Log-concavity for Bernstein-type operators using stochastic orders

Badía, F. G.; Sangüesa, C.

doi:10.1016/j.jmaa.2013.12.014

Cited by 10 publications

(7 citation statements)

References 18 publications

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“…It is worthnoting that Theorem 3.7 of Badía & Sangüesa [5] established similar results to Theorem 2.5. We point out the difference between Theorem 2.5 and Theorem 3.7 of Badía & Sangüesa [5]. First, our proof is different from theirs.…”

Section: Remark 26supporting

confidence: 67%

“…Note that the semi-group property of {X θ , θ ∈ Θ} is equivalent to the property of independent and stationary increments. Hence, Theorem 2.5(i) generalizes Theorem 3.7(c) of Badía & Sangüesa [5] by dropping off the assumption that T (φ, θ) is continuous in θ ∈ R + . However, the results of Theorem 2.5(ii) and (iii) and those of Theorem 3.7(a) and (b) of Badía & Sangüesa [5] do not include each other as they consider a slightly more general class of distributions called the class IPII [IPDI] (independent positive increasing [decreasing] increments) under the additional assumption that T (φ, θ) is continuous in θ ∈ R + .…”

Section: Remark 26mentioning

confidence: 63%

“…First, our proof is different from theirs. Second, Badía & Sangüesa [5] constrains the operator T (φ, θ) to be continuous in θ ∈ R + and the parameter space to be Θ = R + . In order to ensure that T (φ, θ) is continuous in θ ∈ R + , they imposed the assumption of continuity in probability of the process {X θ , θ ∈ R + } as follows: lim s→t P(|X t − X s | > ) = 0, for all > 0 and t > 0.…”

Section: Remark 26mentioning

confidence: 99%

“…An [2] and Bagnoli & Bergstrom [7] are two other reviews of logconcavity in econometrics. For more on log-concavity, we refer the reader to Dharmadhikari & Joag-dev [14], Finner & Roters [15], Finner & Roters [16], Liggett [21], Sengupta & Nanda [31], Wang & Yeh [34], Kahn & Neiman [19], Yu [35], Bobkov & Madiman [9], Badía & Sangüesa [5,6] and references therein.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Preservation of Log-Concavity and Log-Convexity Under Operators

Xia

Mao

2020

Prob. Eng. Inf. Sci.

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Log-concavity [log-convexity] and their various properties play an increasingly important role in probability, statistics, operations research and other fields. In this paper, we first establish general preservation theorems of log-concavity and log-convexity under operator $\phi \longmapsto T(\phi , \theta )=\mathbb {E}[\phi (X_\theta )]$ , θ ∈ Θ, where Θ is an interval of real numbers or an interval of integers, and the random variable $X_\theta$ has a distribution function belonging to the family $\{F_\theta , \theta \in \Theta \}$ possessing the semi-group property. The proofs are based on the theory of stochastic comparisons and weighted distributions. The main results are applied to some special operators, for example, operators occurring in reliability, Bernstein-type operators and Beta-type operators. Several known results in the literature are recovered.

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Section: Remark 26supporting

confidence: 67%

Section: Remark 26mentioning

confidence: 63%

Section: Remark 26mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Preservation of Log-Concavity and Log-Convexity Under Operators

Xia

Mao

2020

Prob. Eng. Inf. Sci.

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“…In applied probability, stochastic orders formalize the concept that one random variable is bigger than another (see [18]) and they have crucial roles in various applications (see e.g. [3], [4], [9], [10], [14], [15], and [16]). Thus, to model the shortened residual lifetimes of the remaining components in a system subject to cascading failures, the concept of stochastic order can be employed.…”

Section: Introductionmentioning

confidence: 99%

On a new stochastic model for cascading failures

Lee

2020

J. Appl. Probab.

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In this paper, to model cascading failures, a new stochastic failure model is proposed. In a system subject to cascading failures, after each failure of the component, the remaining component suffers from increased load or stress. This results in shortened residual lifetimes of the remaining components. In this paper, to model this effect, the concept of the usual stochastic order is employed along with the accelerated life test model, and a new general class of stochastic failure models is generated.

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Inventory models with nonlinear shortage costs and stochastic lead times; applications of shape properties of randomly stopped counting processes

Badía

Sangüesa

2015

Naval Research Logistics

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Abstract:In this article, we study generalizations of some of the inventory models with nonlinear costs considered by Rosling in (Oper. Res. 50 (2002) 797-809). In particular, we extend the study of both the periodic review and the compound renewal demand processes from a constant lead time to a random lead time. We find that the quasiconvexity properties of the cost function (and therefore the existence of optimal (s, S) policies), holds true when the lead time has suitable log-concavity properties. The results are derived by structural properties of renewal delayed processes stopped at an independent random time and by the study of log-concavity properties of compound distributions.

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Log-concavity for Bernstein-type operators using stochastic orders

Cited by 10 publications

References 18 publications

Preservation of Log-Concavity and Log-Convexity Under Operators

Preservation of Log-Concavity and Log-Convexity Under Operators

On a new stochastic model for cascading failures

Inventory models with nonlinear shortage costs and stochastic lead times; applications of shape properties of randomly stopped counting processes

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