Convex transform order of the maximum of independent Weibull random variables

Statistics and Probability Letters 156 (2020) 108597

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Convex transform order of the maximum of independent Weibull random variables Jintang Wu a , Mengshou Wang a , Xiaohu Li b , a b

∗

School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China Department of Mathematical Sciences, Stevens Institute of Technology Hoboken, NJ 07030, USA

article

info

Article history: Received 5 February 2019 Received in revised form 22 July 2019 Accepted 23 August 2019 Available online 28 August 2019

a b s t r a c t The convex transform order on the maximum of independent heterogeneous Weibull distributions with shape parameters less than 1 is built. Together with Theorem 3.2 of Balakrishnan et al. (2018a,b), this outcome enriches the ordering results of Weibull distributions. © 2019 Elsevier B.V. All rights reserved.

MSC: primary 60E15 secondary 90B25 Keywords: Excess wealth order Increasing convex order Increasing hazard rate Skewness

1. Introduction Asymmetric probability distributions are common in statistics and related applied disciplines such as business, economics, engineering, insurance management, and finance. Various measures have been introduced to quantify the skewness of distributions, such as Pearson’s skewness coefficients, Edgeworth’s coefficient, the Groeneveld–Meeden coefficient, L-moments, and coefficients of variation. The probability density of a positively skewed unimodal distribution has a longer right-side tail compared to a symmetric distribution. Although many well-known distributions, such as the chi-squared, exponential, Weibull, log-normal, and gamma distributions, are known to be positively skewed, quantifying their skewness in terms of these commonly used measures is sometimes challenging. Since skewed or heavy-tailed distributions often serve as reasonable models for system lifetimes, insurance claim amounts, financial returns etc. in applied sciences, it is of interest to compare skewness of probability distributions. For example, to assess the income inequality in economics, to control the skewness risk due to using symmetric models for asymmetric financial data, to evaluate the aging degree in reliability and demography etc. Owing to complicated computations and challenging mathematics, researchers are in desperate need for simple and feasible ways to facilitate comparison of skewness of popular probability distributions. Designed to compare skewness of distributions, the convex transform order has not only laid a good foundation for excellent applications in various areas but also motivated several other related orderings such as the star-shaped order, excess wealth order, and Lorenz order etc. The sample maximum has been paid much attention in many areas concerning probability and statistics. In reliability, the lifetime of a parallel system is exactly the maximum of component lifetimes, and to compare the skewness of the ∗ Corresponding author. E-mail address: [email protected] (X. Li). https://doi.org/10.1016/j.spl.2019.108597 0167-7152/© 2019 Elsevier B.V. All rights reserved.

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J. Wu, M. Wang and X. Li / Statistics and Probability Letters 156 (2020) 108597

system lifetime based on component lifetime distributions helps gaining insight of the system reliability. In actuarial science, the maximum of multiple potential losses is critical to an insurance policy, and, undoubtedly, comparing the skewness of the maximum helps to assess the risk of the entire policy. In auction theory, the sample maximum defines the winner’s price of the first-price English auction among multiple bidders. Also, the skewness of bidder value distributions plays a part in determining the winner’s rent in the second-price sealed English auction. Ordering the skewness of sample maximums assists to come up with a better mechanism design. For more related research we refer readers to Li and Li (2015, 2019), Kochar and Xu (2007, 2009, 2014), Khaledi and Kochar (2000), Balakrishnan et al. (2018a,b), Zhang and Zhao (2017), Fang and Li (2015) and Li (2005). The convex transform order of the maximum value of independent Weibull random variables (r.v.’s) has been discussed in the literature. Kochar and Xu (2009) first proved that the maximum of heterogeneous exponential r.v.’s is greater than that of homogeneous ones in the convex transform order. Recently, Balakrishnan et al. (2018a,b) built this order between the maximum of heterogeneous Weibull r.v.’s with a shape parameter less than 1 and that of homogeneous ones with a shape parameter greater than 1. The present work further develops the order between the maximum value of heterogeneous Weibull r.v.’s and that of homogeneous ones where both shape parameters are less than 1. The findings supplement earlier research on the ordering of Weibull distributions. The rest of this paper is organized as follows: Section 2 covers the essential background, and Section 3 builds the convex transform order on the maximum of Weibull r.v.’s. 2. Some preliminaries A random lifetime X with Weibull distribution, denoted X ∼ W (α, λ), has a cumulative distribution function (CDF) given α by F (x) = 1 − e−(λx) for x > 0 and where the scale parameter λ > 0 and the shape parameter α > 0. The hazard rate of W (α, λ) is well known to decrease, remain constant, or increase depending whether α is less than, equal to, or greater than unity, respectively. When α = 1, W (α, λ) reduces to E (λ) — the exponential distribution with rate λ. When considering an absolutely continuous distribution, the hazard rate is a convenient measure for characterizing the aging behavior of the lifetime. In particular, it can be used to compare the aging behaviors of lifetimes in reliability engineering. For example, when considering two different systems with the same expected lifetime, it is reasonable to exclude the one that ages faster. In the literature, an increasing hazard rate (IHR) is extended to the corresponding partial ordering to compare the relative IHR-ness of random lifetimes. For a given CDF F (x), F¯ ≡ 1 − F denotes the survival function and the right continuous inverse F −1 (u) = inf{x ∈ (−∞, +∞) : F (x) ≥ u} for u ∈ (0, 1). Definition 1.

A r.v. X with CDF F is said to be smaller than r.v. Y with CDF G in the

(i) convex transform order, denoted X ≤c Y , if G−1 F (x) is convex on the support of F ; (ii) star-shaped order, denoted X ≤∗ Y , if G−1 F (x)/x is increasing on the support of F . The convex transform order X ≤c Y is a widely used criterion for comparing the skewness of probability distributions. It specifies that the distribution of Y is more skewed to the right than that of X . Since the convexity of G−1 F (x) can be easily verified by using a Q–Q plot in statistics, this ordering has found appeal in the fields of business, economics, engineering, insurance, and finance. However, establishing the convexity of G−1 F (x) mathematically is a challenging task, owing to the complexity of the calculus involved. For further details on the convex transform ordering and the related measures of skewness, we refer readers to Arriaza et al. (2019) and Shaked and Shanthikumar (2007). In reliability studies, a convex G−1 F (x) describes the increasing ratio of hazard rates of G and F at their right continuous inverses. The statement X ≤c Y asserts that X is more IHR and thus ages faster than Y . Notably, (i) neither X nor Y has to be IHR in Definition 1, and (ii) X is IHR if and only if X ≤c Y ∼ E (λ). Several other partial orderings have been proposed to compare the magnitude, variability, and right-skewness of r.v.’s in the context of reliability, economics, and operations research. For a comprehensive account of relative aging, readers may refer to Barlow and Proschan (1981) and Kochar and Wiens (1987). Definition 2.

A r.v. X with CDF F is said to be smaller than r.v. Y with CDF G in the

(i) dispersive order, denoted X ≤disp Y , if F −1 (u) − F −1 (v ) ≤ G−1 (u) − G−1 (v ) for all 0 < v ≤ u < 1; ¯ (ii) usual stochastic order, denoted X ≤st Y , if F¯ (x) ≤ G(x) for all x ∈ (−∞, +∞); (iii) increasing convex order, denoted X ≤icx Y , if (iv) excess wealth order, denoted X ≤ew Y , if

∫ +∞

∫ +∞

t

F −1 (u)

F¯ (x)dx ≤

F¯ (x)dx ≤

∫ +∞

∫ +∞t

G−1 (u)

¯ G(x)dx for any t ∈ (−∞, +∞);

¯ G(x)dx for any u ∈ (0, 1).

Following Proposition 3 of Belzunce (2001), X ≤ew Y implies that X ≤icx Y if they are absolutely continuous and X has a smaller left-endpoint of the support. Further discussion of these orders is given by Shaked and Shanthikumar (1998), Fernandez-Ponce et al. (1998), Kochar et al. (2002, 2007), Li and Shaked (2004), and Shaked and Shanthikumar (2007).

J. Wu, M. Wang and X. Li / Statistics and Probability Letters 156 (2020) 108597

3

3. The convex transform order of the sample maximum Before proceeding to the main result, we introduce three lemmas with their proofs presented in Appendix. The first two build the analytic behavior of two mathematical functions, which are useful for deriving the convex transform order, and the third one concerns a concave function related to the convex transform, to be discussed subsequently. Lemma 1. For bi ∈ (0, 1), i = 1, . . . , n, the function φ (y) = − ln 1 − concave over (0, +∞).

(

Lemma 2.

∏n

i=1 (1

) − byi )1/n is positive, strictly increasing, and

For bi ∈ (0, 1), i = 1, . . . , n, the function ψ (y) = yφ ′ (y) − φ (y) < 0 over (0, +∞).

[

(

Lemma 3. The function − ln 1 − (ii) 0 < α1 ≤ 1 ≤ α2 .

∏n

i=1 (1

α

− e−(λi x) 1 )1/n

)]1/α2

is concave over (0, +∞) whenever (i) 0 < α1 ≤ α2 ≤ 1 or

We now present the main result. Assume α1 > 0, α2 > 0, λ > 0 and λi > 0 for i = 1, . . . , n. For independent α Yi ∼ E (λi 1 ) (i = 1, . . . , n) and independent Yi∗ ∼ E (λα2 ) (i = 1, . . . , n), Kochar and Xu (Theorem 3.1, 2009) proved that max{Y1∗ , . . . , Yn∗ } ≡ Yn∗:n ≤c Yn:n ≡ max{Y1 , . . . , Yn },

for α1 , α2 ∈ (0, +∞).

(3.1)

For independent Xi ∼ W (α1 , λi ) (i = 1, . . . , n) and independent Xi ∼ W (α2 , λ) (i = 1, . . . , n), Kochar and Xu (Corollary 4.1, 2014) also showed that ∗

max{X1∗ , . . . , Xn∗ } ≡ Xn∗:n ≤∗ Xn:n ≡ max{X1 , . . . , Xn },

for α1 = α2 ∈ (0, +∞).

(3.2)

Recently, Balakrishnan et al. (Theorem 3.2, 2018a, 2018b) further showed that Xn∗:n ≤c Xn:n ,

for 0 < α1 ≤ 1 ≤ α2 ,

(3.3)

which not only upgrades the star-shaped order of (3.2) to the convex transform order but also partially generalizes (3.1) to Weibull r.v.’s. Naturally, one may wonder whether (3.3) actually holds for 0 < α1 ≤ α2 . Our main theorem confirms this conjecture for 0 < α1 ≤ α2 ≤ 1. Theorem 1.

Xn∗:n ≤c Xn:n for 0 < α1 ≤ α2 ≤ 1 and λ, λi > 0, i = 1, . . . , n.

Proof. Let hi (x) = x1/αi for x > 0 and α1 , α2 > 0, i = 1, 2. It is straightforward to verify that h1 (Yn:n ) and h2 (Yn∗:n ) have their respective CDF’s, for x ≥ 0, H1 (x) = P Yn:n ≤ xα1 =

(

)

n ∏ [

α1

1 − e−(λi x)

] ,

α2 n

H2 (x) = P Yn∗:n ≤ xα2 = 1 − e−(λx)

(

)

[

]

.

i=1

Clearly, Xn:n and h1 (Yn:n ) have the same CDF, likewise Xn:n and h2 (Yn∗:n ). Therefore, to reach the desired conclusion, we instead prove that h1 (Yn:n ) ≥c h2 (Yn∗:n ) for 0 < α1 ≤ α2 ≤ 1. Note that the concavity of H2−1 H1 (x) =

1

λ

[

)]1/α2 ( n ∏ ( α1 )1/n − ln 1 − 1 − e−(λi x) i=1

over the range [0, +∞) follows directly from Lemma 3.

■

According to Theorem 1, a parallel system ages faster if it has homogeneous, rather than heterogeneous, Weibull components. Because the convex transform order is scale-invariant, it is not surprising that Theorem 1 holds independently of λ, λi , i = 1, . . . , n. Intuitively, the maximum of heterogeneous r.v.’s is more skewed to the right as opposed to that of homogeneous ones. Theorem 1 confirms this fact for Weibull r.v.’s in the context of 0 < α1 ≤ α2 ≤ 1, i.e., when the shape parameter of the heterogeneous sample is smaller than that of the homogeneous one. It is worth mentioning that Theorem 3.2 of Balakrishnan et al. (2018a,b) can also be built based on Lemma 3, similarly to Theorem 1. Example 1. Setting (λ, λ1 , λ2 ) = (1, 2, 10), we check H2−1 H1 (x) for (α1 , α2 ) = (0.3, 0.8) and (α1 , α2 ) = (1.5, 2.5), respectively. For (α1 , α2 ) = (0.3, 0.8), as seen in Fig. 1(a), H2−1 H1 (x) is concave, which confirms the convex transform order when 0 < α1 ≤ α2 ≤ 1. With regard to Fig. 1(b), H2−1 H1 (x) is again concave for (α1 , α2 ) = (1.5, 2.5). However, for (α1 , α2 ) = (6, 8), as depicted in Fig. 1(c), H2−1 H1 (x) is not always concave over (0, ∞). Thus, in the context of 1 ≤ α1 ≤ α2 , Xn∗:n ≤c Xn:n seems to hold for smaller α2 and to fail for greater α1 values. ■ Together with Theorem 3.2 of Balakrishnan et al. (2018a,b), Theorem 1 successfully upgrades Corollary 4.1 of Kochar and Xu (2014) from the star-shaped order to the convex transform order in the context of 0 < α1 ≤ α2 < 1 and

4

J. Wu, M. Wang and X. Li / Statistics and Probability Letters 156 (2020) 108597

Fig. 1. Curves H2−1 H1 (x).

0 < α1 ≤ 1 ≤ α2 . When 1 ≤ α1 ≤ α2 , we conjecture that the convex transform order (3.3) is true at least for smaller α2 ; However, further efforts are necessary to establish this proof convincingly. Finally, we present other ordering results on the maximum of Weibull r.v.’s by considering simple applications. Let α = (α1 , α2 ) and λ = (λ1 , . . . , λn ), and define

γ1 (α, λ) =

( α2 +1 )

Γ

α ( 2 ) Γ α1α+1 1

∑n

i=1 (

∑n ∑

1≤j1 <···
i=1

−

(n)

1 i1/α2 i 1)i+1 r =1

−1)i+1

i

(∑

α

λjr1

)−1/α1 ,

where Γ (x) is the gamma function. The following theorem is a direct consequence of Theorem 1, and the proof is here omitted, given its similarity to Theorems 3.4 and 3.5 of Balakrishnan et al. (2018a,b). Theorem 2. For 0 < α1 ≤ α2 ≤ 1, λ > 0, and λi > 0, i = 1, . . . , n, (i) Xn∗:n ≤ew (≤icx )Xn:n if and only if λ ≥ γ1 (α, λ), and (ii) Xn∗:n ≤disp Xn:n if and only if Xn∗:n ≤st Xn:n . Acknowledgments Dr. Jintang Wu gratefully acknowledges support from the National Natural Science Foundation of China (No. 11701194) and the Promotion Program of Young and Middle-Aged Teachers in Science and Technology Research of Huaqiao University (No. ZQN-PY503). Also, authors would like to thank both the Editor and the reviewer for their valuable comments, helping improve the presentation of the earlier version of this manuscript. Appendix A.1. Proof of Lemma 1 For y ∈ (0, +∞) and bi ∈ (0, 1), i = 1, . . . , n, we have

− byi )1/n · 1n (1 − byj )1/n−1 (−byj ln bj ) ∏n y 1 − i=1 (1 − bi )1/n ∏n n y 1/n y 1 ∑ −bi ln bi i=1 (1 − bi ) ∏ = · y y n 1 − i=1 (1 − bi )1/n n 1 − bi i=1 ∑n − ln bi 1

∑n ∏

φ (y) = ′

j=1

i=1 b−y −1 i

n

= ∏n > 0,

i̸ =j (1

i=1 (1

− byi )−1/n − 1

and

φ (y) = − ′′

[ n ∏

]−1 (1 −

1 y bi ) − n

−1

·

i=1

+

n 1 ∑ − ln bi

n

−y

i=1

bi

−1

1 n

· [∏n

n −y 1 ∑ (− ln bi )2 bi

n

−y (bi i=1 n − ln bi i=1 b−y −1 i

− 1)2

1 y bi ) − n

]2

∑

i=1 (1

−

n ∏

−1

y

1

(1 − bi )− n

i=1

J. Wu, M. Wang and X. Li / Statistics and Probability Letters 156 (2020) 108597

[( =

)2

n 1 ∑ − ln bi

n

−y

bi

i=1

−

n −y 1 ∑ (− ln bi )2 bi

−y

n

−1

i=1

n

n ∏

y

i=1

− 1)2

n ∏

y 1

(1 − bi ) n

i=1

]2 . − byi )−1/n − 1

(A.1)

[∏n

i=1 (1

As regards the AM-GM inequality, we have bi ≤ 1 −

1−

5

)]

y

i=1 (1

n 1∑

(bi

)(

−1/n i=1 (1 − bi )

∏n · [∏n

(

y 1

(1 − bi ) n ,

− byi )

]1/n

≤

1 n

∑n

i=1 (1

− byi ), and hence

for all y ∈ (0, +∞).

(A.2)

i=1

Also, by the Cauchy–Schwarz inequality, we have

(

)2

n 1 ∑ − ln bi

n

−y

i=1

bi

≤

−1

n −y 1 ∑ (− ln bi )2 bi

n

−y

i=1

(bi

− 1)2

n 1∑

·

n

y

bi ,

for all y ∈ (0, +∞).

(A.3)

i=1

Now, based on (A.1), (A.2) and (A.3), we have φ ′′ (y) ≤ 0 for y ∈ (0, +∞) and bi ∈ (0, 1), i = 1, . . . , n. Therefore, φ (y) is positive, strictly increasing, and concave over (0, +∞). ■ A.2. Proof of Lemma 2 y

1−bi −y ln bi

For bi ∈ (0, 1), i = 1, . . . , n, we have limy→0+

= 1, limy→0+ φ (y) = 0, and thus ∏n n y y y i=1 (1 − bi )1/n 1 ∑ −bi ln bi ∏n lim yφ ′ (y) = lim y 1/n · y y→0+ y→0+ 1 − n 1 − bi i=1 (1 − bi ) i=1 = lim y y→0+

= lim y y→0+

=

n ∏ i=1

n ∏

y 1

(1 − bi ) n ·

i=1 n

∏

1

n 1 ∑ − ln bi

n

(−y ln bi ) n ·

i=1 1

(− ln bi ) n ·

n 1∑

n

i=1

y

i=1 n

1 − bi

1 ∑ − ln bi n

y

i=1

1 − bi

−y2 ln bi y y→0+ 1 − b i lim

= 0. As a result, limy→0+ ψ (y) = limy→0+ [yφ ′ (y) − φ (y)] = 0. Now, from Lemma 1, it follows that ψ ′ (y) = yφ ′′ (y) ≤ 0 on (0, +∞), and hence ψ (y) is decreasing. This yields ψ (y) < 0 over (0, +∞). ■ A.3. Proof of Lemma 3 α1

Define ℓ(x) = [φ (xα1 )]1/α2 for x ∈ (0, +∞). Since bi = e−λi ∈ (0, 1) for α1 ∈ (0, 1] and λi > 0 (i = 1, . . . , n), it suffices to show that ℓ′′ (x) ≤ 0 over (0, ∞). ( ) α Note that ℓ′ (x) = α1 xα1 −1 [φ (xα1 )]1/α2 −1 φ ′ (xα1 ) and ℓ′′ (x) = η xα1 [φ (xα1 )]1/α2 −2 , where 2

η(y) = (1 − α1−1 )φ (y)φ ′ (y) + (α2−1 − 1)y[φ ′ (y)]2 + yφ (y)φ ′′ (y),

for y > 0.

As per Lemma 1 we have φ (y) > 0 over (0, +∞). Then, the sign of ℓ′′ (x) always coincides with

[ ] yφ ′ (y) − φ (y) yφ ′′ (y) η(y) = φ (y)φ ′ (y) (α2−1 − 1) + ′ + α2−1 − α1−1 , φ (y) φ (y)

for y > 0.

For 0 < α1 ≤ 1 ≤ α2 , by Lemma 1, we have φ (y) > 0, φ ′ (y) > 0, φ ′′ (y) ≤ 0 and hence η(y) ≤ 0 over (0, +∞). For 0 < α1 ≤ α2 (< 1, by Lemmas 2, we have φ (y) > 0, φ ′ (y) > 0, yφ ′ (y) − φ (y) < 0 and φ ′′ (y) ≤ 0 on (0, +∞). ) yφ ′ (y)−φ (y)1 and yφ ′′ (y) Consequently, α2−1 − 1 + + α2−1 − α1−1 ≤ 0, and hence η(y) ≤ 0 over (0, +∞) again. Therefore, it holds φ (y) φ ′ (y) that ℓ′′ (x) ≤ 0 over (0, +∞) for either 0 < α1 ≤ α2 ≤ 1 or 0 < α1 ≤ 1 ≤ α2 . This completes the proof.

■

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