 2; : : : ; n − m + 1, preserve the log concavity of the parent pdf f(:) (cf.Theorem 3.3), which is a generalization of a result proved by Barlow and Proschan(1966) for m= 1. For normalized simple spacings Dk;n = (n−k + 1)D(1)

k;n; k = 1; 2; : : : ; n,Kochar and Kirmani (1995, p. 56, Remark 3) asserted that if X is log convex, thenDk;n6lr Dk+1; n; k =1; 2; : : : ; n−1. In Section 4, with the help of a counter example, wedemonstrate that this assertion is not true in general and modify this assertion by prov-

ing that if X is log convex, then D(m)k;n 6

lr D(m)k+1; n; k=1; 2; : : : ; n−m, and that, for a ;xed

k; D(m)k;n+16

lr D(m)k;n ; n¿ k + m − 1 (cf. Theorem 4.1). In particular, this demonstrates

that if X is log convex, then although normalized simple spacings may not be orderedin terms of likelihood ratio ordering, but simple spacings (non-normalized) are orderedin terms of likelihood ratio ordering. In Section 4, we also show that if X is DFR, thenD(m)

k;n 6fr D(m)

k+1; n; k=1; 2; : : : ; n−m, and that, for a ;xed k; D(m)k;n+16

fr D(m)k;n ; n¿ k+m−1

(cf. Theorem 4.2). In the process of deriving these results, we also establish someproperties of log concave and log convex distributions.

2. Preliminaries

Let fk;n(x) denote the pdf of the kth order statistic Xk:n; k = 1; 2; : : : ; n. Let f(m)k;n (x);

F (m)k;n (x); GF

(m)k;n (x) = 1 − F (m)

k;n (x) and r(m)k;n (x) = f(m)

k;n (x)= GF(m)k;n (x), respectively, denote the

pdf, the df, the sf, and the failure rate function of D(m)k;n ; k = 1; 2; : : : ; n−m + 1. Then,

for k = 2; 3; : : : ; n− m + 1 and x¿ 0

GF(m)k;n (x) =

∫ ∞

0P(Xk+m−1:n − Xk−1:n ¿x |Xk−1:n = u)fk−1; n(u) du:

From the Markovian property of the order statistics (cf. David, 1981), it is easy to;nd that the sf and the pdf of m-spacings are given by

GF(m)k;n (x) = Ck;m;n

∫ ∞

0

[ GF(x + u)GF(u)

]n−k−m+2

[∫ 1

0zn−k−m+1

(1 − z

GF(x + u)GF(u)

)m−1

dz

]fk−1; n(u) du;

x¿ 0; k = 2; 3; : : : ; n− m + 1; (2.1)

GF(m)1; n (x) = C1;m;n

∫ GF(x)

0zn−m(1 − z)m−1 dz; x¿ 0; (2.2)


f(m)k;n (x) = Ck;m;n

∫ ∞

0

[ GF(x + u)GF(u)

]n−k−m+1 [1 −

GF(x + u)GF(u)

]m−1

f(x + u)GF(u)

fk−1; n(u) du; x¿ 0; k = 2; 3; : : : ; n− m + 1; (2.3)

f(m)1; n (x) = C1;m;n(F(x))m−1( GF(x))n−m; f(x); x¿ 0; (2.4)

where, for k = 1; 2; : : : ; n− m + 1,

Ck;m;n =(n− k + 1)!

(n− k − m + 1)!(m− 1)!:

The following two lemmas will be useful in deriving the results of this paper. Lemma2.2 stated below is an obvious extension of a result given in Lehmann (1986, Problem15, p. 116) and therefore its proof is omitted.

Lemma 2.1. If X is log convex (log concave); then(i) For each 8xed u¿ 0,

u(x) =F(x + u) − F(u)

F(x)

is an increasing (a decreasing) function of x∈ (0;∞).(ii) For 8xed x2 ¿x1 ¿ 0,

�x(u) =F(x2 + u) − F(u)F(x1 + u) − F(u)

is an increasing (a decreasing) function of u∈ [0;∞); here x = (x1; x2)′.

Proof. (i) Since;

@@x

u(x) =F(x)f(x + u) − (F(x + u) − F(u))f(x)

(F(x))2 ;

it is enough to establish that

�1(x) = F(x)f(x + u) − (F(x + u) − F(u))f(x)¿ (6) 0; ∀x¿ 0: (2.5)

Since X is log convex (log concave); we have

f(x + u)f(y)¿ (6)f(y + u)f(x); ∀0¡y¡x; u¿ 0;

which on integrating y on both sides from 0 to x yields inequality (2.5).(ii) For ;xed x2 ¿x1 ¿ 0, we can write

�x(u) =

∫ x2

0 f(t + u) dt∫ x1

0 f(t + u) dt:


We are required to prove that, for u2 ¿u1 ¿ 0,

�2(x1; x2)¿ (6) 0;

where

�2(x1; x2) =∫ x2

0

∫ x1

0f(s+u1)f(t+u2) ds dt−

∫ x1

0

∫ x2

0f(s+u1)f(t+u2) ds dt:

Clearly,

�2(x1; x2) =∫ x2

x1

∫ x1


∫ x1

0

∫ x2

x1

f(s+u1)f(t+u2) ds dt

=∫ x2

x1

∫ x1


∫ x2

x1

∫ x1

0f(s+u2)f(t+u1) ds dt

=∫ x2

x1

∫ x1

0(f(s+u1)f(t+u2)−f(s+u2)f(t+u1)) ds dt:

Since logf(x) is convex (concave), for 0¡s¡x1 ¡t¡x2, the integrand in the aboveintegral is nonnegative (nonpositive), and therefore

�2(x1; x2)¿ (6) 0:

Hence, the result follows.

Remark 2.1. (i) It is well known in the literature (cf. Barlow and Proschan; 1981)that if X is log concave (i.e. log(f(x)) is concave); then F(x) and GF(x) are alsolog concave. Since; for each ;xed u¿ 0; F(u)=F(x) is a decreasing function of x;under the log concavity of f(x); Lemma 2.1(i) implies that; for each ;xed u¿ 0;F(x + u)=F(x) = u(x) + F(u)=F(x) is a decreasing function of x (i.e. F(x) is logconcave). Also; if X is log concave; then; on taking x2 ↑ ∞; Lemma 2.1(ii) impliesthat; for each ;xed x1 ¿ 0; GF(x1 + u)= GF(u) is a decreasing function of u (i.e. GF(x)is log concave). Thus; under the log concavity of f(x); Lemma 2.1 provides muchstronger results than those available in the literature.

(ii) Under the log concavity (log convexity) of f(x), on taking x2 ↑ ∞, Lemma2.1(ii) implies that, for each ;xed x1 ¿ 0; GF(x1 + u)= GF(u) is a decreasing (an increas-ing) function of u (i.e. X is IFR (DFR)).

Lemma 2.2. Let be a subset of the real line R and let X be a nonnegative rv havinga df belonging to the family P = {G(: | !); !∈ }; which is stochastically ordered inthe sense that; for !1; !2 ∈ and !1 ¡!2; we have GG(x | !1)6 (¿) GG(x | !2); ∀x¿ 0.Let "(x; !) be a real valued function de8ned on R× . Then

(i) E!("(X; !)) is an increasing function of !; provided "(x; !) is an increasingfunction of ! and increasing (decreasing) function of x.

(ii) E!("(X; !)) is a decreasing function of !; provided "(x; !) is a decreasing func-tion of ! and decreasing (increasing) function of x.


3. Preservation of log convexity and log concavity by m-Spacings

Barlow and Proschan (1966) established that if X is DFR, then the correspondingsimple spacings D(1)

k;n; k=1; 2; : : : ; n, are also DFR. In the following theorem, we extendthis result by establishing that simple spacings also preserve the log convexity of X .

Theorem 3.1. Suppose that X is log convex. Then D(1)k;n (k = 1; 2; : : : ; n) is also log

convex.

Proof. From (2.4); we have;

log(f(1)1; n (x)) = log(n) + (n− 1) log( GF(x)) + log(f(x)):

Since the log convexity of f(:) also implies the log convexity of GF(:); it follows thatD(1)

1; n is log convex. To establish the log convexity of f(1)k;n (·); for k=2; 3; : : : n; ;x �¿ 0

and consider

R1(!) =f(1)k;n (! + �)

f(1)k;n (!)

; !¿ 0:

From (2.3); for k = 2; 3; : : : ; n and !¿ 0; we get

R1(!) =

∫∞0 (

GF(!+�+u)GF(u)

)n−k f(!+�+u)GF(u)

fk−1; n(u) du∫∞0 (

GF(!+u)GF(u)

)n−k f(!+u)GF(u)

fk−1; n(u) du:

We are required to prove that R1(!) is an increasing function of !∈ [0;∞). To do this;consider a nonnegative rv U1 having a df belonging to the family P1={G1(: | !); !¿ 0};with corresponding pdfs given by

g1(u | !) = c1(!)( GF(! + u)

GF(u)

)n−kf(! + u)

GF(u)fk−1; n(u); u¿ 0; !¿ 0; (3.1)

where c1(!) is the normalizing constant.Then we can write,

R1(!) = E!("1(U1; !));

where

"1(u; !) =( GF(! + � + u)

GF(! + u)

)n−kf(! + � + u)

f(! + u); u¿ 0; !¿ 0:

The log convexity of f(:) implies that, for !2 ¿!1 ¿ 0, the functions GF(!2 + u)=GF(!1 +u) and f(!2 +u)=f(!1 +u) are increasing functions of u∈ [0;∞), and therefore,

the family of pdfs de;ned by (3.1) has the monotone likelihood ratio (mlr) property, i.e.for !2 ¿!1 ¿ 0; g1(u | !2)=g1(u | !1) is an increasing function of u. This, in turn, im-plies that, for each ;xed u¿ 0; GG1(u | !) is an increasing function of !. Also, it follows


that "1(u; !) is an increasing function of u and !. Now, on using Lemma 2.2(i), itfollows that R1(!) is an increasing function of !, which proves the required assertion.

Remark 3.1. If X is exponentially distributed; then one can easily verify that; form¿ 2; the corresponding m-spacings are log concave. Thus we conclude that; form¿ 2; the m-spacings do not preserve the log convexity (or the DFR property) of theparent distribution.

Barlow and Proschan (1966) also demonstrated that if X is IFR then the corre-sponding simple spacings may not be IFR. However, they established that the simplespacings preserve the log concavity of the parent distribution. Now, we will generalizethis result from simple spacings to general m-spacings (16m6 n). To do this, werequire Prekopa’s Theorem (cf. Eaton, 1982, Theorem 5.1), which is reproduced below,and a lemma stated after this.

For a positive integer k, let Rk denote the k-dimensional Euclidean space, and letR+ denote the positive part of the real line. Before stating Prekopa’s Theorem, werecall the de;nition of log concave (log convex) functions on Rp.

De$nition 3.1. A nonnegative function h(:) de;ned on a convex set A ⊆ Rp is saidto be log concave (log convex) if; for all x; y∈A and for all )∈ (0; 1); we have

h()x + (1 − ))y)¿ (6)(h(x)))(h(y))1−): (3.2)

For p = 1 (i.e. on the real line R), (3.2) is equivalent to saying that, for each�¿ 0; h(x + �)=h(x) is a decreasing (increasing) function of x.

Theorem 3.2 (Prekopa’s Theorem). Suppose that h :Rm × Rk → [0;∞] is a logconcave function and

g(x) =∫Rk

h(x; z) dz

is 8nite for each x∈Rm. Then g(:) is log concave on Rm.

Lemma 3.1. Suppose that X is log concave. Then K(x; u) = F(x + u) − F(u) is logconcave on R+ ×R+.

Proof. Clearly; for x¿ 0 and u¿ 0;

K(x; u) =∫ ∞

0h1(x; u; z) dz;

where; for x; u; z¿ 0;

h1(x; u; z) =

{f(u + z) if 0¡z¡x

0 otherwise:


In view of Prekopa’s theorem; it is enough to show that h1(x; u; z) is a log concavefunction of x; u; z ∈R+ ×R+ ×R+. Fix )∈ (0; 1) and xi; yi ∈R+; i = 1; 2; 3. Then

(h1(x)))(h1(y))1−)

=

{(f(x2 + x3)))(f(y2 + y3))1−) if 0¡x3 ¡x1; 0¡y3 ¡y1;

0 otherwise:

6

(f(x2 + x3)))(f(y2 + y3))1−) if 0¡)x3 + (1 − ))y3

¡)x1 + (1 − ))y1;

0 otherwise:

6

f()(x2 + x3) + (1 − ))(y2 + y3)) if 0¡)x3 + (1 − ))y3

¡)x1 + (1 − ))y1;

0 otherwise:

=h1()x + (1 − ))y);

where the last inequality follows on using the log concavity of f(:). This proves therequired result.

Theorem 3.3. Suppose that X is log concave. Then the m-spacings D(m)k;n ; k = 1; 2; : : : ;

n− m + 1; are also log concave.

Proof. The log concavity of D(m)1; n = Xm:n follows from (2.4) on observing that the log

concavity of f(x) implies the log concavity of F(x) and GF(x). For k=2; 3; : : : ; n−m+1;we may write (2.3) as

f(m)k;n (x) =

n!(k − 2)!(n− k − m + 1)!(m− 1)!

∫ ∞

0h2(x; u) du;

where

h2(x; u) = ( GF(x + u))n−k−m+1(F(x + u) − F(u))m−1(F(u))k−2f(x + u)f(u);

x¿ 0; u¿ 0:

On using the log concavity of f(x); F(x) and GF(x); along with Lemma 3.1; it isstraightforward to show that; for )∈ (0; 1) and xi; ui ∈R+; i = 1; 2;

h2()x1+(1−))x2; )u1+(1−))u2)¿ (h2(x1; u1)))(h2(x2; u2))1−);

which proves that h2(:; :) is a log concave function on R+×R+. Now the result followson using Prekopa’s theorem.


4. Stochastic comparisons among m-spacings

For normalized simple spacings Dk;n = (n − k + 1)D(1)k;n; k = 1; 2; : : : ; n, Kochar and

Kirmani (1995, p. 56, Remark 3) asserted that if X is log convex, then

Dk;n6lr Dk+1; n; k = 1; 2; : : : ; n− 1: (4.1)

We observed that assertion (4.1), made by Kochar and Kirmani (1995), is not truein general. To see this, consider

f(x) =e−x + 2e−2x

2; x¿ 0:

Since the mixture of log convex densities is log convex (cf. Marshall and Olkin,1979, p. 452), it follows that f(:), being a mixture of exponential densities, is logconvex. For n = 2, the pdfs of D1;2 and D2;2 are given by

fD1; 2 (x) =e−x + 3e−3x=2 + 2e−2x

4; x¿ 0

and

fD2; 2 (x) =7e−x + 10e−2x

12; x¿ 0;

respectively. Therefore,

S(x) =fD2; 2 (x)fD1; 2 (x)

=13

7e x + 10e x + 3e x=2 + 2

:

It is easy to verify that S ′(x), the derivative of S(x), is negative in the neighborhoodof zero. Thus, it follows that the assertion (8), made by Kochar and Kirmani (1995)is not true in general. The plot of function S(x), in the neighborhood of x = 0, isdisplayed in Fig. 1.

Fig. 1. Plot of S(x).


In the following theorem we modify assertion (4.1), made for normalized simplespacings by Kochar and Kirmani (1995), by proving that if X is log convex, thenthe m-spacings are ordered in terms of likelihood ratio ordering. In particular, thisdemonstrates that if X is log convex, then although normalized simple spacings may notbe ordered in terms of likelihood ratio ordering, but simple spacings (nonnormalized)are ordered in terms of likelihood ratio ordering.

Theorem 4.1. Suppose that X is log convex. Then

(i) D(m)k;n 6

lr D(m)k+1; n; k = 1; 2; : : : ; n− m.

(ii) D(m)k;n+16

lr D(m)k;n ; n¿ k + m− 1; for 8xed k.

Proof. (i) From (2.3); for !¿ 0 and k = 2; 3; : : : ; n− m; we have

f(m)k+1; n(!)

f(m)k;n (!)

=n− k − m + 1

k − 1R2(!);

where

R2(!) = E!("2(U2; !));

"2(u; !) =F(u)

GF(! + u); u¿ 0; !¿ 0

and the nonnegative rv U2 has a df belonging to the family P2 = {G2(: | !); !¿ 0};with corresponding pdfs given by

g2(u | !) = c2(!)( GF(!+u))n−k−m+1(F(!+u)−F(u))m−1(F(u))k−2f(!+u)f(u);

u¿ 0; !¿ 0;

here c2(!) is the normalizing constant.Now on using Lemmas 2.1 and 2.2 and from the kind of arguments used in proving

Theorem 3.1, it follows that R2(!) is an increasing function of !, which proves that

D(m)k;n 6

lr D(m)k+1; n; k = 2; 3; : : : ; n− m:

To prove that D(m)1; n 6

lr D(m)2; n , consider

R3(!) =f(m)

2; n (!)

f(m)1; n (!)

; !¿ 0:

From (2.3) and (2.4), it follows that

R3(!) = (n− m)∫ ∞

0

1GF(! + u)

( GF(! + u)GF(!)

)n−m

×(F(! + u) − F(u)

F(!)

)m−1 f(! + u)f(!)

f(u) du:


Now, on using the log convexity of X and Lemma 2.1(i), it follows that R3(!) is anincreasing function of !. This proves assertion (i).

(ii) For !¿ 0 and k = 2; 3; : : : ; n− m + 1, from (2.3), we have

f(m)k;n (!)

f(m)k;n+1(!)

=n− k − m + 2

n + 1R4(!);

where

R4(!) = E!("4(U4; !)); !¿ 0;

"4(u; !) =1

GF(! + u); u¿ 0; !¿ 0

and the nonnegative rv U4 has a df belonging to the family P4 = {G4(: | !); !¿ 0},with corresponding pdfs given by

g4(u | !) = c4(!)( GF(!+u))n−k−m+2(F(!+u)−F(u))m−1(F(u))k−2f(!+u)f(u);

u¿ 0; !¿ 0;

here c4(!) is the normalizing constant.Now on using Lemmas 2.1 and 2.2 and from the kind of arguments used in proving

Theorem 3.1, it follows that R4(!) is an increasing function of !, which proves thatfor a ;xed 26 k6 n− m + 1,

D(m)k;n+16

lr D(m)k;n ; n¿ k + m− 1: (4.1)

To prove that D(m)1; n+16

lr D(m)1; n , from (2.4), we have

f(m)1; n (!)

f(m)1; n+1(!)

=n + 1 − m

n + 11

GF(!); !¿ 0;

which is clearly an increasing function of !. Therefore

D(m)1; n+16

lr D(m)1; n ; n¿m: (4.2)

Now assertion (ii) follows on combining (4.1) and (4.2).Barlow and Proschan (1966) proved that if X is DFR, then the corresponding succes-

sive normalized simple spacings are stochastically ordered. Kochar and Kirmani (1995)strengthened this result by proving that if X is DFR, then the successive normalizedsimple spacings are ordered in terms of the failure rate. In the following theorem weprove that if X is DFR, then the m-spacings D(m)

k;n ; k = 1; 2; : : : ; n−m+ 1 (for a general16m6 n) are also ordered in terms of the failure rate.


Theorem 4.2. Suppose that X is DFR. Then

(i) D(m)k;n 6

fr D(m)k+1; n; k = 1; 2; : : : ; n− m.

(ii) D(m)k;n+16

fr D(m)k;n ; n¿ k + m− 1; for 8xed k.

Proof. (i) On using (2.1) and (2.3); the failure rate function of D(m)!;n ; for !∈ 1 =

{2; 3; : : : ; n− m + 1}; can be written as

r(m)!;n (x) = E!("x

5(U5; !)); !∈ 1;

where

"x5(u; !) =

f(x + u)GF(x + u)

· (1 − ( GF(x + u)= GF(u)))m−1∫ 10 zn−!−m+1(1 − z( GF(x + u)= GF(u)))m−1 dz

;

u¿ 0; !∈ 1

and the nonnegative rv U5 has a df belonging to the family P5 = {G5(u | !); !∈ 1};with corresponding pdfs given by

g5(u | !) = c5(!)( GF(x + u)

GF(u)

)n−!−m+2

×(∫ 1

0zn−!−m+1

(1 − z

GF(x + u)GF(u)

)m−1

dz

)f!−1; n(u); (4.3)

u¿ 0; !∈ 1; here c5(!) is the normalizing constant.Clearly "x

5(u; !) is a decreasing function of !∈ 1. Since X is DFR, it follows that"x

5(u; !) is also a decreasing function of u. For ! = 2; 3; : : : ; n− m, we have

g5(- | ! + 1)g5(- | !)

=c5(! + 1)c5(!)

n− ! + 1!− 1

F(-)GF(x + -)

R6(-); -¿ 0;

where

R6(-) = E-

(1U6

);

and the nonnegative rv U6 has a df belonging to the family P6 = {G6(: |-); -¿ 0},with corresponding pdfs given by

g6(z |-) = c6(-)zn−!−m+1(

1 − zGF(x + -)

GF(-)

)m−1

; 0¡z¡ 1; -¿ 0;

here c6(-) is the normalizing constant.Since X is DFR, it follows that, for 0¡-1 ¡-2; g6(z |-2)=g6(z |-1) is a decreasing

function of z ∈ (0; 1), which in turn implies that, for each z ∈ [0;∞); GG6(z |-) is adecreasing function of -. Now, on using Lemma 2.2(i), it follows that R6(-) is an in-creasing function of -∈ (0;∞), which in turn implies that the family of pdfs, as de;nedin (4.3), has the mlr property, which further implies that, for each ;xed u¿ 0; GG5(u | !)is an increasing function of !. Now since "x

5(u; !) is a decreasing function of ! and u,


on using Lemma 2.2(ii), it follows that, for each ;xed x¿ 0; r(m)!;n (x) is a decreasing

function of !∈ 1, i.e.

D(m)k;n 6

fr D(m)k+1; n; k = 2; 3; : : : ; n− m: (4.4)

It remains to show that D(m)1; n 6

fr D(m)2; n . For this consider

GF(m)2; n (x)

GF(m)1; n (x)

=

∫∞0 P(Xm+1:n − X1:n ¿x |X1:n = u)f1; n(u) du

P(Xm:n ¿x); x¿ 0:

For a ;xed u¿ 0, let (Si; T ui ); i = 1; 2; : : : be a sequence of independent random vari-

ables, where the Sis are identically distributed with pdf f(:) and the Tui s are identi-

cally distributed with pdf f(x + u)= GF(u). Since X is DFR, it follows that Si6fr Tui ;

i = 1; 2; : : : : Also, on using the Markovian property of the order statistics, we have

GF(m)2; n (x) =

∫ ∞

0P(Tu

m:n−1 ¿x)f1; n(u) du; x¿ 0:

Therefore, for x¿ 0,

GF(m)2; n (x)

GF(m)1; n (x)

=∫ ∞

0

P(Tum:n−1 ¿x)

P(Sm:n ¿x)f1; n(u) du

=∫ ∞

0

P(Tum:n−1 ¿x)

P(Sm:n−1 ¿x)P(Sm:n−1 ¿x)P(Sm:n ¿x)

f1; n(u) du: (4.5)

Since Si6fr Tui ; i = 1; 2; : : : ; on using Theorem 1.B.4 of Shaked and Shanthikumar

(1994, p. 15), it follows that Sm:n−16fr Tum:n−1, which implies that the ;rst ratio in the

integrand in (4.5) is an increasing function of x. Also, on using Theorem 1.B.17 ofShaked and Shanthikumar (1994, p. 22) it follows that Sm:n6fr Sm:n−1, which in turnimplies that the second ratio in the integrand in (4.5) is also an increasing function ofx. On combining these two observations, we conclude that

D(m)1; n 6

fr D(m)2; n : (4.6)

Now the assertion (i) follows on combining (4.4) and (4.6).(ii) For ;xed values of x (¿ 0); n and k (¿ 2), we write the failure rate function

of D(m)k;! (!∈ 2 = {k + m− 1; k + m; : : :}) as

r(m)k;! (x) = E!("x

7(U7; !)); !∈ 2;

where

"x7(u; !) =

f(x + u)GF(x + u)

(1 − ( GF(x + u)= GF(u)))m−1∫ 10 z!−k−m+1(1 − z( GF(x + u)= GF(u)))m−1 dz

;

u¿ 0; !∈ 2


and the nonnegative rv U7 has a df belonging to the family P7 = {G7(u | !); !∈ 2},with corresponding pdfs given by

g7(u | !) = c7(!)( GF(x + u)

F(u)

)!−k−m+2(∫ 1

0z!−k−m+1

(1 − z

GF(x + u)GF(u)

)m−1

dz

)

×fk−1; !(u); u¿ 0; !∈ 2;

here c7(!) is the normalizing constant.As in (i), it can be shown that "x

7(u; !) is a decreasing function of u and in-creasing function of !. Using similar arguments, one can also show that, for ;xedu¿ 0; GG7(u | !) is a decreasing function of !. Now, on using Lemma 2.2(i), it followsthat, for each ;xed x¿ 0 and k¿ 2; r(m)

k;! (x) is an increasing function of !∈ 2, i.e.,for ;xed k¿ 2,

D(m)k;n+16

fr D(m)k;n ; n¿ k + m− 1:

The assertion D(m)1; n+16

frD(m)1; n follows from Theorem 1.B.17 of Shaked and Shanthikumar

(1994, p. 22). This proves the required result.

Remark 4.1. By taking X to be exponentially distributed; one can easily show that theresults parallel to Theorem 4.2 may not hold when X is assumed to be IFR.

5. Concluding remarks

(i) In this paper we have considered the successive (overlapping) m-spacings. How-ever, exactly the same proofs work for establishing these results for the disjointm-spacings E(m)

k;n =Xkm:n−X(k−1)m:n; k=1; 2; : : : ; [m=n], where [x] denotes the largestinteger not exceeding x.

(ii) It will be interesting to extend the results of this paper to some other orderings,such as the mean residual life ordering. Kirmani (1996, 1998) established that ifX1:n has IMRL (increasing mean residual life) then the simple spacings also havethe IMRL and that they are ordered in terms of the mean residual life. Whethersuch a result hold for m-spacings (m¿ 2) is not known. Similarly it will beinteresting to investigate if the results of this paper can also be extended to establishordering between two sets of m-spacings, based on independent observations ontwo diOerent rvs X and Y , which are known to satisfy certain reliability-theoreticordering. Such problems for simple normalized spacings have been considered byKochar (1999) and Khaledi and Kochar (1999), and some results for m-spacingshave been derived by Hu and Wei (2001).

Acknowledgements

The research of the ;rst author was supported by a fellowship of the KatholiekeUniversity Leuven (Belgium), while the research of the second author was supported


by the NATO Research Grant No. CRG 931030. The authors also thank the refereesfor valuable comments.

References

Barlow, R.E., Proschan, F., 1966. Inequalities for linear combinations of order statistics from restrictedfamilies. Ann. Math. Statist. 37, 1574–1592.

Barlow, R.E., Proschan, F., 1981. Statistical Theory of Reliability and Life Testing. Holt, Reinhart andWilson, New York.

Cressie, N., 1976. On the logarithms of high-order spacings. Biometrika 63, 343–355.David, H.A., 1981. Order Statistics, 2nd Edition. Wiley, New York.Dudewicz, E.J., van der Meulen, E.C., 1981. Entropy-based tests of uniformity. J. Amer. Statist. Assoc. 76,

967–974.Eaton, M.L., 1982. A review of selected topics in multivariate probability inequalities. Ann. Statist. 10,

11–43.Hu, T., Wei, Y., 2001. Stochastic comparisons of spacings from restricted families of distributions. Statist.

Probab. Lett. 53, 91–99.Khaledi, B.E., Kochar, S.C., 1999. Stochastic ordering between distributions and their sample spacings.

Statist. Probab. Lett. 44, 161–166.Kirmani, S.N.U.A., 1996. On sample spacings from IMRL distributions. Statist. Probab. Lett. 29, 159–166.Kirmani, S.N.U.A., 1998. Erratum on sample spacings from IMRL distributions. Statist. Probab. Lett. 37,

315.Kirmani, S.N.U.A., Alam, S.N., 1974. On goodness of ;t tests based on spacings. Sankhya Ser. A 36,

197–203.Kochar, S.C., 1999. On stochastic orderings between distributions and their sample spacings. Statist. Probab.

Lett. 42, 345–352.Kochar, S.C., Kirmani, S.N.U.A., 1995. Some results on normalized spacings from restricted families of

distributions. J. Statist. Plann. Inference 46, 47–57.Kochar, S.C., Korwar, R., 1996. Stochastic order for spacings from heterogeneous exponential random

variables. J. Multivariate Anal. 57, 69–83.Kochar, S.C., Rojo, J., 1996. Some new results on stochastic comparisons of spacings from heterogeneous

exponential distributions. J. Multivariate Anal. 59, 272–281.Lehmann, E.L., 1986. Testing Statistical Hypothesis, 2nd Edition. Wiley, New York.Marshall, A.W., Olkin, I., 1979. Inequalities: Theory of Majorization and its Applications. Academic Press,

New York.Pledger, G., Proschan, F., 1971. Comparisons of order statistics and spacings from heterogeneous

distributions. In: Rustagi, J.S. (Ed.), Optimizing Methods in Statistics. Academic Press, New York,pp. 89–113.

Pyke, R., 1965. Spacings. J. Roy. Statist. Soc. B 7, 395–449.Pyke, R., 1972. Spacings revisited. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics

and Probability, Vol. 1, pp. 417–427.Rao, J.S., Kuo, M., 1984. Asymptotic results on Greenwood statistic and some of it’s generalizations.

J. Roy. Statist. Soc. B 46, 228–237.Shaked, M., Shanthikumar, J.G., 1994. Stochastic Orders and their Applications. Academic Press, San Diego,

CA.

Documents

On stochastic properties of m-spacings