The Bivariate Lack-of-Memory Distributions

Gwo Dong Lin; Xiaoling Dou; Satoshi Kuriki

doi:10.1007/s13171-017-0119-1

The Bivariate Lack-of-Memory Distributions

Gwo Dong Lin^*, Xiaoling Dou, Satoshi Kuriki

^*Corresponding author for this work

Center for Data Science

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

We treat all the bivariate lack-of-memory (BLM) distributions in a unified approach and develop some new general properties of the BLM distributions, including joint moment generating function, product moments, and dependence structure. Necessary and sufficient conditions for the survival functions of BLM distributions to be totally positive of order two are given. Some previous results about specific BLM distributions are improved. In particular, we show that both the Marshall–Olkin survival copula and survival function are totally positive of all orders, regardless of parameters. Besides, we point out that Slepian’s inequality also holds true for BLM distributions.

Original language	English
Pages (from-to)	273-297
Number of pages	25
Journal	Sankhya A
Volume	81
Issue number	2
DOIs	https://doi.org/10.1007/s13171-017-0119-1
Publication status	Published - 2019 Dec 1

Keywords

62H05
62H20
62N86
Bivariate lack-of-memory distributions
Block and Basu’s BVE
Copula
Freund’s BVE
Hazard rate order
Lack-of-memory property
Likelihood ratio order
Marshall and Olkin’s BVE
Positive quadrant dependent
Primary: 62N05
Slepian’s lemma/inequality
Survival copula
Survival function
Total positivity
Usual stochastic order

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

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10.1007/s13171-017-0119-1

Cite this

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title = "The Bivariate Lack-of-Memory Distributions",

abstract = "We treat all the bivariate lack-of-memory (BLM) distributions in a unified approach and develop some new general properties of the BLM distributions, including joint moment generating function, product moments, and dependence structure. Necessary and sufficient conditions for the survival functions of BLM distributions to be totally positive of order two are given. Some previous results about specific BLM distributions are improved. In particular, we show that both the Marshall–Olkin survival copula and survival function are totally positive of all orders, regardless of parameters. Besides, we point out that Slepian{\textquoteright}s inequality also holds true for BLM distributions.",

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author = "Lin, {Gwo Dong} and Xiaoling Dou and Satoshi Kuriki",

note = "Funding Information: The authors would like to thank the Editor-in-Chief and two referees for helpful comments and constructive suggestions which improve the presentation of the paper. The paper was presented at the Ibusuki International Seminar, Ibusuki Phoenix Hotel, held from 6th to 8th March 2016 by Waseda University, Japan. The authors thank the organizer Professor Masanobu Taniguchi for his kind invitation and the audiences for their comments and suggestions. Publisher Copyright: {\textcopyright} 2017, Indian Statistical Institute.",

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doi = "10.1007/s13171-017-0119-1",

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AU - Dou, Xiaoling

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N1 - Funding Information: The authors would like to thank the Editor-in-Chief and two referees for helpful comments and constructive suggestions which improve the presentation of the paper. The paper was presented at the Ibusuki International Seminar, Ibusuki Phoenix Hotel, held from 6th to 8th March 2016 by Waseda University, Japan. The authors thank the organizer Professor Masanobu Taniguchi for his kind invitation and the audiences for their comments and suggestions. Publisher Copyright: © 2017, Indian Statistical Institute.

PY - 2019/12/1

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N2 - We treat all the bivariate lack-of-memory (BLM) distributions in a unified approach and develop some new general properties of the BLM distributions, including joint moment generating function, product moments, and dependence structure. Necessary and sufficient conditions for the survival functions of BLM distributions to be totally positive of order two are given. Some previous results about specific BLM distributions are improved. In particular, we show that both the Marshall–Olkin survival copula and survival function are totally positive of all orders, regardless of parameters. Besides, we point out that Slepian’s inequality also holds true for BLM distributions.

AB - We treat all the bivariate lack-of-memory (BLM) distributions in a unified approach and develop some new general properties of the BLM distributions, including joint moment generating function, product moments, and dependence structure. Necessary and sufficient conditions for the survival functions of BLM distributions to be totally positive of order two are given. Some previous results about specific BLM distributions are improved. In particular, we show that both the Marshall–Olkin survival copula and survival function are totally positive of all orders, regardless of parameters. Besides, we point out that Slepian’s inequality also holds true for BLM distributions.

KW - 62H05

KW - 62H20

KW - 62N86

KW - Bivariate lack-of-memory distributions

KW - Block and Basu’s BVE

KW - Copula

KW - Freund’s BVE

KW - Hazard rate order

KW - Lack-of-memory property

KW - Likelihood ratio order

KW - Marshall and Olkin’s BVE

KW - Positive quadrant dependent

KW - Primary: 62N05

KW - Slepian’s lemma/inequality

KW - Survival copula

KW - Survival function

KW - Total positivity

KW - Usual stochastic order

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The Bivariate Lack-of-Memory Distributions

Abstract

Keywords

ASJC Scopus subject areas

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