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Fariborz Maseeh Department of Mathematics and Statistics
6-11-2015
On Stochastic Comparisons of Largest Order Statistics in the Scale
Model
Subhash C. Kochar
Portland State University, [email protected]
Nuria Torrado
University of Coimbra
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Subhash C. Kochar and Nuria Torrado, On stochastic comparisons of largest order statistics in the scale model, to appear in Communications in Statistics - Theory and Methods.
On stochastic comparisons of largest order statistics
in the scale model
Subhash C. Kochar
Fariborz Maseeh Department of Mathematics and Statistics
Portland State University, Portland, OR 97006, USA
Nuria Torrado
Centre for Mathematics, University of Coimbra
Apartado 3008, EC Santa Cruz, 3001-501 Coimbra, Portugal
June 11, 2015
Abstract
Let Xλ1, Xλ2, . . . , Xλn be independent nonnegative random variables with Xλi ∼ F(λit), i = 1, . . . , n, where λi > 0, i = 1, . . . , n and F is an absolutely continuous
distribution. It is shown that, under some conditions, one largest order statistic Xnλ:n
is smaller than another one Xnθ:n according to likelihood ratio ordering. Furthermore, we apply these results when F is a generalized gamma distribution which includes Weibull, gamma and exponential random variables as special cases.
Keywords: likelihood ratio order; reverse hazard rate order; majorization; order statistics
1
Introduction
Let F be the distribution function of some nonnegative random variable X. Then the independent random variables Xλ1, Xλ2, . . . , Xλn follow the scale model if there exists λ1 > 0, . . . , λn >0 such that, Fi(t) =F(λit) for i= 1, . . . , n. F is called the baseline distribution
and the λ′
is are the scale parameters. Recently, Khaledi et al. (2011) studied conditions
under which series and parallel systems consisting of components with lifetimes from the scale family of distributions are ordered in the hazard rate and the reverse hazard rate orderings, respectively. In this paper we revisit this problem and broaden the scope of their results to likelihood ratio ordering, which is stronger than the other orderings.
There is an extensive literature on stochastic orderings among order statistics and spac-ings when the observations follow the exponential distribution with different scale param-eters, see for instance, Kochar and Kirmani (1996), Dykstra et al. (1997), Bon and P˘alt˘anea (1999), Khaledi and Kochar (2000), Kochar and Xu (2009), Joo and Mi (2010), Torrado et al. (2010), Torrado and Lillo (2013) and the references therein. Also see a review paper by Kochar (2012) on this topic. A natural way to extend these works is to consider the scale model since it includes the exponential distribution, among others. The scale model, also known in the literature as the proportional random variables (PRV) model, is of theoretical as well as practical importance in various fields of probability and statistics and has been investigated in Pledger and Proschan (1971), Hu (1995) and Torrado and Veerman (2012), among others.
In this article, we focus on stochastic orders to compare the magnitudes of two largest order statistics from the scale model when one set of scale parameters majorizes the other. The new results obtained here are applied when the baseline distributions are generalized gamma distributions. Recall that a random variableXhas a generalized gamma distribution, denoted by X ∼GG(β, α), when its density function has the following form
f(t) = β Γ(αβ)x
α−1 e−xβ
, t >0,
flexibility in describing lifetime distributions ensuring their applications in survival analysis and reliability theory. It is of great interest in several areas of application, see for example, Manning et al. (2005), Ali et al. (2008) and Chen et al. (2012). It is well known that gener-alized gamma distribution includes many important distributions like exponential, Weibull and gamma as special cases. We also present some new results which strengthen some of those established earlier in the literature by Zhao (2011), Misra and Misra (2013) and Zhao and Balakrishnan (2013) for gamma distributions and Torrado and Kochar (2015) for Weibull distributions. Further results on these subjects are contained in, e.g., Lihong and Xinsheng (2005), Khaledi and Kochar (2007), Zhao and Balakrishnan (2011), Balakrishnan and Zhao (2013), Fang and Zhang (2013). It may be mentioned that Gupta et al. (2006) considered monotonicity of the hazard rate and the reverse hazard rates of series and parallel systems when the components are dependent.
The rest of the paper is organized as follows. In Section 2, we introduce the required definitions. Sections 3 and 4 are devoted to investigate the reverse hazard rate and likelihood ratio orderings of largest order statistics considering the general scale model, respectively.
2
Basic definitions
In this section, we review some definitions and well-known notions of majorization concepts and stochastic orders. Throughout this article increasing and non-decreasing will be used synonymously as decreasing and non-increasing.
We focus attention in this article on nonnegative random variables. We shall also be using the concept of majorization in our discussion. Let {x(1), x(2), . . . , x(n)} denote the increasing arrangement of the components of the vector x= (x1, x2, . . . , xn).
Definition 2.1 The vector x is said to be majorized by the vectory, denoted by x
m ≤y, if j X i=1 x(i)≥ j X i=1 y(i), for j = 1, . . . , n−1 and n X i=1 x(i) = n X i=1 y(i).
Functions that preserve the ordering of majorization are said to be Schur-convex, as one can see in the following definition.
Definition 2.2 A real valued functionϕ defined on a setA ∈ ℜn is said to be Schur-convex
(Schur-concave) on A if
x
m
≤yon A ⇒ϕ(x)≤(≥)ϕ(y).
Replacing the equality in Definition 2.1 by a corresponding inequality leads to the concept of weak majorization. One can majorize from above or below. The following definition addresses majorization from above. It is also called supermajorization.
Definition 2.3 The vector x is said to be weakly majorized by the vector y, denoted by x w ≤y, if j X i=1 x(i) ≥ j X i=1 y(i), for j = 1, . . . , n. It is known that x m ≤y⇒x w
≤y. The converse is, however, not true. For extensive and comprehensive details on the theory of majorization orders and their applications, please refer to the book of Marshall et al. (2011).
Let X and Y be univariate random variables with cumulative distribution functions (c.d.f.’s) F and G, survival functions ¯F(= 1−F) and ¯G(= 1−G), p.d.f.’sf and g, hazard rate functions hF =f /F¯
and hG =g/G¯
, and reverse hazard rate functions rF(= f /F)
and rG(=g/G), respectively. The following definitions introduce stochastic orders, which
are considered in this article, to compare the magnitudes of two random variables. For more details on stochastic comparisons, see Shaked and Shanthikumar (2007).
Definition 2.4 We say that X is smaller than Y in the:
a) usual stochastic order if F¯(t)≤G¯(t) for all t and in this case, we write X ≤stY,
b) reverse hazard rate order if G(t)/F(t) is increasing in t for which the ratio is well defined, or if rF(t)≤rG(t), for all t, denoted by X ≤rh Y,
c) likelihood ratio order if g(t)/f(t) is increasing in t for which the ratio is well defined, for all t, denoted by X ≤lr Y.
3
Reverse hazard rate ordering results
Let Xλ1, Xλ2, . . . , Xλn be independent nonnegative random variables with Xλi ∼ F(λit),
i = 1, . . . , n, where λi > 0, i = 1, . . . , n and F is an absolutely continuous distribution. Let
f,handrbe the density, hazard rate and the reverse hazard rate functions ofF, respectively. The distribution function of Xλ
n:n, the largest order statistic formed from Xλ1, Xλ2, . . . , Xλn is Fλ n:n(t) = n Y i=1 F(λit),
and its reverse hazard rate function is rλn:n(t) =
n
X
i=1
λir(λit). (3.1)
Khaledi et al. (2011) proved the following result on comparing two parallel systems when the underlying random variables follow the scale model and their scale parameters majorize each other.
Theorem 3.1 Let Xλ1, . . . , Xλn be independent nonnegative random variables with Xλi ∼
F(λit), i = 1, . . . , n, where λi > 0, i = 1, . . . , n and F is an absolutely continuous
distribu-tion. Let r be the reverse hazard rate function of F, respectively. If t2r′
(t) is increasing in
t, then
(λ1, . . . , λn) m
≤(θ1, . . . , θn)⇒Xnλ:n≤rh Xnθ:n.
In the next theorem we extend the above result to the case when the two sets of scale parameters weakly majorize each other instead of usual majorization.
Theorem 3.2 Let Xλ1, Xλ2, . . . , Xλn be independent random variables with Xλi ∼ F(λit)
where λi >0, i= 1, . . . , n. If tr(t) is decreasing in t and t2r
′
(t) is increasing in t, then
(λ1, . . . , λn) w
Proof. Fix t >0. Then the reverse hazard rate of Xλ
n:n as given by (3.1) can be rewritten
as rλ n:n(t) = n X i=1 λir(λit) = 1 t n X i=1 ψ(λit),
where ψ(t) =tr(t), t≥ 0. From Theorem A.8 of Marshall et al. (2011) (p. 59) it suffices to show that, for each t >0,rλ
n:n(t) is decreasing in each λi,i= 1, . . . , n, and is a Schur-convex
function of (λ1, . . . , λn). By the assumptions,tr(t) is decreasing int, then the reverse hazard
rate function of Xn:n is decreasing in each λi.
Now, from Proposition C.1 of Marshall et al. (2011) (p. 64), the convexity of ψ(t) is needed to prove Schur-convexity ofrλ
n:n(t). Note that the assumptiont2r
′ (t) is increasing in t is equivalent to r(t) +tr′ (t) is increasing in t since t2r′ (t)′ =t(2r′ (t) +tr′′ (t)) =t[r(t) +tr′ (t)]′, and r(t) +tr′
(t) is increasing in t is equivalent to tr(t) is convex since [tr(t)]′
=r(t) +tr′ (t). Hence, ψ(t) is convex.
Note that the conditions of Theorem 3.2 are satisfied by the generalized gamma distri-bution with parameters β ≤ 1 and α > 0 as Khaledi et al. (2011) proved that t2r′
(t) is an increasing function forX ∼GG(β, α), when β ≤1 andα >0. It is easy to verify that tr(t) is a decreasing function of t when β, α >0.
As one natural application, Theorem 3.2 guarantees that, for parallel systems of compo-nents having independent generalized gamma distributed lifetimes with parameters β ≤ 1 and α >0, the weakly majorized scale parameter vector leads to a larger system’s lifetime in the sense of the reverse hazard rate order.
The generalized gamma distribution includes many important distributions like exponen-tial (β = α = 1), Weibull (β = α) and gamma (β = 1) as special cases. Misra and Misra (2013) proved that in the case of gamma distribution with density function
f(t) = λ α i Γ(α)t α−1 e−λit, t >0,
when α >0 and n≥2,
(λ1, . . . , λn) w
≤(θ1, . . . , θn)⇒Xnλ:n≤rh Xnθ:n. (3.2)
Note that, when 0 < α ≤ 1, (3.2) can be seen as a particular case of Theorem 3.2 since gamma distribution is a particular case of generalized gamma distribution when β = 1.
Recently, Torrado and Kochar (2015) established, in Theorem 4.1, the reverse hazard rate ordering between parallel systems based on two sets of heterogeneous Weibull random variables with a common shape parameter and with scale parameters which are ordered according to a majorization order when the common shape parameter αsatisfies 0 < α≤1. So Theorem 4.1 in Torrado and Kochar (2015) can be seen as a particular case of Theorem 3.2 because Weibull distribution is a particular case of generalized gamma distribution when β =α.
4
Likelihood ratio ordering results
In this section, we investigate whether the result of Theorem 3.2 can be strengthened from reverse hazard rate ordering to likelihood ratio ordering. First, we consider the case whenn= 2 and the scale parameters of the scale model are ordered according to a weekly majorization order.
Theorem 4.1 LetXλ1, Xλ be independent nonnegative random variables withXλ1 ∼F(λ1t)
andXλ ∼F(λt), where λ1, λ >0 andF is an absolutely continuous distribution. LetYλ∗
1, Yλ
be independent nonnegative random variables with Yλ∗
1 ∼ F(λ
∗
1t) and Yλ ∼ F(λt), where
λ∗
1, λ > 0. Let r be the reverse hazard rate function of F. Assume tr(t) and tr
′
(t)/r(t) are both decreasing in t. Suppose λ∗
1 = min(λ, λ1, λ∗1), then (λ1, λ) w ≤(λ∗ 1, λ)⇒ r∗ 2:2(t) r2:2(t) is increasing in t. Proof. Let φ(t) = r ∗ 2:2(t) r2:2(t) = λ ∗ 1r(λ ∗ 1t) +λr(λt) λ1r(λ1t) +λr(λt) ,
and its derivative for t >0 is, φ′ (t) sign= t3 (λ∗ 1) 2 r′ (λ∗ 1t) +λ2r ′ (λt)(λ1r(λ1t) +λr(λt)) −t3(λ∗ 1r(λ ∗ 1t) +λr(λt)) λ21r ′ (λ1t) +λ2r ′ (λt) = λ1λ ∗ 1t3(λ ∗ 1r ′ (λ∗ 1t)r(λ1t)−λ1r(λ ∗ 1t)r ′ (λ1t)) +λ1λt3(λr ′ (λt)r(λ1t)−λ1r(λt)r ′ (λ1t)) +λλ∗ 1t3(λ ∗ 1r ′ (λ∗ 1t)r(λt)−λr(λ ∗ 1t)r ′ (λt)) = λ1λ ∗ 1t2r(λ ∗ 1t)r(λ1t) λ∗ 1t r′ (λ∗ 1t) r(λ∗ 1t) −λ1t r′ (λ1t) r(λ1t) +λ1λt2r(λt)r(λ1t) λtr ′ (λt) r(λt) −λ1t r′ (λ1t) r(λ1t) +λλ∗ 1t2r(λ ∗ 1t)r(λt) λ∗ 1t r′ (λ∗ 1t) r(λ∗ 1t) −λtr ′ (λt) r(λt) = ψ(λ∗ 1t)ψ(λ1t) (−η(λ ∗ 1t) +η(λ1t)) +ψ(λt)ψ(λ1t) (−η(λt) +η(λ1t)) +ψ(λ∗ 1t)ψ(λt) (−η(λ ∗ 1t) +η(λt)) , where ψ(t) =tr(t) and η(t) =−tr ′ (t) r(t). Note that ψ(t) ≥ 0 for all t ≥ 0 and η(t)≥ 0 since r′
(t) ≤ 0 because tr(t) is a decreasing function. By the assumptions, we know that ψ(t) is decreasing and η(t) is increasing in t. If λ∗ 1 = min(λ, λ1, λ∗1) and (λ1, λ) w ≤ (λ∗ 1, λ), then λ ∗ 1 ≤ λ ≤ λ1 or λ∗1 ≤ λ1 ≤ λ. When λ∗ 1 ≤λ ≤λ1, we have φ′ (t) sign= ψ(λ∗ 1t)ψ(λ1t) (−η(λ ∗ 1t) +η(λ1t)) +ψ(λt)ψ(λ1t) (−η(λt) +η(λ1t)) +ψ(λ∗ 1t)ψ(λt) (−η(λ ∗ 1t) +η(λt)) ≥ 0, since η(λ∗ 1t)≤η(λt)≤η(λ1t). Whenλ∗1 ≤λ1 ≤λ, we get φ′ (t) ≥ ψ(λt)ψ(λ1t) (−η(λ ∗ 1t) +η(λ1t)) +ψ(λt)ψ(λ1t) (−η(λt) +η(λ1t)) +ψ(λ1t)ψ(λt) (−η(λ ∗ 1t) +η(λt)) = 2ψ(λt)ψ(λ1t) (−η(λ ∗ 1t) +η(λ1t))≥0.
Therefore r∗
2:2(t)/r2:2(t) is increasing in t.
In the next result, we extend Theorem 3.2 from reverse hazard rate ordering to likelihood ratio ordering for n = 2.
Theorem 4.2 LetXλ1, Xλ be independent nonnegative random variables withXλ1 ∼F(λ1t)
andXλ ∼F(λt), whereλ1, λ >0andF is an absolutely continuous distribution. Letr be the
reverse hazard rate function of F. Let Yλ∗
1, Yλ be independent nonnegative random variables
with Yλ∗
1 ∼ F(λ
∗
1t) and Yλ ∼F(λt), where λ∗1, λ >0. Assume tr(t) and tr′(t)/r(t) are both
decreasing in t and t2r′ (t) is increasing in t. Suppose λ∗ 1 = min(λ, λ1, λ∗1), then (λ1, λ) w ≤(λ∗ 1, λ)⇒X2:2≤lr Y2:2. Proof. From Theorem 4.1, we know that r∗
2:2(t)/r2:2(t) is increasing in t under the given assumptions. By Theorem 3.2, (λ1, λ)
w
≤ (λ∗
1, λ) implies X2:2 ≤rh Y2:2. Thus the required result follows from Theorem 1.C.4 of Shaked and Shanthikumar (2007).
The conditions of Theorem 4.2 hold when the baseline distribution in the scale model is GG(β, α) with parametersα ≤β ≤1. We know from Khaledi et al. (2011) that forα, β >0, the functiontr(t) is decreasing intand forβ ≤1 andα >0, the functiont2r′
(t) is increasing in t. In Lemma 4.3, we show that the function tr′
(t)/r(t) is decreasing in t when α≤β. Lemma 4.3 Let X ∼ GG(β, α), α ≤ β, with reverse hazard rate r(t), then tr′
(t)/r(t) is a decreasing function.
Proof. The reverse hazard rate of GG(β, α) is r(t) = t α−1 e−tβ Rt 0 xα −1 e−xβ dx. From (A.21) in Khaledi et al. (2011), we know
tr ′
(t)
r(t) =α−1−βt
β−tr(t). (4.1)
Differentiating with respect tot, we get
tr ′ (t) r(t) ′ =−β2tβ−1 −r(t)−tr′ (t).
Note that, in general, the derivative of any reverse hazard rate with respect to t is r′ (t) = f ′ (t) F(t) −r 2(t). (4.2)
Combining these observations, we have
tr ′ (t) r(t) ′ = −β2tβ−1 −r(t)−tf ′ (t) F(t) +tr 2(t) = −β2tβ−1 +r(t) tr(t)−1−tf ′ (t) f(t) .
From Khaledi et al. (2011), we know thattr(t) is a decreasing function forβ, α >0 and also that limt→0tr(t) =α and limt→∞tr(t) = 0, then tr(t)≤α for all t >0. Then
tr ′ (t) r(t) ′ ≤ −β2tβ−1 +r(t) α−1−tf ′ (t) f(t) . (4.3) From (4.1) and (4.2), we get
tr ′ (t) r(t) = t r(t) f′ (t) F(t) −r 2(t) =t f′ (t) f(t) −r(t) , then tf ′ (t) f(t) = α−1−βt β −tr(t) +tr(t) = α−1−βtβ.
By replacing the above expression in (4.3), we have
tr ′ (t) r(t) ′ ≤ −β2tβ−1 +r(t) α−1− α−1−βtβ = −β2tβ−1 +βtβr(t) = βtβ −β t +r(t) ≤0 since tr(t)≤α≤β.
Theorem 4.2 says that the lifetime of a parallel system consisting of two types of gener-alized gamma components with parameters α ≤ β ≤ 1 is stochastically larger according to likelihood ratio ordering when the scale parameters are more dispersed according to weakly majorization.
Note that, when 0 < α ≤ 1, Theorem 3.4 in Zhao (2011) can be seen as a particular case of Theorem 4.2 since gamma distribution is a particular case of generalized gamma distribution whenβ = 1.
As an immediate consequence of Theorem 4.2, we have the following result which provides an upper bound of two random variables from a scale model.
Corollary 4.4 LetXλ1, Xλ2 be independent nonnegative random variables withXλi ∼F(λit)
for i = 1,2. Let Y1, Y2 be independent nonnegative random variables with a common
distri-bution Yi ∼ F(λt) for i = 1,2. Assume tr(t) and tr
′
(t)/r(t) are both decreasing in t and
t2r′
(t) is increasing in t. Suppose λ≤min(λ1, λ2), then
λ≤ λ1+λ2
2 ⇒X2:2 ≤lr Y2:2.
Next, we extend the study of likelihood ratio ordering between largest order statistics from the two-variable case to multiple-outlier scale models.
Theorem 4.5 Let X1, . . . , Xn be independent nonnegative random variables such thatXi ∼
F(λ1t) for i = 1, . . . , p and Xj ∼ F(λt) for j = p+ 1, . . . , n, with λ1, λ > 0 and F is
an absolutely continuous distribution. Let Y1, . . . , Yn n be independent nonnegative random
variables with Yi ∼ F(λ
∗
1t) for i = 1, . . . , p and Yj ∼ F(λt) for j = p + 1, . . . , n, with
λ∗
1, λ > 0. Let r be the reverse hazard rate function of F. Assume tr(t) and tr
′
(t)/r(t) are both decreasing in t. Suppose λ∗
1 = min(λ, λ1, λ∗1), then (λ1, . . . , λ1 | {z } p , λ, . . . , λ | {z } q )≤w (λ∗ 1, . . . , λ ∗ 1 | {z } p , λ, . . . , λ | {z } q )⇒ r ∗ n:n(t) rn:n(t) is increasing in t, where q=n−p.
Proof. From (3.1) we get the reverse hazard rate function of Xn:n:
rn:n(t) =pλ1r(λ1t) +qλr(λt), where p+q =n. Let φ(t) = r ∗ n:n(t) rn:n(t) = pλ ∗ 1r(λ ∗ 1t) +qλr(λt) pλ1r(λ1t) +qλr(λt) .
On differentiating φ(t) with respect to t, we get φ′ (t) sign= t3 p(λ∗ 1) 2 r′ (λ∗ 1t) +qλ2r ′ (λt)(pλ1r(λ1t) +qλr(λt)) −t3(pλ∗ 1r(λ ∗ 1t) +qλr(λt)) pλ21r ′ (λ1t) +qλ2r ′ (λt) = p2λ1λ ∗ 1t3(λ ∗ 1r ′ (λ∗ 1t)r(λ1t)−λ1r(λ ∗ 1t)r ′ (λ1t)) +pqλ1λt3(λr ′ (λt)r(λ1t)−λ1r(λt)r ′ (λ1t)) +pqλλ∗ 1t3(λ ∗ 1r ′ (λ∗ 1t)r(λt)−λr(λ ∗ 1t)r ′ (λt)) = p2λ1λ ∗ 1t2r(λ ∗ 1t)r(λ1t) λ∗ 1t r′ (λ∗ 1t) r(λ∗ 1t) −λ1t r′ (λ1t) r(λ1t) +pqλ1λt2r(λt)r(λ1t) λtr ′ (λt) r(λt) −λ1t r′ (λ1t) r(λ1t) +pqλλ∗ 1t2r(λ ∗ 1t)r(λt) λ∗ 1t r′ (λ∗ 1t) r(λ∗ 1t) −λtr ′ (λt) r(λt) = p2ψ(λ∗ 1t)ψ(λ1t) (−η(λ ∗ 1t) +η(λ1t)) +pqψ(λt)ψ(λ1t) (−η(λt) +η(λ1t)) +pqψ(λ∗ 1t)ψ(λt) (−η(λ ∗ 1t) +η(λt)) , where ψ(t) =tr(t) and η(t) =−tr ′ (t) r(t).
Note that ψ(t), η(t) ≥ 0 for all t ≥ 0. By the assumptions, we know that ψ(t) is de-creasing and η(t) is increasing in t. If λ∗
1 = min(λ, λ1, λ∗1) and (λ1, . . . , λ1, λ, . . . , λ) w ≤ (λ∗ 1, . . . , λ ∗ 1, λ, . . . , λ), then λ ∗ 1 ≤ λ ≤ λ1 or λ∗1 ≤ λ1 ≤ λ. When λ∗1 ≤ λ ≤ λ1, it is easy to check thatφ′ (t)≥0 sinceη(λ∗ 1t)≤η(λt)≤η(λ1t). When λ∗1 ≤λ1 ≤λ, we get φ′ (t) ≥ p2ψ(λt)ψ(λ1t) (−η(λ ∗ 1t) +η(λ1t)) +pqψ(λt)ψ(λ1t) (−η(λt) +η(λ1t)) +pqψ(λ1t)ψ(λt) (−η(λ ∗ 1t) +η(λt)) = npψ(λt)ψ(λ1t) (−η(λ ∗ 1t) +η(λ1t))≥0. Therefore r∗ n:n(t)/rn:n(t) is increasing in t.
In the next result, we extend Theorem 4.2 from the two-variable case to multiple-outlier scale models.
Theorem 4.6 Let X1, . . . , Xn be independent nonnegative random variables such thatXi ∼
F(λ1t) for i = 1, . . . , p and Xj ∼ F(λt) for j = p+ 1, . . . , n, with λ1, λ > 0 and F is
an absolutely continuous distribution. Let Y1, . . . , Yn n be independent nonnegative random
variables with Yi ∼ F(λ
∗
1t) for i = 1, . . . , p and Yj ∼ F(λt) for j = p + 1, . . . , n, with
λ∗
1, λ > 0. Let r be the reverse hazard rate function of F. Assume tr(t) and tr
′
(t)/r(t) are both decreasing in t and t2r′
(t) is increasing in t. Suppose λ∗ 1 = min(λ, λ1, λ∗1), then (λ1, . . . , λ1 | {z } p , λ, . . . , λ | {z } q )≤w (λ∗ 1, . . . , λ ∗ 1 | {z } p , λ, . . . , λ | {z } q )⇒Xn:n ≤lrYn:n.
Proof. From Theorem 4.5, we know that r∗
n:n(t)/rn:n(t) is increasing in t when tr(t) and
tr′
(t)/r(t) are both decreasing in t. Since (λ1, . . . , λ1, λ, . . . , λ)
w ≤ (λ∗ 1, . . . , λ ∗ 1, λ, . . . , λ)and t2r′
(t) is increasing in t , then Xn:n ≤rh Yn:n from Theorem 3.2. Thus the required result
follows from Theorem 1.C.4 in Shaked and Shanthikumar (2007).
Note that, when 0 < α≤1, Theorem 3.1 in Zhao and Balakrishnan (2013) can be seen as a particular case of Theorem 4.6 whenλ∗
1 ≤λ1 ≤λsince gamma distribution is a particular case of generalized gamma distribution when β = 1.
Next, we establish the analog of Theorem 4.6 when both the baseline distributions and the scale parameters are different in the multiple-outlier scale models.
Theorem 4.7 Let X1, . . . , Xn be independent nonnegative random variables such thatXi ∼
F(λ1t) for i = 1, . . . , p and Xj ∼ G(λt) for j = p+ 1, . . . , n, with λ1, λ > 0 and F is an
absolutely continuous distribution. Let X∗
1, . . . , X
∗
n be n independent nonnegative random
variables with X∗ i ∼ F(λ ∗ 1t) for i = 1, . . . , p and X ∗ j ∼ G(λt) for j = p+ 1, . . . , n, with λ∗
1, λ > 0. Let rF and rG be the reverse hazard rate functions of F and G, respectively.
Assume trF(t) and tr
′
F(t)/rF(t) are both decreasing in t. Suppose rF(t)/rG(t) is increasing
in t, then λ∗ 1 = min(λ, λ1, λ ∗ 1)⇒Xn:n≤lr X ∗ n:n.
Proof. From (3.1) we get the reverse hazard rate function of Xn:n:
where p+q =n. Similarly the reverse hazard rate function ofX∗ n:n is r∗ n:n(t) =pλ ∗ 1rF(λ1t) +qλrG(λt). Observe that Xj st = X∗
j for j = p+ 1, . . . , n. By the assumptions, we know that trF(t) is
decreasing intandλ∗
1 ≤λ1, then we haveXn:n≤rh Xn∗:nsincern:n(t)≤rn∗:n(t) for allt. From
Theorem 1.C.4 in Shaked and Shanthikumar (2007), it is enough to prove that the ratio of their reverse hazard rate functions is increasing, i.e., we need to show that the function
φ(t) = r ∗ n:n(t) rn:n(t) = pλ ∗ 1rF(λ ∗ 1t) +qλrG(λt) pλ1rF(λ1t) +qλrG(λt)
is increasing in t. On differentiating φ(t) with respect to t, we get φ′ (t) sign= t3 p(λ∗ 1) 2 r′ F(λ ∗ 1t) +qλ2r ′ G(λt) (pλ1rF(λ1t) +qλrG(λt)) −t3(pλ∗ 1rF(λ ∗ 1t) +qλrG(λt)) pλ21r ′ F(λ1t) +qλ 2r′ G(λt) . Let us denote: ψF (t) =trF(t),ηF(t) =−t r′ F(t) rF(t) , ψG(t) =trG(t) and ηG(t) =−t r′ G(t) rG(t) , then the derivative of φ(t) can be rewritten as
φ′ (t) sign= p2ψF(λ ∗ 1t)ψF(λ1t) (−ηF(λ ∗ 1t) +ηF(λ1t)) +pqψG(λt)ψF(λ1t) (−ηG(λt) +ηF(λ1t)) +pqψF(λ ∗ 1t)ψG(λt) (−ηF(λ ∗ 1t) +ηG(λt)) .
The assumption rF(t)/rG(t) is increasing in t is equivalent to ηF(t) ≤ ηG(t) for all t. In
addition, we know that ηF (t) is increasing in t and λ∗1 ≤λ then ηF(λ∗1t)≤ ηF(λt)≤ ηG(λt)
for all t. By the assumptions, we know thatψF(t) is decreasing int and λ∗1 ≤λ1, then
φ′ (t) ≥ p2ψF(λ ∗ 1t)ψF(λ1t) (−ηF(λ ∗ 1t) +ηF(λ1t)) +pqψG(λt)ψF(λ1t) (−ηF(λ ∗ 1t) +ηF(λ1t)) = pψF(λ1t) (−ηF(λ ∗ 1t) +ηF(λ1t)) (pψF(λ ∗ 1t) +qψG(λt))≥0,
since ηF(t) is increasing in t. Therefore r
∗
Acknowledgements
This article is dedicated to our friend Ramesh Gupta for his many contributions and for his encouragement to new researchers. The authors wish to thank the Associate Editor and an anonymous referee for their comments which have greatly improved the initial ver-sion of this manuscript. The research of Nuria Torrado was supported by the Portuguese Government through the Funda¸c˜ao para a Ciˆencia e Tecnologia (FCT) under the grant SFRH/BPD/91832/2012 and partially supported by the Centro de Matem´atica da Univer-sidade de Coimbra (CMUC) under the project PEst-C/MAT/UI0324/2013.
References
Ali, M.M., Woo, J. and Nadarajah, S. (2008). Generalized gamma variables with drought application. Journal of the Korean Statistical Society 37, 37–45.
Balakrishnan, N. and Zhao, P. (2013). Hazard rate comparison of parallel systems with het-erogeneous gamma components. Journal of Multivariate Analysis 113, 153–160.
Bon, J.L. and P˘alt˘anea, E.(1999). Ordering properties of convolutions of exponential random variables. Lifetime Data Analysis 5, 185–192.
Chen, Y., Karagiannidis, G.K., Lu, H. and Cao, N. (2012). Novel Approximations to the Statistics of Products of Independent Random Variables and Their Applications in Wireless Communications IEEE Transactions on Vehicular Technology 61, 443–454.
Dykstra, S.C., Kochar, S.C. and Rojo, J.(1997). Stochastic comparisons of parallel systems of heterogeneous exponential components. Journal of Statistical Planning and Inference 65, 203–211.
Fang, L. and Zhang, X. (2013). Stochastic comparisons of series systems with heterogeneous Weibull components. Statistics and Probability Letters 83, 1649–1653.
Gupta, R.C., Gupta, R.D. and Gupta, P.L. (2006). Monotonicity of the (reversed) hazard rate of the (maximum) minimum in bivariate distributions. Metrika 63, 223–241.
Hu, T.(1995). Monotone coupling and stochastic ordering of order statistics. Syst. Sci. Math. Sci. 8, 209–214.
Joo, S. and Mi, J. (2010). Some properties of hazard rate functions of systems with two compo-nents. Journal of Statistical Planning and Inference 140, 444–453.
Khaledi, B-E., Farsinezhad, S. and Kochar, S.C. (2011). Stochastic comparisons of order statistics in the scale model. Journal of Statistical Planning and Inference 141, 276–286.
Khaledi, B-E.and Kochar, S.C.(2007). Stochastic orderings of order statistics of independent random variables with different scale parameters. Communications in Statistics - Theory and Methods 36, 1441–1449.
Khaledi, B-E.and Kochar, S.C.(2000). Some new results on stochastic comparisons of parallel systems. J. Appl. Probab 37, 1123–1128.
Khaledi, B-E. and Kochar, S.C. (2006). Weibull distribution: some stochastic comparisons results. Journal of Statistical Planning and Inference 136, 3121–3129.
Kochar, S. and Kirmani, S. (1996). Some new results on normalized spacings from restricted families of distributions. Journal of Statistical Planning and Inference 47, 47–57.
Kochar, S.andXu, M.(2009). Comparisons of parallel systems according to the convex transform order. Journal of Applied Probability 46, 342–352.
Kochar, S.C.(2012). Stochastic Comparisons of Order Statistics and Spacings: A Review. ISRN Probability and Statisticsvol. 2012, Article ID 839473, 47 pages, 2012. doi:10.5402/2012/839473.
Lihong, S. and Xinsheng, Z. (2005). Stochastic comparisons of order statistics from gamma distributions. Journal of Multivariate Analysis 93, 112–121.
Manning, W.G., Basu, A. and Mullahy, J. (2005). Generalized modeling approaches to risk adjustment of skewed outcomes data. Journal of Health Economics 24, 465–488.
Marshall, A. W., Olkin, I. and Arnold, B. C. (2011). Inequalities: Theory of Majorization and Its Applications. Springer, New York.
Marshall, A. W. and Olkin, I.(2007). Life distributions. Springer, New York.
Misra, N.and Misra, A.K.(2013). On comparison of reverse hazard rates of two parallel systems comprising of independent gamma components.Statistics and Probability Letters 83, 1567–1570.
Pledger, P. and Proschan, F. (1971). Comparisons of order statistics and of spacings from heterogeneous distributions. In:Rustagi, J.S.(Ed.), Optimizing Methods in Statistics. Academic Press, New York, pp 89–113.
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.
Torrado, N., Lillo, R. E.andWiper, M. P.(2010). On the conjecture of Kochar and Korwar.
Journal of Multivariate Analysis 101, 1274–1283.
Torrado, N.and Veerman, J. J. P.(2012). Asymptotic reliability theory ofk-out-of-nsystems.
Journal of Statistical Planning and Inference 142, 2646–2655.
Torrado, N. and Lillo, R. E.(2013). On stochastic properties of spacings with applications in multiple-outlier models. In : Li, H. and Li, X.(Ed.), Stochastic Orders in Reliability and Risk. Springer Lecture Notes in Statistics, 103–123.
Torrado, N. andKochar, S.C. Stochastic order relations among parallel systems from Weibull distributions. To appear in Journal of Applied Probability 52(1), March 2015.
Zhao, P. (2011). On parallel systems with heterogeneous gamma components. Probability in the Engineering and Informational Sciences 25, 369–391.
Zhao, P. and Balakrishnan, N. (2011). New results on comparisons of parallel systems with heterogeneous gamma components. Statistics and Probability Letters 81, 36–44.
Zhao, P. and Balakrishnan, N. (2013). Comparisons of largest order statistics from multiple-outlier gamma models. Methodol Comput Appl Probab. DOI 10.1007/s11009-013-9377-0.
