Abstract
Building on the seminal work by Shaked and Shanthikumar (Adv Appl Probab 20:427–446, 1988a; Stoch Process Appl 27:1–20, 1988b), Denuit et al. (Eng Inf Sci 13:275–291, 1999; Methodol Comput Appl Probab 2:231–254, 2000; 2001) studied the stochastic s-increasing convexity properties of standard parametric families of distributions. However, the analysis is restricted there to a single parameter. As many standard families of distributions involve several parameters, multivariate higher-order stochastic convexity properties also deserve consideration for applications. This is precisely the topic of the present paper, devoted to stochastic \((s_1,s_2,\ldots ,s_d)\)-increasing convexity of distribution families indexed by a vector \((\theta _1,\theta _2,\ldots ,\theta _d)\) of parameters. This approach accounts for possible correlation in multivariate mixture models.
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Notes
Note, however, that in the main results of [4], non-decreasingness must be replaced with increasingness to ensure that the transformation corresponding to the conditional expectation is one-to-one.
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Acknowledgments
The authors are grateful to an anonymous reviewer for careful reading and constructive comments and suggestions which greatly helped to improve the initial manuscript. Michel Denuit acknowledges the financial support from the contract “Projet d’Actions de Recherche Concertées” No. 12/17-045 of the “Communauté française de Belgique”, granted by the “Académie universitaire Louvain”. Mhamed Mesfioui acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada.
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Appendix: Some Results for the Binomial Distribution
Appendix: Some Results for the Binomial Distribution
Property 6.1
Let \(X_{(n,p)}, n \in \mathbb {N}, p \in (0,1)\), denote a random variable distributed according to the Binomial distribution with mean np and variance \(np(1-p)\). With \(g^{\star }(n,p)=\mathbb {E}[g(X_{(n,p)})]\), define
Then, we have
Proof
We have
Using the well-known identities
we can write
which ends the proof of (6.2). Now, let us show that the second equality is valid. Clearly, one has
Since
we can write
Finally, let us establish the last formula (6.4). First, recall that for any functions \(h_1\) and \(h_2:\mathbb {N}\rightarrow \mathbb {R}\), one has
Considering \(h_1(n)=n\) and \(h_2(n)=\mathbb {E} \left[ \Delta g(X_{(n-1,p)})\right] =(\Delta g)^\star (n-1,p)\), we get
so that (6.4) is indeed valid. \(\square \)
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Denuit, M.M., Mesfioui, M. Multivariate Higher-Degree Stochastic Increasing Convexity. J Theor Probab 29, 1599–1623 (2016). https://doi.org/10.1007/s10959-015-0628-6
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DOI: https://doi.org/10.1007/s10959-015-0628-6
Keywords
- Multivariate higher-degree increasing convex order
- Ordering of mixtures
- Parametric families of distributions
- Upper orthant order
- Orthant convex order