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Some new results on the Rényi quantile entropy Ordering
Accepted Manuscript Some new results on the R´enyi quantile entropy Ordering Lei Yan, Dian-tong Kang PII: DOI: Reference:
S1572-3127(16)30003-X http://dx.doi.org/10.1016/j.stamet.2016.04.003 STAMET 528
To appear in:
Statistical Methodology
Received date: 23 June 2015 Revised date: 31 March 2016 Accepted date: 25 April 2016 Please cite this article as: L. Yan, D.-t. Kang, Some new results on the R´enyi quantile entropy Ordering, Statistical Methodology (2016), http://dx.doi.org/10.1016/j.stamet.2016.04.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Some new results on the R´ enyi quantile entropy Ordering Lei Yan∗ School of Economics and Management, Lanzhou University of Technology, Lanzhou, Gansu 730050, People’s Republic of China E-mail: [email protected]
Dian-tong Kang School of Mathematics and Statistics, Hexi University, Zhangye, Gansu 734000, People’s Republic of China E-mail: [email protected]
Abstract R´enyi (1961) proposed the R´enyi entropy. Ebrahimi and Pellerey (1995) and Ebrahimi (1996) proposed the residual entropy. Recently, Nanda et al. (2014) obtained a quantile version of the R´enyi residual entropy, the R´enyi residual quantile entropy (RRQE). Base on the RRQE function, they defined a new stochastic order, the R´enyi quantile entropy (RQE) order, and studied some properties of this order. In this paper, we focus on further properties of this new order. Some characterizations of the RQE order are investigated, closure and reversed closure properties are obtained, meanwhile, some illustrative examples are shown. As applications of a main result, the preservation of the RQE order in several stochastic models are discussed. Keywords: R´enyi’s residual quantile entropy, R´enyi residual quantile entropy order, closure property, proportional odds model, record valus model MSC: 60E15, 62N05, 62E10
1. Introduction The notion of entropy, originated from thermodynamics and statistical mechanics, is of importance in some scientific and technological areas such as communication theory, physics, probability, statistics and economics. It measures uncertainty of a physical system. The Shannon entropy plays a central role in information theory since it was introduced mathematically by Shannon (1948). Let X be a nonnegative random variable (rv) representing the lifetime of a device or a living thing with absolutely continuous cumulative distribution function FX (x), survival function F X (x) = 1−FX (x) and probability density function fX (x). The Shannon entropy of X is defined by Z HX = −E[ln fX (X)] = −
+∞
fX (x) ln fX (x)dx.
(1.1)
0
HX measures the uncertainty contained in fX (x) about the predictability of an outcome of X. In continuous case, HX is also referred to as the Shannon differential entropy. In recent years, the literature on information theory has grown quite voluminous. It has been found that the Shannon entropy has lot of applications in different areas such as physics, probability and statistics, communication theory and economics. (see, for instance, Taneja (1990, 2001), Kumar and Taneja (2011), Nanda et al. (2014), and Kayal (2015).) ∗
Corresponding author.
1
In the literature several dynamic generalizations of HX have been proposed (see, for example, Khinchin (1957), Taneja (1990), Gupta and Nanda (2002), Nanda and Paul (2006a), Asadi and Zohrevand (2007), Di Crescenzo and Longobardi (2009), Kumar and Taneja (2011), Khorashadizadeh et al. (2013), Nanda et al. (2014), and Kayal (2015)). Ebrahimi and Pellerey (1995) and Ebrahimi (1996) presented the residual entropy as a useful dynamic measure of uncertainty. The residual life of X is defined by Xt = [X − t |X > t] . The residual entropy of X at time t is defined as the differential entropy of Xt . Formally, for all t ≥ 0, the residual entropy of X is given by HX (t) = HXt = −E[ln fXt (Xt )]. Ebrahimi and Pellerey showed that Z +∞ fX (x) fX (x) ln dx. (1.2) HX (t) = − F X (t) F X (t) t
Given that a component survived up to time t, then HX (t) = HXt measures the uncertainty of the residual life Xt . Some authors did their research works about the residual entropy, various results concerning the residual entropy HX (t) have been obtained by Ebrahimi and Kirmani (1996a, b), Asadi and Ebrahimi (2000), Di Crescenzo and Longobardi (2002, 2004), Nanda and Paul (2006b, c), Navarro et al. (2010), Kumar and Taneja (2011), Nanda et al. (2014), and Kayal (2015). Since the work of Shannon (1948), many authors have generalized the Shannon entropy in different directions according to their research interests. One generalization of the Shannon entropy is due to R´enyi (1961). For a nonnegative rv X the R´enyi entropy is defined as Z ∞ 1 X ln (fX (x))β dx, β > 0, β 6= 1, (1.3) Hβ = 1−β 0 which reduces to the Shannon entropy as β → 0+ . For a used item, Abraham and Sankaran (2005) proposed a dynamic version of R´enyi entropy, the residual R´enyi entropy, given by β Z ∞ fX (x) 1 ln dx, β > 0, β 6= 1. (1.4) HβX (t) = 1−β F X (x) t
When X has the age t, for different values of β, HβX (t) gives the spectrum of R´enyi’s information on the residual life of X. Clearly, HβX (0) = HβX . A number of recent studies by the authors have indicated that the quantile function (QF) defined by −1 QX (u) = FX (u) = inf{x | FX (x) ≥ u}, u ∈ [0, 1] (1.5) is a convenient and equivalent alternative to the distribution function in modelling and analysis −1 of statistical data. Sometimes the quantile function FX (·) is also called the right-continuous inverse function of FX (·) (or of X). For the usefulness of QF in dynamic study of entropy, we refer to Sunoj and Sankaran (2012), Sunoj et al. (2013), Nanda et al. (2014) and the references therein.
More recently, Nanda et al. (2014) obtained a quantile version of the R´enyi residual entropy, termed the R´enyis residual quantile entropy (RRQE). They showed that the R´enyi’s residual quantile entropy is efficient and convenient and is an equivalent alternative to the R´enyi’s residual entropy in describing ageing properties of life distributions and in making stochastic comparisons. On the basis of RRQE function, they defined two new nonparametric classes of life distributions, the increasing (decreasing) R´enyi’s residual quantile entropy (IRRQE (DRRQE)) classes, and they defined a new stochastic order, the R´enyi’s residual quantile entropy (RQE) order, to compare the uncertainties of residual lives of two random lives X and Y at the age points with equal survival probabilities. They studied some ageing and ordering properties of HβX (QX (u)). 2
In this paper, we focus on the further properties of the RQE ordering. Nanda et al. (2014) gave the following lemma which will play an important role in the proofs of the paper. R1 Lemma 1.1. Let f (u, x) : [0, 1] × <+ → <+ be such that u f (u, x)dx ≥ 0 for all u ∈ [0, 1], and R1 h(x) be any nonnegative increasing function in x. Then u f (u, x)h(x)dx ≥ 0.
The rest of the paper is organized as follows. In Section 2, we investigate some characterizations of the RQE order. In section 3 we study closure and reversed closure properties of the RQE order under several reliability transformations, mainly including linear transformations, parallel and series operations, increasing convex and increasing concave transformations. Meanwhile, some illustrative examples are shown. As applications of main result Theorem 2.1, in section 4 we deal with the preservation of the RQE order in several stochastic models, including proportional hazard rate model, proportional reversed hazard rate model, proportional odds model, and the record values model. Throughout this paper, the terms increasing and decreasing are used in a non-strict sense. Assume that all integrals involved are finite, and ratios are well defined whenever written. 2. Characterization results In this section we obtain some characterization results of the RQE ordering. Suppose that X is an absolutely continuous nonnegative rv as described in Section 1. From (1.5) we have FX (QX (u)) = u. Denote the density quantile function by fX (QX (u)) and the quantile density function by qX (u) = Q0X (u), where the prime denotes the differentiation. Differentiating both sides of FX (QX (u)) = u yields qX (u)fX (QX (u)) = 1.
(2.1)
Sunoj and Sankaran (2012) introduced a quantile versions of the Shannon residual entropy (1.2). And they showed that ψX (u) = HX [QX (u)] = ln(1 − u) +
1 1−u
Z
1
ln[qX (p)]dp.
u
Recently, Nanda et al. (2014) obtained a quantile version of the R´enyi residual entropy, the R´enyi residual quantile entropy (RQE), defined by ψβX (u) := HβX (QX (u)) , for all u ∈ [0, 1]. They showed that, for all u ∈ [0, 1], (1 − β)HβX (QX (u)) = ln = ln
∞
QX (u)
(fX (x))β dx (1 − u)β
β 1 qX (p)dp (1 − u)qX (p) u Z 1 (qX (p))1−β = ln dp (1 − u)β u
Z
1
Z
3
= ln = ln
Z
Z
1
u
1
u
(fX [QX (p)])β−1 dp. (1 − u)β
(2.2)
(qX (p))1−β dp − β ln(1 − u).
For different values of β, HβX (QX (u)) in (2.2) gives the spectrum of R´enyi information contained in the conditional density about the predictability of an outcome of X until 100(1 − u)% point of distribution. In other words, HβX (QX (u)) measures the uncertainty of residual life XF −1 (u) , −1 that is, HβX (QX (u)) measures the uncertainty of X at age point FX (u).
X
Recently, Nanda et al. (2014) defined a new stochastic order based on the comparison of functions HβX (QX (u)) and HβY (QY (u)) corresponding to two nonnegative random variables X and Y . Definition 2.1. X is said to be smaller than Y in the R´enyi quantile entropy order (written as X ≤RQE Y ), if HβX (QX (u)) ≤ HβY (QY (u)) for all u ∈ [0, 1]. We obtain the following theorem which will play a key role in the proofs of results in this paper. Theorem 2.1. Let X and Y be two absolutely continuous nonnegative random variables with probability density functions fX and gY , and distribution functions FX and GY , respectively. (a) If 0 < β < 1, then X ≤RQE Y if, and only if, # " 1−β Z ∞ fX (x) β − 1 dx ≥ 0, [fX (x)] gY [G−1 t Y (FX (x))] (b) If β > 1, then X ≤RQE Y if, and only if, " 1−β # Z ∞ fX (x) β [fX (x)] 1 − dx ≥ 0, gY [G−1 t Y (FX (x))]
for all t ≥ 0.
(2.3)
for all t ≥ 0.
(2.4)
Proof. (a) If 0 < β < 1. From Definition 2.1 and (2.2) we have that X ≤RQE Y if, and only if, Z
1
u
(fX [QX (p)])
β−1
dp ≤
Z
1
(gY [QY (p)])β−1 dp
(2.5)
u
for all u ∈ [0, 1]. Letting p = FX (x) in two integrands of the above inequality and then letting −1 FX (u) = t in two integrals of both sides, we see that inequality (2.5) is equivalent to inequality (2.3). (b) The proof is similar to that of the case (a) by noting that when β > 1, 1 − β < 0. Thus the proof is complete. Example 2.1. Let X and Y be two exponential random variables with survival functions, respectively, F X (x) = e−λ1 x , GY (x) = e−λ2 x , for all x ≥ 0.
It can be verified that
fX (x) λ1 = . λ gY [G−1 (F (x))] 2 X Y 4
Making use of Theorem 2.1 we get that, for any β > 0, β 6= 1, (a) if 0 < λ2 ≤ λ1 , then X ≤RQE Y ; (b) if 0 < λ1 ≤ λ2 , then Y ≤RQE X. Below we recall several stochastic orders which will be useful in the sequel. For more details about these orders one can see Shaked and Shanthikumar (2007). Definition 2.2. Let X and Y be two nonnegative random variables. (a) X is said to be smaller than Y in the dispersive order (denoted by X ≤disp Y ) if −1 −1 −1 FX (β) − FX (α) ≤ G−1 Y (β) − GY (α),
equivalently,
fX (x) ≥ gY G−1 Y (FX (x)) ,
for all 0 < α < β < 1, for all x ≥ 0.
(2.6)
(b) X is said to be smaller than Y in the convex −1 transform order (denoted by X ≤c Y ) if G−1 Y [FX (x)] is convex, equivalently, fX (x) gY GY (FX (x)) is increasing in x ≥ 0. (c) X is said to be smaller than Y in the usual stochastic order (denoted by X ≤st Y ) if F X (x) ≤ GY (x) for all x ≥ 0. By means of Theorem 2.1 and Definition 2.2 we obtain the following theorem, which gives a sufficient condition for the RQE order. Theorem 2.2. If X ≤disp Y , then X ≤RQE Y . Proof. Suppose that X ≤disp Y . Then from (2.6) we have fX (x) ≥ gY G−1 Y (FX (x)) for x ≥ 0. By Theorem 2.1 the proof is complete. Below we give an illustrative example that meets the RQE order. Example 2.2. Let X and Y be two Pareto random variables having their survival functions, respectively, α1 α2 λ λ F X (x) = , GY (x) = , for all x ≥ 0. λ+x λ+x Denoted by X ∼ P (α1 , λ), Y ∼ P (α2 , λ), where α1 , α2 , λ are positive real numbers. Then the RQE order is determined by the parameters α1 and α2 irrespective of λ > 0. Indeed, the density functions of X and Y are, respectively, α1 fX (x) = λ
λ λ+x
α1 +1
α2 gY (x) = λ
,
λ λ+x
α2 +1
,
for all x ≥ 0.
It can be verified that −1
G−1 Y [FX (x)] = GY Hence,
α x α12 F X (x) = λ 1 + − λ. λ
α α fX (x) x α21 −1 −1 = 1 1+ . α2 λ gY GY (FX (x))
5
Clearly, if α1 ≥ α2 > 0, then X ≤disp Y , and if 0 < α1 ≤ α2 , then X ≥disp Y . By means of Theorem 2.2 we get the following results: for any β > 0, β 6= 1, (a) if 0 < α2 ≤ α1 , then X ≤RQE Y ; (b) if 0 < α1 ≤ α2 , then Y ≤RQE X. From Theorem 2.1 and definition of order ≤c , we get another sufficient condition for the RQE order as follows. Theorem 2.3. If X ≤c Y, and fX (0) ≥ gY (0) > 0, then X ≤RQE Y . Proof. Suppose that X ≤c Y . Then according to the definition of the order X ≤c Y , we have fX (x) the function g G−1 is increasing in x ≥ 0. Thus Y [ Y (FX (x))] fX (x) f (0) −1 ≥ X ≥ 1. gY (0) gY GY (FX (x))
From this fact, by Theorem 2.1 the required result follows.
By means of Theorem 2.1 it is easy to verify that the following Theorems 2.4, 2.5 and 2.6 hold. Theorem 2.4. Let X be a nonnegative continuous random variable and a be any real constant. Then (a) If a ≥ 1, then X ≤RQE aX. (b) If 0 < a ≤ 1, then aX ≤RQE X. Theorem 2.5. Let X and Y be two nonnegative continuous random variables and a be any real constant. Assume that X ≤RQE Y . (a) If a ≥ 1, then X ≤RQE aY . (b) If 0 < a ≤ 1, then aX ≤RQE Y . Theorem 2.6. If X ≤RQE Y, then X ≤RQE Y + c for any real number c. Remark 2.1. Theorem 2.6 shows that the RQE order is location independent. Theorem 2.7. X ≤RQE Y and X ≥RQE Y hold simultaneously if, and only if, X =disp Y , here, X =disp Y means that X =st Y + k, where k is constant. Proof. From Theorem 2.1 we have that X ≤RQE Y and X ≥RQE Y hold simultaneously, if, and only if, " # 1−β Z ∞ f (x) X [fX (x)]β − 1 dx ≡ 0, for all t ≥ 0. gY [G−1 t Y (FX (x))] Differentiating both sides of above equality yields fX (x) −1 = 1, gY GY (FX (x)) 6
for all x ≥ 0,
which is equivalent to that X =disp Y , where X =disp Y means that X =st Y + k, and k is any real constant. And this reasoning is reversible. Therefore the proof is complete. 3. Closure and reversed closure properties of the RQE Order In this section, we investigate the closure and reversed closure properties of the RQE order under several reliability transforms, mainly including linear transformations, parallel and series operations, increasing convex and increasing concave transforms. First we explore the closure property of the RQE order under linear transformations. According to Theorems 2.5 and 2.6, the following theorem is easy. Theorem 3.1. Let X ≤RQE Y . Then for any real numbers a, b, c and d with 0 < a ≤ c, 0 ≤ b ≤ d, we have aX + b ≤RQE cY + d. As a direct consequence we have the following corollary. Corollary 3.1. If X ≤RQE Y . Then for any real numbers a and b with a > 0, b ≥ 0, we have aX + b ≤RQE aY + b. Remark 3.1. Corollary 3.1 indicates that a positive linear system preserves the RQE order. Corollary 3.1 also says that the RQE order has the closure property with respect to a positive linear system. Corollary 3.1 also states that the RQE order has the closure property under a positive linear transformation. Let X and Y be two nonnegative continuous random variables, let X1 , . . ., Xn and Y1 , . . . Yn be independent and identically distributed (iid) copies of X and Y , respectively. Denote by X1:n = min{X1 , . . . , Xn },
Xn:n = max{X1 , . . . , Xn }.
Similarly, Y1:n and Yn:n . Denote the survival function and the density function of X1:n by F X1:n and fX1:n , respectively. Denote the distribution function and the density function of Xn:n by FXn:n and fXn:n , respectively. Similarly, GY1:n , gY1:n , GYn:n , and gYn:n . We have the following theorem. Theorem 3.2. If X ≤RQE Y, then Xn:n ≤RQE Yn:n . Proof. We only give the proof for the case 0 < β < 1, the proof of the case β > 1 is similar and hence is omitted here. Suppose that X ≤RQE Y . Then from (2.3) we have, for all t ≥ 0, " # 1−β Z ∞ fX (x) β − 1 dx ≥ 0, for all t ≥ 0. (3.1) [fX (x)] gY [G−1 t Y (FX (x))] It is easy to see that
n−1
fXn:n (x) = n [FX (x)]
and hence h
fXn:n (x)
gYn:n G−1 Yn:n
· fX (x),
fX (x) i= −1 , gY GY (FX (x)) (FXn:n (x)) 7
for all x ≥ 0.
(3.2)
n−1
Since the function h(x) = (n [FX (x)] )β is nonnegative increasing in x ≥ 0, so, from (3.1), (3.2) and Lemma 1.1 we obtain that !1−β Z +∞ f (x) Xn:n [fXn:n (x)]β − 1 dx ≥ 0, for all t ≥ 0, gYn:n G−1 t Yn:n (FXn:n (x)) again, by Theorem 2.1 we get that Xn:n ≤RQE Yn:n . Therefore the proof is complete.
Remark 3.2. Theorem 3.2 indicates that the parallel systems preserve the RQE order. Theorem 3.2 also says that the RQE order has the closure property with respect to the parallel systems. Example 3.1. Let X and Y be two exponential random variables with survival functions, respectively, F X (x) = e−2x , GY (x) = e−x , for all x ≥ 0.
Then X2:2 and Y2:2 have distribution functions, respectively, FX2:2 (x) = 1 − e−2x
2
,
GY2:2 (x) = 1 − e−x
2
,
for all x ≥ 0.
According to Example 2.1 we have X ≤RQE Y . By Theorem 3.2 we obtain X2:2 ≤RQE Y2:2 . With a similar manner to above Theorem 3.2, one can prove that the following theorem holds. Theorem 3.3. If X1:n ≤RQE Y1:n , then X ≤RQE Y . Remark 3.3. Theorem 3.3 says that series systems reversely preserve the RQE order. Theorem 3.3 also indicates that the RQE order has the reversed closure property under the formation of series systems. Example 3.2. Let X and Y be two Pareto random variables having survival functions, respectively, 2 1 1 , GY (x) = , for all x ≥ 0. F X (x) = 1+x 1+x
Then X1:2 and Y1:2 are also Pareto random variables having survival functions, respectively, F X1:2 (x) =
1 1+x
4
,
GY1:2 (x) =
1 1+x
2
,
for all x ≥ 0.
In view of Example 2.2 we get that X1:2 ≤RQE Y1:2 . On using Theorem 3.3 we obtain that X ≤RQE Y . Let X and Y be two nonnegative continuous random variables, let X1 , X2 , . . . and Y1 , Y2 , . . . be sequences of iid copies of X and Y , respectively. Suppose that N is a positive integer-valued random variable with the probability mass function pN (n) = P {N = n}, n = 1, 2, . . ., and N is independent of Xi ’s and Yi ’s. Denote by XN :N = max{X1 , . . . , XN },
YN :N = max{Y1 , . . . , YN }.
Then the following theorem can be viewed as an extension of Theorem 3.2 from a finite number n to a random number N . The proof of the theorem is similar to that of the following Theorem 3.5 and hence omitted here. 8
Theorem 3.4. If X ≤RQE Y , then XN :N ≤RQE YN :N . Remark 3.4. Theorem 3.4 says that a random parallel system preserves the RQE order. Theorem 3.4 also indicates that the RQE order has the closure property with respect to a random parallel system. Example 3.3. Let X and Y be two exponential random variables with survival functions, respectively, F X (x) = e−2x , GY (x) = e−x , for all x ≥ 0.
Assume that N is a Bernoulli random variable with a probability mass function pN (n) = P {N = n} = 1/2, n = 1, 2, and N is independent of X and Y . Then XN :N and YN :N have distribution functions, respectively, 2 o 1 n , 1 − e−2x + 1 − e−2x 2 2 o 1 n 1 − e−x + 1 − e−x , for all x ≥ 0. GYN :N (x) = 2 In light of Example 2.1 we know X ≤RQE Y . Then from Theorem 3.4 we obtain that XN :N ≤RQE YN :N . FXN :N (x) =
Denote by X1:N = min{X1 , . . . , XN }, Y1:N = min{Y1 , . . . , YN }. On using a method similar to above Theorem 3.4 we have the following result, which can be viewed as an extension of Theorem 3.3 from a finite number n to a random number N . Theorem 3.5. If X1:N ≤RQE Y1:N , then X ≤RQE Y . Proof. We only give the proof for the case 0 < β < 1, the proof of the case β > 1 is similar and hence is omitted here. Suppose that X1:N ≤RQE Y1:N . Then from (2.3) we have 1−β Z ∞ f (x) h X1:N i [fX1:N (x)]β − 1 dx ≥ 0, for all t ≥ 0. (3.3) −1 t gY1:N GY (FX1:N (x)) 1:N
It is easy to see that X1:N and Y1:N have density functions, respectively, # " ∞ X n−1 F X (x) pN (n) , and fX1:N (x) = fX (x) · n n=1
"
gY1:N (x) = gY (x) · n
∞ X
n=1
GY (x)
n−1
#
pN (n) ,
where pN (n) = P(N = n), n = 1, 2, . . . , is the probability mass function of N . Shaked and Shanthikumar (2007) showed that G−1 [FX1:N (x)] = G−1 Y [FX (x)] , Y 1:N
thus we obtain that
gY1:N
h
fX1:N (x) G−1
Y1:N
(FX1:N (x))
i=
f (x) X , gY G−1 Y (FX (x)) 9
for all x ≥ 0.
(3.4)
Since the function h(x) =
!−1 ∞ X n−1 n pN (n) F X (x) n=1
is nonnegative increasing in x ≥ 0, so, from (3.3), (3.4) and Lemma 1.1 we get " # 1−β Z ∞ fX (x) β [fX (x)] − 1 dx ≥ 0, for all t ≥ 0, gY [G−1 t Y (FX (x))] in turn, by (2.3) which asserts that X ≤RQE Y . Therefore the proof follows.
Remark 3.5. Theorem 3.5 indicates that the random series systems reversely preserve the RQE order. Theorem 3.5 also says that the RQE order has the reversed closure property with respect to the random series systems. Example 3.4. Let X and Y be two Pareto random variables having survival functions, respectively, 2 1 1 , GY (x) = F X (x) = , for all x ≥ 0. 1+x 1+x
Assume that N is a geometric random variable with probability mass function pN (n) = P {N = n} = 1/2n , n = 1, 2, . . ., and N is independent of X and Y . It is easy to get that X1:N and Y1:N have survival functions, respectively, F X1:N (x) =
1 2(1 + x)2 − 1
and GY1:N (x) =
1 , 2x + 1
for all x ≥ 0.
In view of (3.4), a direct calculation gives gY1:N
fX1:N (x) f (x) −1X = 2(1 + x) ≥ 1. −1 = gY GY (FX ( x)) GY1:N (FX1:N ( x))
Thus, X1:N ≤disp Y1:N . Hence, by Theorem 2.1 we have X1:N ≤RQE Y1:N . Then from Theorem 3.5 we get X ≤RQE Y . The correctness of this result can also be seen from Example 2.2. Remark 3.6. Theorem 3.5 indicates that the RQE order has the closure property under a positive linear transform. We naturally consider whether the RQE order has the closure property under a general increasing transform. Unfortunately the answer is negative. This conjecture is incorrect in general. However, after adding some growth condition to the increasing transform, such as convexity condition, the RQE order has the closure property under such an increasing transform. The following theorem verifies this conjecture. Theorem 3.6. Let φ(·) be an increasing convex function such that φ(0) = 0. Assume that X ≤st Y . If X ≤RQE Y , then φ(X) ≤RQE φ(Y ). The proof of this theorem is similar to that of upcoming Theorem 3.7 and hence omitted here. Example 3.5. Let X and Y be two exponential random variables with survival functions, respectively, F X (x) = e−2x , GY (x) = e−x , for all x ≥ 0.
Take φ(t) = et − 1 for all t ≥ 0. Then φ(X) = eX − 1 and φ(Y ) = eY − 1 have survival functions, respectively, 2 1 F φ(X) (x) = F X (ln(1 + x)) = , and 1+x 10
Gφ(Y ) (x) = GY (ln(1 + x)) =
1 , 1+x
for all x ≥ 0.
That is, φ(X) and φ(Y ) are two Pareto random variables. Obviously, φ(t) is an increasing convex function such that φ(0) = 0. By observing that F X (x) ≤ GY (x) for all x ≥ 0, we get that X ≤st Y . Moreover, in view of Example 2.1 we know that X ≤RQE Y . Thus, according to Theorem 3.6 we obtain φ(X) ≤RQE φ(Y ). The correctness of this result can be seen from Example 3.4. Remark 3.7. Note that the convexity condition of the function φ(·) in Theorem 3.6 is only a sufficient condition, but not necessary. This is shown by the following counterexample. Counterexample 3.1. Assume X and Y are two exponential random variables with survival functions, respectively, F X (x) = e−2x ,
GY (x) = e−x ,
for all x ≥ 0.
In view of Example 2.1 we know X ≤RQE Y . Take φ(t) = t1/2 for all t ≥ 0. Then φ(X) = X 1/2 and φ(Y ) = Y 1/2 have survival functions, respectively, 2
F φ(X) (x) = e−2x ,
2
Gφ(Y ) (x) = e−x ,
for all x ≥ 0.
That is, φ(X) and φ(Y ) are two Weibull random variables. One can prove that X ≤disp Y . Making use of Theorem 2.2 we obtain that φ(X) ≤RQE φ(Y ). Clearly, φ(t) = t1/2 is an increasing concave function in t ∈ [0, +∞). Hence the condition “φ(·) is a convex function” of Theorem 3.6 is a sufficient condition but not necessary. Let X be a nonnegative continuous random variable, and φ(·) be a nonnegative increasing function defined on [0, ∞) with φ(0) = 0, we call φ(·) as a generalized scale transform. If the function φ(·) is increasing convex with φ(0) = 0, then φ(·) is called a risk preference function, and φ(X) is called the risk preference transform of X. If φ(·) is increasing concave with φ(0) = 0, then φ(·) is called a risk aversion function, and φ(X) is called the risk aversion transform of X. Remark 3.8. Theorem 3.6 says that the RQE order has the the closure property under the risk preference transforms. Theorem 3.6 also says that the RQE order has the the closure property under the convex generalized scale transforms. Using a method similar to above Theorem 3.6 we easily have the following result. Theorem 3.7. Let φ(·) be an increasing concave function such that φ(0) = 0. Assume that X ≤st Y . If φ(X) ≤RQE φ(Y ), then X ≤RQE Y . Proof. We only give the proof for the case 0 < β < 1, the proof of the case β > 1 is similar and hence is omitted here. Suppose that φ(X) ≤RQE φ(Y ). Then from (2.3) we have !1−β Z ∞ f (x) φ(X) − 1 dx ≥ 0, for all t ≥ 0. (3.5) [fX (φ(X))]β gφ(Y ) [G−1 (F (x))] t φ(X) φ(Y ) Also from (2.3), X ≤RQE Y if, and only if, " # 1−β Z ∞ fX (x) β [fX (x)] − 1 dx ≥ 0, gY [G−1 t Y (FX (x))] 11
for all t ≥ 0.
(3.6)
Since fφ(X) (x) =
fX (φ−1 (x)) , φ0 (φ−1 (x))
and gφ(Y ) (x) =
gY (φ−1 (x)) , φ0 (φ−1 (x))
then φ0 G−1 FX (φ−1 (x)) fX (φ−1 (x)) Y · . i = −1 (x))) φ0 (φ−1 (x)) gY G−1 Fφ(X) (x) Y (FX (φ
fφ(X) (x)
h gφ(Y ) G−1 φ(Y )
Hence, from (3.5) we get that φ(X) ≤RQE φ(Y ) if, and only if, for all t ≥ 0, −1 !1−β β Z +∞ 0 −1 −1 φ G F (φ (x)) fX (φ−1 (x)) f (φ (x)) X Y X · · − 1 dx −1 (x))) φ0 (φ−1 (x)) φ0 (φ−1 (x)) gY G−1 t Y (FX (φ Z
φ0 G−1 fX (x) Y (FX (x)) −1 · = [fX (x)]β · φ0 (x) gY GY (FX (x)) φ−1 (t) +∞
!1−β
− 1 dx ≥ 0,
(3.7)
where the integral on the right-hand side follows by substituting variable. In light of X ≤st Y implying G−1 ≥ x, and is increasing concave implies that φ0 (·) is nonnegative Y [FX (x)] φ(·) −1 0 0 decreasing, so φ GY (FX (x)) /φ (x) ≤ 1. Hence, !1−β Z +∞ −1 0 φ G (F (x)) f (x) X Y X · − 1 dx [fX (x)]β · 0 (x) φ gY G−1 (F (x)) φ−1 (t) X Y Z
fX (x) −1 ≤ [fX (x)]β · −1 g G φ (t) Y Y (FX (x)) +∞
!1−β
− 1 dx.
(3.8)
Combining (3.7) and (3.8) we obtain that (3.6) holds, which states that X ≤RQE Y . Therefore the proof is complete. Remark 3.9. Note that the concavity condition of the function φ(·) in Theorem 3.7 is only a sufficient condition, but not necessary. Remark 3.10. Theorem 3.7 says that, under the condition X ≤st Y , the RQE order has reversed closure property under the risk aversion transform. Theorem 3.7 also says that the RQE order has reversed closure property under concave generalized scale transforms. 4. Preservation of the RQE order in several stochastic models As applications of main result Theorem 2.1, in this section we deal with the preservation of the RQE order in the proportional hazard rate and reversed hazard rate models, the proportional odds model and the record values model. First we consider the following proportional hazard rate model. For more details on the proportional hazard rate model, we refer to Nanda and Paul (2006a), Abbasnejad et al. (2010), and Shaked and Shanthikumar (2007). Let X be a nonnegative random variable with survival function F X . For θ > 0, let X(θ) denote a random variable with survival function (F X )θ . Similarly, if Y is a nonnegative random variable with survival function GY , then denote by Y (θ) a random variable with survival function (GY )θ . Then we have the following result. 12
Theorem 4.1. Let X and Y be as described above. (a) Assume θ ≥ 1. If X ≤RQE Y , then X(θ) ≤RQE Y (θ). (b) Assume 0 < θ ≤ 1. If X(θ) ≤RQE Y (θ), then X ≤RQE Y . Proof. Below we only give the proof for the case 0 < β < 1, the proof of the case β > 1 is similar and hence is omitted here. Suppose that X, Y, X(θ) and Y (θ) have the distribution functions FX , GY , FX(θ) and GY (θ) , the density functions fX , gY , fX(θ) and gY (θ) , and the right-continuous −1 −1 −1 inverse functions FX , G−1 Y , FX(θ) and GY (θ) , respectively. It can be proven that, for all x ≥ 0,
θ−1 fX (x); fX(θ) (x) = θ F X (x)
Hence
h gY (θ) G−1 Y (θ)
(4.1)
−1 G−1 Y (θ) FX(θ) (x) = GY [FX (x)] , i θ−1 gY G−1 FX(θ) (x) = θ F X (x) Y (FX (x)) .
f (x) fX (x) h X(θ) i= −1 . gY GY (FX (x)) gY (θ) G−1 F (x) X(θ) Y (θ)
From (2.3) we have that X ≤RQE Y if, and only if, the inequality " # 1−β Z ∞ fX (x) β [fX (x)] − 1 dx ≥ 0, gY [G−1 t Y (FX (x))]
(4.2)
for all t ≥ 0
holds, and that X(θ) ≤RQE Y (θ) if, and only if, the inequality !1−β Z ∞ f (x) X(θ) − 1 dx ≥ 0, [fX(θ) (x)]β gY (θ) [GY−1(θ) (FX(θ) (x))] t
for all t ≥ 0
holds, or equivalently, on using (4.1), (4.2) and (4.4), the inequality " # 1−β Z +∞ θ−1 β fX (x) β θ F X (x) · [fX (x)] − 1 dx ≥ 0 gY [G−1 t Y (FX (x))]
(4.3)
(4.4)
(4.5)
holds for all t ≥ 0.
θ−1 β (a) Assume that X ≤RQE Y . If 0 < θ ≤ 1, then the function h(x) = θ F X (x) is
nonnegative increasing in x ≥ 0. By Lemma 1.1 and (4.3), we see that inequality (4.4) holds. Thus X(θ) ≤RQE Y (θ). θ−1 −β is (b) Assume that X(θ) ≤RQE Y (θ). If θ ≥ 1, then the function h(x) = θ F X (x)
nonnegative increasing in x ≥ 0. Making use of Lemma 1.1 and (4.5), we obtain that inequality (4.3) holds, which asserts that X ≤RQE Y . Now we consider a proportional reversed hazard rate model. For more details on the proportional reversed hazard rate model, we refer to Di Crescenzo (2000), Gupta and Gupta (2007), Di Crescenzo and Longobardi (2009), and Shaked and Shanthikumar (2007). 13
Let X be a nonnegative random variable with distribution function FX . For θ > 0, let X(θ) denote a random variable with distribution function (FX )θ . Similarly, let Y be another nonnegative random variable with distribution function GY , denote by Y (θ) a random variable with distribution function (GY )θ . The following theorem deals with the preservation of the RQE order in a proportional reversed failure rate model. The proof of the result is similar to that of the above Theorem 4.1 and hence omitted. Theorem 4.2. Let X and Y be as described above. (a) Assume θ ≥ 1. If X ≤RQE Y , then X(θ) ≤RQE Y (θ). (b) Assume 0 < θ ≤ 1. If X(θ) ≤RQE Y (θ), then X ≤RQE Y Navarro et al. (2010) studied the following proportional odds model. Let X be a nonnegative continuous random variable with survival function F X and density function fX . The proportional odds random variable, denoted by Xp , is defined by the survival function F Xp (x) =
θF X (x) 1 − (1 − θ)F X (x)
for θ > 0, where θ is proportional constant. Let Y be another nonnegative continuous random variable with survival function GY and density function gY . Similarly define the proportional odds random variable Yp of Y by the survival function GYp (x) =
θGY (x) 1 − (1 − θ)GY (x)
for θ > 0, where the proportional constant θ is the same as above. For this model we obtain the following results. Theorem 4.3. Let X, Y, Xp and Yp be as described above. (a) Assume θ ≥ 1. If X ≤RQE Y , then Xp ≤RQE Yp . (b) Assume 0 < θ ≤ 1. If Xp ≤RQE Yp , then X ≤RQE Y . Proof. We only give the proof for the case 0 < β < 1, the proof of the case β > 1 is similar and hence is omitted here. Denote the function h(u) =
θu , 1 − (1 − θ)u
u ∈ [0, 1],
(4.6)
for any θ > 0. It is easy to see that (a) If θ ≥ 1, then h(u) is increasing concave on [0, 1]. (b) If 0 < θ ≤ 1, then h(u) is increasing convex on [0, 1]. From the definition of Xp and Yp we have F Xp (x) = h[F X (x)]
and GYp (x) = h[GY (x)],
for all x ≥ 0.
Hence the density functions of Xp and Yp are, respectively, fXp (x) = h0 [F X (x)]fX (x)
and gYp (x) = h0 [GY (x)]gY (x),
for all x ≥ 0.
(4.7)
−1 It can be proven that G−1 Yp [FXp (x)] = GY [FX (x)], and thus
h
fXp (x)
gYp G−1 Yp FXp (x)
i = 14
gY
fX (x) . −1 GY (FX (x))
(4.8)
From (2.3) we have that X ≤RQE Y if, and only if, the inequality " # 1−β Z ∞ fX (x) β [fX (x)] − 1 dx ≥ 0 gY [G−1 t Y (FX (x))] holds for all t ≥ 0, and that Xp ≤RQE Yp if, and only if, the inequality !1−β Z ∞ fXp (x) β − 1 dx ≥ 0 [fXp (x)] gYp [G−1 t Yp (FXp (x))]
(4.9)
(4.10)
holds for all t ≥ 0, or equivalently, from (4.7) and (4.8), the inequality Z
+∞
t
holds for all t ≥ 0.
β h [F X (x)] [fX (x)]β 0
"
fX (x) −1 gY [GY (FX (x))]
1−β
#
− 1 dx ≥ 0
(4.11)
β (a) Assume that X ≤RQE Y . If θ ≥ 1, then the function h0 [F X (x)] is nonnegative increasing in x ≥ 0. From (4.9), (4.8) and Lemma 1.1, we see that inequality (4.11) holds, which asserts by Theorem 2.1 that Xp ≤RQE Yp . β (b) Assume that Xp ≤RQE Yp . If 0 < θ ≤ 1, then the function 1/ h0 [F X (x)] is nonnegative increasing in x ≥ 0. By means of (4.11) and Lemma 1.1, we see that inequality (4.9) holds, which asserts by Theorem 2.1 that X ≤RQE Y . Therefore the proof is complete. Below we consider the preservation of the RQE order in the record values model. Chandler (1952) introduced and studied many basic properties of records. For more details about record values and their applications, one may refer to Khaledi et al. (2009), Kundu et al. (2009), Zhao and Balakrishnan (2009), Zarezadeh and Asadi (2010), Li and Zhang (2011), and the references therein. Let {Xi , i ≥ 1} be a sequence of independent and identically distributed random variables (rv’s) from an absolutely continuous nonnegative random variable X with survival function F X (·) and density function fX (·). The rv’s TnX , defined recursively by T1X = 1 and X Tn+1 = min{j > TnX : Xj > XTnX },
n≥1
are called the n-th record times, and the quantities XTnX , denoted by RnX , are termed as the n-th record values. For a detailed discussion on record values one may refer to Arnold et al. (1998). X are given, It can be proven that the probability density function and survival function of Rn,k respectively, by 1 fRnX (x) = Λn−1 (x)fX (x), (4.12) Γ(n) X
F RnX (x) = F X (x)
n−1 X j=0
(ΛX (x))j = Γn (ΛX (x)) j!
(4.13)
for all x ≥ 0, where Γn (·) is the survival function of a Gamma random variable with the shape parameter n and the scale parameter 1, and ΛX (x) = − ln F X (x) is the cumulative failure rate function of X. 15
Let Y be another absolutely continuous nonnegative random variable with survival function GY (·) and density function gY (·), the corresponding n-th record values are denoted by RnY . For the preservation property of the RQE ordering in the record values model, we obtain the following result. Theorem 4.4. Let X and Y be two absolutely continuous and nonnegative random variables, m and n be positive integers. Then (a) X ≤RQE Y =⇒ RnX ≤RQE RnY , for all n ≥ 1. X Y (b) Rm ≤RQE Rm =⇒ RnX ≤RQE RnY , for all n > m ≥ 1. Proof. We only give the proof for the case 0 < β < 1, the proof of the case β > 1 is similar and hence is omitted here. (a) Suppose that X ≤RQE Y . Then from (2.3) we have # " 1−β Z ∞ fX (x) β − 1 dx ≥ 0, [fX (x)] gY [G−1 t Y (FX (x))]
for all t ≥ 0.
(4.14)
for all t ≥ 0. On using (4.12) and (4.13) we can prove that, for all n ≥ 1 and x ≥ 0, gRnY
1 Since function Γ(n) Λn−1 X (x) 1.1 we get that inequality
Z
+∞
t
fRnX (x) fX (x) h −1 . i = −1 g G Y GRY FRnX (x) Y (FX (x))
β
(4.15)
n
is increasing, making use of (4.15), (4.12), (4.14) and Lemma
(fRnX (x))β
gRnY
h
fRnX (x) G−1 Y Rn
1−β
i FRnX (x)
− 1 dx ≥ 0
(4.16)
holds for all t ≥ 0, by (2.3) which asserts, in turn, that RnX ≤RQE RnY .
Y X , then from (2.3) we have ≤DCRQE Rm (b) Suppose that Rm 1−β Z +∞ X (x) fRm β h X (x)) (fRm − 1 dx ≥ 0 i −1 t Y X (x) gRm GRY FRm
(4.17)
m
for all t ≥ 0. On using (4.15) one can prove that, for all positive integers n > m ≥ 1 and real x ≥ 0, X (x) fRnX (x) fRm fX (x) h i= h −1 . (4.18) i = −1 −1 g G Y Y X (x) gRnY GRY FRnX (x) gRm G R Y FR m Y (FX (x)) n
m
From (4.12) we see that the function
fRnX (x) Γ(m) = (ΛX (x))n−m X (x) fRm Γ(n)
is increasing in x ≥ 0.
On using (4.17) (4.18), (4.19) and Lemma 1.1 we obtain that 1−β Z +∞ fRnX (x) h (fRnX (x))β − 1 dx ≥ 0 i −1 t gRnY GRY FRnX (x) n
16
(4.19)
(4.20)
holds for all t ≥ 0. Again, by (2.3) this gives that RnX ≤RQE RnY . Therefore the desired result follows. Acknowledgements. The authors are grateful to the editor-in-chief and two anonymous referees for their careful readings of the manuscript and constructive comments, which improved the presentation of the paper. This work was supported by the social science planning fund program of Gansu Province of China (YB045) and by the soft science research planning fund program of Gansu Province of China (148ZCRA004).
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[16] Gupta, R.C., Gupta, R.D., 2007. Propotional reversed hazard rate model and its applications, J. Statist. Plann. Inference 137, 3525-3536. [17] Kayal, S., 2015. On generalized dynamic survival and failure entropies of order (α, β), Statist. Probab. Lett. 96, 123-132. [18] Khaledi, B.-E., Amiripour, F., Hu, T., Shojaei, S.R., 2009. Some new results on stochastic comparisons of record values. Comm. Statist. Theory Methods 38, 2056-2066. [19] Khinchin, A.J., 1957. Mathematical Foundation of Information Theory. Dover, New York. [20] Khorashadizadeh, M., Roknabadi, A.H.R., Borzadaran, G.R.M., 2013. Doubly truncated (interval) cumulative residual and past entropy. Statist. Probab. Lett. 83, 1464-1471. [21] Kumar, V., Taneja, H.C., 2011. Some characterization results on generalized cumulative residual entropy measure, Statist. Probab. Lett. 81, 1072-1077. [22] Kundu, C., Nanda, A.K., Hu, T., 2009. A note on reversed hazard rate of order statistics and record values. J. Statist. Plann. Inference 139, 1257-1265. [23] Li, X., Zhang, S., 2011. Some new results on R´enyi entropy of residual life and inactivity time. Probab. Engrg. Inform. Sci. 25, 237-250. [24] Nanda, A.K., Paul, P., 2006a. Some results on generalized residual entropy. Inform. Sci. 176, 27-47. [25] Nanda, A.K., Paul, P., 2006b. Some properties of past entropy and their applications. Metrika 64, 47-61. [26] Nanda, A.K., Paul, P., 2006c. Some results on generalized past entropy. J. Statist. Plann. Inference 136, 3659-3674. [27] Nanda, A.K., Sankaran, P.G., Sunoj, S.M., 2014. R´enyis residual entropy: A quantile approach. Statist. Probab. Lett. 85, 114-121. [28] Navarro, J., del Aguila, Y., Asadi, M., 2010. Some new results on the cumulative residual entropy. J. Statist. Plann. Inference 140, 310-322. [29] R´enyi, A., 1961. On measures of entropy and infromation. In: Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, vol. 1. pp. 547-561. [30] Shaked, M., Shanthikumar, J.G., 2007. Stochastic Orders and Their Applications. Springer Press, New York. [31] Shannon, C.E., 1948. A mathematical theory of comunication. Bell Syst. Tech. J. 27, 379423. [32] Sunoj, S.M., Sankaran, P.G., 2012. Quantile based entropy function. Statist. Probab. Lett. 82, 1049-1053. [33] Sunoj, S.M., Sankaran, P.G., Nanda, A.K., 2013. Quantile based entropy function in past lifetime. Statist. Probab. Lett. 83, 366-372. [34] Taneja, I.J., 1990. On generalized entropy with applications. In: Ricciardi, L.M. (ed.), Lectures in applied mathematics and informatics. Manchester University Press, Manchester, pp. 107-169. [35] Taneja, I.J., 2001. Generalized information measures and their applications. http://www.mtm.ufsc.br/∼taneja/book/nodel.html.
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S1572-3127(16)30003-X http://dx.doi.org/10.1016/j.stamet.2016.04.003 STAMET 528
To appear in:
Statistical Methodology
Received date: 23 June 2015 Revised date: 31 March 2016 Accepted date: 25 April 2016 Please cite this article as: L. Yan, D.-t. Kang, Some new results on the R´enyi quantile entropy Ordering, Statistical Methodology (2016), http://dx.doi.org/10.1016/j.stamet.2016.04.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Some new results on the R´ enyi quantile entropy Ordering Lei Yan∗ School of Economics and Management, Lanzhou University of Technology, Lanzhou, Gansu 730050, People’s Republic of China E-mail: [email protected]
Dian-tong Kang School of Mathematics and Statistics, Hexi University, Zhangye, Gansu 734000, People’s Republic of China E-mail: [email protected]
Abstract R´enyi (1961) proposed the R´enyi entropy. Ebrahimi and Pellerey (1995) and Ebrahimi (1996) proposed the residual entropy. Recently, Nanda et al. (2014) obtained a quantile version of the R´enyi residual entropy, the R´enyi residual quantile entropy (RRQE). Base on the RRQE function, they defined a new stochastic order, the R´enyi quantile entropy (RQE) order, and studied some properties of this order. In this paper, we focus on further properties of this new order. Some characterizations of the RQE order are investigated, closure and reversed closure properties are obtained, meanwhile, some illustrative examples are shown. As applications of a main result, the preservation of the RQE order in several stochastic models are discussed. Keywords: R´enyi’s residual quantile entropy, R´enyi residual quantile entropy order, closure property, proportional odds model, record valus model MSC: 60E15, 62N05, 62E10
1. Introduction The notion of entropy, originated from thermodynamics and statistical mechanics, is of importance in some scientific and technological areas such as communication theory, physics, probability, statistics and economics. It measures uncertainty of a physical system. The Shannon entropy plays a central role in information theory since it was introduced mathematically by Shannon (1948). Let X be a nonnegative random variable (rv) representing the lifetime of a device or a living thing with absolutely continuous cumulative distribution function FX (x), survival function F X (x) = 1−FX (x) and probability density function fX (x). The Shannon entropy of X is defined by Z HX = −E[ln fX (X)] = −
+∞
fX (x) ln fX (x)dx.
(1.1)
0
HX measures the uncertainty contained in fX (x) about the predictability of an outcome of X. In continuous case, HX is also referred to as the Shannon differential entropy. In recent years, the literature on information theory has grown quite voluminous. It has been found that the Shannon entropy has lot of applications in different areas such as physics, probability and statistics, communication theory and economics. (see, for instance, Taneja (1990, 2001), Kumar and Taneja (2011), Nanda et al. (2014), and Kayal (2015).) ∗
Corresponding author.
1
In the literature several dynamic generalizations of HX have been proposed (see, for example, Khinchin (1957), Taneja (1990), Gupta and Nanda (2002), Nanda and Paul (2006a), Asadi and Zohrevand (2007), Di Crescenzo and Longobardi (2009), Kumar and Taneja (2011), Khorashadizadeh et al. (2013), Nanda et al. (2014), and Kayal (2015)). Ebrahimi and Pellerey (1995) and Ebrahimi (1996) presented the residual entropy as a useful dynamic measure of uncertainty. The residual life of X is defined by Xt = [X − t |X > t] . The residual entropy of X at time t is defined as the differential entropy of Xt . Formally, for all t ≥ 0, the residual entropy of X is given by HX (t) = HXt = −E[ln fXt (Xt )]. Ebrahimi and Pellerey showed that Z +∞ fX (x) fX (x) ln dx. (1.2) HX (t) = − F X (t) F X (t) t
Given that a component survived up to time t, then HX (t) = HXt measures the uncertainty of the residual life Xt . Some authors did their research works about the residual entropy, various results concerning the residual entropy HX (t) have been obtained by Ebrahimi and Kirmani (1996a, b), Asadi and Ebrahimi (2000), Di Crescenzo and Longobardi (2002, 2004), Nanda and Paul (2006b, c), Navarro et al. (2010), Kumar and Taneja (2011), Nanda et al. (2014), and Kayal (2015). Since the work of Shannon (1948), many authors have generalized the Shannon entropy in different directions according to their research interests. One generalization of the Shannon entropy is due to R´enyi (1961). For a nonnegative rv X the R´enyi entropy is defined as Z ∞ 1 X ln (fX (x))β dx, β > 0, β 6= 1, (1.3) Hβ = 1−β 0 which reduces to the Shannon entropy as β → 0+ . For a used item, Abraham and Sankaran (2005) proposed a dynamic version of R´enyi entropy, the residual R´enyi entropy, given by β Z ∞ fX (x) 1 ln dx, β > 0, β 6= 1. (1.4) HβX (t) = 1−β F X (x) t
When X has the age t, for different values of β, HβX (t) gives the spectrum of R´enyi’s information on the residual life of X. Clearly, HβX (0) = HβX . A number of recent studies by the authors have indicated that the quantile function (QF) defined by −1 QX (u) = FX (u) = inf{x | FX (x) ≥ u}, u ∈ [0, 1] (1.5) is a convenient and equivalent alternative to the distribution function in modelling and analysis −1 of statistical data. Sometimes the quantile function FX (·) is also called the right-continuous inverse function of FX (·) (or of X). For the usefulness of QF in dynamic study of entropy, we refer to Sunoj and Sankaran (2012), Sunoj et al. (2013), Nanda et al. (2014) and the references therein.
More recently, Nanda et al. (2014) obtained a quantile version of the R´enyi residual entropy, termed the R´enyis residual quantile entropy (RRQE). They showed that the R´enyi’s residual quantile entropy is efficient and convenient and is an equivalent alternative to the R´enyi’s residual entropy in describing ageing properties of life distributions and in making stochastic comparisons. On the basis of RRQE function, they defined two new nonparametric classes of life distributions, the increasing (decreasing) R´enyi’s residual quantile entropy (IRRQE (DRRQE)) classes, and they defined a new stochastic order, the R´enyi’s residual quantile entropy (RQE) order, to compare the uncertainties of residual lives of two random lives X and Y at the age points with equal survival probabilities. They studied some ageing and ordering properties of HβX (QX (u)). 2
In this paper, we focus on the further properties of the RQE ordering. Nanda et al. (2014) gave the following lemma which will play an important role in the proofs of the paper. R1 Lemma 1.1. Let f (u, x) : [0, 1] × <+ → <+ be such that u f (u, x)dx ≥ 0 for all u ∈ [0, 1], and R1 h(x) be any nonnegative increasing function in x. Then u f (u, x)h(x)dx ≥ 0.
The rest of the paper is organized as follows. In Section 2, we investigate some characterizations of the RQE order. In section 3 we study closure and reversed closure properties of the RQE order under several reliability transformations, mainly including linear transformations, parallel and series operations, increasing convex and increasing concave transformations. Meanwhile, some illustrative examples are shown. As applications of main result Theorem 2.1, in section 4 we deal with the preservation of the RQE order in several stochastic models, including proportional hazard rate model, proportional reversed hazard rate model, proportional odds model, and the record values model. Throughout this paper, the terms increasing and decreasing are used in a non-strict sense. Assume that all integrals involved are finite, and ratios are well defined whenever written. 2. Characterization results In this section we obtain some characterization results of the RQE ordering. Suppose that X is an absolutely continuous nonnegative rv as described in Section 1. From (1.5) we have FX (QX (u)) = u. Denote the density quantile function by fX (QX (u)) and the quantile density function by qX (u) = Q0X (u), where the prime denotes the differentiation. Differentiating both sides of FX (QX (u)) = u yields qX (u)fX (QX (u)) = 1.
(2.1)
Sunoj and Sankaran (2012) introduced a quantile versions of the Shannon residual entropy (1.2). And they showed that ψX (u) = HX [QX (u)] = ln(1 − u) +
1 1−u
Z
1
ln[qX (p)]dp.
u
Recently, Nanda et al. (2014) obtained a quantile version of the R´enyi residual entropy, the R´enyi residual quantile entropy (RQE), defined by ψβX (u) := HβX (QX (u)) , for all u ∈ [0, 1]. They showed that, for all u ∈ [0, 1], (1 − β)HβX (QX (u)) = ln = ln
∞
QX (u)
(fX (x))β dx (1 − u)β
β 1 qX (p)dp (1 − u)qX (p) u Z 1 (qX (p))1−β = ln dp (1 − u)β u
Z
1
Z
3
= ln = ln
Z
Z
1
u
1
u
(fX [QX (p)])β−1 dp. (1 − u)β
(2.2)
(qX (p))1−β dp − β ln(1 − u).
For different values of β, HβX (QX (u)) in (2.2) gives the spectrum of R´enyi information contained in the conditional density about the predictability of an outcome of X until 100(1 − u)% point of distribution. In other words, HβX (QX (u)) measures the uncertainty of residual life XF −1 (u) , −1 that is, HβX (QX (u)) measures the uncertainty of X at age point FX (u).
X
Recently, Nanda et al. (2014) defined a new stochastic order based on the comparison of functions HβX (QX (u)) and HβY (QY (u)) corresponding to two nonnegative random variables X and Y . Definition 2.1. X is said to be smaller than Y in the R´enyi quantile entropy order (written as X ≤RQE Y ), if HβX (QX (u)) ≤ HβY (QY (u)) for all u ∈ [0, 1]. We obtain the following theorem which will play a key role in the proofs of results in this paper. Theorem 2.1. Let X and Y be two absolutely continuous nonnegative random variables with probability density functions fX and gY , and distribution functions FX and GY , respectively. (a) If 0 < β < 1, then X ≤RQE Y if, and only if, # " 1−β Z ∞ fX (x) β − 1 dx ≥ 0, [fX (x)] gY [G−1 t Y (FX (x))] (b) If β > 1, then X ≤RQE Y if, and only if, " 1−β # Z ∞ fX (x) β [fX (x)] 1 − dx ≥ 0, gY [G−1 t Y (FX (x))]
for all t ≥ 0.
(2.3)
for all t ≥ 0.
(2.4)
Proof. (a) If 0 < β < 1. From Definition 2.1 and (2.2) we have that X ≤RQE Y if, and only if, Z
1
u
(fX [QX (p)])
β−1
dp ≤
Z
1
(gY [QY (p)])β−1 dp
(2.5)
u
for all u ∈ [0, 1]. Letting p = FX (x) in two integrands of the above inequality and then letting −1 FX (u) = t in two integrals of both sides, we see that inequality (2.5) is equivalent to inequality (2.3). (b) The proof is similar to that of the case (a) by noting that when β > 1, 1 − β < 0. Thus the proof is complete. Example 2.1. Let X and Y be two exponential random variables with survival functions, respectively, F X (x) = e−λ1 x , GY (x) = e−λ2 x , for all x ≥ 0.
It can be verified that
fX (x) λ1 = . λ gY [G−1 (F (x))] 2 X Y 4
Making use of Theorem 2.1 we get that, for any β > 0, β 6= 1, (a) if 0 < λ2 ≤ λ1 , then X ≤RQE Y ; (b) if 0 < λ1 ≤ λ2 , then Y ≤RQE X. Below we recall several stochastic orders which will be useful in the sequel. For more details about these orders one can see Shaked and Shanthikumar (2007). Definition 2.2. Let X and Y be two nonnegative random variables. (a) X is said to be smaller than Y in the dispersive order (denoted by X ≤disp Y ) if −1 −1 −1 FX (β) − FX (α) ≤ G−1 Y (β) − GY (α),
equivalently,
fX (x) ≥ gY G−1 Y (FX (x)) ,
for all 0 < α < β < 1, for all x ≥ 0.
(2.6)
(b) X is said to be smaller than Y in the convex −1 transform order (denoted by X ≤c Y ) if G−1 Y [FX (x)] is convex, equivalently, fX (x) gY GY (FX (x)) is increasing in x ≥ 0. (c) X is said to be smaller than Y in the usual stochastic order (denoted by X ≤st Y ) if F X (x) ≤ GY (x) for all x ≥ 0. By means of Theorem 2.1 and Definition 2.2 we obtain the following theorem, which gives a sufficient condition for the RQE order. Theorem 2.2. If X ≤disp Y , then X ≤RQE Y . Proof. Suppose that X ≤disp Y . Then from (2.6) we have fX (x) ≥ gY G−1 Y (FX (x)) for x ≥ 0. By Theorem 2.1 the proof is complete. Below we give an illustrative example that meets the RQE order. Example 2.2. Let X and Y be two Pareto random variables having their survival functions, respectively, α1 α2 λ λ F X (x) = , GY (x) = , for all x ≥ 0. λ+x λ+x Denoted by X ∼ P (α1 , λ), Y ∼ P (α2 , λ), where α1 , α2 , λ are positive real numbers. Then the RQE order is determined by the parameters α1 and α2 irrespective of λ > 0. Indeed, the density functions of X and Y are, respectively, α1 fX (x) = λ
λ λ+x
α1 +1
α2 gY (x) = λ
,
λ λ+x
α2 +1
,
for all x ≥ 0.
It can be verified that −1
G−1 Y [FX (x)] = GY Hence,
α x α12 F X (x) = λ 1 + − λ. λ
α α fX (x) x α21 −1 −1 = 1 1+ . α2 λ gY GY (FX (x))
5
Clearly, if α1 ≥ α2 > 0, then X ≤disp Y , and if 0 < α1 ≤ α2 , then X ≥disp Y . By means of Theorem 2.2 we get the following results: for any β > 0, β 6= 1, (a) if 0 < α2 ≤ α1 , then X ≤RQE Y ; (b) if 0 < α1 ≤ α2 , then Y ≤RQE X. From Theorem 2.1 and definition of order ≤c , we get another sufficient condition for the RQE order as follows. Theorem 2.3. If X ≤c Y, and fX (0) ≥ gY (0) > 0, then X ≤RQE Y . Proof. Suppose that X ≤c Y . Then according to the definition of the order X ≤c Y , we have fX (x) the function g G−1 is increasing in x ≥ 0. Thus Y [ Y (FX (x))] fX (x) f (0) −1 ≥ X ≥ 1. gY (0) gY GY (FX (x))
From this fact, by Theorem 2.1 the required result follows.
By means of Theorem 2.1 it is easy to verify that the following Theorems 2.4, 2.5 and 2.6 hold. Theorem 2.4. Let X be a nonnegative continuous random variable and a be any real constant. Then (a) If a ≥ 1, then X ≤RQE aX. (b) If 0 < a ≤ 1, then aX ≤RQE X. Theorem 2.5. Let X and Y be two nonnegative continuous random variables and a be any real constant. Assume that X ≤RQE Y . (a) If a ≥ 1, then X ≤RQE aY . (b) If 0 < a ≤ 1, then aX ≤RQE Y . Theorem 2.6. If X ≤RQE Y, then X ≤RQE Y + c for any real number c. Remark 2.1. Theorem 2.6 shows that the RQE order is location independent. Theorem 2.7. X ≤RQE Y and X ≥RQE Y hold simultaneously if, and only if, X =disp Y , here, X =disp Y means that X =st Y + k, where k is constant. Proof. From Theorem 2.1 we have that X ≤RQE Y and X ≥RQE Y hold simultaneously, if, and only if, " # 1−β Z ∞ f (x) X [fX (x)]β − 1 dx ≡ 0, for all t ≥ 0. gY [G−1 t Y (FX (x))] Differentiating both sides of above equality yields fX (x) −1 = 1, gY GY (FX (x)) 6
for all x ≥ 0,
which is equivalent to that X =disp Y , where X =disp Y means that X =st Y + k, and k is any real constant. And this reasoning is reversible. Therefore the proof is complete. 3. Closure and reversed closure properties of the RQE Order In this section, we investigate the closure and reversed closure properties of the RQE order under several reliability transforms, mainly including linear transformations, parallel and series operations, increasing convex and increasing concave transforms. First we explore the closure property of the RQE order under linear transformations. According to Theorems 2.5 and 2.6, the following theorem is easy. Theorem 3.1. Let X ≤RQE Y . Then for any real numbers a, b, c and d with 0 < a ≤ c, 0 ≤ b ≤ d, we have aX + b ≤RQE cY + d. As a direct consequence we have the following corollary. Corollary 3.1. If X ≤RQE Y . Then for any real numbers a and b with a > 0, b ≥ 0, we have aX + b ≤RQE aY + b. Remark 3.1. Corollary 3.1 indicates that a positive linear system preserves the RQE order. Corollary 3.1 also says that the RQE order has the closure property with respect to a positive linear system. Corollary 3.1 also states that the RQE order has the closure property under a positive linear transformation. Let X and Y be two nonnegative continuous random variables, let X1 , . . ., Xn and Y1 , . . . Yn be independent and identically distributed (iid) copies of X and Y , respectively. Denote by X1:n = min{X1 , . . . , Xn },
Xn:n = max{X1 , . . . , Xn }.
Similarly, Y1:n and Yn:n . Denote the survival function and the density function of X1:n by F X1:n and fX1:n , respectively. Denote the distribution function and the density function of Xn:n by FXn:n and fXn:n , respectively. Similarly, GY1:n , gY1:n , GYn:n , and gYn:n . We have the following theorem. Theorem 3.2. If X ≤RQE Y, then Xn:n ≤RQE Yn:n . Proof. We only give the proof for the case 0 < β < 1, the proof of the case β > 1 is similar and hence is omitted here. Suppose that X ≤RQE Y . Then from (2.3) we have, for all t ≥ 0, " # 1−β Z ∞ fX (x) β − 1 dx ≥ 0, for all t ≥ 0. (3.1) [fX (x)] gY [G−1 t Y (FX (x))] It is easy to see that
n−1
fXn:n (x) = n [FX (x)]
and hence h
fXn:n (x)
gYn:n G−1 Yn:n
· fX (x),
fX (x) i= −1 , gY GY (FX (x)) (FXn:n (x)) 7
for all x ≥ 0.
(3.2)
n−1
Since the function h(x) = (n [FX (x)] )β is nonnegative increasing in x ≥ 0, so, from (3.1), (3.2) and Lemma 1.1 we obtain that !1−β Z +∞ f (x) Xn:n [fXn:n (x)]β − 1 dx ≥ 0, for all t ≥ 0, gYn:n G−1 t Yn:n (FXn:n (x)) again, by Theorem 2.1 we get that Xn:n ≤RQE Yn:n . Therefore the proof is complete.
Remark 3.2. Theorem 3.2 indicates that the parallel systems preserve the RQE order. Theorem 3.2 also says that the RQE order has the closure property with respect to the parallel systems. Example 3.1. Let X and Y be two exponential random variables with survival functions, respectively, F X (x) = e−2x , GY (x) = e−x , for all x ≥ 0.
Then X2:2 and Y2:2 have distribution functions, respectively, FX2:2 (x) = 1 − e−2x
2
,
GY2:2 (x) = 1 − e−x
2
,
for all x ≥ 0.
According to Example 2.1 we have X ≤RQE Y . By Theorem 3.2 we obtain X2:2 ≤RQE Y2:2 . With a similar manner to above Theorem 3.2, one can prove that the following theorem holds. Theorem 3.3. If X1:n ≤RQE Y1:n , then X ≤RQE Y . Remark 3.3. Theorem 3.3 says that series systems reversely preserve the RQE order. Theorem 3.3 also indicates that the RQE order has the reversed closure property under the formation of series systems. Example 3.2. Let X and Y be two Pareto random variables having survival functions, respectively, 2 1 1 , GY (x) = , for all x ≥ 0. F X (x) = 1+x 1+x
Then X1:2 and Y1:2 are also Pareto random variables having survival functions, respectively, F X1:2 (x) =
1 1+x
4
,
GY1:2 (x) =
1 1+x
2
,
for all x ≥ 0.
In view of Example 2.2 we get that X1:2 ≤RQE Y1:2 . On using Theorem 3.3 we obtain that X ≤RQE Y . Let X and Y be two nonnegative continuous random variables, let X1 , X2 , . . . and Y1 , Y2 , . . . be sequences of iid copies of X and Y , respectively. Suppose that N is a positive integer-valued random variable with the probability mass function pN (n) = P {N = n}, n = 1, 2, . . ., and N is independent of Xi ’s and Yi ’s. Denote by XN :N = max{X1 , . . . , XN },
YN :N = max{Y1 , . . . , YN }.
Then the following theorem can be viewed as an extension of Theorem 3.2 from a finite number n to a random number N . The proof of the theorem is similar to that of the following Theorem 3.5 and hence omitted here. 8
Theorem 3.4. If X ≤RQE Y , then XN :N ≤RQE YN :N . Remark 3.4. Theorem 3.4 says that a random parallel system preserves the RQE order. Theorem 3.4 also indicates that the RQE order has the closure property with respect to a random parallel system. Example 3.3. Let X and Y be two exponential random variables with survival functions, respectively, F X (x) = e−2x , GY (x) = e−x , for all x ≥ 0.
Assume that N is a Bernoulli random variable with a probability mass function pN (n) = P {N = n} = 1/2, n = 1, 2, and N is independent of X and Y . Then XN :N and YN :N have distribution functions, respectively, 2 o 1 n , 1 − e−2x + 1 − e−2x 2 2 o 1 n 1 − e−x + 1 − e−x , for all x ≥ 0. GYN :N (x) = 2 In light of Example 2.1 we know X ≤RQE Y . Then from Theorem 3.4 we obtain that XN :N ≤RQE YN :N . FXN :N (x) =
Denote by X1:N = min{X1 , . . . , XN }, Y1:N = min{Y1 , . . . , YN }. On using a method similar to above Theorem 3.4 we have the following result, which can be viewed as an extension of Theorem 3.3 from a finite number n to a random number N . Theorem 3.5. If X1:N ≤RQE Y1:N , then X ≤RQE Y . Proof. We only give the proof for the case 0 < β < 1, the proof of the case β > 1 is similar and hence is omitted here. Suppose that X1:N ≤RQE Y1:N . Then from (2.3) we have 1−β Z ∞ f (x) h X1:N i [fX1:N (x)]β − 1 dx ≥ 0, for all t ≥ 0. (3.3) −1 t gY1:N GY (FX1:N (x)) 1:N
It is easy to see that X1:N and Y1:N have density functions, respectively, # " ∞ X n−1 F X (x) pN (n) , and fX1:N (x) = fX (x) · n n=1
"
gY1:N (x) = gY (x) · n
∞ X
n=1
GY (x)
n−1
#
pN (n) ,
where pN (n) = P(N = n), n = 1, 2, . . . , is the probability mass function of N . Shaked and Shanthikumar (2007) showed that G−1 [FX1:N (x)] = G−1 Y [FX (x)] , Y 1:N
thus we obtain that
gY1:N
h
fX1:N (x) G−1
Y1:N
(FX1:N (x))
i=
f (x) X , gY G−1 Y (FX (x)) 9
for all x ≥ 0.
(3.4)
Since the function h(x) =
!−1 ∞ X n−1 n pN (n) F X (x) n=1
is nonnegative increasing in x ≥ 0, so, from (3.3), (3.4) and Lemma 1.1 we get " # 1−β Z ∞ fX (x) β [fX (x)] − 1 dx ≥ 0, for all t ≥ 0, gY [G−1 t Y (FX (x))] in turn, by (2.3) which asserts that X ≤RQE Y . Therefore the proof follows.
Remark 3.5. Theorem 3.5 indicates that the random series systems reversely preserve the RQE order. Theorem 3.5 also says that the RQE order has the reversed closure property with respect to the random series systems. Example 3.4. Let X and Y be two Pareto random variables having survival functions, respectively, 2 1 1 , GY (x) = F X (x) = , for all x ≥ 0. 1+x 1+x
Assume that N is a geometric random variable with probability mass function pN (n) = P {N = n} = 1/2n , n = 1, 2, . . ., and N is independent of X and Y . It is easy to get that X1:N and Y1:N have survival functions, respectively, F X1:N (x) =
1 2(1 + x)2 − 1
and GY1:N (x) =
1 , 2x + 1
for all x ≥ 0.
In view of (3.4), a direct calculation gives gY1:N
fX1:N (x) f (x) −1X = 2(1 + x) ≥ 1. −1 = gY GY (FX ( x)) GY1:N (FX1:N ( x))
Thus, X1:N ≤disp Y1:N . Hence, by Theorem 2.1 we have X1:N ≤RQE Y1:N . Then from Theorem 3.5 we get X ≤RQE Y . The correctness of this result can also be seen from Example 2.2. Remark 3.6. Theorem 3.5 indicates that the RQE order has the closure property under a positive linear transform. We naturally consider whether the RQE order has the closure property under a general increasing transform. Unfortunately the answer is negative. This conjecture is incorrect in general. However, after adding some growth condition to the increasing transform, such as convexity condition, the RQE order has the closure property under such an increasing transform. The following theorem verifies this conjecture. Theorem 3.6. Let φ(·) be an increasing convex function such that φ(0) = 0. Assume that X ≤st Y . If X ≤RQE Y , then φ(X) ≤RQE φ(Y ). The proof of this theorem is similar to that of upcoming Theorem 3.7 and hence omitted here. Example 3.5. Let X and Y be two exponential random variables with survival functions, respectively, F X (x) = e−2x , GY (x) = e−x , for all x ≥ 0.
Take φ(t) = et − 1 for all t ≥ 0. Then φ(X) = eX − 1 and φ(Y ) = eY − 1 have survival functions, respectively, 2 1 F φ(X) (x) = F X (ln(1 + x)) = , and 1+x 10
Gφ(Y ) (x) = GY (ln(1 + x)) =
1 , 1+x
for all x ≥ 0.
That is, φ(X) and φ(Y ) are two Pareto random variables. Obviously, φ(t) is an increasing convex function such that φ(0) = 0. By observing that F X (x) ≤ GY (x) for all x ≥ 0, we get that X ≤st Y . Moreover, in view of Example 2.1 we know that X ≤RQE Y . Thus, according to Theorem 3.6 we obtain φ(X) ≤RQE φ(Y ). The correctness of this result can be seen from Example 3.4. Remark 3.7. Note that the convexity condition of the function φ(·) in Theorem 3.6 is only a sufficient condition, but not necessary. This is shown by the following counterexample. Counterexample 3.1. Assume X and Y are two exponential random variables with survival functions, respectively, F X (x) = e−2x ,
GY (x) = e−x ,
for all x ≥ 0.
In view of Example 2.1 we know X ≤RQE Y . Take φ(t) = t1/2 for all t ≥ 0. Then φ(X) = X 1/2 and φ(Y ) = Y 1/2 have survival functions, respectively, 2
F φ(X) (x) = e−2x ,
2
Gφ(Y ) (x) = e−x ,
for all x ≥ 0.
That is, φ(X) and φ(Y ) are two Weibull random variables. One can prove that X ≤disp Y . Making use of Theorem 2.2 we obtain that φ(X) ≤RQE φ(Y ). Clearly, φ(t) = t1/2 is an increasing concave function in t ∈ [0, +∞). Hence the condition “φ(·) is a convex function” of Theorem 3.6 is a sufficient condition but not necessary. Let X be a nonnegative continuous random variable, and φ(·) be a nonnegative increasing function defined on [0, ∞) with φ(0) = 0, we call φ(·) as a generalized scale transform. If the function φ(·) is increasing convex with φ(0) = 0, then φ(·) is called a risk preference function, and φ(X) is called the risk preference transform of X. If φ(·) is increasing concave with φ(0) = 0, then φ(·) is called a risk aversion function, and φ(X) is called the risk aversion transform of X. Remark 3.8. Theorem 3.6 says that the RQE order has the the closure property under the risk preference transforms. Theorem 3.6 also says that the RQE order has the the closure property under the convex generalized scale transforms. Using a method similar to above Theorem 3.6 we easily have the following result. Theorem 3.7. Let φ(·) be an increasing concave function such that φ(0) = 0. Assume that X ≤st Y . If φ(X) ≤RQE φ(Y ), then X ≤RQE Y . Proof. We only give the proof for the case 0 < β < 1, the proof of the case β > 1 is similar and hence is omitted here. Suppose that φ(X) ≤RQE φ(Y ). Then from (2.3) we have !1−β Z ∞ f (x) φ(X) − 1 dx ≥ 0, for all t ≥ 0. (3.5) [fX (φ(X))]β gφ(Y ) [G−1 (F (x))] t φ(X) φ(Y ) Also from (2.3), X ≤RQE Y if, and only if, " # 1−β Z ∞ fX (x) β [fX (x)] − 1 dx ≥ 0, gY [G−1 t Y (FX (x))] 11
for all t ≥ 0.
(3.6)
Since fφ(X) (x) =
fX (φ−1 (x)) , φ0 (φ−1 (x))
and gφ(Y ) (x) =
gY (φ−1 (x)) , φ0 (φ−1 (x))
then φ0 G−1 FX (φ−1 (x)) fX (φ−1 (x)) Y · . i = −1 (x))) φ0 (φ−1 (x)) gY G−1 Fφ(X) (x) Y (FX (φ
fφ(X) (x)
h gφ(Y ) G−1 φ(Y )
Hence, from (3.5) we get that φ(X) ≤RQE φ(Y ) if, and only if, for all t ≥ 0, −1 !1−β β Z +∞ 0 −1 −1 φ G F (φ (x)) fX (φ−1 (x)) f (φ (x)) X Y X · · − 1 dx −1 (x))) φ0 (φ−1 (x)) φ0 (φ−1 (x)) gY G−1 t Y (FX (φ Z
φ0 G−1 fX (x) Y (FX (x)) −1 · = [fX (x)]β · φ0 (x) gY GY (FX (x)) φ−1 (t) +∞
!1−β
− 1 dx ≥ 0,
(3.7)
where the integral on the right-hand side follows by substituting variable. In light of X ≤st Y implying G−1 ≥ x, and is increasing concave implies that φ0 (·) is nonnegative Y [FX (x)] φ(·) −1 0 0 decreasing, so φ GY (FX (x)) /φ (x) ≤ 1. Hence, !1−β Z +∞ −1 0 φ G (F (x)) f (x) X Y X · − 1 dx [fX (x)]β · 0 (x) φ gY G−1 (F (x)) φ−1 (t) X Y Z
fX (x) −1 ≤ [fX (x)]β · −1 g G φ (t) Y Y (FX (x)) +∞
!1−β
− 1 dx.
(3.8)
Combining (3.7) and (3.8) we obtain that (3.6) holds, which states that X ≤RQE Y . Therefore the proof is complete. Remark 3.9. Note that the concavity condition of the function φ(·) in Theorem 3.7 is only a sufficient condition, but not necessary. Remark 3.10. Theorem 3.7 says that, under the condition X ≤st Y , the RQE order has reversed closure property under the risk aversion transform. Theorem 3.7 also says that the RQE order has reversed closure property under concave generalized scale transforms. 4. Preservation of the RQE order in several stochastic models As applications of main result Theorem 2.1, in this section we deal with the preservation of the RQE order in the proportional hazard rate and reversed hazard rate models, the proportional odds model and the record values model. First we consider the following proportional hazard rate model. For more details on the proportional hazard rate model, we refer to Nanda and Paul (2006a), Abbasnejad et al. (2010), and Shaked and Shanthikumar (2007). Let X be a nonnegative random variable with survival function F X . For θ > 0, let X(θ) denote a random variable with survival function (F X )θ . Similarly, if Y is a nonnegative random variable with survival function GY , then denote by Y (θ) a random variable with survival function (GY )θ . Then we have the following result. 12
Theorem 4.1. Let X and Y be as described above. (a) Assume θ ≥ 1. If X ≤RQE Y , then X(θ) ≤RQE Y (θ). (b) Assume 0 < θ ≤ 1. If X(θ) ≤RQE Y (θ), then X ≤RQE Y . Proof. Below we only give the proof for the case 0 < β < 1, the proof of the case β > 1 is similar and hence is omitted here. Suppose that X, Y, X(θ) and Y (θ) have the distribution functions FX , GY , FX(θ) and GY (θ) , the density functions fX , gY , fX(θ) and gY (θ) , and the right-continuous −1 −1 −1 inverse functions FX , G−1 Y , FX(θ) and GY (θ) , respectively. It can be proven that, for all x ≥ 0,
θ−1 fX (x); fX(θ) (x) = θ F X (x)
Hence
h gY (θ) G−1 Y (θ)
(4.1)
−1 G−1 Y (θ) FX(θ) (x) = GY [FX (x)] , i θ−1 gY G−1 FX(θ) (x) = θ F X (x) Y (FX (x)) .
f (x) fX (x) h X(θ) i= −1 . gY GY (FX (x)) gY (θ) G−1 F (x) X(θ) Y (θ)
From (2.3) we have that X ≤RQE Y if, and only if, the inequality " # 1−β Z ∞ fX (x) β [fX (x)] − 1 dx ≥ 0, gY [G−1 t Y (FX (x))]
(4.2)
for all t ≥ 0
holds, and that X(θ) ≤RQE Y (θ) if, and only if, the inequality !1−β Z ∞ f (x) X(θ) − 1 dx ≥ 0, [fX(θ) (x)]β gY (θ) [GY−1(θ) (FX(θ) (x))] t
for all t ≥ 0
holds, or equivalently, on using (4.1), (4.2) and (4.4), the inequality " # 1−β Z +∞ θ−1 β fX (x) β θ F X (x) · [fX (x)] − 1 dx ≥ 0 gY [G−1 t Y (FX (x))]
(4.3)
(4.4)
(4.5)
holds for all t ≥ 0.
θ−1 β (a) Assume that X ≤RQE Y . If 0 < θ ≤ 1, then the function h(x) = θ F X (x) is
nonnegative increasing in x ≥ 0. By Lemma 1.1 and (4.3), we see that inequality (4.4) holds. Thus X(θ) ≤RQE Y (θ). θ−1 −β is (b) Assume that X(θ) ≤RQE Y (θ). If θ ≥ 1, then the function h(x) = θ F X (x)
nonnegative increasing in x ≥ 0. Making use of Lemma 1.1 and (4.5), we obtain that inequality (4.3) holds, which asserts that X ≤RQE Y . Now we consider a proportional reversed hazard rate model. For more details on the proportional reversed hazard rate model, we refer to Di Crescenzo (2000), Gupta and Gupta (2007), Di Crescenzo and Longobardi (2009), and Shaked and Shanthikumar (2007). 13
Let X be a nonnegative random variable with distribution function FX . For θ > 0, let X(θ) denote a random variable with distribution function (FX )θ . Similarly, let Y be another nonnegative random variable with distribution function GY , denote by Y (θ) a random variable with distribution function (GY )θ . The following theorem deals with the preservation of the RQE order in a proportional reversed failure rate model. The proof of the result is similar to that of the above Theorem 4.1 and hence omitted. Theorem 4.2. Let X and Y be as described above. (a) Assume θ ≥ 1. If X ≤RQE Y , then X(θ) ≤RQE Y (θ). (b) Assume 0 < θ ≤ 1. If X(θ) ≤RQE Y (θ), then X ≤RQE Y Navarro et al. (2010) studied the following proportional odds model. Let X be a nonnegative continuous random variable with survival function F X and density function fX . The proportional odds random variable, denoted by Xp , is defined by the survival function F Xp (x) =
θF X (x) 1 − (1 − θ)F X (x)
for θ > 0, where θ is proportional constant. Let Y be another nonnegative continuous random variable with survival function GY and density function gY . Similarly define the proportional odds random variable Yp of Y by the survival function GYp (x) =
θGY (x) 1 − (1 − θ)GY (x)
for θ > 0, where the proportional constant θ is the same as above. For this model we obtain the following results. Theorem 4.3. Let X, Y, Xp and Yp be as described above. (a) Assume θ ≥ 1. If X ≤RQE Y , then Xp ≤RQE Yp . (b) Assume 0 < θ ≤ 1. If Xp ≤RQE Yp , then X ≤RQE Y . Proof. We only give the proof for the case 0 < β < 1, the proof of the case β > 1 is similar and hence is omitted here. Denote the function h(u) =
θu , 1 − (1 − θ)u
u ∈ [0, 1],
(4.6)
for any θ > 0. It is easy to see that (a) If θ ≥ 1, then h(u) is increasing concave on [0, 1]. (b) If 0 < θ ≤ 1, then h(u) is increasing convex on [0, 1]. From the definition of Xp and Yp we have F Xp (x) = h[F X (x)]
and GYp (x) = h[GY (x)],
for all x ≥ 0.
Hence the density functions of Xp and Yp are, respectively, fXp (x) = h0 [F X (x)]fX (x)
and gYp (x) = h0 [GY (x)]gY (x),
for all x ≥ 0.
(4.7)
−1 It can be proven that G−1 Yp [FXp (x)] = GY [FX (x)], and thus
h
fXp (x)
gYp G−1 Yp FXp (x)
i = 14
gY
fX (x) . −1 GY (FX (x))
(4.8)
From (2.3) we have that X ≤RQE Y if, and only if, the inequality " # 1−β Z ∞ fX (x) β [fX (x)] − 1 dx ≥ 0 gY [G−1 t Y (FX (x))] holds for all t ≥ 0, and that Xp ≤RQE Yp if, and only if, the inequality !1−β Z ∞ fXp (x) β − 1 dx ≥ 0 [fXp (x)] gYp [G−1 t Yp (FXp (x))]
(4.9)
(4.10)
holds for all t ≥ 0, or equivalently, from (4.7) and (4.8), the inequality Z
+∞
t
holds for all t ≥ 0.
β h [F X (x)] [fX (x)]β 0
"
fX (x) −1 gY [GY (FX (x))]
1−β
#
− 1 dx ≥ 0
(4.11)
β (a) Assume that X ≤RQE Y . If θ ≥ 1, then the function h0 [F X (x)] is nonnegative increasing in x ≥ 0. From (4.9), (4.8) and Lemma 1.1, we see that inequality (4.11) holds, which asserts by Theorem 2.1 that Xp ≤RQE Yp . β (b) Assume that Xp ≤RQE Yp . If 0 < θ ≤ 1, then the function 1/ h0 [F X (x)] is nonnegative increasing in x ≥ 0. By means of (4.11) and Lemma 1.1, we see that inequality (4.9) holds, which asserts by Theorem 2.1 that X ≤RQE Y . Therefore the proof is complete. Below we consider the preservation of the RQE order in the record values model. Chandler (1952) introduced and studied many basic properties of records. For more details about record values and their applications, one may refer to Khaledi et al. (2009), Kundu et al. (2009), Zhao and Balakrishnan (2009), Zarezadeh and Asadi (2010), Li and Zhang (2011), and the references therein. Let {Xi , i ≥ 1} be a sequence of independent and identically distributed random variables (rv’s) from an absolutely continuous nonnegative random variable X with survival function F X (·) and density function fX (·). The rv’s TnX , defined recursively by T1X = 1 and X Tn+1 = min{j > TnX : Xj > XTnX },
n≥1
are called the n-th record times, and the quantities XTnX , denoted by RnX , are termed as the n-th record values. For a detailed discussion on record values one may refer to Arnold et al. (1998). X are given, It can be proven that the probability density function and survival function of Rn,k respectively, by 1 fRnX (x) = Λn−1 (x)fX (x), (4.12) Γ(n) X
F RnX (x) = F X (x)
n−1 X j=0
(ΛX (x))j = Γn (ΛX (x)) j!
(4.13)
for all x ≥ 0, where Γn (·) is the survival function of a Gamma random variable with the shape parameter n and the scale parameter 1, and ΛX (x) = − ln F X (x) is the cumulative failure rate function of X. 15
Let Y be another absolutely continuous nonnegative random variable with survival function GY (·) and density function gY (·), the corresponding n-th record values are denoted by RnY . For the preservation property of the RQE ordering in the record values model, we obtain the following result. Theorem 4.4. Let X and Y be two absolutely continuous and nonnegative random variables, m and n be positive integers. Then (a) X ≤RQE Y =⇒ RnX ≤RQE RnY , for all n ≥ 1. X Y (b) Rm ≤RQE Rm =⇒ RnX ≤RQE RnY , for all n > m ≥ 1. Proof. We only give the proof for the case 0 < β < 1, the proof of the case β > 1 is similar and hence is omitted here. (a) Suppose that X ≤RQE Y . Then from (2.3) we have # " 1−β Z ∞ fX (x) β − 1 dx ≥ 0, [fX (x)] gY [G−1 t Y (FX (x))]
for all t ≥ 0.
(4.14)
for all t ≥ 0. On using (4.12) and (4.13) we can prove that, for all n ≥ 1 and x ≥ 0, gRnY
1 Since function Γ(n) Λn−1 X (x) 1.1 we get that inequality
Z
+∞
t
fRnX (x) fX (x) h −1 . i = −1 g G Y GRY FRnX (x) Y (FX (x))
β
(4.15)
n
is increasing, making use of (4.15), (4.12), (4.14) and Lemma
(fRnX (x))β
gRnY
h
fRnX (x) G−1 Y Rn
1−β
i FRnX (x)
− 1 dx ≥ 0
(4.16)
holds for all t ≥ 0, by (2.3) which asserts, in turn, that RnX ≤RQE RnY .
Y X , then from (2.3) we have ≤DCRQE Rm (b) Suppose that Rm 1−β Z +∞ X (x) fRm β h X (x)) (fRm − 1 dx ≥ 0 i −1 t Y X (x) gRm GRY FRm
(4.17)
m
for all t ≥ 0. On using (4.15) one can prove that, for all positive integers n > m ≥ 1 and real x ≥ 0, X (x) fRnX (x) fRm fX (x) h i= h −1 . (4.18) i = −1 −1 g G Y Y X (x) gRnY GRY FRnX (x) gRm G R Y FR m Y (FX (x)) n
m
From (4.12) we see that the function
fRnX (x) Γ(m) = (ΛX (x))n−m X (x) fRm Γ(n)
is increasing in x ≥ 0.
On using (4.17) (4.18), (4.19) and Lemma 1.1 we obtain that 1−β Z +∞ fRnX (x) h (fRnX (x))β − 1 dx ≥ 0 i −1 t gRnY GRY FRnX (x) n
16
(4.19)
(4.20)
holds for all t ≥ 0. Again, by (2.3) this gives that RnX ≤RQE RnY . Therefore the desired result follows. Acknowledgements. The authors are grateful to the editor-in-chief and two anonymous referees for their careful readings of the manuscript and constructive comments, which improved the presentation of the paper. This work was supported by the social science planning fund program of Gansu Province of China (YB045) and by the soft science research planning fund program of Gansu Province of China (148ZCRA004).
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