scholarly journals Diagonal sections of copulas, multivariate conditional hazard rates and distributions of order statistics for minimally stable lifetimes

2021 ◽  
Vol 9 (1) ◽  
pp. 394-423
Author(s):  
Rachele Foschi ◽  
Giovanna Nappo ◽  
Fabio L. Spizzichino
Keyword(s):  
Order Statistics ◽  
Hazard Rate ◽  
Joint Distribution ◽  
Survival Function ◽  
Hazard Rates ◽  
Archimedean Copulas ◽  
Absolutely Continuous ◽  
Marginal Distributions ◽  
Special Cases ◽  
Survival Copula

Abstract As a motivating problem, we aim to study some special aspects of the marginal distributions of the order statistics for exchangeable and (more generally) for minimally stable non-negative random variables T 1, ..., Tr. In any case, we assume that T 1, ..., Tr are identically distributed, with a common survival function ̄G and their survival copula is denoted by K. The diagonal sections of K, along with ̄G, are possible tools to describe the information needed to recover the laws of order statistics. When attention is restricted to the absolutely continuous case, such a joint distribution can be described in terms of the associated multivariate conditional hazard rate (m.c.h.r.) functions. We then study the distributions of the order statistics of T 1, ..., Tr also in terms of the system of the m.c.h.r. functions. We compare and, in a sense, we combine the two different approaches in order to obtain different detailed formulas and to analyze some probabilistic aspects for the distributions of interest. This study also leads us to compare the two cases of exchangeable and minimally stable variables both in terms of copulas and of m.c.h.r. functions. The paper concludes with the analysis of two remarkable special cases of stochastic dependence, namely Archimedean copulas and load sharing models. This analysis will allow us to provide some illustrative examples, and some discussion about peculiar aspects of our results.

Characterization of generalized Gamma-Lindley distribution using truncated moments of order statistics

2021 ◽  
Vol 71 (2) ◽  
pp. 455-474
Author(s):  
Dorsaf Laribi ◽  
Afif Masmoudi ◽  
Imen Boutouria
Keyword(s):  
Order Statistics ◽  
Survival Function ◽  
Lindley Distribution ◽  
Underlying Distribution ◽  
First Order ◽  
New Class ◽  
Generalized Gamma ◽  
Special Cases ◽  
Two Parameters

Abstract Having only two parameters, the Gamma-Lindley distribution does not provide enough flexibility for analyzing different types of lifetime data. From this perspective, in order to further enhance its flexibility, we set forward in this paper a new class of distributions named Generalized Gamma-Lindley distribution with four parameters. Its construction is based on certain mixtures of Gamma and Lindley distributions. The truncated moment, as a characterization method, has drawn a little attention in the statistical literature over the great popularity of the classical methods. We attempt to prove that the Generalized Gamma-Lindley distribution is characterized by its truncated moment of the first order statistics. This method rests upon finding a survival function of a distribution, that is a solution of a first order differential equation. This characterization includes as special cases: Gamma, Lindley, Exponential, Gamma-Lindley and Weighted Lindley distributions. Finally, a simulation study is performed to help the reader check whether the available data follow the underlying distribution.

MODELS BASED ON PARTIAL INFORMATION ABOUT SURVIVAL AND HAZARD GRADIENT

2010 ◽  
Vol 24 (4) ◽  
pp. 561-584 ◽  
Author(s):  
Majid Asadi ◽  
Somayeh Ashrafi ◽  
Nader Ebrahimi ◽  
Ehsan S. Soofi
Keyword(s):  
Joint Distribution ◽  
Partial Information ◽  
Survival Function ◽  
Information Criteria ◽  
Hazard Rates ◽  
Optimal Model ◽  
Dynamic Information ◽  
Hazard Gradient ◽  
Other Information ◽  
Gumbel Model

This article develops information optimal models for the joint distribution based on partial information about the survival function or hazard gradient in terms of inequalities. In the class of all distributions that satisfy the partial information, the optimal model is characterized by well-known information criteria. General results relate these information criteria with the upper orthant and the hazard gradient orderings. Applications include information characterizations of the bivariate Farlie–Gumbel–Morgenstern, bivariate Gumbel, and bivariate generalized Gumbel, for which no other information characterization are available. The generalized bivariate Gumbel model is obtained from partial information about the survival function and hazard gradient in terms of marginal hazard rates. Other examples include dynamic information characterizations of the bivariate Lomax and generalized bivariate Gumbel models having marginals that are transformations of exponential such as Pareto, Weibull, and extreme value. Mixtures of bivariate Gumbel and generalized Gumbel are obtained from partial information given in terms of mixtures of the marginal hazard rates.

Shock processes by aftereffects and multivariate lack of memory

1984 ◽  
Vol 21 (04) ◽  
pp. 786-801
Author(s):  
S. G. Ghurye ◽  
Albert W. Marshall
Keyword(s):  
Functional Equation ◽  
Exponential Distribution ◽  
Survival Function ◽  
Practical Interest ◽  
Absolutely Continuous ◽  
Marginal Distributions ◽  
Image Position ◽  
Multivariate Exponential Distribution ◽  
Multivariate Exponential

If the survival function satisfies the functional equation where e = (1, …, 1), and if the marginal distributions are exponential, then is the multivariate exponential distribution of Marshall and Olkin. The functional equation has many solutions if the requirement of exponential marginals is not imposed, but the class of possible marginals is somewhat limited (e.g., marginals must be absolutely continuous). The class of possible solutions of the equation is characterized in this paper, and several examples are obtained from models for dependence that may be of practical interest.

The joint distribution of reference crop evapotranspiration and irrigation water in the irrigation district

2017 ◽  
Vol 18 (2) ◽  
pp. 567-576 ◽  
Author(s):  
Jinping Zhang ◽  
Xiaomin Lin ◽  
Yong Zhao
Keyword(s):  
Irrigation Water ◽  
Joint Distribution ◽  
Water Shortage ◽  
Irrigation District ◽  
Crop Evapotranspiration ◽  
Archimedean Copulas ◽  
Marginal Distributions ◽  
Reference Crop Evapotranspiration ◽  
Best Fitting ◽  
Reference Crop

Abstract For the irrigation district, irrigation water is the manual water supply for the farmland while reference crop evapotranspiration (ET0) reflects water demand. Thus, the joint distribution of irrigation water and ET0 can reveal water shortage risk under the condition of the manual water supply. In order to understand their relationships and overcome the drawbacks of different marginal distributions of hydrological variables, Archimedean copulas are introduced. Based on the data series of ET0 and irrigation water in the Luhun irrigation district of China, the univariate marginal distributions of ET0 and irrigation water are first selected. Then, with the Gumbel–Hougaard copula in the Archimedean copulas, the joint distribution of ET0 and irrigation water is proposed. The results show that the best-fitting marginal distributions of ET0 and irrigation water are generalized extreme values and normal distributions, respectively, but for their joint distribution, the Gumbel–Hougaard copula is the best-fitting one. The water shortage risks with different encounter situations of ET0 and irrigation water are better revealed using the proposed copula-based joint distribution.

Shock processes by aftereffects and multivariate lack of memory

1984 ◽  
Vol 21 (4) ◽  
pp. 786-801 ◽  
Author(s):  
S. G. Ghurye ◽  
Albert W. Marshall
Keyword(s):  
Functional Equation ◽  
Exponential Distribution ◽  
Survival Function ◽  
Practical Interest ◽  
Absolutely Continuous ◽  
Marginal Distributions ◽  
Image Position ◽  
Multivariate Exponential Distribution ◽  
Multivariate Exponential

If the survival function satisfies the functional equation where e = (1, …, 1), and if the marginal distributions are exponential, then is the multivariate exponential distribution of Marshall and Olkin. The functional equation has many solutions if the requirement of exponential marginals is not imposed, but the class of possible marginals is somewhat limited (e.g., marginals must be absolutely continuous). The class of possible solutions of the equation is characterized in this paper, and several examples are obtained from models for dependence that may be of practical interest.

ON THE RELIABILITY PROPERTIES OF SOME WEIGHTED MODELS OF BATHTUB SHAPED HAZARD RATE DISTRIBUTIONS

2012 ◽  
Vol 27 (1) ◽  
pp. 125-140 ◽  
Author(s):  
M. Shafaei Noughabi ◽  
G.R. Mohtashami Borzadaran ◽  
A.H. Rezaei Roknabadi
Keyword(s):  
Order Statistics ◽  
Hazard Rate ◽  
Residual Life ◽  
Rate Function ◽  
Mean Residual Life ◽  
Special Cases ◽  
Weighted Model ◽  
The Times ◽  
The One ◽  
Useful Lifetime

Let F be a bathtub-shaped (BT) hazard rate distribution function. It has been shown that the hazard rate function of the order statistics may be BT, increasing, etc. Then, we have carried out a graphical study for some useful lifetime models.Moreover, we are interested to compare the time that maximizes the mean residual life (MRL) function of F with the one related to a general weighted model in terms of their locations. Also, the times maximizing the conditional reliability proposed by Mi [13] of F have been compared with the corresponding times of a general weighted model. As special cases, we consider order statistics and the proportional hazard rate model.

Load-Sharing Reliability Models with Different Component Sensitivities to Other Components’ Working States

2021 ◽  
Vol 53 (1) ◽  
pp. 107-132
Author(s):  
Tomasz Rychlik ◽  
Fabio Spizzichino
Keyword(s):  
Order Statistics ◽  
Load Sharing ◽  
Single Component ◽  
Generalized Order Statistics ◽  
Hazard Rates ◽  
System Operation ◽  
Reliability Models ◽  
Mixtures Of Distributions ◽  
Failure Times ◽  
Generalized Order

AbstractWe study the distributions of component and system lifetimes under the time-homogeneous load-sharing model, where the multivariate conditional hazard rates of working components depend only on the set of failed components, and not on their failure moments or the time elapsed from the start of system operation. Then we analyze its time-heterogeneous extension, in which the distributions of consecutive failure times, single component lifetimes, and system lifetimes coincide with mixtures of distributions of generalized order statistics. Finally we focus on some specific forms of the time-nonhomogeneous load-sharing model.

scholarly journals On a new distribution based on the arccosine function

2021 ◽  
Author(s):  
Christophe Chesneau ◽  
Lishamol Tomy ◽  
Jiju Gillariose
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Hazard Rate ◽  
Survival Function ◽  
Real Data ◽  
Cumulative Distribution ◽  
Data Sets ◽  
Power Functions ◽  
New Distribution

AbstractThis note focuses on a new one-parameter unit probability distribution centered around the inverse cosine and power functions. A special case of this distribution has the exact inverse cosine function as a probability density function. To our knowledge, despite obvious mathematical interest, such a probability density function has never been considered in Probability and Statistics. Here, we fill this gap by pointing out the main properties of the proposed distribution, from both the theoretical and practical aspects. Specifically, we provide the analytical form expressions for its cumulative distribution function, survival function, hazard rate function, raw moments and incomplete moments. The asymptotes and shape properties of the probability density and hazard rate functions are described, as well as the skewness and kurtosis properties, revealing the flexible nature of the new distribution. In particular, it appears to be “round mesokurtic” and “left skewed”. With these features in mind, special attention is given to find empirical applications of the new distribution to real data sets. Accordingly, the proposed distribution is compared with the well-known power distribution by means of two real data sets.

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