Comparison of order statistics in a random sequence to the same statistics with I.I.D. variables

January 2006, Vol. 10, p. 1–10 DOI: 10.1051/ps:2005020

COMPARISON OF ORDER STATISTICS IN A RANDOM SEQUENCE TO THE SAME STATISTICS WITH I.I.D. VARIABLES

Jean-Louis Bon

and Eugen P˘ alt˘ anea

Abstract. The paper is motivated by the stochastic comparison of the reliability of non-repairable k-out-of-nsystems. The lifetime of such a system with nonidentical components is compared with the lifetime of a system with identical components. Formally the problem is as follows. LetUi, i= 1, ..., n, be positive independent random variables with common distribution F. For λi >0 and µ >0, let considerXi=Ui/λiand Yi=Ui/µ, i= 1, ..., n. Remark that this is no more than a change of scale for each term. Fork ∈ {1,2, ..., n},let us defineXk:n to be the kth order statistics of the random variablesX1, ..., Xn, and similarlyYk:nto be thekth order statistics ofY1, ..., Yn.IfXi, i= 1, ..., n,are the lifetimes of the components of an+1-k-out-of-nnon-repairable system, thenXk:nis the lifetime of the system. In this paper, we give for a fixedk a sufficient condition forXk:n≥stYk:nwherestis the usual ordering for distributions. In the Markovian case (all components have an exponential lifetime), we give a necessary and sufficient condition. We prove thatXk:n is greater thatYk:naccording to the usualstochastic orderingif and only if

n

k

µ^k≥

1≤i1<i2<...<i_k≤n

λi1λi2...λik.

Mathematics Subject Classification. 60E15, 62N05, 62G30, 90B25, 60J27.

Received December 14, 2004. Revised May 25, 2005.

1. Introduction

In reliability theory, a system of n identical independent components is usually saidk-out-of-n when it is functioning if and only if at leastkof the components are functioning. In this case, there aren−kfailures. Such systems are getting more and more frequent in industrial processes. For example, a given parameter (presence or not of a train, temperature, ...) might be controlled by several devices and the decision rule used to fix the value of this parameter is of typek-out-of-n. This reliability notion is, in fact, the same as the order statistics notion. If the lifetimes of the components are independent identically distributed (i.i.d.) random variables X₁, X₂, ..., X_n,then the lifetime of then+1-k-out-of-nsystem is exactly thekth order statistics (denotedX_k:n) of the random variablesX_i, i= 1, ..., n.

Keywords and phrases. Stochastic ordering, Markov system, order statistics,k-out-of-n.

1 Polytech-Lille, USTL, Laboratoire CNRS Painlev´e, 59655 Villeneuve d’Ascq, France;jean-louis.bon@polytech-lille.fr

2 Transilvania University of Bra¸sov, Faculty of Mathematics and Computer Sciences, Romˆania.

c EDP Sciences, SMAI 2006

Article published by EDP Sciences and available at http://www.edpsciences.org/psor http://dx.doi.org/10.1051/ps:2005020

Let us now describe the practical problem which motivated this paper. We consider an+1-k-out-of-nsystem where the n components have independent exponential lifetimes X₁, ..., X_n, but not necessarily identically distributed. We denote λ₁, ..., λ_n the respective parameters of these exponential lifetimes. The problem is to compare this system with an equivalent system with i.i.d. components. Practically, when we have to replace many different components (from different factories) by identical components (from the same factory), we need to guarantee the same quality.

Let us denote byY₁, ..., Y_n,the random lifetimes of the identical components with the common parameterµ.

The problem of comparison is equivalent to the following one. What are the values ofµwhich characterize the stochastic inequalityX_k:n≥stY_k:n,i.e. IP(X_k:n> t)≥IP(Y_k:n> t), ∀t >0 ?

A general result on the stochastic comparison of order statistics was obtained by Pledger and Proschan [6], in connection with theSchur’s majorization(see Marshall and Olkin [4]). In our context, this result provides a sufficient condition which can be written as follows:

µ= λ₁+λ₂+...+λ_n

n ⇒ X_k:n ≥stY_k:n. More recently, Khaledi and Kochar [3] studied the casek=nand proved that

µ= ⁿ

λ₁λ₂...λ_n ⇒ X_n:n ≥hrY_n:n,

where hr denotes the hazard rate ordering. See Shaked and Shantikhumar [7] for an overview of the different notions of ordering.

Here we extend this results referring tostochastic orderingwhich gives the comparison of survival functions.

With the exponential assumption, we propose a necessary and sufficient condition on the parameters for the in- equalityX_k:n ≥stY_k:n, k= 1,2, ..., n.More generally, we give sufficient conditions for the stochastic comparison in the case where the distributionF is not exponential.

2. Elementary symmetrical functions

First, let us introduce some notations and recall some results about elementary symmetrical functions.

Letx= (x₁, x₂, ..., x_n), n >1 a vector with positive components. Forj∈ {1,2, ..., n},

S_j(x) =

1≤i1<i2<...<ij≤n

x_i1x_i2...x_ij (1)

is thejth elementary symmetrical function of the positivex₁, x₂, ..., x_n. As usualS₀(x) = 1,andS_j(x) = 0, forj > n.

For p, q ∈ {1,2, ..., n}, p= q, we denote x^p = (..., x_p−1, x_p+1, ...) ∈(0,∞)ⁿ⁻¹ and x^p,q = (x^p)^q ∈ (0,∞)ⁿ⁻². Therefore

S_j(x^p) =

i₁, ..., i_j∈ {1, ..., n} \ {p}

i₁< i₂< ... < i_j

x_i1x_i2...x_ij (2)

is thejth elementary symmetrical function obtained without the component of numberpand

S_j(x^p,q) =

i₁, ..., i_j∈ {1, ..., n} \ {p, q} i₁< i₂< ... < i_j

x_i1x_i2...x_ij (3)

is the jth elementary symmetrical function made withoutpand q. In the case where all coordinates of xare equal tom, we have

S_k(x) = n

k

m^k. (4)

These functions satisfy the elementary relations:

S_j(x) =x_pS_j−1(x^p) +S_j(x^p) (5)

and

S_j(x) =x_px_qS_j−2(x^p,q) + (x_p+x_q)S_j−1(x^p,q) +S_j(x^p,q). (6) Moreover we define

m_j(x) = n

j ₋₁

S_j(x) ¹_j

=

⎛

⎝ n j

₋₁

|J|=j i∈J

x_i

⎞

⎠

1j

(7) thejth symmetrical mean,j= 1,2, ..., n. These different averages are classical and satisfy the well-known Mac Laurin’s inequalities (see Hardy, Littlewood and P´olya [2]):

m₁(x)≥m₂(x)≥...≥m_n(x).

Two special cases are of interest. If j = 1 and j =n then m₁(x) = ^x1+x2+...+x_n ⁿ and m_n(x) = √ⁿ

x₁x₂...x_n are the arithmetical and geometrical means. In order to compare the elementary symmetrical functions of two different vectors, we shall use the following lemma.

Lemma 1. Letx= (x₁, ..., x_m)andy= (y₁, ..., y_m)be two vectors such that0< x_i≤y_i for alli= 1, ..., m.If r∈ {0,1, ..., m−1} then

S_r+1(x)

S_r(x) ≤S_r+1(y)

S_r(y) · (8)

Moreover, if there existsi₀ so that x_i0< y_i0 then the inequality (8) is strict.

Proof. Letr be an integer, 0≤r < m. It is sufficient to prove that the symmetrical function γ_r: (0,∞)^m→ (0,∞), γr(x) =^S_S^r+1(x)

r(x) is strictly increasing inx₁.

The property is clear for r= 0. Let us considerr >0. Since S_r+1(x) = x₁S_r(x¹) +S_r+1(x¹) and S_r(x) = x₁S_r−1(x¹) +S_r(x¹) we have

∂γ_r

∂x₁(x) =

S_r(x¹)₂

−S_r−1(x¹)S_r+1(x¹) (S_r(x))² · The relation _∂x^∂γr

1(x)>0 is equivalent to:

S_r(x¹)₂

> S_r−1(x¹)S_r+1(x¹),

for x¹= (x₂, ..., x_n). But this is an immediate consequence of the Newton inequalities (see Hardy, Littlewood

and P´olya [2]).

In the sequel, we shall use the following consequence of Lemma 1.

Corollary 1. Let x = (x₁, ..., x_m) and y = (y₁, ..., y_m) be two vectors with m positive components such that α≤ ^y_xⁱ_i ≤β, i= 1,2, ..., m,where0< α < β. Then, for r∈ {0,1, ..., m−1}, the next inequalities are true:

αS_r(y)

S_r(x) ≤ S_r+1(y)

S_r+1(x) ≤β S_r(y)

S_r(x), (9)

and at least one of these inequalities is a strict inequality.

The assertion follows easily by replacing the pair (x,y) by (αx,y) and (y, βx) in the above lemma.

Remark. We give also an elementary result about the comparison of two fractions. The proof is omitted.

Ifa, b, c, d, e, f are positive with ^a_b ≤ ^c_d ≤ ^e_f and ^a_b <_f^e then a+c

b+d< c+e

d+f· (10)

In order to prove our main results, we present a sufficient condition to recognize the minimum value of a symmetrical function. This result is interesting by itself and, to the best of our knowledge, it is new.

Lemma 2. Forn >1, letψ: (0,∞)ⁿ →(0,∞)be a symmetrical and continuously differentiable mapping, and k an integer, 1≤k≤n. Let us assume that, for any vector x= (x₁, ..., x_n)∈(0,∞)ⁿ,with x_p = minx_i and x_q= maxx_i we have:

x_p< x_q ⇒

∂ψ

∂xp(x)

∂Sk

∂xp(x) <

∂ψ

∂xq(x)

∂Sk

∂xq(x)· (11)

Then, for any x= (x₁, ..., x_n)∈(0,∞)ⁿ, the following inequality holds:

ψ(x₁, ..., x_n)≥ψ(m_k(x), ..., m_k(x)

n times

). (12)

Proof. The casek= 1 is well-known (see, for example, Marshall and Olkin [4]).

Now we supposek >1. For a fixed vector x= (x₁, ..., x_n)∈(0,∞)ⁿ let consider a= minx_i, b= maxx_i, m= m_k(x) and m= (m, ..., m

n times

). Inequality (12) is an equality fora=b.

Let us assumea < b. Thenm∈(a, b).We consider the compact subsetK of (0,∞)ⁿ: K={t= (t₁, t₂, ..., t_n)∈[a, b]ⁿ |m_k(t) =m}.

Clearly,x andm belong to K. From Weierstrass’s theorem it follows that the continuous mappingψ reaches an absolute minimum on the compactK on some pointu= (u₁, u₂, ..., u_n)∈K.

Now let us assumeu=m.In this case, there existsp, q∈ {1,2, ..., n}such thata≤u_p= minu_i<maxu_i= u_q≤b.Rewriting condition m_k(u) =mfrom relation (6), yields to

u_pu_qS_k−2(u^p,q) + (u_p+u_q)S_k−1(u^p,q) +S_k(u^p,q) = n

k

m^k.

The equation S_k−2(u^p,q)z²+ 2S_k−1(u^p,q)z+S_k(u^p,q) = n

k

m^k has a positive solution in z which is de- noted z₁. Clearly u_p < z₁ < u_q. For t ∈ [u_p, z₁), let us consider the functiong(t) defined on [u_p, z₁) by the relation:

tg(t)S_k−2(u^p,q) + (t+g(t))S_k−1(u^p,q) +S_k(u^p,q) = n

k

m^k. We haveg(u_p) =u_q and more generally:

g(t) = n

k

m^k−tS_k−1(u^p,q)−S_k(u^p,q)

tS_k−2(u^p,q) +S_k−1(u^p,q) ∈(z₁, u_q], ∀t∈[u_p, z₁).

Let us denote by u(t) the vector with the components u_p(t) = t, u_q(t) = g(t) and u_i(t) = u_i for i ∈ {1,2, ..., n} \ {p, q}. We haveu(t)∈Kand

S_k(u(t)) = n

k

m^k, ∀t∈[u_p, z₁).

The continuously differentiable decreasing functiong has the following derivative:

g(t) =−

∂Sk

∂xp(u(t))

∂Sk

∂xq(u(t))· (13)

Now let us consider the continuously differentiable functionϕ: [u_p, z₁)→IR, ϕ(t) =ψ(u(t)).

From relation (13) we obtain:

ϕ(t) = ∂ψ

∂x_p(u(t)) + ∂ψ

∂x_q(u(t))·g(t) =∂S_k

∂x_p(u(t)) _∂ψ

∂xp(u(t))

∂Sk

∂xp(u(t))−

∂ψ

∂xq(u(t))

∂Sk

∂xq(u(t))

.

But, from assumption (11), it follows that ϕ(u_p)<0. Hence, there exists ε > 0 such that u_p+ε < z₁ and ϕ(t)<0, ∀t∈ [u_p, u_p+ε). Therefore, ψ(u(t))< ψ(u(u_p)) =ψ(u), for anyt ∈(u_p, u_p+ε). This gives the contradiction. Then the unique minimum point ofψonK ismand the relation (12) follows.

Remark. The assumption (11) can be replaced by the following more restrictive assumption:

∀x∈(0,∞)ⁿ ∀i, j∈ {1, ..., n} x_i=x_j ⇒ (x_i−x_j) _∂ψ

∂xi(x)

∂Sk

∂xi(x)−

∂ψ

∂xj(x)

∂Sk

∂xj(x)

>0.

In the case k = 1, one obtains again a well-known sufficient condition of Schur convexity (see Marshall and Olkin [4]).

3. The main results

This section is concerned with the characterizations of the comparison between a system with different lifetime components (X_i) and a system with i.i.d. lifetime components (Y_i).

Formally, letX andY be two random variables with supportIR⁺, having the survival functionsF_X = 1−F_X andF_Y,respectively. F_X andF_Y are assumed to be continuously differentiable.

The variableX is saidstochastically larger thanY (denotedX ≥stY) whenF_X(t)≥F_Y(t), ∀t≥0.

The following theorem provides sufficient conditions for the stochastic comparison between the same order statistics in two sequences of independent random variables.

Theorem 1. Let U₁, U₂, ..., U_n be i.i.d. positive random variables with the common distribution function F having a positive non-increasing hazard rate h(x) =_1−F(x)^F(x) , x∈(0,∞).

For a vector λ= (λ₁, ..., λ_n)with positive components andk∈ {1, ..., n},

m_k(λ) =

⎛

⎝ n k

₋₁

|J|=k i∈J

λ_i

⎞

⎠

k1

is the kth symmetrical mean of λ. Let us define X_i =U_i/λ_i and Y_i = U_i/m_k(λ) for all i ∈ {1, ..., n}. Let us denote by X_k:n (resp. Y_k:n) the kth order statistics of (X₁, ..., X_n) (resp. (Y₁, ..., Y_n)). If _x(1−F(x))^F(x) is an increasing function on (0,∞), then

X_k:n ≥st Y_k:n.

Proof. From the definition, the random variableX_i has the distribution functionF_Xi(t) =F(λ_it), t≥0, i= 1,2, ..., n.

First, let us consider the case k = 1. The random variables X_1:n and Y_1:n have the survival functions F_X1:n(t) = _n

i=1F(λ_it) and F_Y1:n(t) =

F(^λ1+...+λ_n ⁿt)n

respectively. From logF

= −h with h a non- increasing function, it follows that logF is a convex function. Thus, from Jensen’s inequality we get F_X1:n(t)≥ F_Y1:n(t), ∀t≥0,and the conclusion is proved fork= 1.

Let us assume nowk >1. The survival function ofX_k:n is:

F_Xk:n(t) =

k−1

j=0

|J|=j

i∈J

F(λ_it)

i∈J/

F(λ_it)

=

n i=1

F(λ_it)

k−1

j=0

|J|=j

i∈J

F(λ_it) F(λ_it)

. (14)

Let us consider the functions y: (0,∞)→(0,∞), y(x) = ^F(x)_F(x) and ψ: (0,∞)ⁿ →(0,1),

ψ(x₁, ..., x_n)) = _k−1

j=0S_j(y(x₁), ..., y(x_n)) _n

i=1(1 +y(x_i)) · Fort >0 we haveF_Xk:n(t) =ψ(λ₁t, ..., λ_nt).

Similarly,F_Yk:n(t) =ψ(m_k(λt), ..., m_k(λt)

n times

). To obtain the conclusion X_k:n ≥stY_k:n, it is sufficient to prove the following property of the symmetrical and continuously differentiable mappingψ:

ψ(x₁, ..., x_n)≥ψ(m_k(x), ..., m_k(x)

n times

), ∀x= (x₁, ..., x_n)∈(0,∞)ⁿ. (15)

Forx= (x₁, ..., x_n)∈(0,∞)ⁿ, we use the notationy=y(x) = (y(x₁), ..., y(x_n)) andy_i =y(x_i). The function ψhas the following partial derivatives:

∂ψ

∂x_s(x) =

y(x_s)

i=s(1 +y_i)

(1 +y_s)_k−1

j=1S_j−1(y^s)−

1 +_k−1

j=1(y_sS_j−1(y^s) +S_j(y^s)) _n

i=1(1 +y(x_i))²

=− y(x_s) 1 +y(x_s)

S_k−1(y^s) _n

i=1(1 +y_i)

=−h(xs) S_nk−1(y^s)

i=1(1 +y_i), s= 1, ..., n. (16)

We denotex_p = minx_i andx_q = maxx_i. Let us assume thatx_p< x_q. Fork < n, using relation (16), we get:

_∂ψ

∂xp

∂Sk

∂xp

−

∂ψ

∂xq

∂Sk

∂xq

(x) = _n 1

i=1(1 +y_i)

h(x_q)S_k−1(y^q)

S_k−1(x^q) −h(x_p)S_k−1(y^p) S_k−1(x^p)

= 1

_n

i=1(1 +y_i)

h(x_q)y_pS_k−2(y^p,q) +S_k−1(y^p,q)

x_pS_k−2(x^p,q) +S_k−1(x^p,q)−h(x_p)y_qS_k−2(y^p,q) +S_k−1(y^p,q) x_qS_k−2(x^p,q) +S_k−1(x^p,q)

· (17)

Since x→ _x(1−F(x))^F(x) is assumed to be an increasing function on (0,∞), we have _x^yp_p < ^y_x^q

q and y_p

x_p ≤ y_i x_i ≤ y_q

x_q, ∀i∈ {1, ..., n} \ {p, q}.

Corollary 1 can be applied for m=n−2, r=k−2, α= ^y_x^p

p, β= ^y_x^q

q and the following inequalities hold:

y_p x_p

S_k−2(y^p,q)

S_k−2(x^p,q) ≤S_k−1(y^p,q) S_k−1(x^p,q) ≤ y_q

x_q

S_k−2(y^p,q) S_k−2(x^p,q)·

Moreover, at least one of these inequalities is strict. Hence, from relation (10) the next inequality follows:

y_pS_k−2(y^p,q) +S_k−1(y^p,q)

x_pS_k−2(x^p,q) +S_k−1(x^p,q) < y_qS_k−2(y^p,q) +S_k−1(y^p,q) x_qS_k−2(x^p,q) +S_k−1(x^p,q)·

Buthis a positive non-increasing function. Thus, 0< h(x_q)≤h(x_p). Therefore, from (17), we get:

∂ψ

∂xp(x)

∂Sk

∂xp(x) <

∂ψ

∂xq(x)

∂Sk

∂xq(x)·

Hence, inequality (15) may be deduced from Lemma 2 and the conclusion follows.

It is worth to note that equation (17) cannot be used fork=n. But, in this case, using equation (16), we get _∂ψ

∂xp

∂Sk

∂xp

−

∂ψ

∂xq

∂Sk

∂xq

(x) = 1 _n

i=1(1 +y_i)

i=p,q

y_i x_i

h(x_q)y_p

x_p −h(x_p)y_q x_q

< 0.

And the conclusion follows from Lemma 2.

One important field of application concerns the exponential distribution (see below). But the result is more general. We give an example of a distribution function which satisfies the assumptions of Theorem 1.

Example. For a > 1, let us define the distribution function F(x) = 1− _(1+x)¹ a, x ≥ 0. We have h(x) =

F(x)

1−F(x) = _1+x^a , x ≥ 0. The function g : (0,∞) → (0,∞), g(x) = _x(1−F(x))^F(x) = ^(1+x)_x^a−1 has the derivative g(x) = ^1+(1+x)a−1_x2 ^(ax−1). Butv(x) = 1 + (1 +x)^a−1(ax−1) is a positive function on (0,∞), sincev(0) = 0 and v(x) =a(a−1)x(1 +x)^a−2>0, ∀x >0. Thereforehis a decreasing function andg is an increasing function on (0,∞).

Theorem 1 can be naturally applied to the comparisons of Markov systems in reliability. Let us consider a system which is composed of n components and is considered failed when k components are failed. Let us assume that the failure rates of the components are constant. With such properties, the system is an+1-k-out- of-n Markov system. If the system is starting as new, the lifetime of the system is nothing but thekth order statistics of the exponential lifetimes of the components.

The next theorem gives a necessary and sufficient condition for the stochastic comparison of the lifetimes of twon+1-k-out-of-nMarkov systems inkth order statistics language. This result supplements the known results on this subject.

Theorem 2. Let X_i, i = 1, ..., n, be independent exponential random variables with respective parameters λ_i >0, i= 1,2, ..., n. Let Y_i, i = 1,2, ..., n, be independent exponential random variables with the common parameter µ > 0. For k∈ {1,2, ..., n}, let us denote by X_k:n the kth order statistics of the random variables X₁, X₂, ..., X_n, and similarly let us denote byY_k:n thekth order statistics of the random variables Y₁, Y₂, ..., Y_n.

Then

X_k:n ≥stY_k:n if and only if µ≥

⎛

⎝ n k

₋₁

|J|=k i∈J

λ_i

⎞

⎠

1k

·

Proof. We shall denote byF(x) = 1−e^−x, x≥0, the exponential distribution function with parameter 1.

At first, let us assume that X_k:n ≥st Y_k:n, i.e. F_Xk:n(t) ≥ F_Yk:n(t), ∀t > 0. The survival function of the random variableX_k:n can be written as:

F_Xk:n(t) = 1−e^−tni=1λi n j=k

S_j(e^λ1t−1, ...,e^λnt−1).

Using the Taylor’s expansion about 0, we obtain:

F_Xk:n(t) = 1−S_k(λ₁, ..., λ_n)t^k+o(t^k), t→0.

In the same way, we have:

F_Yk:n(t) = 1− n

k

µ^kt^k+o(t^k), t→0.

Therefore,S_k(λ₁, ..., λ_n)≤ n

k

µ^k.This can be rewritten asµ≥m_k(λ₁, ..., λ_n).

Conversely, suppose that µ≥m_k(λ₁, ..., λ_n). Since the survival function ofY_k:n is clearly decreasing inµ, it suffices to prove the assertionF_Xk:n(t)≥F_Yk:n(t), ∀ t >0, forµ=m_k(λ₁, ..., λ_n).

The exponential distribution functionF has a constant hazard rateh(x) = 1, ∀x≥0. Moreover the function

x(1−F(x))F(x) = ^ex_x⁻¹ is increasing on (0,∞). Hence, the exponential distribution F satisfies the assumptions of Theorem 1. Clearly, the distribution function of the random variableX_i isF_Xi(t) =F(λ_it), t≥0, i= 1, ..., n.

Similarly, we haveF_Yi(t) =F(µt), t≥0.Then, applying Theorem 1 we get the conclusion X_k:n≥stY_k:n. The practical interest of this result is to give precise production constraints on the components of ak-out-of-n system. For example, in the case of replacing several components with well-known failure rates (λ_i)_iby identical components, the previous theorem gives an exact valuem_n+1−k(λ) for the characteristic of the new components in order to preserve the reliability. This result was known for k = 2, it has been proved by P˘alt˘anea [5]. In the same spirit, Bon and P˘alt˘anea [1] have obtained necessary and sufficient conditions about comparisons of convolutions of exponential variables.

4. Numerical examples

The previous results can be illustrated as follows. Let us considernexponential independent random variables X_iwith parameterλ_iandX_k:n(λ) thekth order statistics. Let us denote byY_k:n(m_j(λ)) thekth order statistics ofnexponential independent random variablesY_i with common parameter m_j(λ), j= 1, ..., n.

The survival functions ofX_k:n(λ) andY_k:n(m_j(λ)), j= 1, ..., nare plotted in Figure 1. It can be clearly seen that

F_Xk:n(λ) ≥F_Yk:n(mj(λ)) ⇔ j≤k. (18)

In Figure 2, we give an example of a distribution which satisfies the assumptions of Theorem 1 such that this ordering is true again:

F(x) = 1− 1

(x+ 1)², x≥0.

If we consider a distribution function with an strict increasing hazard rate (usually named IFR) then the assumptions of Theorem 1 are not satisfied. Figure 3 refers to an IFR Weilbull distribution. It can be seen that inequality (18) does not hold.

0 200 400 600 800

0.00.20.40.60.81.0

time

Survival functions

m1m2m3

m4

m5

m6

λ

Comparison of 3:6 order statistics in exponential samples

10⁻⁵≤ λi≤10⁻² m1(λ)=0.0046 m2(λ)=0.0042 m3(λ)=0.0036 m4(λ)=0.0027 m5(λ)=0.0016 m6(λ)=9e−04 F(^x)=exp(−x)

Figure 1. The exponential case for n= 6 andk= 3.

0 200 400 600 800

0.00.20.40.60.81.0

time

Survival functions

m1

m2

m3

m4

m5

m6

λ Comparison of 3:6 order statistics

10⁻⁴≤ λi≤10⁻¹ F(x)= 1

(^x+1)²

Figure 2. The case of the distribution function F(x) = 1−(x+ 1)⁻² forn= 6 andk= 3.

0 200 400 600 800

0.00.20.40.60.81.0

time

Survival functions

m1

m2

m3

m4

m5

m6

λ

Comparison of 2:6 order statistics in Weilbull−IFR samples

10⁻⁴≤ λi≤1 F(^x)=exp(−(^{x 100})²)

Figure 3. The case of an IFR Weilbull distribution forn= 6 andk= 2.

Acknowledgements. We would like to thank the anonymous referees for carefull reading of and constructive comments on the first version of this paper.

References

[1] J.-L. Bon and E. P˘alt˘anea, Ordering properties of convolutions of exponential random variables.Lifetime Data Anal.5(1999) 185–192.

[2] G.H. Hardy, J.E. Littlewood and G. P´olya,Inequalities. Cambridge University Press, Cambridge (1934).

[3] B.-E. Khaledi and S. Kochar, Some new results on stochastic comparisons of parallel systems. J. Appl. Probab.37 (2000) 1123–1128.

[4] A.W. Marshall and I. Olkin,Inequalities: Theory of Majorization and Its Applications. Academic Press, New York (1979).

[5] E. P˘alt˘anea. A note of stochastic comparison of fail-safe Markov systems, in 17th Scientific Session on Mathematics and its Aplications, “Transilvania” Univ. Press (2003) 179–182.

[6] P. Pledger and F. Proschan, Comparisons of order statistics and spacing from heterogeneous distributions, in Optimizing Methods in Statistics. Academic Press, New York (1971) 89–113.

[7] M. Shaked and J.G. Shanthikumar,Stochastic Orders and Their Applications. Academic Press, New York (1994).